# Tilted Poincaré Sphere Geodesics

Andrew A. Voitiv1, Mark T. Lusk2 Mark E. Siemens1<EMAIL_ADDRESS>1Department
of Physics and Astronomy, University of Denver, 2112 E. Wesley Avenue, Denver,
CO 80208, USA  2Department of Physics, Colorado School of Mines, 1500 Illinois
Street, Golden, CO 80401, USA

###### Abstract

We provide the first experimental demonstration of geometric phase generated
in association with closed Poincaré Sphere trajectories comprised of geodesic
arcs that do not start, end, or necessarily even include, the north and south
poles that represent pure Laguerre-Gaussian modes. Arbitrarily tilted
(elliptical) single vortex states are prepared with a spatial light modulator,
and Poincaré Sphere circuits are driven by beam transit through a series of
$\pi$-converters and Dove prisms.

When the Poincaré Sphere (PS) for Gaussian modes was introduced, the
connection was immediately made between trajectories produced by linear optics
and the accumulation of geometric phase [1]. Within a closely related setting,
the measurement of geometric phase had already been well-established by then
for the polarization of light [2, 3, 4], as first predicted by Pancharatnam
even before the context of Berry’s seminal work [5, 6]. This parallel offered
a convenient way of interpreting the underlying physics by comparing
polarization states with their orbital angular momentum (OAM) counterparts:
namely, circularly polarized states (Laguerre-Gaussian modes) on the poles,
linearly polarized states (Hermite-Gaussian modes) on the equator, and
mixtures of these states otherwise throughout the rest of the spherical
surface [7]. Likewise, the optics that generate trajectories on the PS are
functionally analogous, such as quarter-waveplates
($\frac{\pi}{2}$-converters) and half-waveplates ($\pi$-converters). A series
of $\frac{\pi}{2}$ mode converters and Dove prisms was used by Galvez et al.
to transform beams from a fundamental Lageurre-Gaussian mode at a pole, to
Hermite-Gaussian modes on the equator, and back to form closed circuits on the
PS [8]. Using a collinear Gaussian beam for interferometric measurements, the
authors extracted a geometric phase that scaled linearly with the rotation of
the transforming optics—just as seen in the polarization counterpart many
years before (e.g., [2]).

Consider the first-order Gaussian mode sphere of Fig. 1, which we will refer
to as the “vortex PS” to emphasize that the beam waists and waist positions
are not fixed on such spheres (particularly between start and end points
through focusing optics). Previous experimentally considered trajectories on
this PS have been confined to geodesic, or “great circle,” paths [9, 1, 8].
Such trajectories are experimentally favorable because the total phase
accumulation measured in the beam is strictly geometric—there is a lack of
dynamic phase specifically associated with the mode transformations (distinct
from propagation and oscillatory phase, also referred to as “dynamic”) [10,
11]. While starting on the poles or equator is convenient, it is also possible
to experimentally access arbitrary starting and ending points on the PS,
something that has yet to be realized. This amounts to starting and ending
with tilted vortices that have an elliptical core structure, which we
anticipate will find applications in related developing fields of geometric
phase accumulation, such as in nonlinear optics, which generally have used
starting points only on the poles [12].

Figure 1: (a) Vortex Poincaré Sphere (PS). Axes are labelled pictorially with
experimental measurements of the mode amplitude (grayscale) and phase (hue),
along with spherical coordinates $(\xi,\theta)$. North and south poles are
also labelled with $\mathcal{I_{+}}$ and $\mathcal{I_{-}}$, respectively, for
“positively” and “negatively” charged circular vortices. Similarly, the
equatorial axes are labelled with $\mathcal{I}_{0}$, $\mathcal{I}_{45}$,
$\mathcal{I}_{90}$, and $\mathcal{I}_{135}$, corresponding with the tilt-angle
(in degrees) of the Hermite-Gaussian modes. An arbitrary field on the PS is
represented by the green dot, $\psi(\xi,\theta)$. (b) Example geodesic
trajectory of an arbitrary state, $\psi_{0}(90\degree,37\degree)$, through a
$\pi$-converter; the final state after the converter is
$\psi_{1}(270\degree,143\degree)$, as described by Eqn. 3. (c) Example
geodesic trajectory of an initial state, $\psi_{1}(270\degree,143\degree)$,
through a Dove prism. The Dove prism is rotated at $\eta=90\degree$,
“sideways” orientation, which picks a point on the equator
$\xi_{\mathrm{Dove}}=2\eta$ to transform the $\xi$ angle about symmetrically.
The $\xi_{1}=270\degree$ of $\psi_{1}$ is transformed to $\xi_{2}=90\degree$
passing through $\xi_{\mathrm{Dove}}=180\degree$ on the PS. See Eqn. 4 for
details. (d) Physical depictions of the orientation $\eta$ and its relation to
$\xi_{\mathrm{Dove}}$ on the PS.

In this Letter, we demonstrate arbitrary geodesic circuits on a vortex PS
utilizing common optical components: cylindrical lenses, Dove prisms, and a
spatial light modulator (SLM). The SLM is used to construct vortex beams that
are “virtually tilted,” allowing for controlled initial positions on the PS
off of the poles and equator. We demonstrate how geodesic trajectories, and
their accumulated geometric phases, can be obtained for arbitrary initial
states of vortices. Our measurements are consistent with the fact that the
geometric phase is invariant for arbitrary initial locations on the PS for a
given set of transforming optics yielding geodesic arcs.

To construct arbitrary vortex states on the PS, use the coordinates of Fig. 1
and a basis in terms of first-order ($\pm 1$) Laguerre-Gaussian (LG) modes.
Then, a natural way to write a given field is

$\psi(\xi,\theta)=\frac{1}{\sqrt{2}}\biggl{[}\cos{\left(\frac{\theta}{2}\right)}e^{+i\xi}\mathrm{LG}_{\mathcal{I}_{+}}+\sin{\left(\frac{\theta}{2}\right)}e^{-i\xi}\mathrm{LG}_{\mathcal{I}_{-}}\biggl{]},$
(1)

where the polar angle $\theta$ tells us about the ratio of the two fundamental
modes (on the poles) and the azimuthal angle $\xi$ shows the relative phase
between them, for making any mode on the PS. In this experiment, however, we
chose to employ the perspective of “virtual tilt,”[13] which works by taking a
vortex on the north pole ($\mathcal{I}_{+}$) and stretching it by angle
$\theta_{\mathrm{tilt}}$ and rotating it about its centroid by
$\xi_{\mathrm{tilt}}$. For clarity, we focus the discussion on only the PS
coordinates of Eqn. 1. See the caption of Fig. 3 for relations between the two
sets of coordinates.

Figure 2:  Schematic. A $\lambda=532$ nm, collimated, single-mode Gaussian
passes through a hologram on a transmission-SLM, from which the first
diffracted order is imaged onto the different possible pairs of
$\pi$-converters and Dove prisms. The SLM is controlled by an Epson 83H
projector [14], and an Andor Zyla sCMOS detector captures transverse images of
the vortex beams. Boxed in red are the gratings used on the SLM, including
four with a superposed collinear Gaussian reference for phase measurements
under different reference phase steps [15]. Boxed in yellow are the first
series of optics encountered by the beam, to bring the vortex from its initial
state on the PS to some intermediate state. Boxed in blue are the second
series of optics used to bring the intermediate state back to the starting
location on the PS, to form closed geodesic circuits. Cylindrical lenses are
separated by two times their focal length and are both locked to focus in the
$y$-axis ($\eta=0\degree$). Dove prisms are either oriented along the same
$y$-axis, or are rotated relatively by $\eta=90\degree$. Inset. One example of
experimental measurements of the complex field of initial and final vortex
states, $\psi_{\textrm{i}}$ and $\psi_{\textrm{f}}$. The rotation of the
“phase $=0\,(2\pi)$” line (red color, $\varphi$ labels) is ultimately how the
geometric phase is measured, similarly to tracking the rotation of
interferograms [8].

A vortex of the form of Eqn. 1 can be constructed experimentally by
programming a hologram to impart the proper phase and amplitude structure onto
an incoming Gaussian laser beam, as shown with a transmission-SLM in Fig. 2.
The hologram of the desired mode is a plane wave grating that has the
structural form

$\mathrm{Hologram}(x,y)=\mathrm{Abs}[e^{i\mathrm{Arg}[\psi]}+e^{ik_{\mathrm{g}}x}]\times\frac{\mathrm{Abs}[\psi]}{\mathrm{Max}[\mathrm{Abs}[\psi]]},$
(2)

for $\psi$ of Eqn. 1, and wave number $k_{\mathrm{g}}$ to set the grating
spacing. From generation at the SLM, the vortex mode is then imaged onto a
series of $\pi$-converters and Dove prisms.

Figure 3: (Left Column) Configurations of $\pi$-converters and Dove prisms
used to measure different settings of geodesic geometric phase. Each optic is
colored to match its associated transformation trajectory on the analytical
plots at right. Relative rotations of the two pairs of optics are denoted by
$\eta_{1}$ and $\eta_{2}$, respectively, as defined in Fig. 1 (d). (Middle
Column) Analytical circuits of vortex transformations on a vortex PS [16].
There are two examples (two different initial conditions, $\psi_{\mathrm{i}}$)
for each set of optics used. The half solid angle enclosed by each loop
(equivalent for a given set of optics) is labeled between the two examples.
Each optic (each cylindrical lens and Dove prism) transforms the vortex
independently, and therefore each optic’s corresponding trajectory on the PS
is denoted by a different color. (Right Column) Experimental measurements of
the geometric phase (vertical axis) as a function of initial starting
locations on the sphere in terms of initial tilt-angles,
$\theta_{0,\mathrm{tilt}}$ (horizontal axis). Data-point colors correspond to
different arrangements of $\pi$-converters and Dove prisms. There are two teal
data points on the row for $\gamma_{\mathrm{geo}}=-180\degree$, which have
$\xi_{\mathrm{tilt},0}=135\degree$ (rather than $45\degree$ like all other
points), which follow the same geodesic circuit as the other points in this
row (as depicted on the spheres in the middle column).Conversions from the
“virtual tilt” angles [13] to PS coordinates of the paper are:
$\theta=\arccos{\biggl{[}\frac{4\cos{(\theta_{\mathrm{tilt}})}}{3+\cos{(2\theta_{\mathrm{tilt}})}}\biggl{]}}$
and $\xi=2\,\xi_{\mathrm{tilt}}$.

The theory we employ to predict the trajectories made on the PS in our
experiments is based on tracking the vortex transformation as a function of
propagation through the optical lenses [16] (or Dove prisms). This theory is
used to plot the PS trajectories seen in the middle column of Fig. 3, which
can be used to determine a solid angle enclosed by the transformations.

In order to enforce geodesic trajectories for an arbitrary initial vortex
state, one must choose a configuration of $\pi$-converters/Dove prisms that
will “maximally change” the vortex state on the PS, following the analogy with
half-waveplates and polarization states [1]. In Fig. 1 (b), one notices that
keeping the $\pi$-converter locked to $\eta=0\degree$ will transform a vortex
state from $\xi=90\degree$ (“diagonally-oriented”) to $\xi=270\degree$
(oppositely “diagonally-oriented”). To move on a geodesic, the relative angle
between optics and vortex orientations must be:
$\eta_{\mathrm{relative}}=|\eta-\frac{1}{2}\,\xi|=45\degree$, for $\eta$ and
$\xi$ defined in Fig. 1. A $\pi$-converter transforms a vortex mode from an
initial position $(\xi_{\mathrm{i}},\theta_{\mathrm{i}})$ to a final position
according to

$\pi\textrm{-converter: }\xi_{\mathrm{f}}=-\xi_{\mathrm{i}},\hskip
7.22743pt\theta_{\mathrm{f}}=180\degree-\theta_{\mathrm{i}},$ (3)

which is constrained to a $180\degree$ arc that goes through at least one pole
of the PS, as depicted in Fig. 1 (b), because of the geodesic-only
restriction.

A Dove prism transforms a vortex by reflecting it about a specific point on
the equator of the PS. This produces a geodesic trace on the PS that does not
need to encompass one of the poles. If $\eta_{\mathrm{relative}}=45\degree$,
then an intermediate state will transform to a final state that coincides with
the initial state on the PS. In particular, the orientation of the Dove prism,
$0\degree\leq\eta\leq 180\degree$, is identified with a point on the PS
equator ($\xi_{\mathrm{Dove}}=2\eta$ as shown in Fig. 1 (d)) and the mode is
reflected about this point. Polar angle, $\theta_{\mathrm{i}}$, is thus
transformed to $\theta_{\mathrm{f}}=180\degree-\theta_{\mathrm{i}}$ (flipping
the topological charge of the vortex). The associated PS trajectory always
crosses the equator at $\xi_{\mathrm{Dove}}$. This Dove reflection results in
total angle of $2\xi_{\mathrm{Dove}}$ between initial and final azimuthal
orientations. A Dove prism trajectory is depicted in Fig. 1 (c), in which to
return to the same initial state shown in (b) one must choose either
$\eta=0\degree$ or $90\degree$. Considering both PS angles, the Dove prism
transforms a vortex mode according to:

$\textrm{Dove prism: }\xi_{\mathrm{f}}=4\eta-\xi_{\mathrm{i}},\hskip
7.22743pt\theta_{\mathrm{f}}=180\degree-\theta_{\mathrm{i}}.$ (4)

Example PS trajectories described by these equations are shown in the middle
column of Fig. 3. The geometric phase is equal to (minus) half the solid angle
made by a complete circuit.

Once a closed circuit on the PS has been completed by transforming a vortex
mode through any of the series of optics used, one can measure the geometric
phase accumulated during the process. This is done fundamentally by comparing
the final state to the initial state, as depicted in the inset of Fig. 2. One
can take the argument of the “overlap” between two complex fields if circuits
are confined to geodesic arcs:
$\gamma_{\mathrm{geo}}=\mathrm{Arg}[\langle\psi_{\mathrm{i}}|\psi_{\mathrm{f}}\rangle]$
[5, 11]. It is also possible to track the rotation of interferograms, after
different PS trajectories, to compare with the initial state interferogram
[8]. Here, we employ a combination of the two approaches. We measure phase
maps of a vortex beam using a collinear Gaussian reference beam with the
technique of phase-shifting digital holography [15]; these measurements are
the typical hue maps of $\mathrm{arg}[\psi]$ that depict the swirling phase
gradients of the vortex. These maps contain information of where the phase
gradients start from zero, $\mathrm{arg}[\psi]=0$. We call this angle
$\varphi$ and it is measured with respect to the positive x-axis of the
transverse plane of the mode.

Lastly, we measure the geometric phase between initial and final states by
taking a pseudo-overlap between the states with these measurements:
$\gamma_{\mathrm{geo}}=\mathrm{arg}[\psi_{\mathrm{f}}]-\mathrm{arg}[\psi_{\mathrm{i}}]=\varphi_{\mathrm{f}}-\varphi_{\mathrm{i}}$.
To report the data in the right column of Fig. 3 quantitatively, we fit the
measured phase maps to a model of the virtually tilted vortex [13], which
returns the angle $\xi_{\mathrm{tilt}}$ as a fitted parameter, which is
identical to $\varphi$. Each data point of Fig. 3 is the average of five
measurements for the same set-up and initial tilt angles. The error bars
associated with each point were calculated by taking one standard deviation of
the measurements of each $\varphi$ and then propagating the deviations
together to be the error of the average [17]. The errors arise from typical
experimental beam drift between camera acquisitions and are insignificant on
the scale of matching the entire set of theoretical predictions.

The Gaussian reference beam used in the construction of the phase maps is
collinear with the vortex mode throughout the entire circuit. Because the
Gaussian itself picks up a phase factor of $\frac{\pi}{2}$ through a
$\pi$-converter [18], any phase accumulation of $\frac{\pi}{2}$ picked up by
the vortex mode through a $\pi$-converter will not be measured because the
phase measurements are all in reference to that Gaussian beam. For this
reason, we subtract $90\degree$ from the measurements of
$\varphi_{\mathrm{f}}-\varphi_{\mathrm{f}}$ for each $\pi$-converter present
in a circuit, to calculate the correct geometric phase accumulated by the
vortex. (The Gouy phase of the vortex mode through the $\pi$-converters [18]
is purely geometric for geodesic circuits.) For Dove prisms, there is no phase
accumulation for a zero-order Gaussian beam. With these corrections, the
measured change in phase, $\varphi_{\mathrm{f}}-\varphi_{\mathrm{i}}$, is
consistent with the geometric phase predicted by the enclosed solid angle from
a closed loop on the PS. The results show that for a given geodesic circuit,
the geometric phase is invariant towards changing the initial location of the
vortex on the PS or even from reorienting the circuit around different areas
of the sphere.

In conclusion, we have demonstrated the measurement of vortex transformations
along a PS with arbitrary vortex initial states, and we have shown how to
experimentally measure the geometric phase associated with a closed circuit
comprised of geodesic arcs taken. This work is the first experimental
realization, to the best of our knowledge, of geodesic geometric phase
measured from arbitrary initial and final states on the PS.

Funding. W. M. Keck Foundation; NSF (1553905)

Disclosures. The authors declare no conflicts of interest.

Data availability. Data underlying all results presented are available from
the authors upon reasonable request.

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# Approximation by Complex Baskakov-Szász-Durrmeyer Operators in Compact Disks

###### Abstract.

In the present paper, we deal with the complex Baskakov-Szász-Durrmeyer mixed
operators and study Voronovskaja type results with quantitative estimates for
these operators attached to analytic functions of exponential growth in
$\mathbb{D}_{R}=\\{z\in\mathbb{C};|z|<R\\}$. Also, the exact order of
approximation is found. The method used allows to construct complex Szász-type
and Baskakov-type approximation operators without to involve the values on
$[0,\infty)$.

Sorin G. Gal
Department of Mathematics and Computer Science, University of Oradea

Str. Universitatii No. 1, 410087 Oradea, Romania

e-mail address<EMAIL_ADDRESS>
and

Vijay Gupta
Department of Mathematics, Netaji Subhas Institute of Technology

Sector 3 Dwarka, New Delhi-110078, India

e-mail address<EMAIL_ADDRESS>
Subject Classification. 30E10, 41A25, 41A28.
Keywords. Complex Baskakov-Szász-Durrmeyer operators, Voronovskaja type
result, exact order of approximation in compact disks, simultaneous
approximation.

##  1\. Introduction

The study of approximation properties for the Szász type operators on
$[0,+\infty)$ was well established in [30] and then generalized in various
ways, see e.g. [2]. Also, very recently, approximation properties for several
real operators including the Szász-Durrmeyer operators are presented in the
book [18]. In order to approximate integrable functions on the positive real
axis, in [19] it was proposed the modifications of the Baskakov operators with
the weights of Szász basis functions under the integral sign. These operators
reproduce only constant functions. Ten years later in [3] it was proposed yet
another sequence of the Baskakov-Szász-Durrmeyer operators which preserve
constant as well as linear functions. Also, generalizations of the Durrmeyer
polynomials were studied in e.g. [1].

In the complex domain, the overconvergence phenomenon holds, that is the
extension of approximation properties from real domain to complex domain. In
this context, the first qualitative kind results were obtained in the papers
[6], [32], [31]. Then, in the books [7], [8] quantitative approximation
results are presented for several type of approximation operators. For Szász-
Mirakjan operator and its Stancu variant in complex domain, we refer the
readers to [4], [5], [21], [9], [23], [24], [29] and [20]. Also for complex
Bernstein-Durrmeyer operators, several papers are available in the literature
(see e.g. [10], [11], [13], [14], [17], [25], [26], [27], [28]), for complex
Szász-Durrmeyer operators see [15], while for complex $q$-Balász-Szabados
operators see [22].

In the present paper, we study the rate of approximation of analytic functions
in a disk $\mathbb{D}_{R}=\\{z\in\mathbb{C};|z|<R\\}$, i.e.
$f(z)=\sum_{k=0}^{\infty}c_{k}z^{k}$, of exponential growth, and the
Voronovskaja type result, for a natural derivation from the complex operator
$L_{n}(f)(z)$ introduced in the case of real variable in [3], and formally
defined as operator of complex variable by

(1.1)
$L_{n}(f)(z):=n\sum_{v=1}^{\infty}b_{n,v}(z)\int_{0}^{\infty}s_{n,v-1}(t)f(t)dt+(1+z)^{-n}f(0),z\in\mathbb{C},$

where

$b_{n,k}(z)=\left(\begin{array}[]{c}n+k-1\\\
k\end{array}\right)\frac{z^{k}}{(1+z)^{n+k}},s_{n,k}(t)=e^{-nt}\frac{(nt)^{k}}{k!}.$

An important relationship used for the quantitative results in approximation
of an analytic function $f$ by the complex operator $L_{n}(f)$ would be
$L_{n}(f)(z)=\sum_{k=0}^{\infty}c_{k}L_{n}(e_{k})(z)$, but which requires some
additional hypothesis on $f$ (because the definition of $L_{n}(f)(z)$ involves
the values of $f$ on $[0,+\infty)$ too) and implies restrictions on the domain
of convergence. This situation can naturally be avoided, by defining directly
the approximation complex operator

$L_{n}^{*}(f)(z)=\sum_{k=0}^{\infty}c_{k}\cdot L_{n}(e_{k})(z),$

whose definition evidently that omits the values of $f$ outside of its disk of
analyticity.

In this paper we deal with the approximation properties of the complex
operator $L_{n}^{*}(f)(z)$.

It is worth noting here that if instead of the above defined $L_{n}(f)(z)$ we
consider any other Szász-type or Baskakov-type complex operator, then for
$L_{n}^{*}(f)(z)$ defined as above, all the quantitative estimates in e.g.
[4], [5], [9], [12], [15], [16], [20], [21], [23] hold true identically,
without to need the additional hypothesis on the values of $f$ on $[0,\infty)$
imposed there.

Everywhere in the paper we denote $\|f\|_{r}=\max\\{|f(z)|;|z|\leq r\\}$.

##  2\. Auxiliary Result

In the sequel, we need the following lemma :

######  Lemma 1.

Denoting $e_{k}(z)=z^{k}$ and $T_{n,k}(z)=L_{n}(e_{k})(z)$, we have the
recurrence formula

$T_{n,k+1}(z)=\frac{z(1+z)}{n}T^{\prime}_{n,k}(z)+\frac{nz+k}{n}T_{n,k}(z).$

Also, $T_{n,k}(z)$ is a polynomial of degree $k$.

###### Proof.

Using $z(1+z)b_{n,\nu}^{\prime}(z)=(\nu-nz)b_{n,\nu}(z)$, we have

$\displaystyle z(1+z)T_{n,k}^{\prime}(z)$ $\displaystyle=$ $\displaystyle
n\sum_{\nu=1}^{\infty}z(1+z)b_{n,\nu}^{\prime}(z)\int_{0}^{\infty}s_{n,\nu-1}(t)t^{k}dt$
$\displaystyle=$ $\displaystyle n\sum_{\nu=1}^{\infty}(\nu-
nz)b_{n,\nu}(z)\int_{0}^{\infty}s_{n,\nu-1}(t)t^{k}dt$ $\displaystyle=$
$\displaystyle
n\sum_{\nu=1}^{\infty}b_{n,\nu}(z)\int_{0}^{\infty}[(\nu-1-nt)+(1+nt-
nz)]s_{n,\nu-1}(t)t^{k}dt$ $\displaystyle=$ $\displaystyle
n\sum_{\nu=1}^{\infty}b_{n,\nu}(z)\int_{0}^{\infty}s_{n,\nu-1}^{\prime}(t)t^{k+1}dt$
$\displaystyle+(1-nz)T_{n,k}(z)+nT_{n,k+1}(z)$

Thus integrating by parts the last integral, we get.

$\displaystyle z(1+z)T_{n,k}^{\prime}(z)$ $\displaystyle=$
$\displaystyle-(k+1)T_{n,k}(z)+(1-nz)T_{n,k}(z)+nT_{n,k+1}(z)$

which completes the proof of the recurrence relation. Taking above step by
step $k=1,2,...,$, by mathematical induction we easily get that $T_{n,k}(z)$
is a polynomial of degree $k$. ∎

##  3\. Main Results

Our first main result is the following theorem for upper bound.

######  Theorem 1.

For $f:\mathbb{D}_{R}\to\mathbb{C}$, $1<R<+\infty$, analytic on
$\mathbb{D}_{R}$, i.e. $f(z)=\sum_{k=0}^{\infty}c_{k}z^{k}$, for all
$z\in\mathbb{D}_{R}$, suppose that there exist $M>0$ and
$A\in(\frac{1}{R},1)$, with the property that $|c_{k}|\leq M\frac{A^{k}}{k!},$
for all $k=0,1,...,$ (which implies $|f(z)|\leq Me^{A|z|}$ for all
$z\in\mathbb{D}_{R}$).

(i) If $1\leq r<\frac{1}{A}$, then for all $|z|\leq r$ and $n\in\mathbb{N}$
with $n>r+2$, $L_{n}^{*}(f)(z)$ is well-defined and we have

$\displaystyle|L_{n}^{*}(f)(z)-f(z)|\leq\frac{C_{r,A,M}}{n},$

where
$C_{r,A,M}=\frac{M(r+2)}{r}\cdot\sum_{k=2}^{\infty}(k+1)\cdot(rA)^{k}<\infty;$

(ii) If $1\leq r<r_{1}<\frac{1}{A}$, then for all $|z|\leq r$ and
$n,p\in\mathbb{N}$ with $n>r+2$, we have

$|[L_{n}^{*}(f)]^{(p)}(z)-f^{(p)}(z)|\leq\frac{p!r_{1}C_{r_{1},A,M}}{n(r_{1}-r)^{p+1}},$

where $C_{r_{1},A,M}$ is given as at the above point (i).

###### Proof.

(i) By using the recurrence relation of Lemma 1, we have

$T_{n,k+1}(z)=\frac{z(1+z)}{n}T^{\prime}_{n,k}(z)+\frac{nz+k}{n}T_{n,k}(z),$

for all $z\in\mathbb{C},k\in\\{0,1,2,....\\},n\in N$. From this we immediately
get the recurrence formula

$\displaystyle T_{n,k}(z)-z^{k}$ $\displaystyle=$
$\displaystyle\frac{z(1+z)}{n}[T_{n,k-1}(z)-z^{k-1}]^{\prime}+\frac{nz+k-1}{n}[T_{n,k-1}(z)-z^{k-1}]$
$\displaystyle+\frac{(k-1)}{n}z^{k-1}(2+z),$

for all $z\in\mathbb{C},k,n\in N$. Now for $1\leq r<R$, if we denote the
norm-$||\cdot||_{r}$ in $C(\overline{\mathbb{D}}_{r})$, where
$\overline{\mathbb{D}}_{r}=\\{z\in\mathbb{C}:|z|\leq r\\}$, then by a linear
transformation, the Bernstein’s inequality in the closed unit disk becomes
$|P_{k}^{\prime}(z)|\leq\frac{k}{r}||P_{k}||_{r}$, for all $|z|\leq r$, where
$P_{k}(z)$ is a polynomial of degree $\leq k$. Thus from the above recurrence
relation, we get

$\displaystyle||T_{n,k}-e_{k}||_{r}$ $\displaystyle\leq$
$\displaystyle\frac{r(1+r)}{n}\cdot||T_{n,k-1}-e_{k-1}||_{r}\frac{k-1}{r}+\frac{nr+k-1}{n}||T_{n,k-1}-e_{k-1}||_{r}$
$\displaystyle+\frac{(k-1)}{n}(2+r)r^{k-1},$

which, by using the notation $\rho=r+2$, implies

$||T_{n,k}-e_{k}||_{r}\leq\left(r+\frac{(2+r)(k-1)}{n}\right)\cdot||T_{n,k-1}-e_{k-1}||_{r}+\frac{(k-1)}{n}(2+r)r^{k-1}$
$=\left(r+\frac{\rho(k-1)}{n}\right)\cdot||T_{n,k-1}-e_{k-1}||_{r}+\frac{(k-1)}{n}\rho\cdot
r^{k-1}.$

In what follows we prove by mathematical induction with respect to $k$ that
for $n\geq\rho$, this recurrence implies

$\displaystyle||T_{n,k}-e_{k}||_{r}$ $\displaystyle\leq$
$\displaystyle\frac{\rho\cdot(k+1)!}{n}\cdot r^{k-1}\ \ \mbox{ for all}\ \
k\geq 1.$

Indeed for $k=1$ it is trivial, as the left-hand side is zero. Suppose that it
is valid for $k$, the above recurrence relation implies that

$\displaystyle||T_{n,k+1}-e_{k+1}||_{r}$ $\displaystyle\leq$
$\displaystyle\left(r+\frac{\rho\cdot
k}{n}\right)\cdot\frac{\rho\cdot(k+1)!}{n}r^{k-1}+\frac{\rho\cdot k}{n}r^{k}.$

It remains to prove that

$\displaystyle\left(r+\frac{\rho\cdot
k}{n}\right)\cdot\frac{\rho\cdot(k+1)!}{n}r^{k-1}+\frac{\rho\cdot
k}{n}r^{k}\leq\frac{\rho\cdot(k+2)!}{n}r^{k},$

or after simplifications, equivalently to

$\displaystyle\left(r+\frac{\rho\cdot
k}{n}\right)\cdot(k+1)!+rk\leq(k+2)!\cdot r,$

for all $k\in\mathbb{N}$ and $r\geq 1$.

Since by $n\geq\rho$, we get

$\left(r+\frac{\rho\cdot
k}{n}\right)\cdot(k+1)!+rk\leq\left(r+k\right)\cdot(k+1)!+rk,$

it is good enough if we prove that

$\left(r+k\right)\cdot(k+1)!+rk\leq(k+2)!\cdot r.$

But this last inequality is obviously valid for all $k\geq 1$ (and fixed
$r\geq 1$).

In conclusion, the required estimate holds.

Now, let us prove that $L_{n}^{*}(f)(z)$ is well-defined for $|z|\leq r$ and
$n>r+2$. Indeed, we have

$|L_{n}^{*}(f)(z)|\leq\sum_{k=0}^{\infty}|c_{k}|\cdot|L_{n}(e_{k})(z)|=\sum_{k=0}^{\infty}|c_{k}|\cdot|T_{n,k}(z)|$
$\leq
M\cdot\sum_{k=0}^{\infty}\frac{A^{k}}{k!}(\|T_{n,k}-e_{k}-e_{k}\|_{r}+r^{k})\leq
Me^{Ar}+M\sum_{k=0}^{\infty}\frac{A^{k}}{k!}\cdot\frac{(r+2)(k+1)!}{n}\cdot
r^{k}$ $\leq Me^{Ar}+M(r+2)\sum_{k=0}^{\infty}(Ar)^{k}(k+1)<+\infty.$

In conclusion, we get

$\displaystyle|L_{n}^{*}(f)(z)-f(z)|$ $\displaystyle\leq$
$\displaystyle\sum_{k=2}^{\infty}|c_{k}|\cdot|T_{n,k}(z)-e_{k}(z)|\leq\sum_{k=2}^{\infty}M\frac{A^{k}}{k!}\frac{(r+2)\cdot(k+1)!}{n}r^{k-1}$
$\displaystyle=$
$\displaystyle\frac{M(r+2)}{rn}\sum_{k=2}^{\infty}(k+1)\cdot(rA)^{k}=\frac{C_{r,A,M}}{n},$

where
$C_{r,A,M}=\frac{M(r+2)}{r}\cdot\sum_{k=2}^{\infty}(k+1)\cdot(rA)^{k}<\infty$
for all $1\leq r<\frac{1}{A},$ taking into account that the series
$\sum_{k=1}^{\infty}(k+1)u^{k}$ is uniformly convergent in any compact disk
included in the open unit disk.

(ii) Denote by $\gamma$ the circle of radius $r_{1}>r$ and center $0$. For any
$|z|\leq r$ and $v\in\gamma$, we have $|v-z|\geq r_{1}-r$ and by the Cauchy’s
formula, for all $|z|\leq r$ and $n>r+2$ it follows

$\displaystyle|[L_{n}^{*}(f)]^{(p)}(z)-f^{(p)}(z)|$ $\displaystyle=$
$\displaystyle\frac{p!}{2\pi}\left|\int_{\gamma}\frac{L_{n}^{*}(f)(v)-f(v)}{(v-z)^{p+1}}dv\right|\leq\frac{C_{r_{1},A,M}}{n}\frac{p!}{2\pi}\frac{2\pi
r_{1}}{(r_{1}-r)^{p+1}}$ $\displaystyle=$
$\displaystyle\frac{C_{r_{1},A,M}}{n}\frac{p!r_{1}}{(r_{1}-r)^{p+1}},$

which proves (ii) and the theorem. ∎

The following Voronovskaja type result holds.

######  Theorem 2.

For $f:\mathbb{D}_{R}\to\mathbb{C}$, $2<R<+\infty$, analytic on
$\mathbb{D}_{R}$, i.e. $f(z)=\sum_{k=0}^{\infty}c_{k}z^{k}$, for all
$z\in\mathbb{D}_{R}$, suppose that there exist $M>0$ and
$A\in(\frac{1}{R},1)$, with the property that $|c_{k}|\leq M\frac{A^{k}}{k!},$
for all $k=0,1,...,$.

If $1\leq r<r+1<\frac{1}{A}$ then for all $|z|\leq r$ and $n\in\mathbb{N}$
with $n>r+2$, we have

$\left|L_{n}^{*}(f)(z)-f(z)-\frac{z(z+2)}{2n}f^{\prime\prime}(z)\right|\leq\frac{C_{r,A,M}(f)}{n^{2}},$

where
$C_{r,A,M}(f)=M\sum_{k=2}^{\infty}\frac{k-1}{k!}[A(r+1)]^{k}B_{k,r}+\frac{4M(r+2)}{r}\cdot\frac{1}{ln^{2}(1/\rho)}\cdot\left(\frac{1}{(1-Ar)^{2}}+\frac{4}{1-Ar}\right)<\infty$
and

$B_{k,r}=(k-1)^{2}(k-2)r^{2}+2(k-1)(k-2)(2k-3)(r+1)+(r+1)(r+2)\cdot(k+1)!.$

###### Proof.

By Theorem 1, (i), $L_{n}^{*}(f)(z)$ is well-defined, for all $|z|\leq r$,
$n>r+2$. We can write

$\frac{z(z+2)f^{\prime\prime}(z)}{2n}=\frac{z(z+2)}{2n}\sum_{k=2}^{\infty}c_{k}k(k-1)z^{k-2}=\frac{1}{2n}\sum_{k=1}^{\infty}c_{k}k(k-1)(z+2)z^{k-1}.$

Thus

$\left|L_{n}^{*}(f)(z)-f(z)-\frac{z(z+2)}{2n}f^{\prime\prime}(z)\right|\leq\sum_{k=1}^{\infty}|c_{k}|\left|T_{n,k}(z)-e_{k}(z)-\frac{k(k-1)(z+2)z^{k-1}}{2n}\right|.$

By Lemma 1, for all $n\in\mathbb{N},z\in\mathbb{C}$ and $k=0,1,2,...$, we have

$T_{n,k+1}(z)=\frac{z(1+z)}{n}T^{\prime}_{n,k}(z)+\frac{nz+k}{n}T_{n,k}(z).$

If we denote

$E_{k,n}(z)=T_{n,k}(z)-e_{k}(z)-\frac{k(k-1)(z+2)z^{k-1}}{2n},$

then it is obvious that $E_{k,n}(z)$ is a polynomial of degree less than or
equal to $k$ and by simple computation and the use of above recurrence
relation, we are led to

$E_{k,n}(z)=\frac{z(1+z)}{n}E_{k-1,n}^{\prime}(z)+\frac{nz+k-1}{n}E_{k-1,n}(z)+X_{k,n}(z),$

where

$X_{k,n}(z)=\frac{z^{k-2}}{2n^{2}}\left[(k-1)^{2}(k-2)z^{2}+2(k-1)(k-2)(2k-3)z+2(k-1)(k-2)(2k-3)\right]$

for all $k\geq 2,n\in\mathbb{N}$ and $|z|\leq r$.

Using the estimate in the proof of Theorem 1, we have

$|T_{n,k}(z)-e_{k}(z)|\leq\frac{(k+1)!(r+2)r^{k-1}}{n},$

for all $k\geq 1,n\geq r+2,|z|\leq r$, with $1\leq r$.

It follows

$|E_{k,n}(z)|\leq\frac{r(r+1)}{n}|E_{k-1,n}^{\prime}(z)|+\left(r+\frac{k-1}{n}\right)|E_{k-1,n}(z)|+|X_{k,n}(z)|.$

Now we shall find the estimation of $|E_{k-1,n}^{\prime}(z)|$ for $k\geq 2$.
Taking into account the fact that $E_{k-1,n}(z)$ is a polynomial of degree
$\leq k-1$, we have

$|E_{k-1,n}^{\prime}(z)|\leq\frac{k-1}{r}||E_{k-1,n}||_{r}$
$\leq\frac{k-1}{r}\left[||T_{n,k-1}(z)-e_{k-1}(z)||_{r}+\left|\left|\frac{(k-1)(k-2)[e_{1}+2]e_{k-2}}{2n}\right|\right|_{r}\right]$
$\leq\frac{k-1}{r}\left[\frac{k!(r+2)r^{k-2}}{n}+\frac{r^{k-2}(k-1)(k-2)(r+2)}{n}\right]$
$\leq\frac{(r+2)r^{k-2}(k-1)(k!+(k-1)(k-2))}{nr}.$

Thus, by the obvious inequality $(k-1)(k!+(k-1)(k-2))\leq(k+1)!$, we get

$\frac{r(1+r)}{n}|E_{k-1,n}^{\prime}(z)|\leq\frac{(r+1)(r+2)\cdot(k+1)!r^{k-2}}{n^{2}}$

and

$|E_{k,n}(z)|\leq\frac{(r+1)(r+2)\cdot(k+1)!r^{k-2}}{n^{2}}+\left(r+\frac{k-1}{n}\right)|E_{k-1,n}(z)|+|X_{k,n}(z)|,$

for all $|z|\leq r,k\geq 2$ and $n>r+2$.

For $k-1\leq n$ (i.e. $k\leq n+1$) and $|z|\leq r$, taking into account that
$r+(k-1)/n\leq r+1$, we get

$|E_{k,n}(z)|\leq\frac{(r+1)(r+2)(k+1)!\cdot
r^{k-2}}{n^{2}}+(r+1)|E_{k-1,n}(z)|+|X_{k,n}(z)|,$

where

$|X_{k,n}(z)|\leq\frac{r^{k-2}}{n^{2}}\left[(k-1)^{2}(k-2)r^{2}+2(k-1)(k-2)(2k-3)r+2(k-1)(k-2)(2k-3)\right]$
$\leq\frac{r^{k-2}}{n^{2}}A_{k,r},$

for all $|z|\leq r,k\geq 1,n>r+2,$ where

$A_{k,r}=(k-1)^{2}(k-2)r^{2}+2(k-1)(k-2)(2k-3)(r+1).$

Thus for all $|z|\leq r,n>r+2$ and $k\leq n+1$,

$|E_{k,n}(z)|\leq(r+1)|E_{k-1,n}(z)|+\frac{r^{k-2}}{n^{2}}B_{k,r},$

where

$B_{k,r}=A_{k,r}+(r+1)(r+2)\cdot(k+1)!.$

But $E_{0,n}(z)=E_{1,n}(z)=0$, for any $z\in\mathbb{C}$ and therefore by
writing last inequality for $2\leq k\leq n+1$, we easily obtain step by step
the following

$|E_{k,n}(z)|\leq\frac{(r+1)^{k}}{n^{2}}\sum_{j=2}^{k}B_{j,r}\leq\frac{(k-1)(r+1)^{k}}{n^{2}}B_{k,r}.$

It follows that

$\left|L_{n}^{*}(f)(z)-f(z)-\frac{z(z+2)}{2n}f^{\prime\prime}(z)\right|\leq\sum_{k=2}^{n+1}|c_{k}|\cdot|E_{k,n}(z)|+\sum_{k=n+2}^{\infty}|c_{k}|\cdot|E_{k,n}(z)|$
$\leq\frac{1}{n^{2}}\sum_{k=2}^{\infty}|c_{k}|(k-1)(r+1)^{k}B_{k,r}+\sum_{k=n+2}^{\infty}|c_{k}|\cdot\left[|T_{n,k}(z)-e_{k}(z)|+\frac{k(k-1)(r+2)r^{k-1}}{2n}\right]$
$\leq\frac{1}{n^{2}}\sum_{k=2}^{\infty}|c_{k}|(k-1)(r+1)^{k}B_{k,r}+\sum_{k=n+2}^{\infty}|c_{k}|\cdot\left[\frac{(r+2)\cdot(k+1)!}{n}\cdot
r^{k-1}+\frac{k(k-1)(r+2)r^{k-1}}{2n}\right]$
$\leq\frac{1}{n^{2}}\sum_{k=2}^{\infty}|c_{k}|(k-1)(r+1)^{k}B_{k,r}+2\sum_{k=n+2}^{\infty}|c_{k}|\cdot\frac{(r+2)\cdot(k+1)!}{n}\cdot
r^{k-1}$
$\leq\frac{1}{n^{2}}\sum_{k=2}^{\infty}|c_{k}|(k-1)(r+1)^{k}B_{k,r}+\frac{2M(r+2)}{nr}\sum_{k=n+2}^{\infty}(k+1)(Ar)^{k}$
$=\frac{1}{n^{2}}\sum_{k=2}^{\infty}|c_{k}|(k-1)(r+1)^{k}B_{k,r}+\frac{2M(r+2)}{nr}\cdot(Ar)^{n+2}\sum_{k=n+2}^{\infty}(k+1)(Ar)^{k-n-2}.$

But, denoting for simplicity $\rho=Ar<1$, we easily obtain

$(Ar)^{n+2}\cdot\sum_{k=n+2}^{\infty}(k+1)(Ar)^{k-n-2}=\rho^{n+2}\cdot\left[\sum_{j=0}^{\infty}j\rho^{j}+(n+3)\sum_{j=0}^{\infty}\rho^{j}\right]$
$=\rho^{n+2}\left[\rho\cdot\left(\sum_{j=0}^{\infty}\rho^{j}\right)^{\prime}+(n+3)\cdot\frac{1}{1-\rho}\right]=\rho^{n+3}\cdot\frac{1}{(1-\rho)^{2}}+\rho^{n+2}\cdot(n+3)\cdot\frac{1}{1-\rho},$

which leads to the estimate

$\left|L_{n}^{*}(f)(z)-f(z)-\frac{z(z+2)}{2n}f^{\prime\prime}(z)\right|$
$\leq\frac{1}{n^{2}}\sum_{k=2}^{\infty}|c_{k}|(k-1)(r+1)^{k}B_{k,r}+\frac{2M(r+2)}{nr}\cdot\left[\frac{(Ar)^{n+3}}{(1-Ar)^{2}}+\frac{(Ar)^{n+2}(n+3)}{1-Ar}\right]$
$\leq M\frac{1}{n^{2}}\sum_{k=2}^{\infty}\frac{k-1}{k!}[A(r+1)]^{k}\cdot
B_{k,r}+\frac{4M(r+2)}{rn^{2}}\cdot\frac{1}{ln^{2}(1/(Ar))}\cdot\left[\frac{1}{(1-Ar)^{2}}+\frac{4}{1-Ar}\right],$

where we used the inequality

$\rho^{n+3}\leq\rho^{n+2}\leq\rho^{n}\leq\frac{2}{ln^{2}(1/(Ar))}\cdot\frac{1}{n^{2}},\mbox{
for all }n\in\mathbb{N},$

applied for $\rho=Ar<1$ and where for $(r+1)A<1$, we obviously have the series
$\sum_{k=2}^{\infty}|c_{k}|(k-1)(r+1)^{k}B_{k,r}$ convergent.

Indeed, by $e^{x}=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+...$, we get
$e^{x}\geq\frac{x^{2}}{2}$, for all $x\geq 0$. Then, by
$(1/\rho)^{n}=e^{nln(1/\rho)}$, it follows
$\frac{1}{\rho^{n}}\geq\frac{n^{2}ln^{2}(1/\rho)}{2}$, for all
$n\in\mathbb{N}$, which immediately implies the above claimed inequality. ∎

The following exact order of approximation can be obtained.

######  Theorem 3.

Suppose that the hypothesis in Theorem 2 hold.

(i) If $f$ is not a polynomial of degree $\leq 1$ then for all $n>r+2$ we have

$\|L_{n}^{*}(f)-f\|_{r}\sim\frac{1}{n}.$

where the constants in the equivalence depend only on $f$, $R$, $A$ and $r$.

(ii) If $1\leq r<r_{1}<r_{1}+1<1/A$ and $f$ is not a polynomial of degree
$\leq p-1,(p\geq 1)$ then

$\|[L_{n}^{*}(f)]^{(p)}-f^{(p)}\|_{r}\sim\frac{1}{n},\mbox{ for all }n>r+2,$

where the constants in the equivalence depend only on $f$, $R$, $A$, $r$,
$r_{1}$ and $p$.

###### Proof.

(i) For all $|z|\leq r$ and $n\in\mathbb{N}$ with $n>r+2$, we can write

$\displaystyle L_{n}^{*}(f)(z)-f(z)$ $\displaystyle=$
$\displaystyle\frac{1}{n}\bigg{[}\frac{z(z+2)}{2}f^{\prime\prime}(z)$
$\displaystyle+\frac{1}{n}\cdot
n^{2}\left(L_{n}^{*}(f)(z)-f(z)-\frac{z(z+2)}{2n}f^{\prime\prime}(z)\right)\bigg{]}$

Applying the inequality

$\displaystyle\|F+G\|_{r}\geq|\,\|F\|_{r}-\|G\|_{r}\,|\geq\|F\|_{r}-\|G\|_{r},$

we obtain

$\|L_{n}^{*}(f)-f\|_{r}\geq\frac{1}{n}\left[\left\|\frac{e_{1}(e_{1}+2)}{2}f^{\prime\prime}\right\|_{r}\right.\left.-\frac{1}{n}\cdot
n^{2}\left\|L_{n}^{*}(f)-f-\frac{e_{1}(e_{1}+2)f^{\prime\prime}}{2n}\right\|_{r}\right].$

Since $f$ is not a polynomial of degree $\leq 1$, we get
$\left\|\frac{e_{1}(e_{1}+2)}{2}f^{\prime\prime}\right\|_{r}>0$. Indeed,
supposing the contrary, it follows that

$\frac{z(z+2)}{2}f^{\prime\prime}(z)=0,\,\mbox{ for all }|z|\leq r.$

The last equality is equivalent to $f(z)=C_{1}z+C_{2}$, with $C_{1},C_{2}$ are
constants, a contradiction with the hypothesis. Now by Theorem 2, we have

$n^{2}\left\|L_{n}^{*}(f)-f-\frac{e_{1}(e_{1}+2)f^{\prime\prime}}{2n}\right\|_{r}\leq
C_{r,A,M}(f),$

for all $n>r+2$.

Thus, there exists $n_{0}>r+2$ (depending on $f$ and $r$ only) such that for
all $n\geq n_{0},$ we have

$\left\|\frac{e_{1}(e_{1}+2)}{2}f^{\prime\prime}\right\|_{r}\left.-\frac{1}{n}\cdot
n^{2}\left\|L_{n}^{*}(f)-f-\frac{e_{1}(e_{1}+2)f^{\prime\prime}}{2n}\right\|_{r}\right]\geq\frac{1}{4}\left\|e_{1}(e_{1}+2)f^{\prime\prime}\right\|_{r},$

which implies that

$\displaystyle\|L_{n}^{*}(f)-f\|_{r}\geq\frac{1}{4n}\left\|e_{1}(e_{1}+2)f^{\prime\prime}\right\|_{r},$

for all $n\geq n_{0}$.
For $r+2<n\leq n_{0}-1$, we get
$\|L_{n}^{*}(f)-f\|_{r}\geq\frac{M_{r,n}(f)}{n}$ with
$M_{r,n}(f)=n\cdot\|L_{n}^{*}(f)-f\|_{r}>0$ (since $\|L_{n}^{*}(f)-f\|_{r}=0$
for a certain $n$ is valid only for $f$ a polynomial of degree $\leq 1$,
contradicting the hypothesis on $f$).

Therefore, finally we have

$||L_{n}^{*}(f)-f||_{r}\geq\frac{C_{r}(f)}{n}$

for all $n>r+2$, where

$C_{r}(f)=\min_{n_{0}-1\geq
n>r+2}\left\\{M_{r,n}(f),....,M_{r,n_{0}-1}(f),\frac{1}{4}\left\|e_{1}(e_{1}+2)f^{\prime\prime}\right\|_{r}\right\\},$

which combined with Theorem 1, (i), proves the desired conclusion.

(ii) The upper estimate is exactly Theorem 1, (ii), therefore it remains to
prove the lower estimate. Denote by $\gamma$ the circle of radius $r_{1}>r$
and center $0$. For any $|z|\leq r$ and $v\in\gamma$, we have $|v-z|\geq
r_{1}-r$ and by the Cauchy’s formula, for all $|z|\leq r$ and $n>r+2$ it
follows

$[L_{n}^{*}(f)]^{(p)}(z)-f^{(p)}(z)=\frac{p!}{2\pi
i}\int_{\gamma}\frac{L_{n}^{*}(f)(v)-f(v)}{(v-z)^{p+1}}dv,$

where $|v-z|\geq r_{1}-r$, for all $v\in\gamma$.

Since for $v\in\gamma$ we get

$L_{n}^{*}(f)(v)-f(v)$
$=\frac{1}{n}\left\\{\frac{v(v+2)}{2}f^{\prime\prime}(v)+\frac{1}{n}\left[n^{2}\left(L_{n}^{*}(f)(v)-f(v)-\frac{v(v+2)f^{\prime\prime}(v)}{2n}\right)\right]\right\\},$

replaced in the Cauchy’s formula implies

$[L_{n}^{*}(f)]^{(p)}(z)-f^{(p)}(z)=\frac{1}{n}\left\\{\frac{p!}{2\pi
i}\int_{\gamma}\frac{v(v+2)f^{\prime\prime}(v)}{2(v-z)^{p+1}}dv\right.$
$+\left.\frac{1}{n}\cdot\frac{p!}{2\pi
i}\int_{\gamma}\frac{n^{2}\left(L_{n}^{*}(f)(v)-f(v)-\frac{v(v+2)f^{\prime\prime}(v)}{2n}\right)}{(v-z)^{p+1}}dv\right\\}$
$=\frac{1}{n}\left\\{\frac{1}{2}\cdot\left[z(z+2)f^{\prime\prime}(z)\right]^{(p)}+\frac{1}{n}\cdot\frac{p!}{2\pi
i}\int_{\gamma}\frac{n^{2}\left(L_{n}^{*}(f)(v)-f(v)-\frac{v(v+2)f^{\prime\prime}(v)}{2n}\right)}{(v-z)^{p+1}}dv\right\\}.$

Passing to the norm $\|\cdot\|_{r}$, we obtain

$\|[L_{n}^{*}(f)]^{(p)}-f^{(p)}\|_{r}$
$\geq\frac{1}{n}\left\\{\frac{1}{2}\left\|\left[e_{1}(e_{1}+2)f^{\prime\prime}\right]^{(p)}\right\|_{r}-\frac{1}{n}\left\|\frac{p!}{2\pi}\int_{\gamma}\frac{n^{2}\left(L_{n}^{*}(f)(v)-f(v)-\frac{v(v+2)f^{\prime\prime}(v)}{2n}\right)}{(v-z)^{p+1}}dv\right\|_{r}\right\\},$

where by Theorem 2, for all $n>r+2$ it follows

$\displaystyle\left\|\frac{p!}{2\pi}\int_{\gamma}\frac{n^{2}\left(L_{n}^{*}(f)(v)-f(v)-\frac{v(v+2)f^{\prime\prime}(v)}{2n}\right)}{(v-z)^{p+1}}dv\right\|_{r}$
$\displaystyle\leq$ $\displaystyle\frac{p!}{2\pi}\cdot\frac{2\pi
r_{1}n^{2}}{K^{p+1}}\left\|L_{n}^{*}(f)-f-\frac{e_{1}(e_{1}+2)f^{\prime\prime}}{2n}\right\|_{r_{1}}$
$\displaystyle\leq$ $\displaystyle
C_{r,A,M}(f)\cdot\frac{p!r_{1}}{(r_{1}-r)^{p+1}}.$

Now, by hypothesis on $f$ we have
$\left\|\left[e_{1}(e_{1}+2)f^{\prime\prime}\right]^{(p)}\right\|_{r}>0$.
Indeed, supposing the contrary it follows that $z(z+2)f^{\prime\prime}(z)$ is
a polynomial of degree $\leq p-1$, which by the analyticity of $f$ obviously
implies that $f$ is a polynomial of degree $\leq p-1$, a contradiction with
the hypothesis.

For the rest of the proof, reasoning exactly as in the proof of the above
point (i), we immediately get the required conclusion. ∎

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