#  Sharp $L^{1}$ estimates for the dyadic maximal operator

Eleftherios N. Nikolidakis  Department of Mathematics, University of Ioannina
<EMAIL_ADDRESS>and  Andreas G. Tolias  Department of Mathematics, University
of Ioannina<EMAIL_ADDRESS>
###### Abstract.

We prove sharp $L^{1}$ inequalities for the dyadic maximal function
${\mathcal{M}}_{{\mathcal{T}}}\phi$ when $\phi$ satisfies certain $L^{1}$ and
$L^{\infty}$ conditions.

##  1\. Introduction

The dyadic maximal operator on $\mathbb{R}^{n}$ is a useful tool in analysis
and is defined by the formula

${\mathcal{M}}_{d}\phi(x)=\sup\left\\{\frac{1}{|S|}\int_{S}|\phi(u)|du:x\in
S,\;S\subset\mathbb{R}^{n}\ \text{ is a dyadic cube}\right\\},$ (1)

for every $\phi\in L^{1}_{\text{loc}}(\mathbb{R}^{n})$, where $|\cdot|$
denotes the Lebesgue measure on $\mathbb{R}^{n}$, and the dyadic cubes are
those formed by the grids $2^{-N}\mathbb{Z}^{n}$, for $N=0,1,2,\ldots$.
It is well known that the operator defined above satisfies the following weak
type $(1,1)$ inequality

$\left|\left\\{x\in\mathbb{R}^{n}:{\mathcal{M}}_{d}\phi(x)>\lambda\right\\}\right|\leq\frac{1}{\lambda}\int_{\left\\{{\mathcal{M}}_{d}\phi>\lambda\right\\}}|\phi(u)|\,du$
(2)

for every $\phi\in L^{1}(\mathbb{R}^{n})$ and every $\lambda>0$, from which it
is easy to get the following $L^{p}$-inequality

$\|{\mathcal{M}}_{d}\phi\|_{p}\leq\frac{p}{p-1}\|\phi\|_{p}$ (3)

for every $p>1$, and every $\phi\in L^{p}(\mathbb{R}^{n})$. It is easy to see
that the weak type inequality (2) is the best possible. For refinements of
this inequality see [7].

It has also been proved that (3) is best possible (see [1] and [2] for general
martingales and [10] for dyadic ones). An approach for the study of the
behaviour of this maximal operator in more depth is the introduction of the so
called Bellman functions which play the role of generalized norms of
${\mathcal{M}}_{d}$. Such functions related to the $L^{p}$-inequality (3) have
been precisely evaluated in [3], [4] and [6]. For the study of the Bellman
functions of ${\mathcal{M}}_{d}$, we use the notation
$\operatorname{Av}_{E}(\psi)=\frac{1}{|E|}\int_{E}\psi$, whenever $E$ is a
Lebesgue measurable subset of $\mathbb{R}^{n}$ of positive measure and $\psi$
is a real valued integrable function defined on $E$. For a fixed dyadic cube
$Q$ the localized maximal operator ${\mathcal{M}}^{\prime}_{d}\phi$ is defined
as in (1) but with the dyadic cubes $S$ being assumed to be contained in $Q$.
Then for every $p>1$ let

$B_{p}(f,F)=\sup\left\\{\frac{1}{|Q|}\int_{Q}({\mathcal{M}}^{\prime}_{d}\phi)^{p}:\operatorname{Av}_{Q}(\phi)=f,\
\operatorname{Av}_{Q}(\phi^{p})=F\right\\}$ (4)

where $\phi$ is non-negative in $L^{p}(Q)$ and the variables $f,F$ satisfy
$0<f^{p}\leq F$. This is the well known Bellman function of two integral
variables of the dyadic maximal operator. By a scaling argument it is easy to
see that (4) is independent of the choice of $Q$, so we may choose $Q$ to be
the unit cube $[0,1]^{n}$. In [3], the function (4) has been precisely
evaluated for the first time. The proof has been given in a much more general
setting of tree-like structures on probability spaces.

More precisely for a non-atomic probability space $(X,{\mathcal{A}},\mu)$ and
${\mathcal{T}}$ a family of measurable subsets of $X$ that has a tree-like
structure similar to the one of the dyadic case (the exact definition is given
in Section 2) the dyadic maximal operator associated to ${\mathcal{T}}$ is
defined by

${\mathcal{M}}_{{\mathcal{T}}}\phi(x)=\sup\left\\{\frac{1}{\mu(I)}\int_{I}|\phi|\,\mathrm{\;}d\mu:\;x\in
I\in{\mathcal{T}}\right\\}$ (5)

for every $\phi\in L^{1}(X)$ and $x\in X$.

This operator is related to the theory of martingales and satisfies
essentially the same inequalities as ${\mathcal{M}}_{d}$ does. Now we define
the corresponding Bellman function of three variables of
${\mathcal{M}}_{{\mathcal{T}}}$, by

$B_{p}^{{\mathcal{T}}}(f,F,k)=\sup\left\\{\int_{K}({\mathcal{M}}_{{\mathcal{T}}}\phi)^{p}\mathrm{\;}d\mu:\;\phi\geq
0,\;\int_{X}\phi\,\mathrm{\;}d\mu=f,\right.\\\
\left.\int_{X}\phi^{p}\,\mathrm{\;}d\mu=F,\ K\subset X\ \text{measurable
with}\ \mu(K)=k\right\\},$ (6)

the variables $f,F,k$ satisfying $0<f^{p}\leq F$, $k\in(0,1]$. The exact
evaluation of (6) is given in [3].

It is well known that in general ${\mathcal{M}}_{{\mathcal{T}}}\phi$ does not
belong to $L^{1}(X)$ when $\phi\in L^{1}(X)$. In [11] it is proved that if
$\phi$ satisfies the condition
$\int\limits_{X}|\phi|\log^{+}|\phi|\mathrm{\;}d\mu<+\infty$ then
${\mathcal{M}}_{{\mathcal{T}}}\phi\in L^{1}(X)$. In [9] it is shown that this
condition is also necessary for the integrability of
${\mathcal{M}}_{{\mathcal{T}}}\phi$. In [5] the corresponding to (6) function
with respect to certain $L\log L$ conditions has been precisely evaluated. As
a matter of fact in [5] more general conditions on $\phi$ have been
considered. An application of this result is the evaluation of the following
Bellman type function

$B_{1}^{{\mathcal{T}}}(f,M)=\sup\left\\{\int_{X}{\mathcal{M}}_{{\mathcal{T}}}\phi\;d\mu:\;\phi\geq
0,\;\int_{X}\phi d\mu=f,\|\phi\|_{\infty}=M\right\\}$ (7)

when $0<f\leq M$.

In this article we refine the above result by inserting an additional
variable, which is $\operatorname{essinf}_{X}\phi$. More precisely we prove
the following:

######  Theorem 1.1.

For all real variables $f,M_{1},M_{2}$ with $M_{1}\geq f>M_{2}\geq 0$ the
following holds:

$\sup\Big{\\{}\int\limits_{X}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu\Big{\\}}=f+(f-M_{2})\log\Big{(}\frac{M_{1}-M_{2}}{f-M_{2}}\Big{)}$ (8)

where the supremum is taken over all measurable $\phi:X\to\mathbb{R}^{+}$ with
$\int\limits_{X}\phi d\mu=f$, $\|\phi\|_{\infty}=M_{1}$ and
$\operatorname{essinf}_{X}(\phi)=M_{2}$.

For the proof of the above theorem we study the respective Hardy operator
problem which is connected to the dyadic maximal operator problem. For the
proof we use a symmetrization principle which appears in [6]. We also refine
(7) to a another direction; more precisely we prove the following.

######  Theorem 1.2.

For all $f,M,k$ that satisfy $0<f\leq M$ and $k\in(0,1]$ it holds that

$\sup\Big{\\{}\int\limits_{K}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu\Big{\\}}=\left\\{\begin{array}[]{l@{\quad} l}kM,&\mbox{ if
}0<k\leq\frac{f}{M}\\\\[11.38109pt] f+f\log(\frac{Mk}{f})&\mbox{ if
}\frac{f}{M}<k\leq 1\end{array}\right.$ (9)

where the supremum is taken over all measurable functions
$\phi:X\to\mathbb{R}^{+}$ with $\int\limits_{X}\phi d\mu=f$,
$\|\phi\|_{\infty}=M$, and all measurable sets $K$ with $\mu(K)=k$.

##  2\. Preliminaries

######  Definition 2.1.

Let $(X,{\mathcal{A}},\mu)$ be a non-atomic probability measure space. We
recall that a collection of measurable sets ${\mathcal{T}}$ is called a tree
in ${\mathcal{A}}$ provided that the following conditions are satisfied:

  1. (i)

$X\in{\mathcal{T}}$ and every $I\in{\mathcal{T}}$ has positive measure.

  2. (ii)

For every $I\in{\mathcal{T}}$ corresponds a countable (finite or infinite)
family $C(I)\subset{\mathcal{T}}$, containing at least two elements, such
that:

    1. (a)

The elements of $C(I)$ are almost pairwise disjoint, i.e. for $J,J^{\prime}\in
C(I)$ with $J\neq J^{\prime}$ we have that $\mu(J\cap J^{\prime})=0$.

    2. (b)

$I=\cup C(I)$.

  3. (iii)

If we define ${\mathcal{T}}_{(0)}={X}$ and
${\mathcal{T}}_{(n+1)}=\cup\\{C(I):\;I\in{\mathcal{T}}_{(n)}\\}$ for all $n$
then ${\mathcal{T}}=\bigcup\limits_{n=0}^{\infty}{\mathcal{T}}_{(n)}$.

  4. (iv)

For $({\mathcal{T}}_{(n)})_{n\in\mathbb{N}}$ as defined above,
$\lim\limits_{n}\big{[}\sup\\{\mu(I):I\in{\mathcal{T}}_{(n)}\\}\big{]}=0$.

The maximal operator ${\mathcal{M}}_{{\mathcal{T}}}$ associated to the tree
${\mathcal{T}}$ corresponds to every measurable function $\phi:X\to\mathbb{R}$
the function ${\mathcal{M}}_{\mathcal{T}}\phi$ defined by the formula

${\mathcal{M}}_{\mathcal{T}}\phi(x)=\sup\big{\\{}\frac{1}{\mu(I)}\int\limits_{I}|\phi|d\mu:\;\;x\in
I\in{\mathcal{T}}\big{\\}}.$

We also recall that for every measurable function $\phi:X\to\mathbb{R}$,
defining $\phi^{*}:(0,1]\to[0,+\infty)$ by the formula

$\phi^{*}(t)=\inf\\{y>0:\;\mu([|\phi|>y])<t\\}$

we have that $\phi^{*}$ is the unique decreasing and left continuous function
on $(0,1]$ that is equimeasurable to $\phi$. The following is proved in [3]:

######  Lemma 2.2.

For every $I\in{\mathcal{T}}$ and every $\alpha$ such that $0<\alpha<1$, there
exists a subfamily ${\mathcal{F}}$ of ${\mathcal{T}}$ consisting of almost
pairwise disjoint subsets of $I$ such that

$\mu\big{(}\bigcup\limits_{J\in{\mathcal{F}}}J\big{)}=\sum\limits_{J\in{\mathcal{F}}}\mu(J)=(1-\alpha)\mu(I).$

In [8] the following is proved:

######  Lemma 2.3.

For any integrable $\phi:X\to\mathbb{R}^{+}$ it holds that

$({\mathcal{M}}_{{\mathcal{T}}}f)^{*}(t)\leq\frac{1}{t}\int_{0}^{t}\phi^{*}(u)du\mbox{
for every }t\in(0,1].$

We will also need the following symmetrization principle which appears in [6].

######  Theorem 2.4.

Let $g:(0,1]\to\mathbb{R}^{+}$ be decreasing and let
$G_{1},G_{2}:[0,\infty)\to[0,+\infty)$ be two increasing functions. Then for
every $k\in(0,1]$ the following holds:

$\sup\big{\\{}\int_{K}(G_{1}\circ{\mathcal{M}}_{{\mathcal{T}}}\phi)(G_{2}\circ\phi)d\mu:\;\phi^{*}=g,\;\mu(K)=k\big{\\}}\\\\[2.84526pt]
=\int_{0}^{k}G_{1}\big{(}(\frac{1}{t}\int_{0}^{t}g(u)du\big{)})G_{2}\big{(}(g(t)\big{)})dt.$

Let us fix some notation. For $M_{1}\geq f>M_{2}\geq 0$, we set

$\displaystyle{\mathcal{C}}_{X,{\mathcal{T}}}(M_{1},f,M_{2})$ $\displaystyle=$
$\displaystyle\\{\phi:X\to\mathbb{R}^{+}\text{ measurable,
}\|\phi\|_{\infty}=M_{1},$ $\displaystyle\;\int\limits_{X}\phi
d\mu=f,\;\operatorname{essinf}_{X}(\phi)=M_{2}\\}$

and

${\mathcal{C}}_{X,{\mathcal{T}}}(M_{1},f)=\\{\phi:X\to\mathbb{R}^{+}\text{
measurable, }\|\phi\|_{\infty}=M_{1},\;\int\limits_{X}\phi d\mu=f\\}.$

In Theorem 1.1 we will compute the quantity

$\sup\big{\\{}\int\limits_{X}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu:\;\phi\in{\mathcal{C}}_{X,{\mathcal{T}}}(M_{1},f,M_{2})\big{\\}}$

for all for all $M_{1}\geq f>M_{2}\geq 0$ while in Theorem 1.2 we will compute
the quantity

$\sup\big{\\{}\int\limits_{K}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu,\;\phi\in{\mathcal{C}}_{X,{\mathcal{T}}}(M_{1},f),\;K\in{\mathcal{A}},\;\mu(K)=k\big{\\}}$

for all $M_{1}\geq f>0$ and all $k\in(0,1]$.

It is worthwhile to notice that the values of the two supremums above are
independent of the probability measure space $(X,{\mathcal{A}},\mu)$ and the
tree ${\mathcal{T}}$.

##  3\.  Proof of Theorem 1.1

For the proof of Theorem 1.1 we work as follows. Let $M_{1}\geq f>M_{2}\geq 0$
be three constants and let ${\mathcal{A}}(M_{1},f,M_{2})$ be the class of
functions

$\displaystyle{\mathcal{A}}(M_{1},f,M_{2})$ $\displaystyle=$
$\displaystyle\Big{\\{}g:[0,1)\to[0,+\infty):\;\;g\mbox{ is decreasing,}$
$\displaystyle\mbox{left continuous, continuous at }0,$ $\displaystyle
g(0)=M_{1},\;\lim\limits_{t\to 1}g(t)=M_{2}\mbox{ and
}\int_{0}^{1}g(t)dt=f\Big{\\}}$

For each $g\in{\mathcal{A}}(M_{1},f,M_{2})$ we set

$I_{g}=\int_{0}^{1}\Big{(}\frac{1}{t}\int_{0}^{t}g(u)du\Big{)}dt.$

Our goal is to maximize $I_{g}$ over all $g\in{\mathcal{A}}(M_{1},f,M_{2})$.

We observe that since $\lim\limits_{t\to
0}\Big{(}\log(t)\int_{0}^{t}g(u)du\Big{)}=0$ for every
$g\in{\mathcal{A}}(M_{1},f,M_{2})$, if we integrate by parts we get that

$I_{g}=\int_{0}^{1}\Big{(}\frac{1}{t}\int_{0}^{t}g(u)du\Big{)}dt=-\int_{0}^{1}\log(t)g(t)dt.$

We consider the function $h:[0,1]\to\mathbb{R}$ defined by the formula

$h(t)=\left\\{\begin{array}[]{l@{\quad} l}t-t\log(t)&\mbox{ if }\;\;0<t\leq
1\\\\[11.38109pt] 0&\mbox{ if }\;\;t=0.\end{array}\right.$

Since $h(0)=0=\lim\limits_{t\to 0}h(t)$, $h^{\prime}(t)=-\log(t)>0$ and
$h^{\prime\prime}(t)=-\frac{1}{t}<0$ for every $t\in(0,1)$, $h$ is continuous,
strictly increasing and strictly concave, thus for every
$g\in{\mathcal{A}}(M_{1},f,M_{2})$ the value
$I_{g}=\int_{0}^{1}g(t)\big{(}-\log(t)\big{)}dt$ may be expressed as the
Riemann-Stieltjes integral $I_{g}=\int_{0}^{1}g(t)dh(t).$

######  Proposition 3.1.

The quantity $\sup\\{I_{g}:\;g\in{\mathcal{A}}(M_{1},f,M_{2})\\}$ is equal to
$f+(f-M_{2})\log(\frac{M_{1}-M_{2}}{f-M_{2}})$. Moreover the supremum is
uniquely attained by the function $g_{0}$ defined as

$g_{0}(t)=\left\\{\begin{array}[]{l@{\quad} l}M_{1}&\mbox{ if }\;\;0\leq t\leq
c\\\\[11.38109pt] M_{2}&\mbox{ if }\;\;c<t\leq 1\end{array}\right.$

where $c=\frac{f-M_{2}}{M_{1}-M_{2}}$.

###### Proof..

We start with the second part. A direct calculation shows that

$\int_{0}^{1}g_{0}(t)dt=M_{1}\frac{f-M_{2}}{M_{1}-M_{2}}+M_{2}(1-\frac{f-M_{2}}{M_{1}-M_{2}})=f,$

while $g_{0}(0)=M_{1}$, $g_{0}(1)=M_{2}$ and $g_{0}$ is left continuous; thus
$g_{0}\in{\mathcal{A}}(M_{1},f,M_{2})$. We also have that

$\displaystyle I_{g_{0}}$ $\displaystyle=$
$\displaystyle-\int_{0}^{1}\log(t)g_{0}(t)dt=\
-\int_{0}^{c}M_{1}\log(t)dt-\int_{c}^{1}M_{2}\log(t)dt$ $\displaystyle=$
$\displaystyle M_{1}\big{(}h(c)-h(0)\big{)}+M_{2}\big{(}h(1)-h(c)\big{)}$
$\displaystyle=$ $\displaystyle
M_{1}\big{(}c-c\log(c))+M_{2}\big{(}1-(c-c\log(c)\big{)}$ $\displaystyle=$
$\displaystyle M_{2}+(M_{1}-M_{2})\;c\;\big{(}1-\log(c)\big{)}$
$\displaystyle=$ $\displaystyle
M_{2}+(M_{1}-M_{2})\frac{f-M_{2}}{M_{1}-M_{2}}\big{(}1-\log(\frac{f-M_{2}}{M_{1}-M_{2}})\big{)}$
$\displaystyle=$ $\displaystyle f+(f-M_{2})\log(\frac{M_{1}-M_{2}}{f-M_{2}}).$

Thus $\sup\\{I_{g}:\;g\in{\mathcal{A}}(M_{1},f,M_{2})\\}\geq
f+(f-M_{2})\log(\frac{M_{1}-M_{2}}{f-M_{2}})$.

We will show now that the number
$f+(f-M_{2})\log(\frac{M_{1}-M_{2}}{f-M_{2}})$ is an upper bound of the set
$\\{I_{g}:\;g\in{\mathcal{A}}(M_{1},f,M_{2})\\}$.

Let $g\in{\mathcal{A}}(M_{1},f,M_{2})$. We will calculate the Riemman
Stieltjes integral $I_{g}=\int_{0}^{1}g(t)dh(t)$ as the limit of the Riemman
Stieltjes sums over the net of all partitions
${\mathcal{P}}=\\{0=t_{0}<t_{1}<t_{2}<\cdots<t_{n}=1\\}$ equipped with the
right boundaries of the intervals $[t_{i-1},t_{i}]$ as intermediate points.

$\displaystyle I_{g}$ $\displaystyle=$ $\displaystyle\int_{0}^{1}g(t)dh(t)$
$\displaystyle=$
$\displaystyle\lim\limits_{{\mathcal{P}}}\sum\limits_{i=1}^{n}g(t_{i})\big{(}h(t_{i})-h(t_{i-1})\big{)}$
$\displaystyle=$
$\displaystyle\lim\limits_{{\mathcal{P}}}\big{(}\sum\limits_{i=1}^{n}g(t_{i})h(t_{i})-\sum\limits_{i=1}^{n}g(t_{i})h(t_{i-1})\big{)}$
$\displaystyle=$
$\displaystyle\lim\limits_{{\mathcal{P}}}\;\;\Big{[}-g(t_{1})h(t_{0})+\sum\limits_{i=1}^{n-1}h(t_{i})\big{(}g(t_{i})-g(t_{i+1})\big{)}+g(t_{n})h(t_{n})\Big{]}$
$\displaystyle=$
$\displaystyle\lim\limits_{{\mathcal{P}}}\;\;\Big{[}M_{2}+\sum\limits_{i=0}^{n-1}h(t_{i})\big{(}g(t_{i})-g(t_{i+1})\big{)}\Big{]}$

(we took into account that $h(t_{0})=h(0)=0$, $h(t_{n})=h(1)=1$ and
$g(t_{n})=g(1)=M_{2}$).

Since $g$ is nonincreasing, for every partition
${\mathcal{P}}=\\{0=t_{0}<t_{1}<t_{2}<\cdots<t_{n}=1\\}$ we have that
$g(t_{i})-g(t_{i+1})\geq 0$ for $i=0,1,\ldots,n-1$ while
$\sum\limits_{i=0}^{n-1}\big{(}g(t_{i})-g(t_{i+1})\big{)}=g(0)-g(1)=M_{1}-M_{2}$
and thus the numbers
$\big{(}\frac{g(t_{i})-g(t_{i+1})}{M_{1}-M_{2}}\big{)}_{i=0}^{n-1}$ serve as
coefficients of a convex combination. Since

$I_{g}=M_{2}+(M_{1}-M_{2})\lim\limits_{{\mathcal{P}}}\sum\limits_{i=0}^{n-1}h(t_{i})\frac{g(t_{i})-g(t_{i+1})}{M_{1}-M_{2}}$

the fact that the function $h$ is concave and continuous yields

$I_{g}\leq
M_{2}+(M_{1}-M_{2})h\Big{(}\lim\limits_{{\mathcal{P}}}\sum\limits_{i=0}^{n-1}\frac{g(t_{i})-g(t_{i+1})}{M_{1}-M_{2}}t_{i}\Big{)}$
(10)

But also

$\displaystyle\lim\limits_{{\mathcal{P}}}\sum\limits_{i=0}^{n-1}\big{(}g(t_{i})-g(t_{i+1})\big{)}t_{i}$
$\displaystyle=\lim\limits_{{\mathcal{P}}}\Big{(}g(t_{0})t_{0}+\sum\limits_{i=0}^{n-1}g(t_{i})(t_{i}-t_{i-1})-g(t_{n})t_{n-1}\Big{)}$
$\displaystyle=M_{1}\cdot 0+\int_{0}^{1}g(t)dt-M_{2}\cdot 1=f-M_{2}$

and thus

$\displaystyle I_{g}$ $\displaystyle\leq$ $\displaystyle
M_{2}+(M_{1}-M_{2})h(\frac{f-M_{2}}{M_{1}-M_{2}})$ $\displaystyle=$
$\displaystyle
M_{2}+(M_{1}-M_{2})\frac{f-M_{2}}{M_{1}-M_{2}}(1-\log\frac{f-M_{2}}{M_{1}-M_{2}})$
$\displaystyle=$ $\displaystyle
M_{2}+(f-M_{2})+(f-M_{2})\log\frac{M_{1}-M_{2}}{f-M_{2}}$ $\displaystyle=$
$\displaystyle f+(f-M_{2})\log\frac{M_{1}-M_{2}}{f-M_{2}}.$

Up to this pont we have shown that

$\sup\\{I_{g}:g\in{\mathcal{A}}(M_{1},f,M_{2})\\}=f+(f-M_{2})\log\frac{M_{1}-M_{2}}{f-M_{2}}$

and that the supremum is attained by the function $g_{0}$. It remains to prove
that $g_{0}$ is the unique function in ${\mathcal{A}}(M_{1},f,M_{2})$ with
this property. It is enough to show that for each
$g\in{\mathcal{A}}(M_{1},f,M_{2})$ with $g\neq g_{0}$ we have that
$I_{g}<f+(f-M_{2})\log\frac{M_{1}-M_{2}}{f-M_{2}}$.

Consider such a $g$. Since $g\neq g_{0}$ there exists $y_{0}$ a point of
continuity of $g$ such that $M_{1}>g(y_{0})>M_{2}$. We set $M=g(y_{0})$. In
the proof that is presented above we may consider only partitions of $[0,1]$
containing the point $y_{0}$ i.e. partitions of the form

${\mathcal{P}}=\\{0=t_{0}<t_{1}<t_{2}<\cdots<t_{k-1}<t_{k}=y_{0}<t_{k+1}<\cdots<t_{n}=1\\}.$

Then

$\displaystyle I_{g}$ $\displaystyle=$ $\displaystyle
M_{2}+(M_{1}-M_{2})\lim\limits_{{\mathcal{P}}}\Big{(}\sum\limits_{i=0}^{n}h(t_{i})\frac{g(t_{i})-g(t_{i+1})}{M_{1}-M_{2}}\Big{)}$
$\displaystyle=$ $\displaystyle
M_{2}+(M_{1}-M_{2})\lim\limits_{{\mathcal{P}}}\Big{(}\frac{M_{1}-M}{M_{1}-M_{2}}\sum\limits_{i=0}^{k-1}h(t_{i})\frac{g(t_{i})-g(t_{i+1})}{M_{1}-M}$
$\displaystyle\qquad+\frac{M-M_{2}}{M_{1}-M_{2}}\sum\limits_{i=k}^{n-1}h(t_{i})\frac{g(t_{i})-g(t_{i+1})}{M-M_{2}}\Big{)}$

Since $\sum\limits_{i=0}^{k-1}\frac{g(t_{i})-g(t_{i+1})}{M_{1}-M}=1$ and
$\sum\limits_{i=k}^{n-1}\frac{g(t_{i})-g(t_{i+1})}{M-M_{2}}=1$, using the fact
that $h$ is concave we get that

$\displaystyle I_{g}$ $\displaystyle\leq$ $\displaystyle
M_{2}+(M_{1}-M_{2})\lim\limits_{{\mathcal{P}}}\Big{(}\frac{M_{1}-M}{M_{1}-M_{2}}h\big{(}\sum\limits_{i=0}^{k-1}\frac{g(t_{i})-g(t_{i+1})}{M_{1}-M}t_{i}\big{)}$
$\displaystyle\qquad+\frac{M-M_{2}}{M_{1}-M_{2}}h\big{(}\sum\limits_{i=k}^{n-1}\frac{g(t_{i})-g(t_{i+1})}{M-M_{2}}t_{i}\big{)}\Big{)}$

But

$\displaystyle\lim\limits_{{\mathcal{P}}}\sum\limits_{i=0}^{k-1}\big{(}g(t_{i})-g(t_{i+1})\big{)}t_{i}$
$\displaystyle=$
$\displaystyle\lim\limits_{{\mathcal{P}}}\;\;\Big{[}g(t_{0})t_{0}+\sum\limits_{i=1}^{k-1}g(t_{i})(t_{i}-t_{i-1})-g(t_{k})t_{k-1}\Big{]}$
$\displaystyle=$ $\displaystyle M_{1}\cdot 0+\int_{0}^{y_{0}}g(t)dt-
My_{0}=\int_{0}^{y_{0}}g(t)dt-My_{0}$

and

$\displaystyle\lim\limits_{{\mathcal{P}}}\sum\limits_{i=k}^{n-1}\big{(}g(t_{i})-g(t_{i+1})\big{)}t_{i}$
$\displaystyle=$
$\displaystyle\lim\limits_{{\mathcal{P}}}\;\;g(t_{k})t_{k}+\sum\limits_{i=k+1}^{n-1}g(t_{i})(t_{i}-t_{i-1})-g(t_{n})t_{n-1}$
$\displaystyle=$ $\displaystyle M\cdot y_{0}+\int_{y_{0}}^{1}g(t)dt-M_{2}.$

as $g$ is continuous at $y_{0}$ and
$t_{k-1}\stackrel{{\scriptstyle{\mathcal{P}}}}{{\to}}y_{0}$.

Thus we get that

$\displaystyle I_{g}\leq
M_{2}+(M_{1}-M_{2})\Big{[}\frac{M_{1}-M}{M_{1}-M_{2}}h\Big{(}\frac{\int_{0}^{y_{0}}g(t)dt-
My_{0}}{M_{1}-M}\Big{)}$
$\displaystyle+\frac{M-M_{2}}{M_{1}-M_{2}}h\Big{(}\frac{My_{0}+\int_{y_{0}}^{1}g(t)dt-
M_{2}}{M-M_{2}}\Big{)}\Big{]}$

We observe that the coefficients $a=\frac{M_{1}-M}{M_{1}-M_{2}}$ and
$\beta=\frac{M-M_{2}}{M_{1}-M_{2}}$ are positive, $a+\beta=1$ while,

$\frac{\int_{0}^{y_{0}}g(t)dt-
My_{0}}{M_{1}-M}<y_{0}<\frac{My_{0}+\int_{y_{0}}^{1}g(t)dt-M_{2}}{M-M_{2}}$

where we took into account that the function $g$ is continuous at the points
$y_{0}$ and $1$ and $M_{1}>g(y_{0})=M>M_{2}$.

Therefore the fact that $h$ is strictly concave yields

$\displaystyle
I_{g}<M_{2}+(M_{1}-M_{2})h\Big{(}\frac{M_{1}-M}{M_{1}-M_{2}}\cdot\frac{\int_{0}^{y_{0}}g(t)dt-
My_{0}}{M_{1}-M}$
$\displaystyle+\frac{M-M_{2}}{M_{1}-M_{2}}\cdot\frac{My_{0}+\int_{y_{0}}^{1}g(t)dt-
M_{2}}{M-M_{2}}\Big{)}$
$\displaystyle=M_{2}+(M_{1}-M_{2})h\Big{(}\frac{\int_{0}^{1}g(t)dt-
M_{2}}{M_{1}-M_{2}}\Big{)}$
$\displaystyle=M_{2}+(M_{1}-M_{2})h\big{(}\frac{f-M_{2}}{M_{1}-M_{2}}\big{)}$
$\displaystyle=f+(f-M_{2})\log\big{(}\frac{M_{1}-M_{2}}{f-M_{2}}\big{)}.$

The proof is complete. ∎

Similar arguments to those used in the previous proof lead to the following.

######  Proposition 3.2.

If $({g}_{n})_{n\in\mathbb{N}}$ is a sequence in
${\mathcal{A}}(M_{1},f,M_{2})$ such that
$\lim\limits_{n}I_{g_{n}}=f+(f-M_{2})\log(\frac{M_{1}-M_{2}}{f-M_{2}})$ then
$g_{n}\stackrel{{\scriptstyle a.e.}}{{\longrightarrow}}g_{0}$ (and therefore
also $g_{n}\stackrel{{\scriptstyle L_{1}}}{{\longrightarrow}}g_{0}$), where
$g_{0}$ is the function defined in the statement of Proposition 3.1.

We are now in position to prove Theorem 1.1.

###### Proof of Theorem 1.1.

First we observe that for
$\phi\in{\mathcal{C}}_{X,{\mathcal{T}}}(M_{1},f,M_{2})$ we have that
$\phi^{*}\in{\mathcal{A}}(M_{1},f,M_{2})$. Also from [8] we have that
$({\mathcal{M}}_{{\mathcal{T}}}\phi)^{*}(t)\leq\frac{1}{t}\int_{0}^{t}\phi^{*}(s)ds$
for every $t\in(0,1]$. Since the functions ${\mathcal{M}}_{{\mathcal{T}}}\phi$
and $({\mathcal{M}}_{{\mathcal{T}}}\phi)^{*}$ are equimeasurable we get that

$\displaystyle\int{\mathcal{M}}_{{\mathcal{T}}}\phi d\mu$ $\displaystyle=$
$\displaystyle\int_{0}^{1}({\mathcal{M}}_{{\mathcal{T}}}\phi)^{*}(t)dt$
$\displaystyle\leq$
$\displaystyle\int_{0}^{1}\Big{(}\frac{1}{t}\int_{0}^{t}\phi^{*}(s)ds\Big{)}dt$
$\displaystyle=$ $\displaystyle I_{\phi^{*}}\leq
f+(f-M_{2})\log(\frac{M_{1}-M_{2}}{f-M_{2}}).$

where the last inequality follows from Proposition 3.1.

The sharpness of the above inequality is a consequence of Theorem 2.4 for
$G_{1}(t)=t$, $G_{2}(t)=1$, $t\in[0,+\infty)$ and $k=1$, where $g=g_{0}$ is
the function in the statement of Proposition 3.1 and the fact that since the
probability measure space $(X,{\mathcal{A}},\mu)$ is nonatomic we may easily
find a measurable function $\phi:X\to\mathbb{R}$ such that $\phi^{*}=g_{0}$.
The proof of Theorem 1.1 is complete. ∎

##  4\. Proof of Theorem 1.2

In this section we will prove Theorem 1.2. We start with the following lemma.

######  Lemma 4.1.

Let $g_{1},g_{2}:[0,+\infty)\to[0,+\infty)$ and $0<k\leq 1$ such that:
(a) $g_{1},g_{2}$ are decreasing.
(b) $g_{1}(t)\leq g_{2}(t)$ for every $t\in[0,+\infty)$.
(c) $g_{1}$ is right continuous and $g_{2}$ is left continuous.
(d) $\lim\limits_{t\to+\infty}g_{2}(t)=0$ and $g_{2}(0)=1$. Then there exists
$u\in[0,+\infty)$ such that $g_{1}(u)\leq k\leq g_{2}(u)$.

###### Proof..

We set $u=\sup\;g_{2}^{-1}\Big{(}[k,+\infty)\Big{)}$. Since $g_{2}(0)=1$ the
set $g_{2}^{-1}\Big{(}[k,+\infty)\Big{)}$ is nonempty, while is bounded from
above since $g_{2}$ is decreasing and $\lim\limits_{t\to+\infty}g_{2}(t)=0$.
Thus $u$ is a well defined real number.

The left continuity of $g_{2}$ yields $g_{2}(u)\geq k$, thus it suffices to
show that $g_{1}(u)\leq k$. If $g_{1}(u)>k$ from the right continuity of
$g_{1}$ we get that $g_{1}(t)>k$ for all $t\in[u,u+\varepsilon)$ for some
$\varepsilon>0$. Thus $k<g_{1}(t)\leq g_{2}(t)$ for all
$t\in[u,u+\varepsilon)$, which contradicts the definition of $u$. ∎

######  Lemma 4.2.

Let $(X,{\mathcal{A}},\mu)$ be a measure space, let $g:X\to[0,+\infty)$ be a
measurable function and $u\geq 0$. Let also $D$ be a measurable set such that
$[g>u]\subset D\subset[g\geq u]$. Then for every measurable set $K$ such that
$\mu(D)=\mu(K)$ we have that $\int\limits_{K}gd\mu\leq\int\limits_{D}gd\mu$.

###### Proof..

Let $\nu$ denote the indefinite integral of $g$ with respect to $\mu$, i.e.
the measure defined by the formula $\nu(A)=\int\limits_{A}gd\mu$ for all
$A\in{\mathcal{A}}$.

We set $V_{1}=[g>u]$ and $V_{2}=[g\geq u]$. We have that $\nu(K)=\nu(K\cap
V_{1})+\nu(K\setminus V_{1})=\nu(V_{1})-\nu(V_{1}\setminus K)+\nu(K\setminus
V_{1})$ and taking into account that $V_{1}\subset D\subset V_{2}$ we get that

$\nu(K)-\nu(D)=-\nu(D\setminus V_{1})-\nu(V_{1}\setminus K)+\nu(K\setminus
V_{1})$ (11)

Since for every $x\in D\setminus V_{1}$ we have that $x\in V_{2}$ and hence
$g(x)\geq u$ and since $V_{1}\subset D$ we get that

$-\nu(D\setminus V_{1})\leq-u\mu(D\setminus V_{1})=-u\mu(D)+u\mu(V_{1})$ (12)

For every $x\in V_{1}\setminus K$ we have that $g(x)>u$, thus

$-\nu(V_{1}\setminus K)\leq-u\mu(V_{1}\setminus K)=-u\mu(V_{1})+u\mu(K\cap
V_{1})$ (13)

Finally for every $x\in K\setminus V_{1}$ we have that $g(x)\leq u$ thus

$\nu(K\setminus V_{1})\leq u\mu(K\setminus V_{1})=u\mu(K)-u\mu(K\cap V_{1})$
(14)

Therefore from (11), (12), (13), (14) we get that

$\nu(K)-\nu(D)\leq u\mu(K)-u\mu(D)=0$

i.e. $\int\limits_{K}gd\mu\leq\int\limits_{D}gd\mu$. ∎

###### Proof of Theorem 1.2..

For the case $k\leq\frac{f}{M}$ it is obvious that for every
$\phi\in{\mathcal{C}}_{X,{\mathcal{T}}}(M,f)$ and $K\in{\mathcal{A}}$ with
$\mu(K)=k$ it holds that $\int\limits_{K}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu\leq M\mu(K)=kM$.

We examine next the case where $\frac{f}{M}<k\leq 1$. Let
$\phi\in{\mathcal{C}}_{X,{\mathcal{T}}}(M,f)$. The functions
$g_{1},g_{2}:X\to[0,+\infty]$ defined as
$g_{1}(t)=\mu([{\mathcal{M}}_{{\mathcal{T}}}\phi>t])$ and
$g_{2}(t)=\mu([{\mathcal{M}}_{{\mathcal{T}}}\phi\geq t])$ are decreasing,
$g_{1}$ is right continuous, $g_{2}$ is left continuous, $g_{1}(t)\leq
g_{2}(t)$ for all $t$, $\lim\limits_{t\to+\infty}g_{2}(t)=0$ and $g_{2}(0)=1$.
By Lemma 4.1 there exists $u\geq 0$ such that $g_{1}(u)\leq k\leq g_{2}(u)$,
i.e. $\mu([{\mathcal{M}}_{{\mathcal{T}}}\phi>u])\leq
k\leq\mu([{\mathcal{M}}_{{\mathcal{T}}}\phi\geq u])$. The fact that the
probability measure space $(X,{\mathcal{A}},\mu)$ is non-atomic yields the
existence of a $D\in{\mathcal{A}}$ with
$[{\mathcal{M}}_{{\mathcal{T}}}\phi>u]\subset
D\subset[{\mathcal{M}}_{{\mathcal{T}}}\phi\geq u]$ such that $\mu(D)=k$. Lemma
4.2 implies that for every $K\in{\mathcal{A}}$ with $\mu(D)=\mu(K)$ the
inequality $\int\limits_{K}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu\leq\int\limits_{D}{\mathcal{M}}_{{\mathcal{T}}}\phi d\mu$ holds.

We set $V_{1}=[{\mathcal{M}}_{{\mathcal{T}}}\phi>u]$ and
$V_{2}=[{\mathcal{M}}_{{\mathcal{T}}}\phi\geq u]$. Since $V_{1}\subset
D\subset V_{2}$ and thus $\mu(V_{1})\leq\mu(D)\leq\mu(V_{2})$, there exists
unique $s\in[0,1]$ such that $k=\mu(D)=s\mu(V_{1})+(1-s)\mu(V_{2})$ hence
$\mu(D\setminus V_{1})=(1-s)\mu(V_{2}\setminus V_{1})$.

######  Claim.

$\int\limits_{D}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu=s\int\limits_{V_{1}}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu+(1-s)\int\limits_{V_{2}}{\mathcal{M}}_{{\mathcal{T}}}\phi d\mu$

###### Proof..

$\displaystyle\int\limits_{D}{\mathcal{M}}_{{\mathcal{T}}}\phi d\mu$
$\displaystyle=$
$\displaystyle\int\limits_{V_{1}}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu+\int\limits_{D\setminus V_{1}}{\mathcal{M}}_{{\mathcal{T}}}\phi d\mu$
$\displaystyle=$
$\displaystyle\int\limits_{V_{1}}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu+u\mu(D\setminus V_{1})$ $\displaystyle=$
$\displaystyle\int\limits_{V_{1}}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu+u(1-s)\mu(V_{2}\setminus V_{1})$ $\displaystyle=$ $\displaystyle
s\int\limits_{V_{1}}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu+(1-s)\int\limits_{V_{1}}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu+(1-s)\int\limits_{V_{2}\setminus V_{1}}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu$ $\displaystyle=$ $\displaystyle
s\int\limits_{V_{1}}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu+(1-s)\int\limits_{V_{2}}{\mathcal{M}}_{{\mathcal{T}}}\phi d\mu.$

∎

The set $V_{1}=[{\mathcal{M}}_{{\mathcal{T}}}\phi>u]$ can be written as
$V_{1}=\bigcup\limits_{i\in A}I_{i}$ where $(I_{i})_{i\in A}$ is a pairwise
almost disjoint family of sets in ${\mathcal{T}}$, such that for each $i\in I$
the set $I_{i}$ is a maximal set of the family
$\\{I\in{\mathcal{T}}:\;\frac{1}{\mu(I)}\int\limits_{I}\phi d\mu>u\\}$.

We may also assume that the set $V_{2}=[{\mathcal{M}}_{{\mathcal{T}}}\phi\geq
u]$ can be written $V_{2}=\bigcup\limits_{j\in B}J_{j}$ where $(J_{j})_{j\in
B}$ is a pairwise almost disjoint family of sets in ${\mathcal{T}}$, such that
for each $j\in J$ the set $J_{j}$ is a maximal set of the family
$\\{J\in{\mathcal{T}}:\;\frac{1}{\mu(J)}\int\limits_{J}\phi d\mu\geq u\\}$.

This is possible because we may assume that $\phi$ satisfies the following
additional property:

$\forall x\in X\;\;\exists
I_{x}\in{\mathcal{T}}:\;{\mathcal{M}}_{{\mathcal{T}}}\phi(x)=\frac{1}{\mu(I_{x})}\int_{I_{x}}\phi
d\mu.$ (15)

This is due to the following:

######  Remark.

Let $\phi:X\to\mathbb{R}^{+}$ be a measurable function such that $\int_{X}\phi
d\mu=f$ and $\|\phi\|_{\infty}=M$. If we define the sequence
$({\phi}_{n})_{n\in\mathbb{N}}$ of functions as follows

$\phi_{n}=\sum\limits_{I\in{\mathcal{T}}_{(n)}}\operatorname{Av}_{I}(\phi)\chi_{I}$

(where $\chi_{I}$ denotes the characteristic function of $I$), then
$\int_{X}\phi_{n}d\mu=f$, $\|\phi_{n}\|_{\infty}\leq M$ and each $\phi_{n}$
satisfies (15). Moreover,
$({\mathcal{M}}_{{\mathcal{T}}}\phi_{n})_{n\in\mathbb{N}}$ is an increasing
sequence of functions that converges pointwise to
${\mathcal{M}}_{{\mathcal{T}}}\phi$. This implies that
$\int_{B}{\mathcal{M}}_{{\mathcal{T}}}\phi_{n}\to\int_{B}{\mathcal{M}}_{{\mathcal{T}}}\phi$
for every measurable subset $B$ of $X$.

For each $i\in A$ we set ${\mathcal{A}}_{i}=\\{I\in{\mathcal{A}}:\;I\subset
I_{i}\\}$, ${\mathcal{T}}_{i}=\\{I\in{\mathcal{T}}:\;I\subset I_{i}\\}$ and we
define $\mu_{i}(I)=\frac{1}{\mu(I_{i})}\mu(I)$ for every
$I\in{\mathcal{A}}_{i}$. Then $(I_{i},{\mathcal{A}}_{i},\mu_{i})$ is a non-
atomic probability measure space and ${\mathcal{T}}_{i}$ is a tree in
${\mathcal{A}}_{i}$. Denoting by $\phi_{i}$ the restriction of $\phi$ on
$I_{i}$, by $M_{i}$ the essential supremum of $\phi_{i}$ and setting
$f_{i}(\phi)=\int\limits\phi_{i}d\mu_{i}=\frac{1}{\mu(I_{i})}\int\limits_{I_{i}}\phi
d\mu$ we observe that $0\leq f_{i}(\phi)\leq M_{i}\leq M$ and
$\phi_{i}\in{\mathcal{C}}_{I_{i},{\mathcal{T}}_{i}}(M_{i},f_{i}(\phi),M_{2,i})$
for some $M_{2,i}$ with $0\leq M_{2,i}\leq f_{i}(\phi)$. In the case where
$M_{2,i}<f_{i}(\phi)$, Theorem 1.1 yields

$\int\limits_{I_{i}}{\mathcal{M}}_{{\mathcal{T}}_{i}}\phi_{i}d\mu_{i}\leq
f_{i}(\phi)+\big{(}f_{i}(\phi)-M_{2,i}\big{)}\log\big{(}\frac{M_{i}-M_{2,i}}{f_{i}(\phi)-M_{2,i}}\big{)}$

which implies that

$\int\limits_{I_{i}}{\mathcal{M}}_{{\mathcal{T}}_{i}}\phi_{i}d\mu_{i}\leq
f_{i}(\phi)+f_{i}(\phi)\log\big{(}\frac{M_{i}}{f_{i}(\phi)}\big{)}$

while in the case $M_{2,i}=f_{i}(\phi)$ the last inequality is obvious. The
fact that each $I_{i}$ is a maximal set of the family
$\\{I\in{\mathcal{T}}:\;\frac{1}{\mu(I)}\int\limits_{I}\phi d\mu>u\\}$ imlpies
that
${\mathcal{M}}_{{\mathcal{T}}_{i}}\phi_{i}(x)={\mathcal{M}}_{{\mathcal{T}}}\phi(x)$
for every $x\in I_{i}$. Thus we get that

$\frac{1}{\mu(I_{i})}\int\limits_{I_{i}}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu\leq f_{i}(\phi)+f_{i}(\phi)\log(\frac{M_{i}}{f_{i}(\phi)})\;\;\mbox{ for
every }i\in A.$

By defining in a similar way $f_{j}(\phi)$ and $M_{j}$ for every $j\in B$, we
get that

$\frac{1}{\mu(J_{j})}\int\limits_{J_{j}}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu\leq f_{j}(\phi)+f_{j}(\phi)\log(\frac{M_{j}}{f_{j}(\phi)})\;\;\mbox{ for
every }j\in B.$

From the inequalities above we get that

$\displaystyle\int\limits_{V_{1}}{\mathcal{M}}_{{\mathcal{T}}}\phi d\mu$
$\displaystyle=$ $\displaystyle\sum\limits_{i\in
A}\int\limits_{I_{i}}{\mathcal{M}}_{{\mathcal{T}}}\phi d\mu$
$\displaystyle\leq$ $\displaystyle\sum\limits_{i\in
A}\mu(I_{i})\Big{(}f_{i}(\phi)+f_{i}(\phi)\log(\frac{M_{i}}{f_{i}(\phi)})\Big{)}.$

and

$\displaystyle\int\limits_{V_{2}}{\mathcal{M}}_{{\mathcal{T}}}\phi d\mu$
$\displaystyle=$ $\displaystyle\sum\limits_{j\in
B}\int\limits_{J_{j}}{\mathcal{M}}_{{\mathcal{T}}}\phi d\mu$
$\displaystyle\leq$ $\displaystyle\sum\limits_{j\in
B}\mu(J_{j})\Big{(}f_{j}(\phi)+f_{j}(\phi)\log(\frac{M_{j}}{f_{j}(\phi)})\Big{)}.$

Let now an arbitrary $K\in{\mathcal{A}}$ with $\mu(K)=k$. We have that

$\displaystyle\int\limits_{K}{\mathcal{M}}_{{\mathcal{T}}}\phi d\mu$
$\displaystyle\leq$
$\displaystyle\int\limits_{D}{\mathcal{M}}_{{\mathcal{T}}}\phi d\mu$
$\displaystyle=$ $\displaystyle
s\int\limits_{V_{1}}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu+(1-s)\int\limits_{V_{2}}{\mathcal{M}}_{{\mathcal{T}}}\phi d\mu$
$\displaystyle\leq$ $\displaystyle s\sum\limits_{i\in
A}\mu(I_{i})\Big{(}f_{i}(\phi)+f_{i}(\phi)\log(\frac{M_{i}}{f_{i}(\phi)})\Big{)}$
$\displaystyle+(1-s)\sum\limits_{j\in
B}\mu(J_{j})\Big{(}f_{j}(\phi)+f_{j}(\phi)\log(\frac{M_{j}}{f_{j}(\phi)})\Big{)}$
$\displaystyle=$ $\displaystyle s\sum\limits_{i\in
A}\mu(I_{i})f_{i}(\phi)+(1-s)\sum\limits_{j\in B}\mu(J_{j})f_{j}(\phi)$
$\displaystyle+s\sum\limits_{i\in
A}\mu(I_{i})f_{i}(\phi)\log(\frac{M_{i}}{f_{i}(\phi)})+(1-s)\sum\limits_{j\in
B}\mu(J_{j})f_{j}(\phi)\log(\frac{M_{j}}{f_{j}(\phi)}).$

Therefore

$\displaystyle\int\limits_{K}{\mathcal{M}}_{{\mathcal{T}}}\phi d\mu$
$\displaystyle\leq$ $\displaystyle s\int\limits_{V_{1}}\phi
d\mu+(1-s)\int\limits_{V_{2}}\phi d\mu$ $\displaystyle+s\sum\limits_{i\in
A}\mu(I_{i})f_{i}(\phi)\log(\frac{M_{i}}{f_{i}(\phi)})+(1-s)\sum\limits_{j\in
B}\mu(J_{j})f_{j}(\phi)\log(\frac{M_{j}}{f_{j}(\phi)}).$

We observe that since $V_{1}\subset V_{2}$ we have that

$s\int\limits_{V_{1}}\phi d\mu+(1-s)\int\limits_{V_{2}}\phi
d\mu\leq\int\limits_{V_{2}}\phi d\mu\leq\int\limits_{X}\phi d\mu=f$ (16)

Also, since $M_{i}\leq M$ for each $i\in A$ we get

$\sum\limits_{i\in
A}\mu(I_{i})f_{i}(\phi)\log(\frac{M_{i}}{f_{i}(\phi)})\leq\mu(V_{1})\sum\limits_{i\in
A}\frac{\mu(I_{i})}{\mu(V_{1})}f_{i}(\phi)\log(\frac{M}{f_{i}(\phi)}).$

Since $\sum\limits_{i\in A}\frac{\mu(I_{i})}{\mu(V_{1})}=1$ and the function
$h_{0}$ defined as $h_{0}(t)=t\log(\frac{M}{t})$ is concave we obtain

$\displaystyle\sum\limits_{i\in
A}\frac{\mu(I_{i})}{\mu(V_{1})}f_{i}(\phi)\log(\frac{M}{f_{i}(\phi)})$
$\displaystyle=\sum\limits_{i\in
A}\frac{\mu(I_{i})}{\mu(V_{1})}h_{0}(f_{i}(\phi))$ $\displaystyle\leq
h_{0}\Big{(}\sum\limits_{i\in
A}\frac{\mu(I_{i})}{\mu(V_{1})}f_{i}(\phi)\Big{)}$
$\displaystyle=h_{0}\Big{(}\frac{1}{\mu(V_{1})}\int\limits_{V_{1}}\phi
d\mu\Big{)}$

Thus

$\sum\limits_{i\in
A}\mu(I_{i})f_{i}(\phi)\log(\frac{M_{i}}{f_{i}(\phi)})\leq\mu(V_{1})h_{0}\Big{(}\frac{1}{\mu(V_{1})}\int\limits_{V_{1}}\phi
d\mu\Big{)}.$

Similarly it is shown that

$\sum\limits_{j\in
B}\mu(J_{j})f_{j}(\phi)\log(\frac{M_{j}}{f_{j}(\phi)})\leq\mu(V_{2})h_{0}\Big{(}\frac{1}{\mu(V_{2})}\int\limits_{V_{2}}\phi
d\mu\Big{)}.$

Therefore

$\displaystyle s\sum\limits_{i\in
A}\mu(I_{i})f_{i}(\phi)\log(\frac{M_{i}}{f_{i}(\phi)})+(1-s)\sum\limits_{j\in
B}\mu(J_{j})f_{j}(\phi)\log(\frac{M_{j}}{f_{j}(\phi)})$ $\displaystyle\leq
s\mu(V_{1})h_{0}\Big{(}\frac{1}{\mu(V_{1})}\int\limits_{V_{1}}\phi
d\mu\Big{)}+(1-s)\mu(V_{2})h_{0}\Big{(}\frac{1}{\mu(V_{2})}\int\limits_{V_{2}}\phi
d\mu\Big{)}.$

Taking into account that the number $s$ has been seleceted such that
$s\mu(V_{1})+(1-s)\mu(V_{2})=\mu(D)=k$ and using again the fact that the
function $h_{0}$ is concave we get that the quantity above is

$\displaystyle=k\Big{(}\frac{s\mu(V_{1})}{k}h_{0}(\frac{\int\limits_{V_{1}}\phi
d\mu}{\mu(V_{1})})+\frac{(1-s)\mu(V_{2})}{k}h_{0}(\frac{\int\limits_{V_{2}}\phi
d\mu}{\mu(V_{2})})\Big{)}$ $\displaystyle\leq
kh_{0}\Big{(}\frac{s}{k}\int\limits_{V_{1}}\phi
d\mu+\frac{(1-s)}{k}\int\limits_{V_{2}}\phi d\mu\Big{)}$

We set $\Gamma=\frac{s}{k}\int\limits_{V_{1}}\phi
d\mu+\frac{(1-s)}{k}\int\limits_{V_{2}}\phi d\mu$. Until this point we have
shown that

$\int\limits_{K}{\mathcal{M}}_{{\mathcal{T}}}\phi d\mu\leq
k\Gamma+kh_{0}(\Gamma)=k\Gamma+k\Gamma\log(\frac{Mk}{k\Gamma}).$

We consider the function $h_{1}(t)=t+t\log(\frac{Mk}{t})$, $t\in(0,Mk]$. The
function $h_{1}$ is increasing, since $h_{1}^{\prime}(t)=\log(\frac{Mk}{t})>0$
for every $t\in(0,Mk)$. Since from (16) we have that $k\Gamma\leq f$ we get
that $h_{1}(k\Gamma)\leq h_{1}(f)$ (remember that we are studying the case
where $\frac{f}{M}<k\leq 1$), thus
$k\Gamma+k\Gamma\log(\frac{Mk}{k\Gamma})\leq f+f\log(\frac{Mk}{f})$.

Therefore

$\int\limits_{K}{\mathcal{M}}_{{\mathcal{T}}}\phi d\mu\leq
f+f\log(\frac{Mk}{f}).$

Up to this point we have shown that the left hand side of (9) in the statement
of Theorem 1.2 is less than or equal to the right side. It remains to show
that equality holds. We will again discern the cases $0<k\leq\frac{f}{M}$ and
$\frac{f}{M}<k\leq 1$.

Let’s first consider the case $0<k\leq\frac{f}{M}$. We select a family
$(I_{i})_{i\in A}$ of almost pairwise disjoint sets in ${\mathcal{T}}$ such
that $\sum\limits_{i\in A}\mu(I_{i})=k$. We consider the function
$\phi:X\to\mathbb{R}$ defined as

$\phi(x)=\left\\{\begin{array}[]{l@{\quad} l}M&\mbox{ if
}\;\;x\in\bigcup\limits_{i\in A}I_{i}\\\\[11.38109pt] \frac{f-Mk}{1-k}&\mbox{
if }\;\;x\in X\setminus\bigcup\limits_{i\in A}I_{i}\end{array}\right.$

The function $\phi$ is measurable, while since $k\leq\frac{f}{M}$ we have
$\phi\geq 0$. Also, since $f\leq M$ we have $\frac{f-Mk}{1-k}\leq M$ thus
$\|\phi\|_{\infty}=M$. Finally

$\displaystyle\int\limits_{X}\phi d\mu$ $\displaystyle=$ $\displaystyle
M\mu(\bigcup\limits_{i\in
A}I_{i})+\frac{f-Mk}{1-k}\mu(X\setminus\bigcup\limits_{i\in A}I_{i})$
$\displaystyle=$ $\displaystyle Mk+\frac{f-Mk}{1-k}(1-k)=f$

thus $\phi\in{\mathcal{C}}_{X,{\mathcal{T}}}(M,f)$.

Since

$\int\limits_{\bigcup\limits_{i\in A}I_{i}}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu=Mk$

the proof in the case where $0<k\leq\frac{f}{M}$ is complete.

We treat now the case $\frac{f}{M}\leq k\leq 1$. In this case it is enough to
show that for every $\delta>0$ there exists a $K\in{\mathcal{A}}$ with
$\mu(K)=k$ and a $\phi\in{\mathcal{C}}_{X,{\mathcal{T}}}(M,f)$ such that

$\int\limits_{K}{\mathcal{M}}_{{\mathcal{T}}}\phi
d\mu\geq(1-\delta)\big{(}f+f\log(\frac{Mk}{f})\big{)}.$

As in the first case we select a family $(I_{i})_{i\in A}$ of almost pairwise
disjoint sets in ${\mathcal{T}}$ such that $\sum\limits_{i\in A}\mu(I_{i})=k$.
We set $K=\bigcup\limits_{i\in A}I_{i}$, $f^{\prime}=\frac{f}{k}$ (notice that
$f^{\prime}\leq M$) and for every $i\in A$ we set
${\mathcal{A}}_{i}=\\{I\in{\mathcal{A}}:\;I\subset I_{i}\\}$,
${\mathcal{T}}_{i}=\\{I\in{\mathcal{T}}:\;I\subset I_{i}\\}$ and we define
$\mu_{i}(I)=\frac{1}{\mu(I_{i})}\mu(I)$ for every $I\in{\mathcal{A}}_{i}$. For
each $i\in A$, Theorem 1.1 yields the existence of a
$\phi_{i}\in{\mathcal{C}}_{I_{i},{\mathcal{T}}_{i}}(M,f^{\prime},0)$ such that
$\int{\mathcal{M}}_{{\mathcal{T}}_{i}}\phi_{i}d\mu_{i}>(1-\delta)(f^{\prime}+f^{\prime}\log(\frac{M}{f^{\prime}}))$.

We consider $\phi:X\to\mathbb{R}$ with $\phi|I_{i}=\phi_{i}$ for each $i\in I$
and $\phi|(X\setminus\bigcup\limits_{i\in A}I_{i})=0$. Then $\phi\geq 0$,
$\|\phi\|_{\infty}=M$ and

$\int_{X}\phi d\mu=\sum\limits_{i\in A}\int\limits_{I_{i}}\phi
d\mu=\sum\limits_{i\in A}\mu(I_{i})f^{\prime}=k\frac{f}{k}=f$

hence $\phi\in{\mathcal{C}}_{X,{\mathcal{T}}}(M,f)$.

Finally we have that

$\displaystyle\int\limits_{K}{\mathcal{M}}_{{\mathcal{T}}}\phi d\mu$
$\displaystyle=$ $\displaystyle\sum\limits_{i\in
A}\int\limits_{I_{i}}{\mathcal{M}}_{{\mathcal{T}}}\phi d\mu$
$\displaystyle\geq$ $\displaystyle\sum\limits_{i\in
A}\int\limits_{I_{i}}{\mathcal{M}}_{{\mathcal{T}}_{i}}\phi_{i}d\mu$
$\displaystyle=$ $\displaystyle\sum\limits_{i\in
A}\mu(I_{i})\int\limits_{I_{i}}{\mathcal{M}}_{{\mathcal{T}}_{i}}\phi_{i}d\mu_{i}$
$\displaystyle\geq$ $\displaystyle\sum\limits_{i\in
A}\mu(I_{i})(1-\delta)(f^{\prime}+f^{\prime}\log(\frac{M}{f^{\prime}}))$
$\displaystyle=$ $\displaystyle
k(1-\delta)(\frac{f}{k}+\frac{f}{k}\log(\frac{Mk}{f})$ $\displaystyle=$
$\displaystyle(1-\delta)(f+f\log(\frac{Mk}{f})).$

Letting $\delta\to 0^{+}$ we obtain our result. ∎

## References

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# Accurate benchmark results of Blasius boundary layer problem using a leaping
Taylor’s series that converges for all real values.

Anil Lal S  Milin Martin

###### Abstract

Blasius boundary layer solution is a Maclaurin series expansion of the
function $f(\eta)$, which has convergence problems when evaluating for higher
values of $\eta$ due to a singularity present at $\eta\approx-5.69$. In this
paper we are introducing an accurate solution to $f(\eta)$ using Taylor’s
series expansions with progressively shifted centers of expansion(Leaping
centers). Each series is solved as an IVP with the three initial values
computed from solution of the previous series, so the gap between the centers
of two consecutive expansions is selected from within the convergence disc of
the first series. The last series is formed such that it is convergent for a
reasonable high $\eta$ value needed for implementing the boundary condition at
infinity. The present methodology uses Newton-Raphson method to compute the
value of the unknown initial condition viz. $f^{\prime\prime}(0)$ in an
iterative manner. Benchmark accurate results of different parameters of flat
plate boundary layer with no slip and slip boundary conditions have been
reported in this paper.

###### keywords:

Blasius function , Leaping Taylor’s series , Slip flow , Newton-Raphson method

††journal:

[inst1]organization=Department of Mechanical Engineering,addressline=Amrita
Vishwa Vidyapeetham, city=Amritapuri Campus, postcode=690525, state=Kerala,
country=India

[inst2]organization=Department of Mechanical Engineering,addressline=College
of Engineering, Trivandrum, city=Sreekariyam, postcode=695016, state=Kerala,
country=India

##  1 Introduction

Blasius series solution [1, 2] is well known as the accurate solution for flat
plate boundary layer flow. It is not only a remarkable achievement in the
history of fluid dynamics but also a groundbreaking mathematical exercise in
which the non-linear differential equation
$f^{\prime\prime\prime}+\frac{1}{2}ff^{\prime\prime}=0$ with the known
conditions $f(0)=0,f^{\prime}(0)=0,f^{\prime}(\infty)=1$, called the Blasius
boundary layer equation is solved. It is not an initial value problem because
$f^{\prime\prime}(0)$ is not known and this poses certain difficulties in
determining its solution. The function $f(\eta)$ represents the value of
stream function and is popularly known as the Blasius function. Blasius
determined the solution in the form of a power series expansion [3] as,

$f(\eta)=\frac{\kappa}{2}{\eta}^{2}-\frac{{\kappa}^{2}}{240}{\eta}^{5}+\frac{11}{161280}{\kappa}^{3}{\eta}^{8}-\frac{5}{4257792}{\kappa}^{4}{\eta}^{11}+...$
(1)

where $f^{\prime\prime}(0)=\kappa$. The above series with center of expansion
at $\eta=0$ is found to have convergence problems at certain values of $\eta$
and due to this the value of $f^{\prime\prime}(0)$ is often found with a
lesser accuracy. Understanding the mathematical nature of Blasius function
including convergence properties, obtaining better accurate solutions and
extension to cases of slip boundary conditions etc, have prompted the
researchers to further investigate and solve the problem analytically [4, 5]
and numerically [6, 7].

Along with analytical methods, there have been efforts to use finite-order
accurate numerical techniques such as Runge-Kutta method and shooting method
for getting the unkknown initial condition etc., due to the ease of
computation. The shooting method [8] is used extensively, such that it has
literally become a part of classroom exercises. The shooting method is a
technique for solving a BVP by reducing it into an IVP. The method consists of
solving IVP with different initial conditions until the boundary condition of
the BVP is satisfied. Another technique used to deduce the initial condition
is called Töpfer transformation ($g={\lambda}^{-1/3}f$) [9, 10]. Töpfer
transformation leaves the initial conditions unchanged along with the
introduction of an additional initial condition, leading to the computation of
the Blasius function and its derivatives through an inverse transformation. A
number of previous investigations have reported the use of the Runge-Kutta
scheme along with the shooting method [6, 11] to determine the unknown initial
condition of the problem. One of the historical literature on numerical method
adopted for solving the Blasius equation is by Howarth [12] in which the value
of $f^{\prime\prime}(0)$, accurate upto five decimal places is reported. A
recent study by Majid et al.[13] uses a predictor-corrector two-point block
method to solve the Blasius equation numerically. The idea revolves around the
development of a technique to bypass the step of reducing the BVP to an IVP,
thus it makes use of a multiple shooting method that converges faster .

Previous studies on solutions to Blasius function have also been attempted
using analytical or semi-analytical methods with an aim to get the solution
with higher accuracy and with out convergence difficulties. Adomian
Decomposition Method (ADM) is a semi-analytical method that uses ”Adomain
Polynomials” to represent the non-linear portion of the function. Abbaoui et
al. [14] have given a proof for the convergence of the series solutions
obtained from Adomaian Decomposition Method. Wang [15] has shown the
application of ADM to obtain an approximate analytical solution to Blasius
function. Hashim [16] presents a modified technique that uses a combined ADM
and Pade approach to report the improvement in the accuracy of the solution.
Boyd [17] reported that series solution alone is not enough and combining it
with the Pade approximation is an effective approach. Pade approximation
provides very close estimate of the function by writing it in the form of a
rational function of a given order. It is reported that Pade Approximation to
Blasius equation with $\eta=6$ provides convergence to the solution with a
relative error of about $0.15\%$ in predicting the accurate value of
$f^{\prime\prime}(0)$. For $\eta>8$, the convergence is found to be slow and
there is a need for higher values of Pade order (N) for getting a better
solution. A recent study by Andrianov et al. [18] introduced the Pade
approximation for solving the boundary layer problem and for the problem of
rotating fluid of Ekman layer type.

Differential Transformation Method (DTM) is a systematic framework that gives
transformations which becomes the coefficients of terms of the series
expansion of a function or functions presented in the form of linear, non-
linear and simultaneous ordinary differential equations [19, 20, 21, 22]. It
may be noted that Blasius has derived his series accurately in the same manner
but without resorting to DTM framework. Yu et. al. [19], used DTM by
converting the Blasius problem into a set of three IVPs. A sub-domain approach
is found to overcome the convergence problems of the series solution and
values accurate up to $6$ decimal places are reported. Arikoglu and Ibrahim
[23] have presented an inner-outer matching solution for the Blasius problem
in which $f^{\prime\prime}(0)$ is evaluated by equating two series expansions
with one series expanded about $\eta=0$ and the other one about a higher value
of $\eta=m$ of about $10$. The matching of the solutions is done at a value of
$\eta$ in between $4$ and $5$ to determine $f^{\prime\prime}(0)$ and the
reported value is found to have a difference starting from the fourth decimal
place onward. Lal and Neeraj [22] have used DTM along with the scaled
transformation for converting the boundary value of Blasius function to
initial value and reported results accurate up to 21 decimal points with a
larger radius of convergence.

In the present work, we report a method in which a set of leaping series
solutions that converge for all positive real values of $\eta$ are developed,
with out resorting to approximations. Application of Newton-Raphson method
enables determination of any unknown initial value in the problem from a
suitable boundary condition, and still obtain Taylor’s series solution.
Currently used methods such as Töpfer transformation, truncated boundary
approximation, iterative transformation method [9, 10] are expected to pose
difficulties in deriving the initial conditions from boundary conditions for
the class problems governed by simultaneous differential equations and
equations with more number of unknown initial values. We report benchmark
solutions to boundary layers with no-slip and slip flow over the wall surface,
nature of convergence of series expansions, unknown initial values and
parameters of asymptotic variation with a greater accuracy in this paper.

##  2 Differential Transform

The Taylor series expansion of a function $f(\eta)$ about $\eta=\eta_{i}$ is,

$f(\eta)=\sum_{k=0}^{\infty}\frac{\left.f^{k}(\eta)\right|_{\eta=\eta_{i}}}{k!}(\eta-\eta_{i})^{k}=\sum_{k=0}^{\infty}F(k)(\eta-\eta_{i})^{k}$
(2)

Where, $F(k)=\frac{\left.f^{k}(\eta)\right|_{\eta=\eta_{i}}}{k!}$,
$\forall~{}k\in\mathbb{W}$ is called the differential transform of $f(\eta)$
and $f^{k}(\eta)$ is the $k^{th}$ derivative of $f$ with respect to $\eta$. We
denote differential transform operation of $f(\eta)$ as $DT(f(\eta))$ and the
result of the transform as $F(k)$. Let $U(k)$ and $V(k)$ are the differential
transforms of the functions $u(\eta)$ and $v(\eta)$. Some useful theorems of
differential transforms with $U(k)$ and $V(k)$ [24, 25, 26] that becomes handy
in finding the solutions of IVPs are given here.

###### Theorem 1

If $f(\eta)=u(\eta)\pm v(\eta)$, then $F(k)=U(k)\pm V(k)$

###### Theorem 2

If $f(\eta)=\alpha u(\eta)$, then $F(k)=\alpha U(k)$

###### Theorem 3

If $f(\eta)=\frac{du(\eta)}{d\eta}$, then $F(k)=(k+1)U(k+1)$

###### Theorem 4

If $f(\eta)=\frac{d^{m}u(\eta)}{d\eta^{m}}$, then
$F(k)=(k+1)(k+2)...(k+3)U(k+m)$

###### Theorem 5

If $f(\eta)=u(\eta)v(\eta)$, then $F(k)=\sum_{l=0}^{k}V(l)U(k-l)$

Making use of these theorems the terms in the Blasius equation namely
$f^{\prime\prime\prime}(\eta),f^{\prime\prime}(\eta)$ and $f(\eta)$ can be
differential transformed as

$DT\left[f^{\prime\prime\prime}(\eta)\right]=(k+1)(k+2)(k+3)F(k)$ (3)
$DT\left[f(\eta)f^{\prime\prime}(\eta)\right]=\sum_{l=0}^{k}(l+1)(l+2)F(l)F(k-l)$
(4)

By knowing $F(0)$, $F(1)$, the value of $F(2)$ taken based on generated values
of $f^{\prime\prime}(0)$, the differential transform for any value of $k$ is
found using

$F(k+3)=-\frac{\sum_{l=0}^{k}(l+1)(l+2)F(l)F(k-l)}{2(k+1)(k+2)(k+3)}$ (5)

For $\eta_{i}=0$, we have $F(0)=0$, $F(1)=0$ and $F(2)=\kappa/2$, where
$\kappa=f^{\prime\prime}(0)$.

##  3 Solution Methodology

###  3.1 Series expansion about leaping centers

It is well known that Taylor’s series given in equation (1) with center of
expansion at $\eta=0$ does not converge for $\eta>5.69$ [3]. But in
computations on computers, convergence up to $\eta=5.69$ is rarely possible
due to accuracy problems and restricted summation over a finite number of
terms. This leads to approximating the boundary condition
$f^{\prime}(\infty)=1$ as $f^{\prime}(5.69)=1$, which results in lesser
accurate results. In the present work, we propose an approach of combining
more number of series expansions of the Blasius function by leaping the center
of expansion to different values of $\eta$ to circumvent the convergence
problems associated with approximation of $f^{\prime}(\infty)=1$. We have
approximated the infinity by a larger value of $\eta$ and applied the boundary
condition as $f^{\prime}(\eta)=1$, where $\eta\geq 18$. Such an approximation
is found to provide a more accurate solution to the Blasius function.

Figure 1: Taylor series expansions about leaping centers at
$\eta_{0}=0,\eta_{1},\eta_{2},....,\eta_{n}$. The symbol $n$ denotes the
number of expansions.

The idea used in the present approach is to generate a few Taylor series
expansions for $f(\eta)$ with centers of expansion denoted as
$\eta_{0}=0,\eta_{1},\eta_{2},....,\eta_{n}$, as illustrated in Figure(1). The
corresponding Taylor’s series expansions are denoted as
$s_{0},~{}s_{1},\ldots,s_{n}$ for the functions
$f_{s_{0}},~{}f_{s_{1}},\ldots,f_{s_{n}}$ , respectively. It may be noted that
$f_{s_{0}}=f_{s_{1}}=\ldots=f_{s_{n}}$, because all the series expansions are
for the same function, but the difference is only in their convergence
properties. We use Differential Transform method described in the above
section to compute the coefficients of the Taylor’s series. The coefficients
of the series expansions are denoted by $F_{s_{i}}(k)$ for all the values of
$k\in\mathbb{W}$. The initial condition for $s_{0}$ are $F_{s_{0}}(0)=0$,
$F_{s_{0}}(1)=0$ for no-slip flow and $F_{s_{0}}(1)=v_{slip}$, a prescribed
values of slip flow. We keep $F_{s_{0}}(2)=\kappa^{*}$, a variable which is
varied during the computation of Taylor’s series coefficients.

The initial conditions for computing any series with center of expansion at
$\eta_{i}$, namely, $s_{i}$ are taken from the already obtained series
$s_{i-1}$ as,

$\begin{split}f_{s_{i}}(\eta_{i})&=f_{s_{i-1}}(\eta_{i})\\\
f_{s_{i}}^{\prime}(\eta_{i})&=f^{\prime}_{s_{i-1}}(\eta_{i})\\\
f_{s_{i}}^{\prime\prime}(\eta_{i})&=f^{\prime\prime}_{s_{i-1}}(\eta_{i})\end{split}$
(6)

Note that $f_{s_{i-1}}(\eta)$ should converge at $\eta=\eta_{i}$. So fixing an
$\eta_{i}$ for expanding a series $s_{i}$ depends on the convergence of
$s_{i-1}$ at $\eta_{i}$. By following this procedure, it is possible to
develop any number of series expansions. But an accurate solution can be
obtained from as low as three series expansions, which is going to be
demonstrated in the section on results.

###  3.2 Newton-Raphson Iteration

We first develop the selected number of function expansions, $s_{i}$,
$i\in\mathbb{W}$ and $i\leq n$, by using a guessed value for $\kappa^{*}$, say
$\kappa^{*}=0.1$. We then develop one more set of expansion namely,
$s^{\prime}_{i}$, corresponding to a slightly different value for
$\kappa^{*}$, denoted as $\kappa^{*}+\delta\kappa$, where $\delta\kappa$ is a
very small value. A typical value for $\delta\kappa$ used in the present
computations is $10^{-18}$. It is found that $f^{\prime}(\infty)=1$ is a
strong boundary condition for the problem compared to
$f^{\prime\prime}(\infty)=0$. So we solve the equation
$f^{\prime}(\infty)-1=0$ for finding the converged solution for $\kappa^{*}$.
The converged solution of $\kappa^{*}$ is denoted as $\kappa$. In the present
computations the condition at infinity is changed to the one corresponding to
a very high value of $\eta$ denoted as $\tilde{\infty}$. So we use
$f^{\prime}(\tilde{\infty})-1=0$ as the equation solved for the determination
of $\kappa$. We have used a value for $\tilde{\infty}$ in the range $5-30$,
where $\tilde{\infty}$ is a finite value of $\eta$ at which $f^{\prime}(\eta)$
becomes almost equal to 1. The approximate gradient for Newton-Raphson method
is computed as,

$g=\frac{f^{\prime}_{s^{\prime}_{n}}(\tilde{\infty})-f^{\prime}_{s_{n}}(\tilde{\infty})}{\delta\kappa}$
(7)

$\kappa^{*}$ is now updated as,

$\kappa^{*}_{new}=\kappa^{*}-\frac{(f^{\prime}(\tilde{\infty})-1.)}{g}$ (8)

Now, $\kappa^{*}_{new}$ is assigned as the new value of $\kappa^{*}$ and the
computation of series is continued till $f^{\prime}(\tilde{\infty})$ becomes
equal to 1 and this stage the value of $\kappa^{*}$ is treated as the
converged value for $\kappa$. The number of Newton-Raphson iterations applied
to reach the solution for $\kappa$ is kept as $50$.

##  4 Results and discussion

The methodology for the solution has been implemented in FORTRAN with variable
type for computations set using ”IMPLICIT REAL*16(A-H,O-Z)” so as to get
accuracy up to 32 decimal places in the floating point operations. All the
results we report here are obtained from summing $2000$ terms of the Taylor’s
series. We find no difference in the converged value of $\kappa$ with further
increase in the number of terms of Taylor’s series summed.

###  4.1 Flow with no-slip

Figure 2: Convergence properties of three series expansions, $f_{s_{0}}$,
$f_{s_{1}}$ and $f_{s_{2}}$ having centers of expansion $\eta_{0}=0$,
$\eta_{1}=5$ and $\eta_{2}=10$.

(a) $f(\eta)$ vs. $\eta$

(b) $f^{{}^{\prime}}(\eta)$ and $f^{{}^{\prime\prime}}(\eta)$ vs. $\eta$

Figure 3: The accurate solutions (a) $f(\eta)$, (b) $f^{\prime}(\eta)$ and $f^{\prime\prime}(\eta)$ for $\eta\in[0,30]$. n | $\tilde{\infty}$ | $f^{\prime\prime}(0)$
---|---|---
1 | 5 | 0.3361523439045316853836761393072
2 | 10 | 0.3320573372037024527094809622944
3 | 15 | 0.33205733621519629893842022877817
4 | 20 | 0.33205733621519629893718006201058
5 | 25 | 0.33205733621519629893718006201058
6 | 30 | 0.33205733621519629893718006201058
Table 1: Effect of number of expansion $n$ on the converged solution of the
value of $f^{\prime\prime}(0)$.

Table 1 shows the effect of number of expansions $n$ on the converged solution
of $\kappa$ by leaping the center of expansion by $\triangle\eta=5$. Which
means for every $s_{i}$, the center of expansion $\eta_{i}=\eta_{i-1}+5$, with
$\eta_{0}=0$. The number of iterations of Newton-Raphson step is kept as $50$
and this is found to guarantee the convergence of $f^{\prime\prime}(0)$, up to
$32$ decimal places. It can be seen that the converged solution is independent
of $n$ for $n\geq 4$. This implies that the boundary condition at infinity can
be well approximated at $\tilde{\infty}=20$. More precisely, we find that the
value of $\kappa$ does not change from $\tilde{\infty}=18$ onward. We have
also examined the effect of change of separation between the center of
expansions and the observations are tabulated in Table 2. It can be seen that
the result is independent on the step size of leaping of the center of
expansion and is only dependent on the point $\tilde{\infty}$, where the
boundary condition $f^{\prime}(\eta)=1$ is applied.

The value of $\kappa$ resulted from the present computation has 32 decimal
points accuracy, the accuracy is better than that of 0.33205733621519630 as
given by Boyd in [3] and 0.332057336215196299183 as given by Lal and Neeraj in
[22].

$\eta_{i}-\eta_{i-1}$ | n | $\tilde{\infty}$ | $f^{\prime\prime}(0)$
---|---|---|---
2 | 10 | 20 | 0.33205733621519629893718006201058
3 | 7 | 21 | 0.33205733621519629893718006201058
4 | 5 | 20 | 0.33205733621519629893718006201058
5 | 4 | 20 | 0.33205733621519629893718006201058
Table 2: Effect of change of leaping distance of consecutive centers of
expansion

The convergence properties of the series expansions for $s_{0}$, $s_{1}$ and
$s_{2}$ having centers of expansion at $\eta_{0}=0$, $\eta_{0}=5$ and
$\eta_{0}=10$ are depicted in Figure 2 for $\eta\in[0,15]$. It can be seen
that the series $s_{0}$ converges smoothly up to $\eta=5.69$, which is close
to the value corresponding to the singularity at $\eta=-5.69$ as reported by
Boyd in [3]. The series $s_{1}$ is found to be converging for
$\eta\in[0,10.365]$ and diverges for $\eta>10.365$. The series $s_{2}$ is
converging for $\eta\in[2.4,15]$ and diverges for $\eta<2.4$. The radius of
convergence increases with the increase in distance of center of expansion
from $\eta=0$. These observations on the convergence implies that a single
series can not represent the expansion of Blasius function for a larger
domain.

By considering the convergence properties, we propose to use a combination of
leaping series expansions to represent Blasius function as demonstrated in
Figure (3). It consists of $6$ expansions, namely $s_{0}$ to $s_{5}$ with
centers of expansion at $\eta_{0}=0$, $\eta_{1}=5,~{}\ldots$,
$\eta_{i}=\eta_{i-1}+5$ for determining the convergence properties upto
$\eta=30$. The series expansion $s_{i}$ is used to determine the value of the
function in the converging interval, $\eta\in[\eta_{i},\eta_{i}+5]$. Figure
(3) demonstrates the expansion of the function in the interval
$\eta\in[0,30]$. In the same manner, by combining more number of series
expansions, it is possible to expand the function for any positive real value.
As per the results obtained in the present study $\eta=20$ found to be an
optimum point where the boundary condition $f^{\prime}(\infty)=1$ can be
applied. From $\eta=18$ onward the asymptotic variation of the function is as
$f(\eta)=-1.720787657520502819605438159825+\eta$. The parameter in the
asymptotic variation has more decimal points accuracy than the one reported by
Boyd in [3]. Tables 3 and 4 gives the benchmark results of the present
accurate solutions of $f(\eta)$, $f^{\prime}(\eta)$, $f^{\prime\prime}(\eta)$
and $f(^{\prime\prime\prime}\eta)$ for $\eta\in[0,15]$. It can be seen that
the data satisfy the equation $f^{\prime\prime\prime}+ff^{\prime\prime}/2=0$
almost accurately.

Table 3: Benchmark solutions of $f(\eta)$ and $f^{\prime}(\eta)$ for
$\eta\in[0,15]$ of no-slip Blasius boundary layer.

$\eta$ | $f(\eta)$ | $f^{\prime}(\eta)$
---|---|---
0. | 0.00000000000000000000000000000000 | 0.00000000000000000000000000000000
1. | 0.16557172578927971994607139866464 | 0.32978003124966696806286485058647
2. | 0.65002436993528859325840008060347 | 0.62976573650238585970789214001077
3. | 1.39680823087034546785665318114741 | 0.84604444365799349725098240374192
4. | 2.30574641846207117902246203651341 | 0.95551822981069424686181417757482
5. | 3.28327366515631417796776890035073 | 0.99154190016439334277083771840218
6. | 4.27962092251384908051614632858349 | 0.99897287243586052111975290945127
7. | 5.27923881102910880195475622555865 | 0.99992160414795010955867185310631
8. | 6.27921343134607434184668676342406 | 0.99999627453530080913338909825509
9. | 7.27921237064524251498448401453620 | 0.99999989045379928945150717872348
10. | 8.27921234293432582800203409779852 | 0.99999999801539054606962551624450
11. | 9.27921234248405790426225183944242 | 0.99999999997791640300822743404095
12. | 10.27921234247952546306776851892884 | 0.99999999999984941269916627913591
13. | 11.27921234247949728852564004547502 | 0.99999999999999937182753490370178
14. | 12.27921234247949718064881057910890 | 0.99999999999999999839911378268159
15. | 13.27921234247949718039492878137463 | 0.99999999999999999999751013528420

Table 4: Benchmark solutions of $f^{\prime\prime}(\eta)$ and
$f^{\prime\prime\prime}(\eta)$ for $\eta\in[0,15]$ of no-slip Blasius boundary
layer.

$\eta$ | $f^{\prime\prime}(\eta)$ | $f^{\prime\prime\prime}(\eta)$
---|---|---
0. | 0.33205733621519629893718006201058 | 0.00000000000000000000000000000000
1. | 0.32300711668694284746919034374890 | -0.02674042287603818941557199292502
2. | 0.26675154569727845645745365043572 | -0.08669750271056888580038083842490
3. | 0.16136031954087853155687842360081 | -0.11269471123528408855663373868942
4. | 0.06423412109169066819653982932531 | -0.07405379732511237191225366739570
5. | 0.01590679868531816315507141064886 | -0.02611318661022410269690678475316
6. | 0.00240203984375727885991930092577 | -0.00513990998602777383188838650274
7. | 0.00022016895527113462513633196741 | -0.00058116224682555289777061646404
8. | 0.00001224092624325319054622340463 | -0.00003843169423937603885233795627
9. | 0.00000041278790156878232013889098 | -0.00000150238539977608548717786204
10. | 0.00000000844291586701753292682574 | -0.00000003495034662828381184818483
11. | 0.00000000010473955482006946773795 | -0.00000000048595028491633710186547
12. | 0.00000000000078810075727866872385 | -0.00000000000405052751566817613004
13. | 0.00000000000000359671084986261923 | -0.00000000000002028403270505018845
14. | 0.00000000000000000995593526876784 | -0.00000000000000006112552161659052
15. | 0.00000000000000000001671519173257 | -0.00000000000000000011098229018106

###  4.2 Flow with slip

(a) $f^{\prime\prime}(0)$ Vs $\%$ slip

(b) $f^{\prime\prime}(0)$ Vs $K_{1}$

Figure 4: Variation of $f^{\prime\prime}(0)$ with $\%$slip and the rarefaction
parameter $K_{1}$.

The present methodology for computing $f^{\prime\prime}(0)$ using Newton-
Raphson method has enabled computation of the same for the cases with velocity
slip boundary condition at the wall. We have varied the slip velocity,
$v_{slip}\in[0,1]$, with the extreme values $0$ and $1$ imply no slip and
uniform velocity, respectively. Figure 4(a) shows the plot showing the
variation of $f^{\prime\prime}(0)$ with the percentage slip. The curve has a
monotonic decreasing non-linear variation. It can be found that [27] has
considered the ratio of slip to wall shear stress as $K_{1}$, which represents
the rarefaction of the flow medium, surface properties etc., as given in
equation(9). We have computed the solution corresponding to $K_{1}\in[0,50]$
and the result is presented in Figure 4(b) as the variation of
$f^{\prime\prime}(0)$ with $K_{1}$. It is evident that with increase of
$K_{1}$, the value of $f^{\prime\prime}(0)$ reduces and slope of the variation
reduces continuously and is expected to reach zero asymptotically.

$K_{1}=\frac{2-\sigma}{\sigma}Kn_{x}{Re_{x}}^{1/2}$ (9)

Where, $\sigma$ is called tangential accommodation coefficient, $Kn_{x}$ and
$Re_{x}$ are the local Knudsen and Reyonld numbers. Table (5) provides
benchmark numerical values of $f^{\prime\prime}(0)$ for five typical values of
the slip. Figure 5 shows the variations of $f^{\prime}(\eta)$ and
$f^{\prime\prime}(\eta)$ for the case with slip$=0.5$.

(a) $f^{\prime}(\eta)$ Vs $\eta$

(b) $f^{\prime\prime}(\eta)$ Vs $\eta$

Figure 5: $f^{\prime}(\eta)$ and $f^{\prime\prime}(\eta)$ plotted for
different values of $\eta$ when the slip is 0.5

##  5 Conclusions

In order to overcome the convergence problems of a single series expansion of
Blasius function, a combination of series expansions with different
convergence properties, that are expanded about different centers called the
leaping centers is demonstrated. Though development of series expansion of
functions governed by differential equations requires all the required
conditions as initial values, we demonstrated deduction of series solutions
for boundary value problems using Newton-Raphson method in this work. The
major conclusions from this study are:

  1. 1.

The solution is found to be independent on the point of application of the
boundary condition $f^{\prime}(\infty=1.)$ for $\eta\geq 18$.

  2. 2.

The series with the center of expansion at $\eta=0$ diverges for $\eta>5.68$.
The series with the center of expansion at $\eta=5$ diverges for $\eta>10.36$.
The series with the center of expansion at $\eta=10$ has found to exhibit
divergence for $\eta<2.32$.

  3. 3.

The solution is found to be independent of the leaping distance between the
center of expansions for leaping distances less than $5.68$, corresponding to
the divergence of the series with the center of expansion at $\eta=0$.

  4. 4.

The variation of $f^{\prime\prime}(0)$ with boundary slip is found to be
monotonically decreasing. The variation of $f^{\prime\prime}(0)$ with the
rarefaction parameter has a decreasing tend having an asymptotic behaviour.

  5. 5.

The accuracy of the variations and all the related parameters of Blasius
function are obtained with a better accuracy than those atready available in
the literature.

  6. 6.

The method can be used for solving differential equations of functions having
a singularity on the left side of the origin.

Table 5: Variation of $f^{\prime\prime}(0)$ with wall slip condition. Slip | $f^{\prime\prime}(0)$
---|---
0.2 | 0.31335775171380422946976167600464
0.4 | 0.26523233119785271052478203825874
0.5 | 0.23245507775976810752799593194116
0.6 | 0.19462617044304093400347255410066
0.8 | 0.10538860552764686718424828804120

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