# Group Analysis via Weak Symmetries For
Benjamin-Bona-Mahony Equation

M. Nadjafikhah<EMAIL_ADDRESS>F. Ahangari<EMAIL_ADDRESS>S. Dodangeh<EMAIL_ADDRESS>Corresponding author: School of
Mathematics, Iran University of Science and Technology, Narmak, Tehran
1684613114, Iran.

###### Abstract

In this paper, weak symmetries of the Benjamin-Bona-Mahony (BBM) equation have
been investigated. Indeed, this method has been performed by applying the non-
classical symmetries of the BBM equation and the infinitesimal generators of
the classical symmetry algebra of the KdV equation as the starting
constraints. Similarity reduced equations as well as some exact solutions of
the BBM equation are obtained via this method.

###### keywords:

Weak symmetry, Non-classical symmetry, Similarity reduced Equation, Benjamin-
Bona-Mahony Equation.

##  1 Introduction

The Benjamin-Bona-Mahony equation

$\displaystyle{\rm BBM}\;:\;\;u_{t}+u_{x}+uu_{x}-u_{xxt}=0,$ (1.1)

used to model an approximation for surface water waves in a uniform channel
[1]. If we note the KdV type equation

$\displaystyle u_{t}+u_{x}+uu_{x}+u_{xxx}=0,$ (1.2)

then we find out the likeness between these equations. Indeed, this similarity
is not stochastic. Both of them used to model the waves appear in liquids,
compressible fluids, cold plasma and enharmonic crystals which are of surface,
hydro-magnetics, acoustic-gravity and acoustic types, respectively. The
interesting point is that the main difference between equations (1.1) and
(1.2) occurs in the case of short waves (Find more information in [1, 3]).

The physical applications and mathematical properties of the BBM equation
(1.1) have been motivated many investigations such as obtaining the exact
solutions via finite difference discrete process, global attractor and etc.

In this paper, we find the similarity reduced ODEs as well as resulted
similarity solutions of this equation via weak symmetry implementation.
Indeed, the organization of the present paper is as follows: Some historical
information on the weak symmetry method are given in section 2. In section 3,
we follow [10] in order to describe the theory of weak symmetries. Section 4
is devoted to performing this new class of symmetry methods using the
invariant surface condition of the BBM equation (which is indeed the non-
classical symmetry method) and infinitesimal generators of the classical
symmetry algebra of the KdV type equation as the starting points in the weak
symmetry method implementation. Finally, we have compared our results with
those related papers using the classical symmetry method in order to clarify
the advantages and disadvantages of the both strong and weak symmetry methods.

##  2 Background

Symmetry methods for differential equations, was originally developed by S.
Lie [7]. These methods without any doubt are very useful and algorithmic for
analyzing and solving linear and non-linear differential equations.
Classification of differential equations as well as linearization of them are
some other important applications of the symmetry transformation approach.
First G.W. Bluman and J.D. Cole introduced the notion of the non-classical
symmetry group of differential equations specially for the heat equation in
1969 (Find more information in [2]). For the non-classical method, we seek the
invariance of both the original equation and its invariant surface condition,
exactly this constraint (i.e invariance surface condition) causes the non-
classical solutions which are more general than the classical ones. There are
various implementations for performing the non-classical symmetry method, for
example, using the compatibility condition has been suggested by G. Cai and X.
Ling [5].

First the weak symmetries have been introduced by P.J. Olver, and P. Rosenau
in 1986 as a generalization of the non-classical symmetries with motivation of
finding every solutions of the given system. In principle, not only the
invariant solutions corresponding to arbitrary transformation groups can be
found by the reduction method, but also every possible solution of the system
can be found by using some transformation groups. In other words, there are no
conditions that need to be placed on the transformation group in order to
apply the basic reduction procedure (Find more information in [10]). In the
next section, we have an attempt to explain the notation and implementation of
the weak symmetry method by considering the BBM equation as an example in
order to prepare an appropriate setting.

##  3 On the weak symmetry method

Symmetry groups of a system of partial differential equations can be defined
in two types (see [10]).

#### Definition

Let $\Delta$ be a system of partial differential equations. A strong symmetry
group of $\Delta$ is a group of transformations $G$ on the space of
independent and dependent variables which has the following two properties:

  * a)

The elements of $G$ transform solutions of the system to other solutions of
the system.

  * b)

The $G-$invariant solutions of the system are found from a reduced system of
differential equations involving a fewer number of independent variables than
the original system $\Delta$.

#### Definition

A weak symmetry group of the system $\Delta$ is a group of transformations
which satisfies the reduction property (b), but no longer transforms solutions
to solutions.

Indeed, there are several transformation groups which don’t transform
solutions of given equations again to solutions, but their differential
invariants enable us to reduce them. In continuation we would illustrate the
procedure of performing this method. For this purpose, first consider an
arbitrary one-parameter transformation group, then substitute its related
differential invariants and their derivatives into the original equation,
finally, you will encounter with three different possible cases which in
continuation have been illustrated for the (BBM) equation using an appropriate
one-parameter transformation group.

###  3.1 Reduced equation has no parametric variables

Consider the one-parameter group

$(x,t,u)\mapsto(x+\lambda,t+\lambda,u),$

So, we have the characteristic equation $dx=dt=du/0$. By substituting the
resulted differential invariants i.e. $r=x-t$ and $w=u$, into equation (1.1),
we have $w_{rrr}+ww_{r}=0$. As we see, this equation has no parameter
variable.

###  3.2 Reduced equation isn’t incompatible and has parametric variables

Consider the one-parameter group

$(x,t,u)\mapsto(\lambda x,t,\lambda u).$

So, the characteristic equation is $dx/x=dt/0=du/u$. By substituting the
resulted differential invariants $r=t$ and $w=u/x$, in equation (1.1), we have
$x(w^{2}+w_{r})+w=0$, where $w=0$ is it’s solution and this equation has $x$
as the parametric variable.

###  3.3 Reduced equation is incompatible and has parametric variables

Consider the one-parameter group

$(x,t,u)\mapsto(x+2\lambda t+\lambda^{2},t+\lambda,u+8\lambda
t+4\lambda^{2}).$

By substituting the resulted differential invariants i.e. $r=x-t^{2}$ and
$w=u-4t^{2}$, in equation (1.1), we have
$ww_{r}+w_{r}+(8-2w_{rrr}-2w_{r})t+4w_{r}t^{2}$, where this equation has $t$
as the parametric variable and it is incompatible. Indeed, from the
coefficient of $t^{2}$ we have $w_{r}=0$ and from the coefficient of $t$ we
have $w_{rrr}+w_{r}=4$, this means that these equations are incompatible.

##  4 Implementation of the weak symmetry method for the BBM equation

Since, the weak symmetry method is based on conjecture, so here, the several
ideas of performing this method as well as some of its aspects are presented.

###  4.1 Non-classical symmetries of the BBM equation

There are several implementations to find the non-classical symmetries. Here,
we follow the procedure presented by G. Cai et al. which they obtained the
non-classical symmetries of the Burgers-Fisher equation based on the
compatibility conditions [4].

Consider the following one-parameter group:

$\displaystyle{}\tilde{x}$ $\displaystyle=$ $\displaystyle
x+\varepsilon\xi(x,t,u)+O(\varepsilon^{2}),$ $\displaystyle\tilde{t}$
$\displaystyle=$ $\displaystyle t+\varepsilon\eta(x,t,u)+O(\varepsilon^{2}),$
(4.3) $\displaystyle\tilde{u}$ $\displaystyle=$ $\displaystyle
u+\varepsilon\varphi(x,t,u)+O(\varepsilon^{2}),$

Assume that the equation $\Delta_{1}(x,u^{(n)}):=\mbox{eq}(\ref{eq:1})$ is
invariant under the transformation group (4.1) with the following invariant
surface condition:

$\displaystyle{}\Delta_{2}(x,u^{(n)}):=\eta u_{t}+\xi u_{x}-\varphi=0$ (4.4)

This means that ${X^{(4)}\Delta_{1}}|_{\Delta_{1}=0,\Delta_{2}=0}=0$, where

$X=\xi(x,t,u)\partial_{x}+\eta(x,t,u)\partial_{t}+\varphi(x,t,u)\partial_{u},$

is the infinitesimal generator of (4.1), and

$X^{(4)}=X+\varphi^{x}\partial_{u_{x}}+...+\varphi^{tttt}\partial_{u_{tttt}},$

is the fourth prolongation of $X$, with the coefficients defined as
$\varphi^{J}=D_{J}Q+\xi u_{Jx}+\eta u_{Jt}$, where $Q=\varphi-\xi u_{x}-\eta
u_{t}$ is the Lie characteristic and
$D_{J}=\sum_{i=0}u_{Ji}\,\partial_{u_{J}}$ is the total derivative w.r.t. $J$
(Find more information in [8, 9])

Without loss of generality in condition (4.4), two cases $\eta=0$ and $\eta=1$
must be considered.

Case I $\eta=1$: In this case we have $u_{t}=\varphi-\xi u_{x}$. Substituting
this expression in (1.1) we have $D_{t}(\varphi-\xi
u_{x})=D_{t}(u_{xxt}-u_{x}-uu_{x})$, where $D_{t}$ is total derivative w.r.t.
$t$. By substituting $\xi u_{xx}$ in both sides of above, we find

$\displaystyle\varphi^{t}$ $\displaystyle=$ $\displaystyle
u_{xxtt}-u_{xt}-u_{t}u_{x}-uu_{xt}+\xi u_{xx}$ $\displaystyle=$ $\displaystyle
D_{xxt}(u_{t})-(u+1)D_{x}(u_{t})+(\xi u_{x}-\varphi)u_{x}+\xi u_{xx},$
$\displaystyle=$ $\displaystyle D_{xxt}(\varphi-\xi
u_{x})-(u+1)D_{x}(\varphi-\xi u_{x})+(\xi u_{x}-\varphi)u_{x}+\xi u_{xx},$
$\displaystyle=$ $\displaystyle\varphi^{xxt}-\xi
u_{xxxt}-(u+1)\varphi^{x}+(u+1)\xi u_{xx}+(\xi u_{x}-\varphi)u_{x}+\xi
u_{xx},$

By virtue of $D_{x}(u_{t})=D_{x}(u_{xxt}-u_{x}-uu_{x})$, we have
$u_{xt}=u_{xxxt}-u_{xx}-uu_{xx}-u_{x}^{2}$. Finally, we find the following
governing equation:

$\displaystyle\varphi^{t}=\varphi^{xxt}-(u+1)\varphi^{x}-\varphi u_{x},$ (4.6)

where $\varphi^{t}=D_{t}(\varphi-\xi u_{x})+\xi u_{xt}$,
$\varphi^{x}=D_{x}(\varphi-\xi u_{x})+\xi u_{xx}$, and

$\varphi^{xxt}=D_{xxt}(\varphi-\xi u_{x})+\xi u_{xxxt}.$

By substituting the coefficient functions
$\varphi^{t},\varphi^{x},\varphi^{xxt}$ into invariance condition (4.6), we
are left with a polynomial equation involving the various derivatives of
$u(x,t)$ whose coefficients are certain derivatives of $\xi$ and $\varphi$.
Since, $\xi$ and $\varphi$ depend only on $x$, $t$, $u$ we can equate the
individual coefficients to zero, leading to these complete set of determining
equations: $\xi_{x}=\xi_{t}=\xi_{u}=0$, $\varphi=0$. So, we have $\xi=c_{1}$,
$\varphi=0$. So, we find the infinitesimal generators of the non-classical
symmetries using the above results as follows, when $c_{1}=1$, we have
$\sigma_{1}=u_{x}+u_{t}$, and for $c_{1}\neq 0$ the symmetries are
$\sigma_{2}=u_{x}$, $\sigma_{3}=u_{t}$. As a result we can state the following
proposition:

#### Proposition

The non-classical symmetries of the BBM equation in the case of $\eta=1$,
spanned by

$\displaystyle\sigma_{1}=u_{x}+u_{t},\qquad\sigma_{2}=u_{x},\qquad\sigma_{3}=u_{t}.$
(4.7)

As a result of above proposition we have the following group-invariant
solutions:

  * 1)

For $\mathbf{\sigma_{1}}=u_{x}+u_{t}$, substituting it into $\sigma_{1}(u)$ we
find $u=F(x-t)$, where $F$ must satisfy in:
$FF^{\prime}-F^{\prime\prime\prime}=0$

  * 2)

For $\mathbf{\sigma_{2}}=u_{x}$, substituting it into $\sigma_{2}(u)=0$ we
find $u=F(t)$ for an arbitrary $F$, so from equation (1.1) we obtain: $u=0$.

  * 3)

For $\mathbf{\sigma_{3}}=u_{t}$, substituting it into $\sigma_{3}(u)=0$ we
find $u=F(x)$, where from equation (1.1) $F$ satisfies this equation:
$F^{\prime}+FF^{\prime}+F^{\prime\prime\prime}=0$.

Case II $\eta=0$: In this case, without lose of generality we can let $\xi=1$,
so we have: $u_{x}=\varphi$. Using this we can deduce
$A(x,t,u)=\varphi_{xt}-\varphi-u\varphi$. Subsisting this in the determining
equation $A\varphi_{u}+\varphi_{t}-A_{u}\varphi-A_{x}=0$, we obtain:

$\displaystyle\varphi_{xt}\varphi_{u}-2\varphi\varphi_{u}-u\varphi\varphi_{u}+\varphi_{t}=\varphi_{xtu}\varphi+u\varphi_{u}\varphi+2\varphi^{2}+\varphi_{xxt}+\varphi_{x}+u\varphi_{x}.$
(4.8)

By assuming $\varphi=\varphi(x,t)$ above equation changes into

$\varphi_{t}-2\varphi^{2}-\varphi_{xxt}-\varphi_{x}-u\varphi_{x}=0.$

So we have: $\varphi=1/(c-2x)$. As a result, we deduce that
$u(x,t)=x/(c-2t)+F(t)$ (where $F$ is an arbitrary function) is a solution of
(1.1).

###  4.2 Using the classical symmetries of KdV type equation (1.2)

Since the appearance forms of equation (1.1) and (1.2) are similar, we want to
try our chance in order to obtain new similarity reduced ODEs for BBM equation
through infinitesimal generators of the classical symmetries (CS) of KdV type
equation as the starting constraint. For the classical symmetries of the KdV
type equation using Lie classical symmetry we have the next theorem (Since,
the proof is computational, to keep scope we don’t present it here. Find more
information in [8, 9]).

#### Theorem

If we consider
$X=\xi(x,t,u)\partial_{x}+\eta(x,t,u)\partial_{t}+\eta(x,t,u)\partial_{u}$ as
the infinitesimal generator of the classical symmetry group of the KdV type
equation (1.2), then we have

$\displaystyle\eta=c_{1}t+c_{2},\qquad\xi=\frac{1}{3}c_{1}(x+2t)+c_{3}t+c_{4},\qquad\varphi=-\frac{2}{3}c_{1}u+c_{3},$
(4.9)

where $c_{1}$, $c_{2}$, $c_{3}$ and $c_{4}$ are arbitrary constants.

Hence the next corollary could be stated:

#### Corollary

The classical symmetries of equation (1.2) i.e. KdV type equation, spanned by:

$\displaystyle
X_{1}=(x+2t)\partial_{x}+3t\partial_{t}-2u\partial_{u},\;\;X_{2}=\partial_{t},\;\;X_{3}=t\partial_{x}+\partial_{u},\;\;X_{4}=\partial_{x}.$
(4.10)

So, we can consider any linear combinations of given vector fields in the
above corollary as the starting constraint of the weak symmetry method. In
continuation, we will illustrate the weak symmetry method using some linear
combination of $X_{1}$, $X_{2}$, $X_{3}$ and $X_{4}$ as the starting point.

#### Example

Consider the one-parameter transformation group with the infinitesimal
generator $X_{2}+X_{3}=t\partial_{x}+\partial_{t}+\partial_{u}$. The
characteristic equation is $dx/t=dt=du$. So, we find the differential
invariants as $r=t^{2}-2x$, $w=u/t$. By substituting these new variables in
the original equation (1.1) we deduce
$(2w_{r}-2ww_{r}-w_{rrr})t^{2}=2w_{r}t+4w_{rr}+w$, where $t$ can be considered
as the differential parameter. Note that solving the above ODE doesn’t give
new solution.

#### Example

Consider the one-parameter transformation group with the infinitesimal
generator $X_{3}=t\partial_{x}+\partial_{u}$. The characteristic equation is
$dx/t=dt/0=du$. So, we can obtain the differential invariants as $r=t$,
$w=u-x/t$. By substituting these new variables in the original equation (1.1)
we find: $rw_{r}+w-1=0,$, solving this reduced equation we obtain $w=r/(r+c)$.
So we can find $u=(tx+x^{2}+cx)/(t(x+c))$ as the solution of equation (1.1).

###  4.3 Some other suggestions

Some other ideas may be useful to reach other solutions of the BBM equation.
For example, non-classical potential symmetry method or using classical and
non-classical symmetries of other equations which have the similar forms as
the BBM equation. Meanwhile, Physical knowledge of the model framework can be
so effective in order to reach favorite solutions via weak symmetries. For
example if you know your desired solution may be invariant under some scale of
specific variables then the weak symmetry method can be started with an
appropriate scaling transformation. Since the main goal of this paper was
introducing weak symmetry method for BBM equation, we lay away performing of
above approaches.

##  5 More discussions

Now, we want to compare our results with other related papers. Paper [6] is
concentrated on the classical symmetries and optimal Lie system of the BBM
equation. Comparing with [6], we deduce that in this paper by applying the
weak symmetry method we have obtained more similarity solutions and other
useful suggestions are presented in order to reach more other solutions.
Taking into account the sections 2 and 3 of [6], the next theorem can be
resulted (Find more information in ([8], Chapter 3).

#### Theorem

If $u=f(x,t)$ is solution of the BBM equation (1.1), so are the functions

$\displaystyle u=f(x-\varepsilon,t),\quad
u=f(x-\alpha\varepsilon,t-\varepsilon),\quad
u=e^{(u+1)\varepsilon}f(x-\alpha\varepsilon,e^{-\varepsilon t}t),$

where $\varepsilon\ll 1$ and $\alpha$ are arbitrary constants.

Indeed, above theorem characterizes the invariant solutions of the BBM
equation, for instance if $u=c$ is a solution of equation (1.1), then from
this theorem we obtain $u=ce^{\varepsilon(u+1)}$ as a solution of the BBM
equation. For another example, if we consider the solution
$u=(tx+x^{2}+cx)/(t(x+c))$ of equation (1.1), from this theorem we deduce that

$u=\displaystyle{\frac{(t-\varepsilon)(x-\alpha\varepsilon)+(x-\alpha\varepsilon)^{2}+c(x-\alpha\varepsilon)}{(t-\varepsilon)(x-\alpha\varepsilon+c)}},$

(where $\varepsilon\ll 1$ and $\alpha$, $c$ are arbitrary constants), is again
a solution of BBM equation. By using such approach, we are enable to obtain
more new solutions for the BBM equation.

## Conclusions

In this paper, we have presented a comprehensive explanation of the weak
symmetry method as the generalization of the classical Lie symmetry method.
Indeed, we have performed the weak symmetry method for the BBM equation which
has been fulfilled by applying the non-classical symmetries of the BBM
equation and using the classical symmetries of the KdV type equation as the
starting constraints. Also, the similarity reduced equations as well as some
exact solutions of the BBM equation are obtained via this method. Finally, we
have compared our results with papers using the classical symmetry method.
Other suggestions for finding new exact solutions are also presented.

## References

  * [1] Benjamin, T.B. and Bona, J.L. and Mahony, J.J., Model equations for long waves in nonlinear dispersive systems Phil. Trans. R. Soc. 1972, 272 47 78.
  * [2] Bluman, G.W. and Cole, J.D., The general similarity solution of the heat equation, J. Math. Mech. 18 A969X 1025-1042.
  * [3] Bona, J.L. and Bryant, P.J., A mathematical model for long waves generated by wave makers in nonlinear dispersive systems Proc. Cambridge Phil. Soc. 1973, 73 391 405.
  * [4] Cai, G. and Wang, Y. and Zhang, F., Nonclassical symmetries and group invariant solutions of Burgers-Fisher equations. World Journal of Modelling and Simulation Vol. 3 (2007) No. 4, pp. 305-309.
  * [5] Cai, G. and Ling, X., Nonclassical symmetries of a class of nonlinear partial differential equations and compatibility. World Journal of Modelling and Simulation Vol. 3. 2007. No. 1, pp. 51-57.
  * [6] Karaca, M.A. and Hizel, E., Similarity reductions of Benjamin-Bona-Mahony equation, Applied Mathematical Sciences, Vol. 2, 2008, no. 10, 463 - 469
  * [7] Lie, S., Uber die integration durch bestimmte integrale von einer klasse linear partieller differentialgleichung, Arch, for Math. 6 A881), pp. 328-368; also Gesammelte Abhandlungen, vol. 3, B.G. Teubner, Leipzig, 1922, pp. 492-523.
  * [8] Olver, P.J., Applications of Lie groups to differential equations, New York, Springer, 1986.
  * [9] Olver, P.J., Equivalence, invariants and symmetry, Cambridge University Press, 1995.
  * [10] Olver, P.J. and Rosenau, P., Group-invariant solutions of differential equations Siam J., Applied Mathematics, Vol 47, No 2, April 1987\.


††footnotetext:  File: main.tex, printed: 2024-9-1, 4.16

# Radius of concavity for certain class of functions

Molla Basir Ahamed  Molla Basir Ahamed, Department of Mathematics, Jadavpur
University, Kolkata-700032, West Bengal, India.
<EMAIL_ADDRESS>and  Rajesh Hossain  Rajesh Hossain,
Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal,
India<EMAIL_ADDRESS>
###### Abstract.

Let $\mathcal{S}(p)$ be the class of all meromorphic univalent functions
defined in the unit disc $\mathbb{D}$ of the complex plane with a simple pole
at $z=p$ and normalized by the conditions $f(0)=0$ and $f^{\prime}(0)=1$. In
this paper, we find radius of concavity and compute the same for functions in
$\mathcal{S}(p)$ and for some other well-known classes of functions on unit
disk. We explore general linear combinations
$F(z):=\lambda_{1}f_{1}(z)+\cdots+\lambda_{2n}f_{2n}(z),\;\lambda_{j}\in\mathbb{C}$,
$n\in\mathbb{N}$, of functions belonging to the class $\mathcal{S}(p)$ and
some other classes of functions of analytic univalent functions and
investigate their radii of univalence, convexity and concavity.

###### Key words and phrases:

Meromorphic functions, Convex functions, Concave functions, Radius of
univalence, Radius of convexity, Radius of concavity

###### 2020 Mathematics Subject Classification:

30C45, 30C55

##  1\. Introduction

In the article, we denote the open unit disk of the complex plane $\mathbb{C}$
by $\mathbb{D}:=\\{z\in\mathbb{C}:|z|<1\\}$. We consider the class
$\mathcal{A}(p)$, where $p\in(0,1)$ consisting of all meromorphic functions in
$\mathbb{D}$ with a simple pole at $z=p$ and normalized by the condition
$f(0)=0=f^{\prime}(0)-1$. Let $\mathcal{S}(p)$ be the class of all univalent
functions in $\mathcal{A}(p)$. In [10], Fenchel have obtained the sharp lower
bound of $|f(z)|$ for $f\in\mathcal{S}(p)$ and the upper bound was established
by Kirwan and Schober (see [17]). It is well-known that if
$f\in\mathcal{S}(p)$, then

(1.1) $\displaystyle|k_{p}(-r)|\leq|f(z)|\leq|k_{p}(z)|,\;\;|z|=r<1,$

where

$\displaystyle
k_{p}(z):=\frac{-pz}{(z-p)(1-pz)}\in\mathcal{S}(p)\;\mbox{for}\;z\in\mathbb{D}.$

Let ${\rm Co}(p)$ be a subclass of $\mathcal{S}(p)$ consisting of functions
$f$ such that $\overline{\mathbb{D}}\setminus f(\mathbb{D})$ is a bounded
convex set, where $\overline{\mathbb{C}}:=\mathbb{C}\cup\\{\infty\\}$. In
[26], it is proved that $f\in{\rm Co}(p)$ if, and only if,
$f\in\mathcal{S}(p)$ and there exists a holomorphic function $P_{f}$ in
$\mathbb{D}$ such that

$\displaystyle{\rm
Re}\;P_{f}(z)>0,\;z\in\mathbb{D},\;P_{f}(p)=\frac{1+p^{2}}{1-p^{2}}\;\mbox{and}\;P_{f}(0)=1,$

where

(1.2) $\displaystyle
P_{f}(z)=-\left(1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}+\frac{z+p}{z-p}-\frac{1+pz}{1-pz}\right).$

In this paper, our main focus is on the classical radius problem in geometric
function theory. To describe the problem and the results obtained in this
area, we will first introduce the following classes of functions. Let
$\mathcal{A}$ be the set of all analytic functions in $\mathbb{D}$, normalized
by the conditions $f(0)=0$ and $f^{\prime}(0)=1$. Let $\mathcal{S}$ be the
class of univalent functions in $\mathcal{A}$. If a function $f\in\mathcal{A}$
maps $\mathbb{D}$ conformally onto a convex domain, then we call $f$ a convex
function. Similarly, if a function $f\in\mathcal{A}$ maps $\mathbb{D}$
conformally onto a domain that is starlike with respect to the origin, then we
call $f$ a starlike function. Let $\mathcal{C}$ and $\mathcal{S}^{*}$ be the
sub-classes of $\mathcal{S}$ consisting of all convex and starlike functions,
respectively. The radius of convexity (or starlikeness) of a subset
$\mathcal{A}_{1}$ of $\mathcal{A}$ is defined as the largest number
$r\in(0,1]$ such that each function $f\in\mathcal{A}_{1}$ is convex (or
starlike) in $\mathbb{D}_{r}=\\{z\in\mathbb{C}:|z|<r\\}$. In 1920, Nevanlinna
(see [24]) proved that the radius of convexity of $\mathcal{S}$ is
$2-\sqrt{3}$, and in 1934, Grunsky (see [15]) proved that the radius of
starlikeness of $\mathcal{S}$ is $\tan(\pi/4)$.

Recently, in [5] the radius of concavity is defined. Henceforth, it is worth
observing that $g(z):=r^{-1}f(rz)$, $z\in\mathbb{D}$ does not belong to ${\rm
Co}(p)$, whenever $f\in{\rm Co}(p)$. Consequently, it cannot be concluded that
if $\overline{\mathbb{D}}\setminus f(\mathbb{D})$ is convex, then
$\overline{\mathbb{D}}\setminus f(\mathbb{D}_{r})$ is convex for each $0<r\leq
1$. Because of this fact, the radius of concavity (w.r.t ${\rm Co}(p)$) of
subset of $\mathcal{A}(p)$ in the following way.

######  Definition 1.1.

[5] The radius of concavity (w.r.t ${\rm Co}(p)$) of a subset
$\mathcal{A}_{1}(p)$ of $\mathcal{A}(p)$ is the largest number $R_{{\rm
Co}(p)}\in(0,1]$ such that for each function $f\in\mathcal{A}_{1}(p)$, ${\rm
Re}\;P_{f}(z)>0$ for all $|z|<R_{\rm Co(p)}$, where $P_{f}(z)$ is defined in
(1.2).

In this paper we find a lower bound of the radius of concavity $R_{\rm Co(p)}$
of the class $\mathcal{S}(p)$. Now, we consider functions $f$ in $\mathcal{A}$
that map $\mathbb{D}$ conformally onto a domain whose complement with respect
to $\mathbb{C}$ is convex and that satisfy the normalization $f(1)=\infty$. We
will denote these families of functions by ${\rm Co}(A)$ and $f\in{\rm Co}(A)$
is, and only if, ${\rm T_{f}(z)}>0$ for every $z\in\mathbb{D}$, where
$f(0)=f^{\prime}(0)-1$ and

(1.3) $\displaystyle
T_{f}(z)=\frac{2}{A-1}\left(\frac{(A+1)}{2}\frac{1+z}{1-z}-1-z\frac{f^{\prime\prime}(z)}{f^{\prime}(z)}\right).$

######  Definition 1.2.

[5] The radius of concavity (w.r.t ${\rm Co(A)}$) of a subset
$\mathcal{A}_{1}$ of $\mathcal{A}$ is the largest number
$\mathrm{R}_{\mathrm{Co(A)}}\in(0,1]$ such that for each function
$f\in\mathcal{A}_{1}$, ${\rm Re}\;T_{f}(z)>0$ for all
$|z|<\mathrm{R}_{\mathrm{Co(A)}}$, where $T_{f}(z)$ is defined in (1.3).

Our aim in this paper is to find the sharp radius of concavity
$\mathrm{R}_{\mathrm{Co(A)}}$ of $\mathcal{S}$. Henceforth, we now discuss
about the general linear combinations of univalent functions which will be
required to demonstrate our results. To be more precise, we will examine the
radius of concavity of the general (other that convex or similar) combinations
of functions of the form

(1.4) $\displaystyle
F(z):=\lambda_{1}f_{1}(z)+\cdots+\lambda_{2n}f_{2n}(z),\;\lambda_{j}\in\mathbb{C}$

in $\mathcal{S}$ and ${\rm Co}(A)$ with complex coefficients. Moreover, we
determine radii of univalence and convexity of the general linear combinations
of functions in ${\rm Co}(A)$ as defined in (1.4). We use new method to prove
our results and out study may be applied for other classes of functions and
similar study can be conducted. In fact, our result are established in a such
was that they will not generalize several existing results but also broaden
the scope of study in this direction.

###  1.1. Motivation

Till date, radius of convexity or concavity is obtained only for convex
combinations of univalent functions however, it will be interesting to know
whether we can find a radius of concavity or convexity for general linear
combinations of univalent functions. In this paper, inspired by the results of
Bhowmik and Biswas [5], we obtained the follow result finding the radius of
convexity or concavity for the class $\mathcal{S}$. Firstly, we study on
Univalent functions and read [29],[5] many others paper. Lemma 1 of Stump in
[29], and Lemma 4 of Bhowmik and Biswas in [5], motivated us to establish a
general result for the general linear combination of functions. We find out a
way how to constitute the combination of functions and we bravely constitute
the function as well.

###  1.2. Some auxiliary results

We use a result from [12, P.65] and using our Lemma 1.1, then we have found
the lower bound for the class $\mathcal{S}$.

######  Lemma A.

[29, Lemma 1] If $|w-a|<d$, where $a$ and $d$ are real numbers such that
$a>d\geq 0$, and $w_{0}$ is a given complex number, then

$\displaystyle\mbox{Re}\;(ww_{0})\geq|w_{0}|(a\cos(\arg\;w_{0})-d).$

In $1971$, Robert K. Stump (see [29, Lemma 2]) obtained lower bound of
$\mbox{Re}\;(ww_{0})$ as $|w_{0}|(a\cos(\arg\;w_{0})-d)$. Recently, Bhowmik
and Biswas ( see [5, Lemma 4]) found the upper bound of $\mbox{Re}\;(ww_{0})$
as $|w_{0}|(a\cos(\arg\;w_{0})+d)$ and obtained the following lemma.

######  Lemma B.

[5, Lemma 4] If $|u-a|\leq d$ and $|v-a|\leq d$, where $a$ and $d$ are real
and $a>d\geq 0$, and

$\displaystyle w=\frac{u}{1+be^{i\alpha}}+\frac{v}{1+b^{-1}e^{-i\alpha}}$

with $b\in(0,\infty)$ and $\alpha\in[0,\pi)$, then

$\displaystyle a-d\sec\frac{\alpha}{2}\leq\mbox{Re}(w)\leq
a+d\sec\frac{\alpha}{2}.$

The radii of univalence, concavity, and convexity of general linear
combinations in ${\rm Co}(A)$. To establish our results, we need the following
lemma proved by Robert K. Stump in 1971, which will also be use later to
obtain results related to the radius problem for different subclasses of
univalent functions.

######  Lemma C.

[29, Lemma 3] If ${\rm Re}\;P_{F(z)}$>0 for $|z|<\rho<1$ and $P(0)=1$, then

$\displaystyle\vline
P(z)-\frac{1+\frac{r^{2}}{\rho^{2}}}{1-\frac{r^{2}}{\rho^{2}}}\vline\leq\frac{\frac{2r}{\rho}}{1-\frac{r^{2}}{\rho^{2}}}.$

Considering the class $\mathcal{S}$ and examining the radius of concavity of
general linear combination of functions in $\mathcal{S}$, we first prove the
following lemma which we will use later in the proof of the main results. We
note that the Lemma 1.1 is a general form of the Lemma B.

######  Lemma 1.1.

If $|u_{j}-a|\leq d$ and $|v_{j}-a|\leq d$ , $(j=1,2,\ldots,n)$, where $a$ and
$d$ are real numbers such that $a>d\geq 0$ and

(1.5) $\displaystyle
w=\sum_{j=1}^{n}\frac{u_{j}}{1+be^{i\alpha_{j}}}+\sum_{j=1}^{n}\frac{v_{j}}{1+b^{-1}e^{-i\alpha_{j}}}$

with $b_{j}\in(0,\infty)$ and $\alpha_{j}\in[0,\pi)$, then

$\displaystyle an-d\sum_{j=1}^{n}\sec\left(\frac{\alpha_{j}}{2}\right)\leq{\rm
Re}\;(w)\leq an+d\sum_{j=1}^{n}\sec\left(\frac{\alpha_{j}}{2}\right).$

######  Remark 1.1.

This lemma is generalization and a compact form of [5, Lemma 4], as well as
[29, Lemma 2].

###### Proof.

For $k\in\\{1,2,\ldots,n\\}$, by Lemma A, we have

(1.6)
$\displaystyle\mbox{Re}\left(\frac{u_{k}}{1+be^{i\alpha_{k}}}\right)\geq\vline\frac{1}{1+be^{i\alpha_{k}}}\vline\left(a\cos\left(\arg\left(\frac{1}{1+be^{i\alpha_{k}}}\right)\right)-d\right).$

Assuming

(1.7) $\displaystyle\frac{1}{1+be^{i\alpha_{k}}}=re^{i\theta_{k}}$

we have

(1.8) $\displaystyle\vline\frac{1}{1+be^{i\alpha_{k}}}\vline\;\;\
=\frac{1}{(1+2b\cos\alpha_{k}+b^{2})^{\frac{1}{2}}}.$

Thus it follows from (1.7) that

$\displaystyle\frac{1+b\cos\alpha_{k}-i\sin\alpha_{k}}{1+2b\cos\alpha_{k}+b^{2}}=r\cos\theta_{k}+ir\sin\theta_{k}.$

Comparing real part from both side of the above estimate, we obtain that

(1.9) $\displaystyle
r\cos\theta_{k}=\frac{1+b\cos\alpha_{k}}{1+2b\cos\alpha_{k}+b^{2}}.$

In view of (1.7), (1.8) and (1.9), we obtain

$\displaystyle\cos\theta_{k}=\frac{1+b\cos\alpha_{k}}{(1+2b\cos\alpha_{k}+b^{2})^{\frac{1}{2}}}.$

From (1.6), for any $k\in\\{1,2,\ldots,n\\}$, we see that

(1.10)
$\displaystyle\mbox{Re}\left(\frac{u_{k}}{1+be^{i\alpha_{k}}}\right)\geq\frac{1}{(1+2b\cos\alpha_{k}+b^{2})^{\frac{1}{2}}}\left(\frac{a(1+b\cos\alpha_{k})}{(1+2b\cos\alpha_{k}+b^{2})^{\frac{1}{2}}}-d\right).$

For $k\in\\{1,2,\ldots,n\\}$, in view of Lemma A, a routine computation shows
that

(1.11)
$\displaystyle\mbox{Re}\left(\frac{-u_{k}}{1+be^{i\alpha_{k}}}\right)\geq\vline\frac{1}{1+be^{i\alpha_{k}}}\vline\left(a\cos\left(\arg\left(\frac{-1}{1+be^{i\alpha_{k}}}\right)\right)-d\right).$

Assuming

(1.12) $\displaystyle\frac{-1}{1+be^{i\alpha_{k}}}=re^{i\theta_{k}},$

we see that

(1.13) $\displaystyle\vline\frac{1}{1+be^{i\alpha_{k}}}\vline\;\;\
=\frac{1}{(1+2b\cos\alpha_{k}+b^{2})^{\frac{1}{2}}}.$

From (1.12), we see that

$\displaystyle\frac{-1-be^{-i\alpha_{k}}}{(1+be^{i\alpha_{k}})(1+be^{-i\alpha_{k}})}=re^{i\theta_{k}},$

i.e.,

$\displaystyle\frac{-1-b\cos\alpha_{k}+i\sin\alpha_{k}}{1+2b\cos\alpha_{k}+b^{2}}=r\cos\theta_{k}+ir\sin\theta_{k}.$

Comparing real part from both sides, we obtain

(1.14) $\displaystyle
r\cos\theta_{k}=\frac{-1-b\cos\alpha_{k}}{1+2b\cos\alpha_{k}+b^{2}}.$

In view of (1.12),(1.13) and (1.14), we obtain

$\displaystyle\cos\theta_{k}=\frac{-1-b\cos\alpha_{k}}{(1+2b\cos\alpha_{k}+b^{2})^{\frac{1}{2}}}.$

Thus from (1.11), for any $k\in\\{1,2,\ldots,n\\}$, we get

(1.15) $\displaystyle{\rm
Re}\left(\frac{u_{k}}{1+be^{i\alpha_{k}}}\right)\leq\frac{1}{(1+2b\cos\alpha_{k}+b^{2})^{\frac{1}{2}}}\left(\frac{a(1+b\cos\alpha_{k})}{(1+2b\cos\alpha_{k}+b^{2})^{\frac{1}{2}}}+d\right).$

Similarly, we obtain

(1.16) $\displaystyle{\rm
Re}\left(\frac{v_{k}}{1+b^{-1}e^{-i\alpha_{k}}}\right)\geq\frac{b}{(1+2b\cos\alpha_{k}+b^{2})^{\frac{1}{2}}}\left(\frac{a(b+\cos\alpha_{k})}{(1+2b\cos\alpha_{k}+b^{2})^{\frac{1}{2}}}-d\right)$

and

(1.17) $\displaystyle{\rm
Re}\left(\frac{v_{k}}{1+b^{-1}e^{-i\alpha_{k}}}\right)\leq\frac{b}{(1+2b\cos\alpha_{k}+b^{2})^{\frac{1}{2}}}\left(\frac{a(b+\cos\alpha_{k})}{(1+2b\cos\alpha_{k}+b^{2})^{\frac{1}{2}}}+d\right).$

Therefore, combining (1.10), (1.15), (1.16) and (1.17), we obtain

$\displaystyle\sum_{j=1}^{n}$
$\displaystyle\frac{a(1+b\cos\alpha_{j}+b(b+\cos\alpha_{j}))}{(1+2b\cos\alpha_{j}+b^{2})}-d\sum_{j=1}^{n}\frac{(1+b)}{(1+2b\cos\alpha_{j}+b^{2})^{1/2}}\leq\mbox{Re}\;{w}$
(1.18)
$\displaystyle\leq\sum_{j=1}^{n}\frac{a(1+b\cos\alpha_{j}+b(b+\cos\alpha_{j}))}{(1+2b\cos\alpha_{j}+b^{2})}+d\sum_{j=1}^{n}\frac{(1+b)}{(1+2b\cos\alpha_{j}+b^{2})^{1/2}}.$

This shows that

$\displaystyle
na-d\sum_{j=1}^{n}\frac{(1+b)}{(1+2b\cos\alpha_{j}+b^{2})^{1/2}}\leq\mbox{Re}\;{w}\leq
na+d\sum_{j=1}^{n}\frac{(1+b)}{(1+2b\cos\alpha_{j}+b^{2})^{1/2}}.$

To make the computation easy, for any $k\in\\{1,\ldots,n\\}$, we define

$\displaystyle G_{k}(b):=\frac{(1+b)}{(1+2b\cos\alpha_{j}+b^{2})^{1/2}}.$

Differentiating $G_{k}$ with respect to $b$, we obtain

$\displaystyle\frac{dG_{k}(b)}{db}=\frac{(\cos\alpha_{k}-1)(b-1)}{(1+2b\cos\alpha_{k}+b^{2})}.$

For critical point of $G_{k}$,

$\displaystyle\frac{dG_{k}(b)}{db}=0\implies(\cos\alpha_{k}-1)(b-1)=0.\implies
b=1.$

Again differentiating $\frac{dG_{k}(b)}{db}$ with respect to $b$ and at $b=1$,
we have

$\displaystyle\frac{d^{2}G_{k}(b)}{db^{2}}=\frac{(\cos\alpha_{k}-1)}{(2+2\cos\alpha_{k})^{\frac{5}{2}}}<0\;\;\mbox{for
all}\;\;\alpha_{k}\in\left(0,\pi\right).$

Thus, the maximum value of $G_{k}$ is attained at $b=1$, hence

$\displaystyle
G_{k}(b)\leq\frac{2}{\left(2+2\cos\alpha_{j}\right)^{\frac{1}{2}}}.$

Therefore, we have

$\displaystyle
an-d\sum_{j=1}^{n}\frac{2}{(2+2\cos\alpha_{j})^{\frac{1}{2}}}\leq\mbox{Re}\;{w}\leq
an+d\sum_{j=1}^{n}\frac{2}{(2+2\cos\alpha_{j})^{\frac{1}{2}}}$

which can be written as

$\displaystyle an-d\sum_{j=1}^{n}\mbox{sec}\frac{\alpha_{j}}{2}\leq{\rm
Re}\;{w}\leq an+d\sum_{j=1}^{n}\mbox{sec}\frac{\alpha_{j}}{2}.$

Hence, completes the proof. ∎

###  1.3. Some results on radius of concavity for the classes $\mathcal{S}$,
$\mathcal{S}(p)$, ${\rm Co}(A)$

In $1971$, Robert K. Stump (see[29, Theorem 1]) obtained the radius of
starlikeness for the linear combination of two convex functions. Recently, in
a similar approach to Stump [5], a lower bound for the radius of concavity was
determined for the linear combination of two analytic univalent functions
$f_{1}$ and $f_{2}$.

######  Theorem A.

[5, Theorem 5] Let $f_{j}\in\mathcal{S}$, $j=1,2$ and

$\displaystyle F(z)=\lambda f_{1}(z)+(1-\lambda)f_{2}(z),$

where $\lambda\in\mathbb{C}\;\;\mbox{such
that}\;\;0\leq\alpha:=\arg[\lambda/(1-\lambda)]<\pi.$ Then $\mathrm{Re}$
$T_{F}(z)$ $>0$ for $|z|<\mathrm{R}_{\mathrm{Co(A)}}$, where,
$\mathrm{R}_{\mathrm{Co(A)}}$ is the least value of
$r\in(0,\sin(\pi-\alpha)/8))$ satisfying the equation $\phi(r)=0$, where

$\displaystyle\phi_{1}(r):=r^{2}-\left(\frac{2(A+1)+8\sec(\alpha/2+4\sin^{-1}r)}{A-1}\right)r+1.$

Authors in [5] have considered the class $\mathcal{S}(p)$ of meromorphic
univalent functions and obtained the radii of univalence, convexity, and
concavity of the linear combination $\lambda f_{1}(z)+(1-\lambda)f_{2}(z)$ of
functions in $\mathcal{S}(p)$ and determined a lower bound for the radius of
univalence in the following theorem.

######  Theorem B.

[5, Theorem 9] Let $f_{j}\in\mathcal{S}(p)$, $j=1,2$ and $F(z)=\lambda
f_{1}(z)+(1-\lambda)f_{2}(z)$, where $\lambda\in\mathbb{C}\;\;\mbox{such
that}\;\;\lambda\in\mathbb{C}\;\;\mbox{such
that}\;\;0\leq\alpha:=\arg[\lambda/(1-\lambda)]<\pi.$ Then $F$ is univalent in
$|z|<R_{u}$, where

$\displaystyle
R_{u}:=r_{1}\left(\sec\frac{\alpha}{2}-\tan\frac{\alpha}{2}\right)$

and $r_{1}$ is the smallest value of $r\in(0,p)$ that satisfies the equation
$\phi_{2}(r)=0$,

$\displaystyle\phi_{2}(r):=p(k+1)r^{2}-k(1+p^{2})r+p(k-1)=0$

with $k=e^{\pi/4}$.

We recall here a result [5, Theorem 3] which provides a lower bound for the
radius of concavity of the class $\mathcal{S}(p)$ and it is determined the
same for linear combinations of functions in $\mathcal{S}(p)$.

######  Theorem C.

[5, Theorem 10] Let $f_{j}\in\mathcal{S}(p),j=1,2$ and $F(z)=\lambda
f_{1}(z)+(1-\lambda)f_{2}(z)$, where $\lambda\in\mathbb{C}$ and
$0\leq\alpha:=\arg[\lambda/(1-\lambda)]<\pi$. Then ${\rm
Re}\left(1+\frac{zF^{\prime\prime}(z)}{F^{\prime}(z)}\right)>0$ for
$|z|<R_{c}$, where $R_{c}$ is the smallest value of $r\in(0,r_{1})$ satisfying
the equation

$\displaystyle\phi_{3}(r):=$ $\displaystyle
pr^{4}-(1+p^{2})(1+2\sec{\beta_{1}})r^{3}+2p(1+4\sec{\beta_{1}})r^{2}$
$\displaystyle-(1+p^{2})(1+2\sec{\beta_{1}})r+p=0$

and $r_{1}$ is the smallest value of $r\in(0,p)$ satisfying the equation

$\displaystyle\xi(r):=p(k+1)r^{2}-k(1+p^{2})r+p(k-1)=0,$

with

$\displaystyle
k=e^{\frac{\pi-\alpha}{4}}and\;\;\beta_{1}=\frac{\alpha}{2}+\log\frac{p^{2}(1-r^{2})^{2}}{(p-r)^{2}(1-pr)^{2}}.$

In [5], a lower bound for the radius of univalence of the linear combination
of two concave functions is determined using a technique similar to that used
by Stump in [29].

######  Theorem D.

[5, Theorem 6] Let $f_{j}\in{\rm Co}(A)$, $j=1,2$ and $F(z)=\lambda
f_{1}(z)+(1-\lambda)f_{2}(z)$, where $\lambda\in\mathbb{C}\;\;\mbox{such
that}\;\;0\leq\alpha:=\arg[\lambda/(1-\lambda)]\leq\pi.$ Then $F$ is univalent
in $|z|<R_{u}$, where,

$\displaystyle
R_{u}:=\sin\left(\frac{\pi}{4A}\right)\left(\sec\frac{\alpha}{2}-\tan\frac{\alpha}{2}\right).$

In the following theorem, (see [5]), a lower bound for the radius of concavity
of the linear combination of two concave functions in ${\rm Co}(A)$ is
determined and the following result is obtained.

######  Theorem E.

[5, Theorem 7] Let $f_{j}\in{\rm Co}(A)$, $j=1,2$ and $F(z)=\lambda
f_{1}(z)+(1-\lambda)f_{2}(z)$, where, $\lambda\in\mathbb{C}\;\;\mbox{such
that}\;\;0\leq\alpha:=\arg[\lambda/(1-\lambda)]\leq\pi.$ Then ${\rm
Re}\;T_{F(z)}>0$ for $|z|<R_{Co(A)}$, where $R_{Co(A)}$ is the least value of
$r<\sin\left((\pi-\alpha)/(4A)\right)$ satisfying the equation
$\phi_{4}(r)=0$,

$\displaystyle\phi_{4}(r):=r^{2}-\left(\frac{2(A+1)+4A\sec(\alpha/2+2A\sin^{-1}r)}{A-1}\right)r+1.$

Every function in ${\rm Co}(A)$ is a close-to-convex function of order $A-1$.
It is also proved that the radius of convexity of ${\rm Co}(A)$ is
$A-\sqrt{A^{2}-1}$; (see for instance [4, Cor. 2.13]). In [5], it is
determined a lower bound for the radius of convexity of the linear combination
of functions in ${\rm Co}(A)$ and the following result is obtained.

######  Theorem F.

[5, Theorem 8] Let $f_{j}\in{\rm Co}(A)$, $j=1,2$ and $F(z)=\lambda
f_{1}(z)+(1-\lambda)f_{2}(z)$, where,
$\lambda\in\mathbb{C}\;\;\mbox{and}\;\;0\leq\alpha:=\arg[\lambda/(1-\lambda)]<\pi$.
Then

$\displaystyle\mbox{Re}\left(1+\frac{zF^{\prime\prime}(z)}{F^{\prime}(z)}\right)>0$

for $|z|<R_{c}$, where, $R_{c}$ is the smallest value of
$r<\sin\left((\pi-\alpha)/(4A)\right)$, satisfying the equation
$\phi_{5}(r)=0$,

$\displaystyle\phi_{5}(r)=r^{2}-\left(2A\sec\left(\frac{\alpha}{2}+2A\sin^{-1}r\right)\right)r+1.$

In [5, Theorem 2], the radius of concavity of $\mathcal{S}(p)$ using
Definition 1.1 is determined. In [5], a lower bound of the linear combination
of functions in $\mathcal{S}(p)$ is obtained.

######  Theorem G.

[5, Theorem 11] Let $f_{j}\in\mathcal{S}(p),\;j=1,2$and $F(z)=\lambda
f_{1}(z)+(1-\lambda)f_{2}(z)$, where, $\lambda\in\mathbb{C}$ and
$0\leq\alpha:=\arg[\lambda/(1-\lambda)]<\pi$. Then ${\rm Re}\;P_{F}(z)$>0 for
$|z|<R_{\rm Co(p)}$, where, $R_{\rm Co(p)}$ is the smallest value of
$r\in(0,r_{1})$ satisfying the equation $\phi_{6}(r)=0$,

$\displaystyle\phi_{6}(r)$
$\displaystyle=p^{2}r^{6}-2p(1+p^{2})(1+\sec\alpha_{1})r^{5}-(3p^{2}-p^{4}-1+2(1-p^{2})^{2}\sec\beta_{1})r^{4}$
$\displaystyle+4p(1+p^{2})(1+\sec\alpha_{1})r^{3}-(3p^{2}-p^{4}-1+2(1-p^{2})^{2}\sec\beta_{1})r^{2}$
$\displaystyle-2p(1+p^{2})(1+\sec\beta_{1})r+p^{2}$

and $r_{1}$ is the smallest value of $r\in(0,p)$ satisfying the equation

$\displaystyle\xi(r)=p(k+1)r^{2}-k(1+p^{2})r+p(k-1)=0$

with

$\displaystyle
k=e^{\frac{(\pi-\alpha)}{4}}\;\;\;and\;\;\;\beta_{1}=\frac{\alpha}{2}+\log\frac{p^{2}(1-r^{2})^{2}}{(p-r)^{2})(1-pr)^{2}}.$

Since Lemma 1.1 generalizes the Lemma B and it is mainly applied to prove
Theorems A, B, C, D, E, F and G. It is worth noting that although there has
been significant research on the radius problem for convex combinations or
linear combinations of two functions $f_{1}$ and $f_{2}$ for different
classes, there has been far less attention given to the study of general
convex combinations or linear combinations of functions from those same
classes. Only a few results have been found in [30] but for harmonic
quasiconformal mappings. This lack of attention serves as the primary
motivation for this paper. Our main objective is to address this gap in the
existing literature and contribute to the understanding of the radius problem
for certain class of analytic functions. Therefore, it is natural to ask the
following question for further investigation.

######  Question 1.1.

Can we establish Theorems A to G for general linear combinations for functions
from the corresponding classes?

In this paper, we aim to answer the Question 1.1 affirmatively and organize
the results of the paper as follows: In Section 2, we state the main results
for general linear combination of functions for the classes $\mathcal{S}$,
$\mathcal{S}(p)$, ${\rm Co}(A)$ and present corollaries of each results. In
Section 3, we discuss the proof of the main results.

##  2\. Main results

After carefully examining the proof of Theorems $A$ to $G$, we have come to
understand that the result can be proved not only for a linear combination of
two analytic functions, but also for a general linear combination of analytic
functions. In this paper, we aim to determine both the lower and upper bounds
for the radius of concavity for the general linear combination
$F(z)=\lambda_{1}f_{1}(z)+\cdots+\lambda_{2n}f_{2n}(z)$, where
$n\in\mathbb{N}$ of analytic univalent functions $f_{1}$, $\ldots$, $f_{2n}$
for the classes $\mathcal{S}$, $\mathcal{S}(p)$, ${\rm Co}(A)$. These results
will have implications for several other results [5]. However, the following
formulation will be used in the proof of the main results of this paper.

Given that $F(z)=\lambda_{1}f_{1}(z)+\cdots+\lambda_{2n}f_{2n}(z)$, where
$n\in\mathbb{N}$, and in view of this general combination of functions, we see
that ${zF^{\prime\prime}(z)}/{F^{\prime}(z)}$ can be expressed as

$\displaystyle\frac{zF^{\prime\prime}(z)}{F^{\prime}(z)}$
$\displaystyle=\frac{z(\lambda_{1}f^{\prime\prime}_{1}(z)+\lambda_{2}f^{\prime\prime}_{2}(z)+\cdots+\lambda_{2n}f^{\prime\prime}_{2n}(z))}{\lambda_{1}f^{\prime}_{1}(z)+\lambda_{2}f^{\prime}_{2}(z)+\cdots+\lambda_{2n}f^{\prime}_{2n}(z)}\vspace{2mm}$
(2.1)
$\displaystyle=\dfrac{\dfrac{zf^{\prime\prime}_{1}(z)}{f^{\prime}_{1}(z)}+\dfrac{\lambda_{2}}{\lambda_{1}}\dfrac{zf^{\prime\prime}_{2}(z)}{f^{\prime}_{1}(z)}+\cdots+\dfrac{\lambda_{2n}}{\lambda_{1}}\dfrac{zf^{\prime\prime}_{2n}(z)}{f^{\prime}_{1}(z)}}{1+\dfrac{\lambda_{2}}{\lambda_{1}}\frac{f^{\prime}_{2}(z)}{f^{\prime}_{1}(z)}+\cdots+\dfrac{\lambda_{2n}}{\lambda_{1}}\dfrac{f^{\prime}_{2n}(z)}{f^{\prime}_{1}(z)}}.$

The expression (2) can be further written as

$\displaystyle\frac{zF^{\prime\prime}(z)}{F^{\prime}(z)}$
$\displaystyle=\frac{zf^{\prime\prime}_{1}(z)}{f^{\prime}_{1}(z)}\left[1+\left(\frac{\lambda_{1}}{\lambda_{2}}\frac{f^{\prime}_{1}(z)}{f^{\prime}_{2}(z)}\right)^{-1}+\left(\frac{\lambda_{1}}{\lambda_{3}}\frac{f^{\prime}_{1}(z)}{f^{\prime}_{3}(z)}\right)^{-1}+\cdots+\left(\frac{\lambda_{1}}{\lambda_{2n}}\frac{f^{\prime}_{1}(z)}{f^{\prime}_{2n}(z)}\right)^{-1}\right]^{-1}$
$\displaystyle+\frac{zf^{\prime\prime}_{2}(z)}{f^{\prime}_{2}(z)}\left[\left(\frac{\lambda_{2}}{\lambda_{1}}\frac{f^{\prime}_{2}(z)}{f^{\prime}_{1}(z)}\right)^{-1}+1+\left(\frac{\lambda_{2}}{\lambda_{3}}\frac{f^{\prime}_{2}(z)}{f^{\prime}_{3}(z)}\right)^{-1}+\cdots+\left(\frac{\lambda_{2}}{\lambda_{2n}}\frac{f^{\prime}_{2}(z)}{f^{\prime}_{2n}(z)}\right)^{-1}\right]^{-1}$
$\displaystyle+\cdots$ (2.2)
$\displaystyle+\frac{zf^{\prime\prime}_{2n}(z)}{f^{\prime}_{2n}(z)}\left[\left(\frac{\lambda_{2n}}{\lambda_{1}}\frac{f^{\prime}_{2n}(z)}{f^{\prime}_{1}(z)}\right)^{-1}+\left(\frac{\lambda_{2n}}{\lambda_{2}}\frac{f^{\prime}_{2n}(z)}{f^{\prime}_{2}(z)}\right)^{-1}+\cdots+\left(\frac{\lambda_{2n}}{\lambda_{2n-1}}\frac{f^{\prime}_{2n}(z)}{f^{\prime}_{2n-1}(z)}\right)^{-1}+1\right]^{-1}.$

For brevity, for $t=1,2,\ldots,2n$, we define

$\displaystyle
A_{1t}(z):=\frac{\lambda_{1}}{\lambda_{t}}\frac{f^{\prime}_{1}(z)}{f^{\prime}_{t}(z)},\;A_{2t}(z):=\frac{\lambda_{2}}{\lambda_{t}}\frac{f^{\prime}_{2}(z)}{f^{\prime}_{t}(z)},\ldots,A_{(2n)t}(z):=\frac{\lambda_{2n}}{\lambda_{t}}\frac{f^{\prime}_{2n}(z)}{f^{\prime}_{t}(z)}.$

Hence, (2) takes the following form

$\displaystyle\frac{zF^{\prime\prime}(z)}{F^{\prime}(z)}$
$\displaystyle=\frac{zf^{\prime\prime}_{1}(z)}{f^{\prime}_{1}(z)}\left(1+A_{12}^{-1}+A_{13}^{-1}+\cdots+A_{1(2n)}^{-1}\right)^{-1}$
$\displaystyle+\frac{zf^{\prime\prime}_{2}(z)}{f^{\prime}_{2}(z)}\left(A_{21}^{-1}+1+A_{23}^{-1}+\cdots+A_{2(2n)}^{-1}\right)^{-1}$
$\displaystyle+\cdots+\frac{zf^{\prime\prime}_{2n}(z)}{f^{\prime}_{2n}(z)}\left(A_{(2n)1}^{-1}+A_{(2n)2}^{-1}+\cdots+A_{(2n)(n-1)}^{-1}+1\right)^{-1}.$

We obtain the following result for the class $\mathcal{S}$ by finding the
radius of concavity for general linear combinations of functions.

######  Theorem 2.1.

Let $f_{j}\in\mathcal{S}$, $j=1,2,\ldots,2n$ and
$F(z)=\sum_{j=1}^{2n}\lambda_{j}f_{j}(z)$, where $\lambda_{j}\in\mathbb{C}$
such that $0\leq\alpha_{j}<\pi$. Then ${\rm Re}\;T_{F(z)}>0$ for
$|z|<R^{*}_{\rm{Co}(A),n}$, where $R^{*}_{\rm{Co}(A),n}$ is the least value of
$r\in(0,1)$ satisfying the equation $\psi_{1,n}(r)=0$, where

$\displaystyle\psi_{1,n}(r)=\frac{(A+3-4n)}{(A-1)}r^{2}-\frac{2}{(A-1)}\left(A+1+4\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)r+1.$

######  Remark 2.1.

For combination of two functions $f_{1},f_{2}\in\mathcal{S}$, we have the
following corollary of Theorem 2.1. Corollary 2.1 generalizes Theorem A.

######  Corollary 2.1.

Let $f_{j}\in\mathcal{S}$, $j=1,2$ and
$F(z)=\lambda_{1}f_{1}(z)+\lambda_{2}f_{2}(z)$, where
$\lambda_{j}\in\mathbb{C}$ such that $0\leq\alpha_{j}<\pi$. Then ${\rm
Re}\;T_{F(z)}>0$ for $|z|<R^{*}_{\rm{Co}(A),1}$, where $R^{*}_{\rm{Co}(A),1}$
is the least value of $r\in(0,1)$ satisfying the equation $\psi_{1,1}(r)=0$,
where

$\displaystyle\psi_{1,1}(r)=r^{2}-\frac{2}{(A-1)}\left(A+1+4\sum_{j=1}^{2}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)r+1.$

We will consider the class $\mathcal{S}(p)$ of meromorphic univalent
functions. In this class, we will obtain the radii of univalence, convexity,
and concavity of the general linear combination
$\lambda_{1}f_{1}(z)+\cdots+\lambda_{2n}f_{2n}(z)$ of functions in
$\mathcal{S}(p)$. Furthermore, we will determine a lower bound for the radius
of univalence in the following theorems.

######  Theorem 2.2.

Let $f_{j}\in\mathcal{S}(p)$, $j=1,2,\ldots,2n$ and
$F(z)=\sum_{j=1}^{2n}\lambda_{j}f_{j}(z)$, where $\lambda_{j}\in\mathbb{C}$
such that $0\leq\alpha_{j}<\pi$. Then $F$ is univalent in $|z|<R^{*}_{u,n}$,
where

$\displaystyle
R^{*}_{u,n}=\frac{\rho}{n}\left(\sum_{j=1}^{n}{\sec}\;\left(\frac{\alpha_{j}}{2}\right)-\sqrt{\left(\sum_{j=1}^{n}{\sec}\;\left(\frac{\alpha_{j}}{2}\right)\right)^{2}-n^{2}}\right).$

######  Remark 2.2.

For a combination of two functions $f_{1}$ and $f_{2}$ in $\mathcal{S}(p)$, we
have the following corollary of Theorem 2.2. Corollary 2.2 generalizes Theorem
B.

######  Corollary 2.2.

Let $f_{j}\in\mathcal{S}(p)$, $j=1,2$ and
$F(z)=\lambda_{1}f_{1}(z)+\lambda_{2}f_{2}(z)$, where
$\lambda_{j}\in\mathbb{C}$ such that $0\leq\alpha_{j}<\pi$. Then $F$ is
univalent in $|z|<R^{*}_{u,1}$, where

$\displaystyle R^{*}_{u,1}$
$\displaystyle=\rho\left({\sec}\;\left(\frac{\alpha_{1}}{2}\right)-\sqrt{\left({\sec}\;\left(\frac{\alpha_{1}}{2}\right)\right)^{2}-1}\right)$
$\displaystyle=\rho\left({\sec}\;\left(\frac{\alpha_{1}}{2}\right)-\tan\left(\frac{\alpha_{1}}{2}\right)\right).$

In the case of analytic functions, the radius of convexity of $\mathcal{S}$
was found using a basic estimate (see [12, Theorem 2.13]). Motivated by [5,
Theorem 10], we have determined the lower bound for the radius of convexity of
general linear combinations of concave functions in $\mathcal{S}(p)$.

######  Theorem 2.3.

Let $f_{j}\in\mathcal{S}(p)$, $j=1,2,\ldots,2n$ and
$F(z)=\sum_{j=1}^{2n}\lambda_{j}f_{j}(z)$, where $\lambda_{j}\in\mathbb{C}$
such that $0\leq\alpha_{j}<\pi$ . Then

$\displaystyle{\rm
Re}\left(1+\frac{zF^{\prime\prime}(z)}{F^{\prime}(z)}\right)>0$

for $|z|<R^{*}_{c,n}$, where $R^{*}_{c,n}$ is the smallest value of
$r\in(0,1)$ satisfying the equation $\psi_{3,n}(r)=0$,

$\displaystyle\psi_{3,n}(r)$
$\displaystyle=p\left(2n-1\right)r^{4}+\left(1-2n+p^{2}-2np^{2}+(1+p^{2})\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)r^{3}$
$\displaystyle+2p\left(n+4\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)r^{2}-\left(\left(1+p^{2}\right)\left(1+2\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)\right)r+p=0.$

######  Remark 2.3.

For combination of two functions $f_{1},f_{2}\in\mathcal{S}(p)$, we have the
following corollary of Theorem 2.3. Note that Corollary 2.3 is a general form
of Theorem C.

######  Corollary 2.3.

Let $f_{j}\in\mathcal{S}(p)$, $j=1,2$ and
$F(z)=\lambda_{1}f_{1}(z)+\lambda_{2}f_{2}(z)$, where
$\lambda_{j}\in\mathbb{C}$ such that $0\leq\alpha_{j}<\pi$ . Then

$\displaystyle{\rm
Re}\left(1+\frac{zF^{\prime\prime}(z)}{F^{\prime}(z)}\right)>0$

for $|z|<R^{*}_{c,1}$, where $R^{*}_{c,1}$ is the smallest value of
$r\in(0,1)$ satisfying the equation $\psi_{3,1}(r)=0$,

$\displaystyle\psi_{3,1}(r)$
$\displaystyle=pr^{4}+\left(1+p^{2}\right)\left(-1+\sec\;\left(\frac{\alpha_{1}}{2}\right)\right)r^{3}$
$\displaystyle+2p\left(1+4\sec\;\left(\frac{\alpha_{1}}{2}\right)\right)r^{2}-\left(1+p^{2}\right)\left(1+2\sec\;\left(\frac{\alpha_{1}}{2}\right)\right)r+p=0.$

In this paper, we will determine a lower bound for the radius of univalence of
the general linear combination of concave functions in ${\rm Co}(A)$.

######  Theorem 2.4.

Let $f_{j}\in{\rm Co}(A)$, $j=1,2,\ldots,2n$ and
$F(z)=\sum_{j=1}^{2n}\lambda_{j}f_{j}(z)$, where $\lambda_{j}\in\mathbb{C}$
such that $0\leq\alpha_{j}<\pi$. Then $F$ is univalent in $|z|<R^{\\#}_{u,n}$,
where

$\displaystyle
R^{\\#}_{u,n}=\frac{\rho}{n}\left(\sum_{j=1}^{n}{\sec}\;\left(\frac{\alpha_{j}}{2}\right)-\sqrt{\left(\sum_{j=1}^{n}{\sec}\;\left(\frac{\alpha_{j}}{2}\right)\right)^{2}-n^{2}}\right).$

######  Remark 2.4.

For combination of two functions $f_{1},f_{2}\in{\rm Co}(A)$, we have the
following corollary of Theorem 2.4. Note that Corollary 2.4 is a general form
of Theorem D.

######  Corollary 2.4.

Let $f_{j}\in{\rm Co}(A)$, $j=1,2$ and
$F(z)=\lambda_{1}f_{1}(z)+\lambda_{2}f_{2}(z)$, where
$\lambda_{j}\in\mathbb{C}$ such that $0\leq\alpha_{j}<\pi$. Then $F$ is
univalent in $|z|<R^{\\#}_{u,1}$, where

$\displaystyle R^{\\#}_{u,1}$
$\displaystyle=\rho\left({\sec}\;\left(\frac{\alpha_{1}}{2}\right)-\sqrt{\left({\sec}\;\left(\frac{\alpha_{1}}{2}\right)\right)^{2}-1}\right)$
$\displaystyle=\rho\left({\sec}\;\left(\frac{\alpha_{1}}{2}\right)-\tan\left(\frac{\alpha_{1}}{2}\right)\right).$

In the following theorem, we will determine a lower bound for the radius of
concavity of the general linear combination of concave functions in ${\rm
Co(A})$.

######  Theorem 2.5.

Let $f_{j}\in{\rm Co(A})$, $j=1,2,\ldots,2n$ and
$F(z)=\sum_{j=1}^{2n}\lambda_{j}f_{j}(z)$, where $\lambda_{j}\in\mathbb{C}$
such that $0\leq\alpha_{j}<\pi$. Then ${\rm Re}\;T_{F(z)}>0$ for
$|z|<R^{**}_{\rm Co(A),n}$, where, $R^{**}_{\rm Co(A),n}$ is the smallest
value of $r\in(0,1)$ satisfying the equation $\psi_{5,n}(r)=0$, where

$\displaystyle\psi_{5,n}(r)=\left(\frac{A-5}{A-1}\right)r^{2}-\frac{2}{A-1}\left(A+1+2A\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)r+1.$

######  Remark 2.5.

For combination of two functions $f_{1},f_{2}\in{\rm Co}(A)$, we have the
following corollary of Theorem 2.5. We see that Corollary 2.5 is a general
form of Theorem E.

######  Corollary 2.5.

Let $f_{j}\in{\rm Co(A})$, $j=1,2$ and
$F(z)=\lambda_{1}f_{1}(z)+\lambda_{2}f_{2}(z)$, where
$\lambda_{j}\in\mathbb{C}$ such that $0\leq\alpha_{j}<\pi$. Then ${\rm
Re}\;T_{F(z)}>0$ for $|z|<R^{**}_{\rm Co(A),1}$, where, $R^{**}_{\rm Co(A),1}$
is the smallest value of $r\in(0,1)$ satisfying the equation
$\psi_{5,1}(r)=0$, where

$\displaystyle\psi_{5,1}(r)=\left(\frac{A-5}{A-1}\right)r^{2}-\frac{2}{A-1}\left(A+1+2A\sec\;\left(\frac{\alpha_{1}}{2}\right)\right)r+1.$

Every function in ${\rm Co}(A)$ is a function of order $A-1$ that is close-to-
convex. It has been proven that the radius of convexity of ${\rm Co}(A)$ is
$A-\sqrt{A^{2}-1}$ (see, for instance, [4, Cor. 2.13]). Now, we will calculate
a lower bound for the radius of convexity of the general linear combination of
functions in ${\rm Co}(A)$.

######  Theorem 2.6.

Let $f_{j}\in{\rm Co}(A)$, $j=1,2,\ldots,2n$ and
$F(z)=\sum_{j=1}^{2n}\lambda_{j}f_{j}(z)$, where $\lambda_{j}\in\mathbb{C}$
and $0\leq\alpha_{j}<\pi$. Then

$\displaystyle{\rm
Re}\left(1+\frac{zF^{\prime\prime}(z)}{F^{\prime}(z)}\right)>0$

for $|z|<R^{\\#}_{c,n}$, where, $R^{\\#}_{c,n}$ is the smallest value of
$r<1$, satisfying the equation $\psi_{6,n}(r)=0$, where

$\displaystyle\psi_{6,n}(r)=\left(2n-1\right)r^{2}-2A\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)r+1.$

######  Remark 2.6.

For combination of two functions $f_{1},f_{2}\in{\rm Co}(A)$, we have the
following corollary of Theorem 2.6. It is easy to see that Corollary 2.6 is a
general form of Theorem F.

######  Corollary 2.6.

Let $f_{j}\in{\rm Co}(A)$, $j=1,2$ and
$F(z)=\lambda_{1}f_{1}(z)+\lambda_{2}f_{2}(z)$, where
$\lambda_{j}\in\mathbb{C}$ and $0\leq\alpha_{j}<\pi$. Then

$\displaystyle{\rm
Re}\left(1+\frac{zF^{\prime\prime}(z)}{F^{\prime}(z)}\right)>0$

for $|z|<R^{\\#}_{c,1}$, where, $R^{\\#}_{c,1}$ is the smallest value of
$r<1$, satisfying the equation $\psi_{6,1}(r)=0$, where

$\displaystyle\psi_{6,1}(r)=r^{2}-2A\sec\;\left(\frac{\alpha_{1}}{2}\right)r+1.$

We have discussed the radius of concavity of class $\mathcal{S}(p)$ using
Definition 1.1. We will now use this definition to obtain a lower bound for
the general linear combination of functions belonging to class
$\mathcal{S}(p)$.

######  Theorem 2.7.

Let $f_{j}\in\mathcal{S}(p)$, $j=1,2,\ldots,2n$ and
$F(z)=\sum_{j=1}^{2n}\lambda_{j}f_{j}(z)$, where, $\lambda_{j}\in\mathbb{C}$
such that $0\leq\alpha_{j}<\pi$. Then ${\rm Re}\;P_{F}(z)$>0 for
$|z|<R^{\\#}_{\rm Co(p),n}$, where, $R^{\\#}_{\rm Co(p),n}$ is the smallest
value of $r\in(0,r_{2,n})$ satisfying the equation $\psi_{7,n}(r)=0$, where

$\displaystyle\psi_{7,n}(r):$
$\displaystyle=(3-2n)p^{2}r^{6}-2\left((p^{3}+p)(1+\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right))\right)r^{5}$
$\displaystyle\quad+\left(\left(2n-1-2\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)-p^{2}\left(3-4\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)+2n-1\right)r^{4}$
$\displaystyle\quad+\left(\left(4+4\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)\left(p^{3}+p\right)\right)r^{3}$
$\displaystyle\quad+\left(\left(p^{4}+1\right)\left(1-2\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)-p^{2}\left(1+2n-4\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)\right)r^{2}$
$\displaystyle\quad-2p\left((1+p^{2}+2\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right))r+p^{2}.$

######  Remark 2.7.

For combination of two functions $f_{1},f_{2}\in\mathcal{S}(p)$, we have the
following corollary of Theorem 2.7. Clearly, Corollary 2.7 is a general form
of Theorem G.

######  Corollary 2.7.

Let $f_{j}\in\mathcal{S}(p)$, $j=1,2$ and
$F(z)=\lambda_{1}f_{1}(z)+\lambda_{2}f_{2}(z)$, where,
$\lambda_{j}\in\mathbb{C}$ such that $0\leq\alpha_{j}<\pi$. Then ${\rm
Re}\;P_{F}(z)$>0 for $|z|<R^{\\#}_{\rm Co(p),1}$, where, $R^{\\#}_{\rm
Co(p),1}$ is the smallest value of $r\in(0,r_{2,1})$ satisfying the equation
$\psi_{7,1}(r)=0$, where

$\displaystyle\psi_{7,1}(r):$
$\displaystyle=p^{2}r^{6}-2\left(\left(p^{3}+p\right)\left(1+\sec(\frac{\alpha_{1}}{2}\right)\right)r^{5}$
$\displaystyle\quad+\left(\left(1-2\sec\;\left(\frac{\alpha_{1}}{2}\right)\right)-p^{2}\left(3-4\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)+1\right)r^{4}$
$\displaystyle\quad+\left(\left(4+4\sec\;\left(\frac{\alpha_{1}}{2}\right)\right)\left(p^{3}+p\right)\right)r^{3}$
$\displaystyle\quad+\left(\left(p^{4}+1\right)\left(1-2\sec\;\left(\frac{\alpha_{1}}{2}\right)\right)-p^{2}\left(3-4\sec\;\left(\frac{\alpha_{1}}{2}\right)\right)\right)r^{2}$
$\displaystyle\quad-2p\left((1+p^{2}+2\sec\;\left(\frac{\alpha_{1}}{2}\right)\right))r+p^{2}.$

##  3\. Proof of the main results

###### Proof of Theorem 2.1.

Since $f_{j}\in\mathcal{S}$ for $j=1,2,\ldots,2n$, from [12, Theorem 2.4], and
in view of (2) and (2), we have

$\displaystyle\vline\;\frac{zf^{\prime\prime}_{j}(z)}{f^{\prime}_{j}(z)}-\frac{2r^{2}}{1-r^{2}}\;\vline\leq\frac{4r}{1-r^{2}}\;\;\mbox{for}\;\;|z|=r<1.$

In view of Lemma 1.1, we have the estimate

$\displaystyle\mbox{Re}\;(w)$ $\displaystyle\leq
an+d\sum_{j=1}^{n}\mbox{sec}\;\left(\frac{\alpha_{j}}{2}\right)$
$\displaystyle=\frac{2nr^{2}}{1-r^{2}}+\left(\frac{4r}{1-r^{2}}\right)\sum_{j=1}^{n}\mbox{sec}\;\left(\frac{\alpha_{j}}{2}\right)$
$\displaystyle=\frac{\displaystyle
2nr^{2}+4r\sum_{j=1}^{n}\mbox{sec}\;\left(\frac{\alpha_{j}}{2}\right)}{1-r^{2}}.$

Thus we see that

$\displaystyle\mbox{Re}\;T_{F(z)}$
$\displaystyle\geq\mbox{Re}\left[\frac{2}{A-1}\left(\frac{(A+1)}{2}\frac{1-z}{1+z}-1-\frac{zF^{\prime\prime}(z)}{F^{\prime}(z)}\right)\right]$
$\displaystyle=\frac{2}{A-1}\left[\frac{(A+1)}{2}\frac{1-r}{1+r}-1-\frac{2nr^{2}}{1-r^{2}}-\left(\frac{4r}{1-r^{2}}\right)\sum_{j=1}^{n}\mbox{sec}\;\left(\frac{\alpha_{j}}{2}\right)\right]$
$\displaystyle=\frac{1}{(A-1)(1-r^{2})}\bigg{[}r^{2}(A+3-4n)-2r\left(A+1+4\sum_{j=1}^{n}\mbox{sec}\;\left(\frac{\alpha_{j}}{2}\right)\right)$
$\displaystyle\quad+A-1\bigg{]}$
$\displaystyle=\frac{1}{(1-r^{2})}\bigg{[}\frac{(A+3-4n)}{(A-1)}r^{2}-\frac{2}{(A-1)}\left(A+1+4\sum_{j=1}^{n}\mbox{sec}\;\left(\frac{\alpha_{j}}{2}\right)\right)r+1\bigg{]}.$

Our aim is to show that $\mbox{Re}\;T_{F(z)}>0$ for
$|z|<R^{*}_{\rm{Co}(A),n}$. In this purpose, for $\alpha_{j}\in[0,\pi)$ and
$A\in(1,2]$, we define a function $\psi_{1,n}(r)$ as follows

$\displaystyle\psi_{1,n}(r):=\frac{(A+3-4n)}{(A-1)}r^{2}-\frac{2}{(A-1)}\left(A+1+4\sum_{j=1}^{n}{\sec}\;\left(\frac{\alpha_{j}}{2}\right)\right)r+1.$

In fact, we show that $R^{*}_{\rm{Co}(A),n}$ is a root in $(0,1)$ of
$\psi_{1,n}(r)=0$. It is easy to see that $\psi_{1,n}(r)$ is a real-valued
differentiable function with $\psi_{1,n}(0)=1>0$ and

$\displaystyle\psi_{1,n}(1)=-2\left(\frac{2n-(A+1)}{A-1}\right)-\left(\frac{\left(A+1+4\sum_{j=1}^{n}{\sec}\;\left(\frac{\alpha_{j}}{2}\right)\right)}{(A-1)}\right)<0.$

Because $\psi_{1,n}(r)$ is a real-valued differentiable function on $(0,1)$,
the Intermediate Value Theorem (IVT) ensures that ${\rm Re}\;T_{F(z)}>0$ for
$|z|<R^{*}_{\rm{Co}(A),n}$, where $R^{*}_{\rm{Co}(A),n}$ is the smallest
positive value of $r\in(0,1)$ satisfying the equation $\psi_{1,n}(r)=0$. By
the intermediate value theorem, there exists at least one root in $(0,1)$.
Hence, $R^{*}_{\rm{Co}(A),n}$ exists in $(0,1)$ for each
$\alpha_{j}\in[0,\pi)$. ∎

###### Proof of Theorem 2.2.

Let $f_{j}\in\mathcal{S}(p)$, $j=1,2,\ldots,2n$. Given that
$F(z)=\sum_{j=1}^{2n}\lambda_{j}f_{j}(z)$, using this we see that

$\displaystyle F^{\prime}(z)$
$\displaystyle=\lambda_{1}f^{\prime}_{1}(z)+\lambda_{2}f^{\prime}_{2}(z)+\cdots+\lambda_{2n}f^{\prime}_{2n}(z)$
$\displaystyle=f^{\prime}_{1}(z)\left[1+\left(\frac{\lambda_{1}}{1-\lambda_{1}}\right)^{-1}\right]^{-1}+f^{\prime}_{2}(z)\left[1+\left(\frac{\lambda_{2}}{1-\lambda_{2}}\right)^{-1}\right]^{-1}$
$\displaystyle\quad+\cdots+f^{\prime}_{2n}(z)\left[1+\left(\frac{\lambda_{2n}}{1-\lambda_{2n}}\right)^{-1}\right]^{-1}.$

By Lemma C, we have

$\displaystyle\vline
f^{\prime}_{j}(z)-\frac{1+\frac{r^{2}}{\rho^{2}}}{1-\frac{r^{2}}{\rho^{2}}}\vline\leq\frac{\frac{2r}{\rho}}{1-\frac{r^{2}}{\rho^{2}}}\;\;\mbox{for
each}\;j=1,2,\cdots,2n.$

In view of Lemma 1.1, a simple computation shows that

$\displaystyle\mbox{Re}\;(F^{\prime}(z))$ $\displaystyle\geq
an-d\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)$
$\displaystyle=\frac{n\left(1+\frac{r^{2}}{\rho^{2}}\right)}{\left(1-\frac{r^{2}}{\rho^{2}}\right)}-\frac{1}{1-\frac{r^{2}}{\rho^{2}}}\left({\frac{2r}{\rho}}\right)\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)$
$\displaystyle=\frac{1}{\left(\rho^{2}-r^{2}\right)}\left(nr^{2}-2r\rho\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)+n\rho^{2}\right).$

Assume that

(3.1)
$\displaystyle\psi_{2,n}(r)=nr^{2}-2r\rho\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)+n\rho^{2}.$

Consequently,

$\displaystyle\mbox{Re}\;(F^{\prime}(z))>0\;\;\;\mbox{for}\;\;|z|<R^{*}_{u,n},$

where, $R^{*}_{u,n}$ is given in the statement. Thus, by the Noshiro-
Warschawski Theorem (see [12, Theorem 2.16]), $F$ is univalent in
$|z|<R^{*}_{u,n}$. ∎

###### Proof of Theorem 2.3.

Since $f_{j}\in\mathcal{S}(p)$ for $j=1,2,\ldots,2n$, from [5, Lemma 1], in
view of (2) and (2), we have

$\displaystyle\vline\;\frac{zf^{\prime\prime}_{j}(z)}{f^{\prime}_{j}(z)}-\frac{2r^{2}}{1-r^{2}}\;\vline\leq\frac{2r}{1-r^{2}}\left(\frac{p-r}{1-pr}+\frac{1-pr}{p-r}\right)\;\;\mbox{for}\;\;|z|=r<p.$

By Lemma 1.1, we have

$\displaystyle\mbox{Re}\left(\frac{zF^{\prime\prime}(z)}{F^{\prime}(z)}\right)$
$\displaystyle\geq\frac{2nr^{2}}{1-r^{2}}-\frac{2r}{1-r^{2}}\left(\frac{p-r}{1-pr}+\frac{1-pr}{p-r}\right)\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right).$

We have for $|z|=r<r^{*}_{1}$

$\displaystyle\mbox{Re}\left(1+\frac{zF^{\prime\prime}(z)}{F^{\prime}(z)}\right)\geq
1+\frac{2nr^{2}}{1-r^{2}}-\frac{2r}{1-r^{2}}\left(\frac{p-r}{1-pr}+\frac{1-pr}{p-r}\right)\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right),$

where $r^{*}_{1}$ is the smallest value of $r\in(0,p)$ satisfying the equation
$\psi_{3,n}(r)=0$,

$\displaystyle\psi_{3,n}(r)=1+\frac{2nr^{2}}{1-r^{2}}-\frac{2r}{1-r^{2}}\left(\frac{p-r}{1-pr}+\frac{1-pr}{p-r}\right)\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right).$

Thus, we see that ${\rm
Re}\left(1+\frac{zF^{\prime\prime}(z)}{F^{\prime}(z)}\right)>0$ for
$|z|<R^{*}_{c,n}$, where $R^{*}_{c,n}$ is the smallest value of
$r\in(0,r^{*}_{1})$ satisfying the equation $\psi_{3,n}(r)=0$. We investigate
the existence of $r^{*}_{1}$ for each $p\in(0,1)$. The function
$\psi_{3,n}(r)=0$ is continuous on $[0,r^{*}_{1})$ with

$\displaystyle\psi_{3,n}(0)=1>0\;\;\;\mbox{and}\;\;\;\lim_{r\rightarrow{r^{*}_{1}}^{-}}\psi_{3,n}(r)=-\infty.$

Therefore, by the intermediate value theorem $\psi_{3,n}(r)$ has at-least one
root in $(0,p)$. Hence $r^{*}_{1}$ exists for each $p\in(0,1)$. Therefore, we
have $\psi_{3,n}(r)=0$, i.e.,

$\displaystyle
1+\frac{2nr^{2}}{1-r^{2}}-\frac{2r}{1-r^{2}}\left(\frac{p-r}{1-pr}+\frac{1-pr}{p-r}\right)\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)=0.$

Clearly, $r^{*}_{1}$ is a root of the equation

$\displaystyle\left(2n-1\right)pr^{4}+\left(1-2n+p^{2}-2np^{2}+(1+p^{2})\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)r^{3}$
$\displaystyle+2p\left(n+4\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)r^{2}-\left(\left(1+p^{2}\right)\left(1+2\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)\right)r+p=0.$

This completes the proof. ∎

###### Proof of Theorem 2.4.

Let $f_{j}\in{\rm Co}(A)$, $j=1,2,\ldots,2n$. Given that
$F(z)=\sum_{j=1}^{2n}\lambda_{j}f_{j}(z)$, using this we see that

$\displaystyle F^{\prime}(z)$
$\displaystyle=\lambda_{1}f^{\prime}_{1}(z)++\cdots+\lambda_{2n}f^{\prime}_{2n}(z)$
$\displaystyle=f^{\prime}_{1}(z)\left[\frac{1}{1+\left(\frac{1}{\lambda_{1}}-1\right)}\right]+\cdots+f^{\prime}_{2n}(z)\left[\frac{1}{1+\left(\frac{1}{\lambda_{2n}}-1\right)}\right]$
$\displaystyle=f^{\prime}_{1}(z)\left[\frac{1}{1+\left(\frac{\lambda_{1}}{1-\lambda_{1}}\right)^{-1}}\right]+\cdots+f^{\prime}_{2n}(z)\left[\frac{1}{1+\left(\frac{\lambda_{2n}}{1-\lambda_{2n}}\right)^{-1}}\right].$

We assume that

$\displaystyle\lambda_{j}=\frac{1}{1+b_{j}e^{i\alpha_{j}}}\implies\alpha_{j}=-i\ln\left(\frac{1-\lambda_{j}}{\lambda_{j}b_{j}}\right)\;\;\mbox{for}\;\;\lambda_{j}\neq
0,1.$

By Lemma C, we have

$\displaystyle\vline
F^{\prime}(z)-\frac{1+\frac{r^{2}}{\rho^{2}}}{1-\frac{r^{2}}{\rho^{2}}}\vline\leq\frac{\frac{2r}{\rho}}{1-\frac{r^{2}}{\rho^{2}}}\;\;\;\mbox{for}\;j=1,2,\ldots,2n.$

In view of Lemma 1.1, we have

$\displaystyle\mbox{Re}\;(w)$ $\displaystyle\geq
an-d\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)$
$\displaystyle=\frac{n\left(1+\frac{r^{2}}{\rho^{2}}\right)}{\left(1-\frac{r^{2}}{\rho^{2}}\right)}-\frac{\left({\frac{2r}{\rho}}\right)\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)}{1-\frac{r^{2}}{\rho^{2}}}$
$\displaystyle=\frac{1}{\left(\rho^{2}-r^{2}\right)}\left(nr^{2}-2r\rho\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)+n\rho^{2}\right).$

Our aim is to show that $\mbox{Re}\;(F^{\prime}(z))>0$, i.e.,

$\displaystyle
nr^{2}-2r\rho\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)+n\rho^{2}>0\;\;\
\mbox{for}\;\ r<R^{\\#}_{u,n}.$

We define

$\displaystyle\psi_{4,n}(r)=nr^{2}-2r\rho\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)+n\rho^{2}=0.$

Differentiating $\psi_{4,n}(r)$ w.r.t. $r$, we obtain

$\displaystyle\frac{d\psi_{4,n}(r)}{dr}=2nr-2\rho\sum_{j=1}^{n}\mbox{sec}\;\left(\frac{\alpha_{j}}{2}\right).$

For critical point of $\psi_{4,n}(r)$,

$\displaystyle\frac{d\psi_{4,n}(r)}{dr}=0\implies
r=\frac{\rho}{n}\sum_{j=1}^{n}\mbox{sec}\;\left(\frac{\alpha_{j}}{2}\right).$

Again, differentiating $\frac{d\psi_{4,n}(r)}{dr}$, w.r.t. $r$, we obtain

$\displaystyle\frac{d^{2}\psi_{4,n}(r)}{dr^{2}}=2n>0.$

It is clear that the function $\psi_{4,n}(r)=0$ has minima at
$r=(\rho/n)\sum_{j=1}^{n}\mbox{sec}\;\left({\alpha_{j}}/{2}\right)$.
Consequently, we have $\mbox{Re}\;F^{\prime}(z)$>0 for $|z|<R^{\\#}_{u,n}$.
Thus by the Nishiro-Warschawski (see [12, Theorem 2.16]), the function $F$ is
univalent in $|z|<R^{\\#}_{u,n}$. This completes the proof. ∎

###### Proof of Theorem 2.5.

Since $f_{j}\in{\rm Co}(A)$ for $j=1,2,\ldots,2n$, from [8, page 65], in view
of (2) and (2), we have

$\displaystyle\vline\;\frac{zf^{\prime\prime}_{j}(z)}{f^{\prime}_{j}(z)}-\frac{2r^{2}}{1-r^{2}}\;\vline\leq\frac{2Ar}{1-r^{2}}\;\;\mbox{for}\;\;|z|<r.$

By Lemma 1.1, we have

$\displaystyle\mbox{Re}\left(\frac{zF^{\prime\prime}(z)}{F^{\prime}(z)}\right)$
$\displaystyle\leq\frac{2nr^{2}}{1-r^{2}}+\frac{2Ar}{1-r^{2}}\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right).$

A simple computation gives that

$\displaystyle\mbox{Re}\;T_{F(z)}$
$\displaystyle\geq\mbox{Re}\left[\frac{2}{A-1}\left(\frac{(A+1)}{2}\frac{1-z}{1+z}-1-\frac{zF^{\prime\prime}(z)}{F^{\prime}(z)}\right)\right]$
$\displaystyle=\frac{2}{A-1}\left[\frac{(A+1)}{2}\frac{1-r}{1+r}-1-\frac{2nr^{2}}{1-r^{2}}-\left(\frac{2Ar}{1-r^{2}}\right)\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right]$
$\displaystyle=\frac{1}{1-r^{2}}\left(r^{2}\left(\frac{A+3-4n}{A-1}\right)-2r\left(\frac{A+1+2A\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)}{A-1}\right)\right)+1.$

Clearly, ${\rm Re}\;T_{F(z)}>0$ for $|z|<R^{**}_{\rm Co(A),n}$, where
$R^{**}_{\rm Co(A),n}$ is the smallest positive value of the $r\in(0,1)$
satisfying the equation $\psi_{5,n}(r)=0$. We define

$\displaystyle\psi_{5,n}(r)$
$\displaystyle:=\left(\frac{A+3-4n}{A-1}\right)r^{2}-\frac{2}{A-1}\left(A+1+2A\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)r+1.$

We now investigate the existence of $R^{**}_{\rm Co(A),n}$ for each
$\alpha_{j}\in[0,\pi)$ and $A\in(1,2]$. We consider the function
$\psi_{5,n}(r)$ as in the statement of the result for $r\in(0,1)$. Clearly,
$\psi_{5,n}(r)$ is a real-valued differentiable function on $\in(0,1)$ with

$\displaystyle\psi_{5,n}(0)=1>0\;\;\mbox{and}\;\;\psi_{5,n}(1)=\frac{-4}{A-1}\left(n+A\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)<0.$

By Intermediate Value Theorem, $\psi_{5,n}(r)$ has at-least one root in
$(0,1)$ which we denote by $R^{**}_{\rm Co(A),}$. Hence, $R^{**}_{\rm
Co(A),n}$ exists for each $\alpha_{j}\in[0,\pi)$ and $A\in(1,2]$. ∎

###### Proof of Theorem 2.6.

Since $f_{j}\in{\rm Co}(A)$ for $j=1,2,\ldots,2n$, from [8, page 65], using
(2) and (2), we have

$\displaystyle\vline\;\frac{zf^{\prime\prime}_{j}(z)}{f^{\prime}_{j}(z)}-\frac{2r^{2}}{1-r^{2}}\;\vline\leq\frac{2Ar}{1-r^{2}}\;\;\mbox{for}\;\;|z|=r<1.$

Using Lemma 1.1, we have

$\displaystyle\mbox{Re}\left(\frac{zF^{\prime\prime}(z)}{F^{\prime}(z)}\right)$
$\displaystyle\geq\frac{2nr^{2}}{1-r^{2}}-\left(\frac{2Ar}{(1-r^{2})}\right)\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right).$

It is easy to see that

$\displaystyle\mbox{Re}\left(1+\frac{zF^{\prime\prime}(z)}{F^{\prime}(z)}\right)$
$\displaystyle\geq
1+\frac{2nr^{2}}{1-r^{2}}-\left(\frac{2Ar}{1-r^{2}}\right)\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)$
$\displaystyle=\frac{1}{1-r^{2}}\left(r^{2}\left(2n-1\right)-2Ar\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)+1\right).$

Thus, we see that

$\displaystyle{\rm
Re}\left(1+\frac{zF^{\prime\prime}(z)}{F^{\prime}(z)}\right)>0\;\mbox{for}\;|z|<R^{\\#}_{c,n},$

where $R^{\\#}_{c,n}$ is the smallest positive number of $r\in(0,1)$, that
satisfy the equation $\psi_{6,n}(r)=0$. We define

$\displaystyle\psi_{6,n}(r)=\left(2n-1\right)r^{2}-\left(2A\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)r+1.$

We investigate the existence of $R^{\\#}_{c,n}$ for each
$\alpha_{j}\in[0,\pi)$ and $A\in(1,2]$. Then $\psi_{6,n}(r)$ being a real-
valued differentiable function on $\in[0,1)$, with $\psi_{6,n}(0)=1>0$ and

$\displaystyle\psi_{6,n}(1)=2\left(n-A\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)<0,$

by Intermediate Value Theorem $\psi_{6,n}(r)$ has at-least one root in
$(0,1)$. Hence, $R^{\\#}_{c,n}$ exists for each $\alpha_{j}\in[0,\pi)$ and
$A\in(1,2]$. ∎

###### Proof of Theorem 2.7.

Since $f_{j}\in\mathcal{S}(p)$ for $j=1,2,\ldots,2n$, from [5, Lemma 1], using
(2) and (2), we have

$\displaystyle\vline\;\frac{zf^{\prime\prime}_{j}(z)}{f^{\prime}_{j}(z)}-\frac{2r^{2}}{1-r^{2}}\;\vline\leq\frac{2r}{1-r^{2}}\left(\frac{p-r}{1-pr}+\frac{1-pr}{p-r}\right)\;\;\mbox{for}\;\;|z|=r<p.$

By Lemma 1.1, we have

$\displaystyle\mbox{Re}\left(\frac{zF^{\prime\prime}(z)}{F^{\prime}(z)}\right)$
$\displaystyle\leq\frac{2nr^{2}}{1-r^{2}}+\left[\frac{2r}{1-r^{2}}\left(\frac{p-r}{1-pr}+\frac{1-pr}{p-r}\right)\right]\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right).$

A tedious computation gives

$\displaystyle\mbox{Re}P_{F}(z)$
$\displaystyle\geq\frac{2p(1-r^{2})}{(p+r)(1+pr)}-1-\frac{2nr^{2}}{1-r^{2}}+\left[\frac{2r}{1-r^{2}}\left(\frac{p-r}{1-pr}+\frac{1-pr}{p-r}\right)\right]\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)$
$\displaystyle=(3-2n)p^{2}r^{6}-2\left((p^{3}+p)(1+\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right))\right)r^{5}$
$\displaystyle\quad+\left(\left(2n-1-2\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)-p^{2}\left(3-4\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)+2n-1\right)r^{4}$
$\displaystyle\quad+\left(\left(4+4\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)\left(p^{3}+p\right)\right)r^{3}$
$\displaystyle\quad+\left(\left(p^{4}+1\right)\left(1-2\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)-p^{2}\left(1+2n-4\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right)\right)r^{2}$
$\displaystyle\quad-2p\left((1+p^{2}+2\sum_{j=1}^{n}\sec\;\left(\frac{\alpha_{j}}{2}\right)\right))r+p^{2}.$

Thus, we see that $\mbox{Re}P_{F}(z)>0$ for $|z|<\min\\{r_{2},R^{\\#}_{\rm
Co(p),n}\\}=R^{\\#}_{\rm Co(p),n}$, where, $R^{\\#}_{\rm Co(p),n}$ is the
smallest value of $r\in(0,r_{2})$ satisfying the equation $\psi_{7,n}(r)=0$,
where $\psi_{7,n}(r)$ is given as in the statement of the theorem. Next we
consider the function $\psi_{7,n}(r)$ for $r\in(0,r_{2})$. Then
$\psi_{7,n}(r)$ is differentiable on $\in(0,r_{2})$ with

$\displaystyle\psi_{7,n}(0)=p^{2}>0\;\;\;\;\
\mbox{and}\;\;\;\;\;\lim_{r\rightarrow r^{-}_{2}}\psi_{7,n}(r)=-\infty.$

By the intermediate value theorem, $\psi_{7,n}(r)$ has at least one root in
$(0,r_{2})$. Hence, $R^{\\#}_{\rm Co(p),n}$ exists for each $p\in(0,1)$. ∎

Compliance of Ethical Standards:

Conflict of interest. The authors declare that there is no conflict of
interest regarding the publication of this paper.

Data availability statement. Data sharing is not applicable to this article as
no datasets were generated or analyzed during the current study.

## References

  * [1] F. G. Avkhadiev, Ch. Pommerenke  and K.-J Wirths, On the coefficients of concave univalent functions, Math. Nachr. 271, 3–9 (2004).
  * [2] F. G. Avkhadiev and K.-J.Wirths , A proof of the Livingston conjecture, Forum Math. 19, 149-158 (2007).
  * [3] F. G. Avkhadiev  and K. -J. Wirths, Concave schlicht functions with bounded opening angle at infinity, Lobachevskii J. Math 17, 3-10 (2005).
  * [4] B. Bhowmik.: On concave univalent functions, Math. Nachr. 285(5-6), 606-612(2012).
  * [5] B. Bhowmik and S. Biswas, Distortion, radius of concavity and several other radii results for certain classes of functions, Comput. Method Func. Theory (2024), https://doi.org/10.1007/s40315-024-00525-8.
  * [6] B. Bhowmik and S. Ponnusamy, Coefficient inequalities for concave and meromorphically starlike univalent functions, Ann. Polon. Math. 93(2), 177-186 (2008).
  * [7] B. Bhowmik , S. Ponnusamy  and K. -J. Wirths, Concave functions, Blaschke products and polygonal mappings, Siberian Math. J. 50 609-615, (2009).
  * [8] B. Bhowmik, S. Ponnusamy and K. -J. Wirths, Characterization and the pre-Schwarzian norm estimate for concave univalent functions, Monatsh Math. 161, 59-75 (2010) .
  * [9] B. Bhowmik S. Ponnusamy and K. -J. Wirths, On the Fekete-Szegö problem for concave univalent functions, J. Math. Anal. Appl. 373(2), 432-438 (2011).
  * [10] W. Fenchel, Bemerkungen über die im Einkeitskreis meromorphen schlichten Funktionen, Freuss. Akad. Wiss. Phys. Math. Kl. 22(23)(1931), 431-436.
  * [11] L. Cruz and Ch. Pommerenke, On concave univalent functions, Complex Var. Elliptic Equ. 52, 153-159 (2011).
  * [12] P. T. Duren, Univalent Functions, Springer-Verlag, New York Inc (1983).
  * [13] W. Fenchel, Bemerkungen über die im Einkeitskreis meromorphen schlichten Funktionen, Freuss. Akad. Wiss. Phys. Math. Kl. 22(23), 431-436 (1931).
  * [14] C. Gao and S. Xia, Some theorems on meromorphic univalent functions, Complex Variables, 47(3), 177-181 (2002).
  * [15] H. Grunsky, Zwei Bemerkungen zur konformen Abbildung, Jber. Deutsch. Math. Verein, 43, 140-143 (1934).
  * [16] J. A. Jenkins, On meromorphic univalent functions, Complex variables. Theory Appl 7, 83-87 (1986).
  * [17] W. E. Kirwan and G. Schober, Extremal problems for meromorphic univalent functions, J. Anal. Math. 30, 330-348 (1976).
  * [18] Y. Komatu, Note on the theory of conformal representation by meromorphic functions, I and II. Proc. Japan Acad 21, 269-277 (1945).
  * [19] G. Labelle and I. Q. Rahman, Remarque sur la moyenne arithmetique de fonctions univalentes convexese, Can. J. Math. 21, 977-981 (1969).
  * [20] A. E. Livingston, Convex meromorphic mappings, Ann. Polonici Math. 59, 275-291 (1994).
  * [21] T. H. MacGregor, The univalence of a linear combination of convex mappings, J. Lond. Math. Soc. 44, 210-212 (1969).
  * [22] J. Miller, Convex and starlike meromorphic functions,  Proc. Am. Math. Soc. 80, 608-613 (1980).
  * [23] J. Miller, Convex meromorphic mappings and related functions, Proc. Am. Math. Soc. 25, 220-228 (1970).
  * [24] R. Nevanlinna, Über die konforme Abbildung von Sterngebieten, Översikt av Finska Vetenskaps-Soc. Förh. 63(A), 1-21 (1920-21).
  * [25] M. Obradovi´c and S. Ponnusamy, Where is $f(z)/f^{\prime}(z)$ univalent? J.Anal. 22, 131-143 (2014).
  * [26] J. Pfaltzgraff and B. Pinchuk , A variational method for classes of meromorphic functions, J. Anal. Math. 24, 101-150 (1971).
  * [27] G. Pólya and G. Szeg˝o , Aufgaben und Lehrsätze aus der Analysis, vol. 1. Springer, Berlin (1954).
  * [28] W. C. Royster, Convex meromorphic functions, Mathematical Essays Dedicated to A. J. Macintyre, Ohio Univ. Press, Athens, Ohio, pp 331–339 (1970).
  * [29] R. K. Stump, Linear combinations of univalent functions with complex coefficients, Can. J. Math. XXIII(4), 712-717, (1971).
  * [30] Y. Sun, A. Rasila and Y. -P. Jiang, Linear combinations of harmonic quasiconformal mappings convex in one direction,  Kodai Math. J. 39 (1971), 366-377.
  * [31] M.A. Wang-Cang, Schiffer differential equation of meromorphic univalent functions and its applications, J. Math. Quart. (China), 3, 13-19, (1988) .
  * [32] Y. Zhang, Some questions of meromorphic univalent functions, J. Northwest Univ. (1980).