Approximation Theorems for Complex $\alpha$-Bernstein–Kantorovich Operators

Abstract

In this paper, we introduce the complex form of $\alpha$-Bernstein–Kantorovich operators. Respectively, upper quantitative estimates for the complex $\alpha$-Bernstein–Kantorovich operator and its derivatives, Voronovskaya type result and the exact order of approximation of these operators are studied.

1 Introduction

The approximation of functions with positive linear operators is one of the most important research area of applied mathematics. Especially, Bernstein polynomials play an significant role in approximation theory, thanks to their simple structure and advantages in computation. In 1912, Bernstein was introduced the classical Bernstein polynomials as follows:

$$\begin{aligned} B_{\eta }\left( \varphi ;\tau \right) =\sum _{\rho =0}^{\eta }\varphi \left( \frac{\rho }{\eta }\right) \left( {\begin{array}{c}\eta \\ \rho \end{array}}\right) \tau ^{\rho }(1-\tau )^{\eta -\rho },\text { }\eta \in {\mathbb {N}} \text { and }\tau \in \left[ 0,1\right] \end{aligned}$$

(1)

for $\varphi \in {\mathbb {C}} [0,1].$ For many years, many researchers focused on the discovery and modifications of Bernstein polynomials to get better convergence. Recently, Chen et al. [1] have constructed a generalization of the Bernstein operator, which depends on $\alpha$ as follows.

$$\begin{aligned} T_{\eta ,\alpha }(\varphi ;\tau )=\sum _{\rho =0}^{\eta }\varphi _{\rho }~p_{\eta ,\rho }^{\left( \alpha \right) }(\tau ), \end{aligned}$$

(2)

where $\varphi _{\rho }=\varphi (\frac{\rho }{\eta })$, $\alpha \in \left[ 0,1\right]$ and

$$\begin{aligned} p_{\eta ,\rho }^{\left( \alpha \right) }(\tau )= & {} \left[ \left( {\begin{array}{c}\eta -2\\ \rho \end{array}}\right) \left( 1-\alpha \right) \tau +\left( {\begin{array}{c}\eta -2\\ \rho -2\end{array}}\right) \left( 1-\alpha \right) \left( 1-\tau \right) \right. \\{} & {} \quad \left. +\left( {\begin{array}{c}\eta \\ \rho \end{array}}\right) \alpha \tau (1-\tau )\right] \tau ^{\rho -1}\left( 1-\tau \right) ^{\eta -\rho -1}. \end{aligned}$$

Because the $\alpha$-Bernstein operators are suitable structures for continuous functions on [0, 1], authors constructed the following $\alpha$-Bernstein–Kantorovich operators for integrable functions on [0, 1] and investigated their many approximation properties (See [2, 3]).

$$\begin{aligned} K_{\eta ,\alpha }(\varphi ;\tau )=\sum _{\rho =0}^{\eta }p_{\eta ,\rho }^{\left( \alpha \right) }(\tau )\underset{\frac{\rho }{\eta +1}}{\overset{\frac{\rho +1}{\eta +1}}{ {\displaystyle \int } }}\varphi (t)dt. \end{aligned}$$

(3)

In addition to all these, recently, new definitions and studies have been made due to the increasing interest of researchers in complex operators in approximation problems [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. One of them was introduced and studied by Nursel Çetin. In Çetin [1], introduced the $\alpha$-Bernstein operator in the complex domain as follows:

$$\begin{aligned} T_{\eta ,\alpha }(\varphi ;\zeta )=\sum _{\rho =0}^{\eta }\varphi _{\rho }~p_{\eta ,\rho }^{\left( \alpha \right) }(\zeta ), \end{aligned}$$

(4)

where $\varphi _{\rho }=\varphi (\frac{\rho }{\eta })$, $\alpha \in \left[ 0,1\right]$, $\zeta \in {\mathbb {C}}$.

The above-mentioned studies motivated us to define the $\alpha$-Bernstein–Kantorovich operator in the complex region as follows:

$$\begin{aligned} K_{\eta ,\alpha }\left( \varphi ;\zeta \right) = {\displaystyle \sum _{\rho =0}^{\eta }} p_{\eta ,\rho }^{\left( \alpha \right) }(\zeta ) {\displaystyle \int \limits _{0}^{1}} \varphi \left( \frac{\rho +t}{\eta +1}\right) dt. \end{aligned}$$

(5)

where $\alpha \in \left[ 0,1\right]$ and $\zeta \in {\mathbb {C}}$. For $\alpha =1$, these operators are reduced to the classical complex Bernstein–Kantorovich operators [6]. Throughout the paper, ${\mathbb {D}}_{R}:=\left\{ \zeta \in {\mathbb {C}}:\left| \zeta \right| <R\right\}$ be a disc in the complex plane ${\mathbb {C}}$ and the space of all analytic functions on ${\mathbb {D}}_{R}$ denote by $H\left( {\mathbb {D}} _{R}\right) .$ Also, for $\varphi \in H\left( {\mathbb {D}}_{R}\right) ,$ we assume that $\varphi \left( \zeta \right) =\sum _{\mu =0}^{\infty }a_{\mu } \zeta ^{\mu }$.

The paper is organized as follows. In Sect. 2, we give some auxilary results for the proofs of the main theorems. In Sect. 3, we obtain respectively, upper quantitative estimates for the $K_{\eta ,\alpha }\left( \varphi ;\zeta \right)$ and its derivatives on compact disks. Finally, we obtain Voronovskaja type results and the exact degrees of approximation $\alpha$-Bernstein–Kantorovich operator and their derivatives.

2 Auxiliary Results

Lemma 1

For all $\mu \in \mathbb {N\cup }\left\{ 0\right\} ,\eta \in {\mathbb {N}},\alpha \in \left[ 0,1\right]$ and $\zeta \in {\mathbb {C}}$, we have

$$\begin{aligned} K_{\eta ,\alpha }\left( e_{\mu };\zeta \right) =\sum _{j=0}^{\mu }\left( \begin{array}{c} \mu \\ j \end{array} \right) \frac{\eta ^{j}}{\left( \eta +1\right) ^{\mu }\left( \mu -j+1\right) }T_{\eta ,\alpha }\left( e_{j};\zeta \right) , \end{aligned}$$

(6)

where $e_{\mu }\left( \zeta \right) =\zeta ^{\mu }$.

Proof

From 5, we can write,

$$\begin{aligned} K_{\eta ,\alpha }\left( e_{\mu };\zeta \right)&= {\displaystyle \sum _{\rho =0}^{\eta }} p_{\eta ,\rho }^{\left( \alpha \right) }\left( \zeta \right) \underset{0}{\overset{1}{ {\displaystyle \int } }} \genfrac(){}{}{\rho +t}{\eta +1} ^{\mu }dt\\&= {\displaystyle \sum _{\rho =0}^{\eta }} p_{\eta ,\rho }^{\left( \alpha \right) }\left( \zeta \right) \sum _{j=0}^{\mu }\int _{0}^{1}\left( {\begin{array}{c}\mu \\ j\end{array}}\right) \frac{t^{\mu -j}\rho ^{j}}{\left( \eta +1\right) ^{\mu }}dt\\&=\frac{1}{\left( \eta +1\right) ^{\mu }}\sum _{\rho =0}^{\eta }p_{\eta ,\rho }^{\left( \alpha \right) }\left( \zeta \right) \sum _{j=0}^{\mu }\left( {\begin{array}{c}\mu \\ j\end{array}}\right) \rho ^{j}\int _{0}^{1}t^{\mu -j}dt\\&=\frac{1}{\left( \eta +1\right) ^{\mu }}\sum _{j=0}^{\mu }\frac{\eta ^{j} }{\left( \mu -j+1\right) }T_{\eta ,\alpha }\left( e_{j};\zeta \right) . \end{aligned}$$

$\square$

Lemma 2

For all $\zeta \in {\mathbb {D}}_{r}$, $\alpha \in \left[ 0,1\right]$ and $1\le r$, we have

$$\begin{aligned} \left| K_{\eta ,\alpha }\left( e_{\mu };\zeta \right) \right| \le r^{\mu },\ \ \ \ \mu ,\eta \in {\mathbb {N}}. \end{aligned}$$

(7)

Proof

We know that (see [1])

$$\begin{aligned} \left| T_{\eta ,\alpha }\left( e_{j};\zeta \right) \right| \le r^{j}. \end{aligned}$$

From 6, we obtain

$$\begin{aligned} \left| K_{\eta ,\alpha }\left( e_{\mu };\zeta \right) \right|&\le \sum _{j=0}^{\mu }\left( \begin{array}{c} \mu \\ j \end{array} \right) \frac{\eta ^{j}}{\left( \eta +1\right) ^{\mu }\left( \mu -j+1\right) }\left| T_{\eta ,\alpha }\left( e_{j};\zeta \right) \right| \\&\le \frac{1}{\left( \eta +1\right) ^{\mu }}\sum _{j=0}^{\mu }\left( \begin{array}{c} \mu \\ j \end{array} \right) \eta ^{j}r^{\mu }=r^{\mu } \end{aligned}$$

$\square$

Lemma 3

For all $\alpha \in \left[ 0,1\right]$ and $\eta ,\mu \in {\mathbb {N}}$, $\zeta \in {\mathbb {C}}$, we have

$$\begin{aligned} K_{\eta ,\alpha }\left( e_{\mu };\zeta \right)&=\frac{\zeta \left( 1-\zeta \right) }{\eta }\left( K_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) \right) ^{^{\prime }}+\left[ \zeta +\frac{\left( 1-\zeta \right) }{\eta }\right] K_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) \nonumber \\&\quad -\frac{\alpha \left( 1-\zeta \right) }{\eta }K_{\eta }\left( e_{\mu -1};\zeta \right) \nonumber \\&\quad +\frac{1}{\left( \eta +1\right) ^{\mu }}\sum _{j=0}^{\mu }\left( \begin{array}{c} \mu \\ j \end{array} \right) \frac{\eta ^{j-1}}{\left( \mu -j+1\right) }\left( \frac{\eta \mu -j\left( \eta +1\right) }{\mu }\right) T_{\eta ,\alpha }\left( e_{j} ;\zeta \right) \nonumber \\&\quad +\frac{1}{\left( \eta +1\right) ^{\mu -1}}\sum _{j=0}^{\mu -1}\left( \begin{array}{c} \mu -1\\ j \end{array} \right) \left( \frac{1-\alpha }{\eta }\right) \left( \frac{\eta -1}{\eta }\right) ^{j}\frac{\eta ^{j}}{\left( \mu -j\right) }\left\{ B_{\eta -1}\left( e_{j+1};\zeta \right) \right. \nonumber \\&\quad \left. -B_{\eta -1}\left( e_{j};\zeta \right) \right\} \end{aligned}$$

(8)

where $K_{\eta }\left( e_{\mu };\zeta \right)$ complex Bernstein–Kantorovich operator (see [9]).

Proof

From [1], we have

$$\begin{aligned} \frac{\zeta \left( 1-\zeta \right) }{\eta }\left( T_{\eta ,\alpha }\left( e_{j};\zeta \right) \right) ^{^{\prime }}&=T_{\eta ,\alpha }\left( e_{j+1};\zeta \right) -\left[ \zeta +\frac{\left( 1-\zeta \right) }{\eta }\right] T_{\eta ,\alpha }\left( e_{j};\zeta \right) \nonumber \\&\quad -\left( \frac{1-\alpha }{\eta }\right) \left( \frac{\eta -1}{\eta }\right) ^{j}\left\{ B_{\eta -1}\left( e_{j+1};\zeta \right) -B_{\eta -1}\left( e_{j};\zeta \right) \right\} \nonumber \\&\quad +\frac{\alpha \left( 1-\zeta \right) }{\eta }B_{\eta }\left( e_{j} ;\zeta \right) . \end{aligned}$$

(9)

Differentiatin g $K_{\eta ,\alpha }\left( e_{\mu };\zeta \right)$ with respect to $\zeta \ne 0$ and using (9), we have

$$\begin{aligned} \frac{\zeta \left( 1-\zeta \right) }{\eta }\left( K_{\eta ,\alpha }\left( e_{\mu };\zeta \right) \right) ^{^{\prime }}&=\frac{1}{\left( \eta +1\right) ^{\mu }}\sum _{j=0}^{\mu }\left( \begin{array}{c} \mu \\ j \end{array} \right) \frac{\eta ^{j}}{\left( \mu -j+1\right) }\frac{\zeta \left( 1-\zeta \right) }{\eta }T\left( _{\eta ,\alpha }\left( e_{j};\zeta \right) \right) ^{^{\prime }}\\&=\frac{1}{\left( \eta +1\right) ^{\mu }}\sum _{j=0}^{\mu }\left( \begin{array}{c} \mu \\ j \end{array} \right) \frac{\eta ^{j}}{\left( \mu -j+1\right) }\Bigg \{ T_{\eta ,\alpha }\left( e_{j+1};\zeta \right) \\&\quad \left. -\left[ \zeta +\frac{\left( 1-\zeta \right) }{\eta }\right] T_{\eta ,\alpha }\left( e_{j};\zeta \right) \right. \\&\quad \left. -\left( \frac{1-\alpha }{\eta }\right) \left( \frac{\eta -1}{\eta }\right) ^{j}\left\{ B_{\eta -1}\left( e_{j+1}\right) -B_{\eta -1}\left( e_{j}\right) \right\} \right. \\&\quad +\frac{\alpha \left( 1-\zeta \right) }{\eta }B_{\eta }\left( e_{j};\zeta \right) \Bigg \} . \end{aligned}$$

by some calculations we get

$$\begin{aligned}&K_{\eta ,\alpha }\left( e_{\mu +1};\zeta \right) =\frac{\zeta \left( 1-\zeta \right) }{\eta }\left( K_{\eta ,\alpha }\left( e_{\mu };\zeta \right) \right) ^{^{\prime }}+\left[ \zeta +\frac{\left( 1-\zeta \right) }{\eta }\right] K_{\eta ,\alpha }\left( e_{\mu };\zeta \right) \\&\quad +\sum _{j=0}^{\mu +1}\left( \begin{array}{c} \mu +1\\ j \end{array} \right) \frac{\eta ^{j}}{\left( \eta +1\right) ^{\mu +1}\left( \mu -j+2\right) }T_{\eta ,\alpha }\left( e_{j};\zeta \right) \\&\quad +\sum _{j=0}^{\mu }\left( \begin{array}{c} \mu \\ j \end{array} \right) \left( \frac{1-\alpha }{\eta }\right) \left( \frac{\eta -1}{\eta }\right) ^{j}\frac{\eta ^{j}}{\left( \eta +1\right) ^{\mu }\left( \mu -j+1\right) }\left\{ B_{\eta -1}\left( e_{j+1}\right) -B_{\eta -1}\left( e_{j}\right) \right\} \\&\quad -\sum _{j=1}^{\mu +1}\left( \begin{array}{c} \mu \\ j-1 \end{array} \right) \frac{\eta ^{j-1}}{\left( \eta +1\right) ^{\mu }\left( \mu -j+2\right) }T_{\eta ,\alpha }\left( e_{j};\zeta \right) -\frac{\alpha \left( 1-\zeta \right) }{\eta }K_{\eta }\left( e_{\mu } ;\zeta \right) \\&=\frac{\zeta \left( 1-\zeta \right) }{\eta }\left( K_{\eta }^{\left( \alpha \right) }\left( e_{\mu };\zeta \right) \right) ^{^{\prime }}+\left[ \zeta +\frac{\left( 1-\zeta \right) }{\eta }\right] K_{\eta ,\alpha }\left( e_{\mu };\zeta \right) +\frac{1}{\left( \mu +2\right) \left( \eta +1\right) ^{\mu +1}}\\&\quad +\sum _{j=1}^{\mu +1}\left( \begin{array}{c} \mu +1\\ j \end{array} \right) \frac{\eta ^{j-1}}{\left( \eta +1\right) ^{\mu }\left( \mu -j+2\right) }\left\{ \frac{\left( \mu +1\right) \eta -j\left( \eta +1\right) }{\left( \mu +1\right) \left( \eta +1\right) }\right\} T_{\eta ,\alpha }\left( e_{j};\zeta \right) \\&\quad +\sum _{j=0}^{\mu }\left( \begin{array}{c} \mu \\ j \end{array} \right) \left( \frac{1-\alpha }{\eta }\right) \left( \frac{\eta -1}{\eta }\right) ^{j}\frac{\eta ^{j}}{\left( \eta +1\right) ^{\mu }\left( \mu -j+1\right) }\left\{ B_{\eta -1}\left( e_{j+1}\right) -B_{\eta -1}\left( e_{j}\right) \right\} \\&\quad -\frac{\alpha \left( 1-\zeta \right) }{\eta }K_{\eta }\left( e_{\mu } ;\zeta \right) \\&=\frac{\zeta \left( 1-\zeta \right) }{\eta }\left( K_{\eta ,q}\left( e_{\mu };\zeta \right) \right) ^{^{\prime }}+\left[ \zeta +\frac{\left( 1-\zeta \right) }{\eta }\right] K_{\eta ,\alpha }\left( e_{\mu };\zeta \right) \\&\quad +\sum _{j=0}^{\mu +1}\left( \begin{array}{c} \mu +1\\ j \end{array} \right) \frac{\eta ^{j-1}}{\left( \eta +1\right) ^{\mu +1}\left( \mu -j+2\right) }\left\{ \frac{\left( \mu +1\right) \eta -j\left( \eta +1\right) }{\left( \mu +1\right) }\right\} T_{\eta ,\alpha }\left( e_{j};\zeta \right) \\&\quad +\sum _{j=0}^{\mu }\left( \begin{array}{c} \mu \\ j \end{array} \right) \left( \frac{1-\alpha }{\eta }\right) \left( \frac{\eta -1}{\eta }\right) ^{j}\frac{\eta ^{j}}{\left( \eta +1\right) ^{\mu }\left( \mu -j+1\right) }\left\{ B_{\eta -1}\left( e_{j+1}\right) -B_{\eta -1}\left( e_{j}\right) \right\} \\&\quad -\frac{\alpha \left( 1-\zeta \right) }{\eta }K_{\eta }\left( e_{\mu } ;\zeta \right) . \end{aligned}$$

Here we used the identity

$$\begin{aligned} \left( \begin{array}{c} \mu \\ j-1 \end{array} \right) =\left( \begin{array}{c} \mu +1\\ j \end{array} \right) \frac{j}{\left( \mu +1\right) }. \end{aligned}$$

$\square$

For $\eta \in {\mathbb {N}},\zeta \in {\mathbb {C}}$, $\mu \in \mathbb {N\cup }\left\{ 0\right\}$ and $\alpha \in \left[ 0,1\right]$, if we denote

$$\begin{aligned} E_{\eta ,\mu ,\alpha }\left( \zeta \right) :=K_{\eta ,\alpha }\left( e_{\mu };\zeta \right) -e_{\mu }\left( \zeta \right) -\dfrac{1-2\zeta }{2\left( \eta +1\right) }\mu \zeta ^{\mu -1}-\dfrac{\zeta \left( 1-\zeta \right) }{2\left( \eta +1\right) }\mu \left( \mu -1\right) \zeta ^{\mu -2}\text {,} \end{aligned}$$

then it is clear that degree$\left( E_{\eta ,\mu ,\alpha }\left( \zeta \right) \right) \le \mu .$ Using the above recurrence and by simple calculations, we get the following Lemma.

Lemma 4

For all $\alpha \in \left[ 0,1\right]$ and $\eta ,\mu \in {\mathbb {N}}$, we have

$$\begin{aligned} E_{\eta ,\mu ,\alpha }\left( \zeta \right)&=\frac{\zeta \left( 1-\zeta \right) }{\eta }\left( K_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) -e_{\mu -1}\left( \zeta \right) \right) ^{^{\prime }}+\zeta E_{\eta ,\mu -1,\alpha }\left( \zeta \right) \\&\quad +\frac{(\mu -1)}{\eta (\eta +1)}\zeta ^{\mu -1}\left( 1-\zeta \right) -\frac{1-2\zeta }{2\left( \eta +1\right) }\zeta ^{\mu -1}+\frac{\left( 1-\zeta \right) }{\eta }K_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) \\&\quad +\sum _{j=0}^{\mu }\left( \begin{array}{c} \mu \\ j \end{array} \right) \frac{\eta ^{j}}{\left( \eta +1\right) ^{\mu }\left( \mu -j+1\right) }\left\{ \frac{\mu \eta -j\left( \eta +1\right) }{\eta }\right\} T_{\eta ,\alpha }\left( e_{j};\zeta \right) \\&\quad +\sum _{j=0}^{\mu -1}\left( \begin{array}{c} \mu -1\\ j \end{array} \right) \left( \frac{\eta -1}{\eta }\right) ^{j}\left( \frac{1-\alpha }{\eta }\right) \frac{\eta ^{j}}{\left( \eta +1\right) ^{\mu -1}\left( \mu -j\right) }\left\{ B_{\eta -1}\left( e_{j+1}\right) \right. \\&\quad \left. -B_{\eta -1}\left( e_{j}\right) \right\} -\frac{\alpha \left( 1-\zeta \right) }{\eta }K_{\eta }\left( e_{\mu -1} ;\zeta \right) \end{aligned}$$

Proof

It is immediate that $E_{\eta ,0,\alpha }\left( \zeta \right) =E_{\eta ,1,\alpha }\left( \zeta \right) =0$.

Using the formula (8) we get

$$\begin{aligned} E_{\eta ,\mu ,\alpha }\left( \zeta \right)&=\frac{\zeta \left( 1-\zeta \right) }{\eta }\left( K_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) -e_{\mu -1}\left( \zeta \right) \right) ^{^{\prime }}+\frac{\left( \mu -1\right) }{\eta }\zeta ^{\mu -1}\left( 1-\zeta \right) \\&\quad +\left[ \zeta +\frac{\left( 1-\zeta \right) }{\eta }\right] \left( E_{\eta ,\mu -1,\alpha }\left( \zeta \right) +\frac{1-2\zeta }{2\left( \eta +1\right) }\left( \mu -1\right) \zeta ^{\mu -2}\right. \\&\quad \left. +\frac{\zeta ^{\mu -2}\left( 1-\zeta \right) \left( \mu -1\right) \left( \mu -2\right) }{2\left( \eta +1\right) }\right) \\&\quad +\sum _{j=0}^{\mu -1}\left( \begin{array}{c} \mu -1\\ j \end{array} \right) \left( \frac{1-\alpha }{\eta }\right) \left( \frac{\eta -1}{\eta }\right) ^{j}\frac{\eta ^{j}}{\left( \eta +1\right) ^{\mu -1}\left( \mu -j\right) }\left\{ B_{\eta -1}\left( e_{j+1}\right) \right. \\&\quad \left. -B_{\eta -1}\left( e_{j}\right) \right\} -\frac{\alpha \left( 1-\zeta \right) }{\eta }K_{\eta }\left( e_{\mu -1} ;\zeta \right) \\ {}&\quad +\sum _{j=0}^{\mu }\left( \begin{array}{c} \mu \\ j \end{array} \right) \frac{\eta ^{j}}{\left( \eta +1\right) ^{\mu }\left( \mu -j+1\right) }\left\{ \frac{\mu \eta -j\left( \eta +1\right) }{\eta }\right\} T_{\eta ,\alpha }\left( e_{j};\zeta \right) \\&\quad -\frac{1-2\zeta }{2\left( \eta +1\right) }\mu \zeta ^{\mu -1}-\frac{\mu \left( \mu -1\right) \zeta ^{\mu -1}\left( 1-\zeta \right) }{2\left( \eta +1\right) }, \end{aligned}$$

With a simple calculation, we get the following relation.

$$\begin{aligned} E_{\eta ,\mu ,\alpha }\left( \zeta \right)&=\frac{\zeta \left( 1-\zeta \right) }{\eta }\left( K_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) -e_{\mu -1}\left( \zeta \right) \right) ^{^{\prime }}+\zeta E_{\eta ,\mu -1,\alpha }\left( \zeta \right) \\&\quad +\frac{(\mu -1)}{\eta (\eta +1)}\zeta ^{\mu -1}\left( 1-\zeta \right) -\frac{1-2\zeta }{2\left( \eta +1\right) }\zeta ^{\mu -1}+\frac{\left( 1-\zeta \right) }{\eta }K_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) \\&\quad +\sum _{j=0}^{\mu }\left( \begin{array}{c} \mu \\ j \end{array} \right) \frac{\eta ^{j}}{\left( \eta +1\right) ^{\mu }\left( \mu -j+1\right) }\left\{ \frac{\mu \eta -j\left( \eta +1\right) }{\eta }\right\} T_{\eta ,\alpha }\left( e_{j};\zeta \right) \\&\quad +\sum _{j=0}^{\mu -1}\left( \begin{array}{c} \mu -1\\ j \end{array} \right) \left( \frac{1-\alpha }{\eta }\right) \left( \frac{\eta -1}{\eta }\right) ^{j}\frac{\eta ^{j}}{\left( \eta +1\right) ^{\mu -1}\left( \mu -j\right) }\left\{ B_{\eta -1}\left( e_{j+1}\right) \right. \\&\quad \left. -B_{\eta -1}\left( e_{j}\right) \right\} -\frac{\alpha \left( 1-\zeta \right) }{\eta }K_{\eta }\left( e_{\mu -1} ;\zeta \right) . \end{aligned}$$

$\square$

3 Approximation by Complex $\alpha$-Bernstein–Kantorovich-Type Operator

In this section, we start with the upper quantitative estimates for the new operators and its derivatives attached to an analytic function in a compact disk of radius $1<R$ and center 0.

Theorem 5

Suppose that $\alpha \in \left[ 0,1\right]$ and $\varphi \in H({\mathbb {D}}_{R}).$(i) Let $r\in \left[ 1,R\right)$ be arbitrary fixed. For all $\left| \zeta \right| \le r$ and $\eta \in {\mathbb {N}}$, we have

$$\begin{aligned} \left| K_{\eta ,\alpha }\left( \varphi ;\zeta \right) -\varphi \left( \zeta \right) \right| \le \frac{M_{r}(\varphi )}{\eta }, \end{aligned}$$

(10)

where

$$\begin{aligned} 0<M_{r}(\varphi )=(4+\alpha )\sum _{\mu =1}^{\infty }\left| a_{\mu }\right| \mu \left( \mu +1\right) r^{\mu } \end{aligned}$$

(ii) If $1\le r<r_{1}<R$ are arbitrary fixed and $p\in {\mathbb {N}}$, then for all $\left| \zeta \right| \le r$ and $\eta \in {\mathbb {N}}$, we have

$$\begin{aligned} \left| K_{\eta ,\alpha }^{(p)}\left( \varphi ;\zeta \right) -\varphi ^{(p)}\left( \zeta \right) \right| \le \frac{M_{r_{1}}(\varphi )p!r_{1} }{\eta (r_{1}-r)^{p+1}}, \end{aligned}$$

where $M_{r_{1}}(\varphi )$ is given as in (i).

Proof

(i) To estimate $\left| K_{\eta ,\alpha }\left( \varphi ;\zeta \right) -\varphi \left( \zeta \right) \right|$, firstly, using above recurrence 8 we get

$$\begin{aligned} K_{\eta ,\alpha }\left( e_{\mu };\zeta \right) -e_{\mu }\left( \zeta \right)&=\frac{\zeta \left( 1-\zeta \right) }{\eta }\left( K_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) \right) ^{^{\prime }}+\zeta \left( K_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) -e_{\mu -1}\left( \zeta \right) \right) \nonumber \\&\quad -\frac{\alpha \left( 1-\zeta \right) }{\eta }K_{\eta }\left( e_{\mu -1} ;\zeta \right) +\frac{\left( 1-\zeta \right) }{\eta }K_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) \nonumber \\&\quad +\left( \frac{1-\alpha }{\eta }\right) \left( \frac{\eta -1}{\eta }\right) ^{\mu -1}\frac{\eta ^{\mu -1}}{\left( \eta +1\right) ^{\mu -1}}\left\{ B_{\eta -1}\left( e_{\mu }\right) -B_{\eta -1}\left( e_{\mu -1}\right) \right\} \nonumber \\&\quad +\sum _{j=0}^{\mu }\left( \begin{array}{c} \mu \\ j \end{array} \right) \frac{\eta ^{j-1}}{\left( \eta +1\right) ^{\mu }\left( \mu -j+1\right) }\left\{ \frac{\mu \eta -j\left( \eta +1\right) }{\mu \left( \eta +1\right) }\right\} T_{\eta ,\alpha }\left( e_{j};\zeta \right) \nonumber \\&\quad +.\sum _{j=0}^{\mu -2}\frac{\mu -1}{\mu -1-j}\left( \begin{array}{c} \mu -2\\ j \end{array} \right) \left( \frac{1-\alpha }{\eta }\right) \left( \frac{\eta -1}{\eta }\right) ^{j}\frac{\eta ^{j}}{\left( \eta +1\right) ^{\mu -1}\left( \mu -j\right) }\nonumber \\&\quad \left\{ B_{\eta -1}\left( e_{j+1}\right) -B_{\eta -1}\left( e_{j}\right) \right\} \end{aligned}$$

(11)

and secondly, we can estimate the above two sums (11) as:

$$\begin{aligned}&\left| \frac{1}{\left( \eta +1\right) ^{\mu }}\sum _{j=0}^{\mu }\left( \begin{array}{c} \mu \\ j \end{array} \right) \frac{\eta ^{j}}{\left( \mu -j+1\right) }\left( 1-\frac{j}{\mu }-\frac{j}{\mu \eta }\right) T_{\eta ,\alpha }\left( e_{j};\zeta \right) \right| \\&\quad \le \frac{1}{\left( \eta +1\right) ^{\mu }}\left( \sum _{j=0}^{\mu -1}\left( \begin{array}{c} \mu -1\\ j \end{array} \right) \frac{\mu }{\mu -j}\frac{\eta ^{j}}{\mu -j+1}\left| 1-\frac{j}{\mu }-\frac{j}{\mu \eta }\right| \right) \left| T_{\eta ,\alpha }\left( e_{j};\zeta \right) \right| \\&\qquad +\frac{\eta ^{\mu -1}}{\left( \eta +1\right) ^{\mu }}r^{\mu }\\&\quad \le \frac{2\mu \left( \eta +1\right) ^{\mu -1}+\eta ^{\mu -1}}{\left( \eta +1\right) ^{\mu }}r^{\mu }\le \frac{2\mu +1}{\left( \eta +1\right) }r^{\mu } \end{aligned}$$

and

$$\begin{aligned}&\left| \sum _{j=0}^{\mu -2}\frac{\mu -1}{\mu -1-j}\left( \begin{array}{c} \mu -2\\ j \end{array} \right) \left( \frac{1-\alpha }{\eta }\right) \left( \frac{\eta -1}{\eta }\right) ^{j}\frac{\eta ^{j}}{\left( \eta +1\right) ^{\mu -1}\left( \mu -j\right) }\left\{ B_{\eta -1}\left( e_{j+1}\right) \right. \right. \\&\quad \left. \left. -B_{\eta -1}\left( e_{j}\right) \right\} \right| \\&\le \left( \frac{1-\alpha }{\eta }\right) \frac{1}{\left( \eta +1\right) ^{\mu -1}}\left( \sum _{j=0}^{\mu -2}\left( \begin{array}{c} \mu -2\\ j \end{array} \right) \frac{\mu -1}{\mu -1-j}\frac{\left( \eta -1\right) ^{j}}{\left( \mu -j\right) }\right) \left\{ \left| B_{\eta -1}\left( e_{j+1}\right) \right| \right. \\&\quad \left. +\left| B_{\eta -1}\left( e_{j}\right) \right| \right\} \\&\le \frac{\left( 1-\alpha \right) \left( \mu -1\right) \left( \eta -1+1\right) ^{\mu -2}}{\eta \left( \eta +1\right) ^{\mu -1}}\left( r^{\mu -1}+r^{\mu -2}\right) \le \frac{\left( 1-\alpha \right) \left( \mu -1\right) }{\eta }\left( r^{\mu -1}+r^{\mu -2}\right) . \end{aligned}$$

Also, it is known that, from Theorem 1.0.8 in [9], we have

$$\begin{aligned} \left| P_{\mu }^{\prime }\left( \zeta \right) \right| \le \frac{\mu }{r}\left\| P_{\mu }\right\| _{r},\ \ \ \text {for all\ \ }\left| \zeta \right| \le r,\ \ r\ge 1, \end{aligned}$$

where $P_{\mu }\left( \zeta \right)$ is a complex polynomial in $\zeta$ of degree $\le \mu$.

Therefore, from the recurrence formula 11 we get

$$\begin{aligned} \left| K_{\eta ,\alpha }\left( e_{\mu };\zeta \right) -e_{\mu }\left( \zeta \right) \right|&\le \frac{\left| \zeta \right| \left| 1-\zeta \right| }{\eta }\left| \left( K_{\eta ,\alpha }\left( e_{\mu };\zeta \right) \right) ^{^{\prime }}\right| +\left| \zeta \right| \left| K_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) -e_{\mu -1}\left( \zeta \right) \right| \\&\quad +\frac{\alpha \left| 1-\zeta \right| }{\eta }\left| K_{\eta }\left( e_{\mu -1};\zeta \right) \right| +\frac{\left| 1-\zeta \right| }{\eta }\left| K_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) \right| \\&\quad +\frac{2\left( 1-\alpha \right) }{\eta }\left( r^{\mu }+r^{\mu -1}\right) +\frac{2\mu +1}{\left( \eta +1\right) }r^{\mu }\\&\quad +\frac{\left( 1-\alpha \right) \left( \mu -1\right) }{\eta }\left( r^{\mu -1}+r^{\mu -2}\right) \\&\le \frac{r\left( 1+r\right) }{\eta }\frac{\mu -1}{r}\left\| K_{\eta ,\alpha }\left( e_{\mu -1}\right) \right\| _{r}+r\left| K_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) -e_{\mu -1}\left( \zeta \right) \right| \\&\quad +\frac{\alpha \left( 1+r\right) }{\eta }r^{\mu -1}+\frac{\left( 1+r\right) }{\eta }r^{\mu -1}+\frac{\left( 1-\alpha \right) }{\eta }\left( r^{\mu } +r^{\mu -1}\right) \\&\quad +\frac{2\mu +1}{\left( \eta +1\right) }r^{\mu }+\frac{2\left( 1-\alpha \right) \left( \mu -1\right) }{\eta }\left( r^{\mu -1}+r^{\mu -2}\right) \\&\le \frac{2\left( \mu -1\right) }{\eta }r^{\mu }+r\left| K_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) -e_{\mu -1}\left( \zeta \right) \right| +\frac{2\alpha }{\eta }r^{\mu }+\frac{2}{\eta }r^{\mu }\\&\quad +\frac{2\left( 1-\alpha \right) }{\eta }r^{\mu }+\frac{2\mu +1}{\left( \eta +1\right) }r^{\mu }+\frac{2\left( 1-\alpha \right) \left( \mu -1\right) }{\eta }r^{\mu }\\&=r\left| K_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) -e_{\mu -1}\left( \zeta \right) \right| +\frac{4\mu +2\alpha +1+2\mu \left( 1-\alpha \right) }{\eta }r^{\mu } \end{aligned}$$

Now, by taking $\mu =1,2,...,$ in the inequality

$$\begin{aligned}&\left| K_{\eta ,\alpha }\left( e_{\mu };\zeta \right) -e_{\mu }\left( \zeta \right) \right| \nonumber \\&\quad \le \frac{4\mu +2\alpha +1+2\mu \left( 1-\alpha \right) }{\eta }r^{\mu }\nonumber \\&\quad \quad +r\frac{4\left( \mu -1\right) +2\alpha +1+2\left( \mu -1\right) \left( 1-\alpha \right) }{\eta }r^{\mu -1}\nonumber \\&\quad \quad \left. +r^{2}\frac{4\left( \mu -2\right) +2\alpha +1+2\left( \mu -2\right) \left( 1-\alpha \right) }{\eta }r^{\mu -2}\right. \nonumber \\&\quad \quad +...+r^{\mu -1} \frac{4+2\alpha +1+2\left( 1-\alpha \right) }{\eta }r\nonumber \\&\le \frac{4+2\left( 1-\alpha \right) }{\eta }r^{\mu }\left( \mu +\mu -1+...+1\right) +\frac{\mu \left( 2\alpha +1\right) }{\eta }r^{\mu }\nonumber \\&\quad =\frac{\left\{ 2+\left( 1-\alpha \right) \right\} \mu \left( \mu +1\right) +\mu \left( 2\alpha +1\right) }{\eta }r^{\mu }\nonumber \\&\quad \le \frac{\left( 4+\alpha \right) \mu \left( \mu +1\right) }{\eta }r^{\mu }. \end{aligned}$$

(12)

Since $K_{\eta ,\alpha }\left( \varphi ;\zeta \right)$ is analytic in ${\mathbb {D}}_{R},$ we can write

$$\begin{aligned} K_{\eta ,\alpha }\left( \varphi ;\zeta \right) =\sum _{\mu =0}^{\infty }a_{\mu }K_{\eta ,\alpha }\left( e_{\mu };\zeta \right) ,\ \ \ \ \zeta \in {\mathbb {D}} _{R}, \end{aligned}$$

which together with (12) immediately implies for all $\left| \zeta \right| \le r$

$$\begin{aligned} \left| K_{\eta ,\alpha }\left( \varphi ;\zeta \right) -\varphi \left( \zeta \right) \right|{} & {} \le \sum _{\mu =0}^{\infty }\left| a_{\mu }\right| \left| K_{\eta ,\alpha }\left( e_{\mu };\zeta \right) -e_{\mu }\left( \zeta \right) \right| \\{} & {} \le \frac{(4+\alpha )}{\eta }\sum _{\mu =1}^{\infty }\left| c_{\mu }\right| \mu \left( \mu +1\right) r^{\mu }. \end{aligned}$$

(ii) For the simultaneous approximation, denoting by $\gamma$ the circle of radius $r<r_{1}$ and center 0 (where $r_{1}>r\ge 1)$, by the Cauchy’s formulas it follows that for all $\left| \zeta \right| \le r$ and $\eta \in {\mathbb {N}}$ we have

$$\begin{aligned} K_{\eta ,\alpha }^{(p)}\left( \varphi ;\zeta \right) -\varphi ^{(p)}\left( \zeta \right) \le \frac{p!}{2\pi i}\underset{\gamma }{ {\displaystyle \int } }\frac{K_{\eta ,\alpha }\left( \varphi ;v\right) -\varphi \left( v\right) }{\left( v-\zeta \right) ^{p+1}}dv, \end{aligned}$$

which by (10) and by the inequality $\left| v-\zeta \right| \ge r_{1}-r$ valid for all $\left| \zeta \right| \le r$ and $v\in \gamma ,$ we have

$$\begin{aligned} \left| K_{\eta ,\alpha }^{(p)}\left( \varphi ;\zeta \right) -\varphi ^{(p)}\left( \zeta \right) \right|&\le \frac{p!}{2\pi }\underset{\gamma }{ {\displaystyle \int } }\frac{\left| K_{\eta ,\alpha }\left( \varphi ;v\right) -\varphi \left( v\right) \right| }{\left| v-\zeta \right| ^{p+1}}\left| dv\right| \\&\le \frac{p!}{2\pi }\frac{M_{r_{1}}(\varphi )}{\eta }\frac{2\pi r_{1}}{(r_{1}-r)^{p+1}}=\frac{M_{r_{1}}(\varphi )p!r_{1}}{\eta (r_{1}-r)^{p+1}} \end{aligned}$$

which proves (ii). $\square$

In the next Theorem, we obtain the Voronovskaja type formula with a quantitative estimate for complex $\alpha$-Bernstein–Kantorovich operators.

Theorem 6

Let $r\in \left[ 1,R\right)$ and $\varphi \in H\left( {\mathbb {D}}_{R}\right) .$ Then for all $\eta \in {\mathbb {N}}$ and $\alpha \in \left[ 0,1\right]$, we have

$$\begin{aligned}&\left| K_{\eta ,\alpha }\left( \varphi ;\zeta \right) -\varphi \left( \zeta \right) -\frac{1-2\zeta }{2\left( \eta +1\right) }\varphi ^{\prime }\left( \zeta \right) -\frac{\zeta \left( 1-\zeta \right) }{2\left( \eta +1\right) }\varphi ^{\prime \prime }\left( \zeta \right) \right| \\&\quad \le \frac{1}{\eta ^{2}}\sum _{\mu =2}^{\infty }\left| a_{\mu }\right| \left\{ \left( 43-7\alpha \right) \mu (\mu -1)^{2}\right\} r^{\mu }+\frac{5}{\eta ^{2}}\sum _{\mu =1}^{\infty }\mu (\mu +1)r^{\mu }. \end{aligned}$$

Proof

Using reccurrence formula (8) and simple calculation leads us to the following relationship:

$$\begin{aligned}&E_{\eta ,\mu ,\alpha }\left( \zeta \right) =\frac{\zeta \left( 1-\zeta \right) }{\eta }\left( K_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) -e_{\mu -1}\left( \zeta \right) \right) ^{^{\prime }}+\zeta E_{\eta ,\mu -1,\alpha }\left( \zeta \right) \nonumber \\&\quad +\frac{\left( \mu -1\right) }{\eta \left( \eta +1\right) }\zeta ^{\mu -1}\left( 1-\zeta \right) \nonumber \\&\quad +\frac{1}{\left( \eta +1\right) }\left( \zeta ^{\mu }-T_{\eta ,\alpha }\left( e_{\mu };\zeta \right) \right) +\frac{1}{\left( \eta +1\right) }\left( 1-\frac{\eta ^{\mu -1}}{\left( \eta +1\right) ^{\mu -1}}\right) T_{\eta ,\alpha }\left( e_{\mu };\zeta \right) \nonumber \\&\quad -\frac{1}{2\left( \eta +1\right) }\left( 1-\frac{\eta ^{\mu -1}}{\left( \eta +1\right) ^{\mu -1}}\right) T_{\eta ,\alpha }\left( e_{\mu -1} ;\zeta \right) +\frac{1}{2\left( \eta +1\right) }\left( T_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) -\zeta ^{\mu -1}\right) \nonumber \\&\quad -\frac{\left( \mu -1\right) \eta ^{\mu -2}}{2\left( \eta +1\right) ^{\mu } }T_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) +\frac{1}{\left( \eta +1\right) ^{\mu }}\sum _{j=0}^{\mu -2}\left( \begin{array}{c} \mu \\ j \end{array} \right) \nonumber \\&\quad \frac{\eta ^{j}}{\left( \mu -j+1\right) }\frac{\mu \eta -j\left( \eta +1\right) }{\mu \eta }T_{\eta ,\alpha }\left( e_{j};\zeta \right) \nonumber \\&\quad -\left( K_{\eta }\left( e_{\mu -1};\zeta \right) -e_{\mu -1}\left( \zeta \right) \right) \left( \frac{\alpha \left( 1-\zeta \right) }{\eta }\right) +\frac{\left( 1-\zeta \right) }{\eta }\left( K_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) -e_{\mu -1}\left( \zeta \right) \right) \nonumber \\&\quad -\frac{1-\alpha }{\eta }\left( \frac{\eta -1}{\eta }\right) ^{\mu -1} \frac{\eta ^{\mu -1}}{\left( \eta +1\right) ^{\mu -1}}\left( B_{\eta -1}\left( e_{\mu -1}\right) -e_{\mu -1}\left( \zeta \right) \right) \nonumber \\&\quad {+}\frac{\zeta ^{\mu -1}\left( 1{-}\alpha \right) }{\eta }\left( 1{-}\frac{\left( \eta -1\right) ^{\mu -1}}{\left( \eta {+}1\right) ^{\mu -1}}\right) {+}\frac{1{-}\alpha }{\eta }\left( \frac{\eta -1}{\eta }\right) ^{\mu -1}\frac{\eta ^{\mu -1}}{\left( \eta +1\right) ^{\mu -1}}\left( B_{\eta -1}\left( e_{\mu }\right) {-}e_{\mu }\left( \zeta \right) \right) \nonumber \\&\quad -\left( 1-\frac{\left( \eta -1\right) ^{\mu -1}}{\left( \eta +1\right) ^{\mu -1}}\right) \frac{\zeta ^{\mu }\left( 1-\alpha \right) }{\eta }+\sum _{j=0}^{\mu -2}\left( \begin{array}{c} \mu -1\\ j \end{array} \right) \left( \frac{1-\alpha }{\eta }\right) \left( \frac{\eta -1}{\eta }\right) ^{j}\nonumber \\ {}&\quad \frac{\eta ^{j}}{\left( \eta +1\right) ^{\mu -1}\left( \mu -j\right) }\left\{ B_{\eta -1}\left( e_{j+1}\right) -B_{\eta -1}\left( e_{j}\right) \right\} \nonumber \\&\quad :=\sum _{\rho =1}^{16}I_{\rho }. \end{aligned}$$

(13)

Firstly, the estimate of $I_{3}$ are $I_{8}$ as follows

$$\begin{aligned} \left| I_{3}\right|&\le \frac{\left( \mu -1\right) }{\eta \left( \eta +1\right) }r^{\mu -1}\left( 1+r\right) ,\nonumber \\ \left| I_{8}\right|&\le \frac{\left( \mu -1\right) }{2\left( \eta +1\right) ^{2}}\left| T_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) \right| \le \frac{\left( \mu -1\right) }{2\left( \eta +1\right) ^{2} }r^{\mu -1}, \end{aligned}$$

(14)

Secondly using the known inequality

$$\begin{aligned} 1-\prod \limits _{\rho =1}^{\mu }\tau _{\rho }\le \sum \limits _{\rho =1}^{\mu }\left( 1-\tau _{\rho }\right) ,\ 0\le \tau _{\rho }\le 1, \end{aligned}$$

to estimate $I_{5},I_{6},I_{9},I_{13},I_{15},I_{16}.$

$$\begin{aligned} \left| I_{5}\right|&\le \frac{1}{\eta +1}\left( 1-\frac{\eta ^{\mu -1}}{\left( \eta +1\right) ^{\mu -1}}\right) \left| T_{\eta ,\alpha }\left( e_{\mu };\zeta \right) \right| \le \frac{\mu -1}{\left( \eta +1\right) ^{2}}r^{\mu },\nonumber \\ \left| I_{6}\right|&\le \frac{1}{2\left( \eta +1\right) }\left( 1-\frac{\eta ^{\mu -1}}{\left( \eta +1\right) ^{\mu -1}}\right) \left| T_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) \right| \le \frac{\mu -1}{2\left( \eta +1\right) ^{2}}r^{\mu -1},\nonumber \\ \left| I_{9}\right|&\le \frac{1}{\left( \eta +1\right) ^{\mu } }\sum _{j=0}^{\mu -2}\left( \begin{array}{c} \mu -2\\ j \end{array} \right) \frac{\mu \left( \mu -1\right) }{\left( \mu -j\right) \left( \mu -j-1\right) }\frac{\eta ^{j}}{\left( \mu -j+1\right) }\left( 1-\frac{j}{\mu }-\frac{j}{\mu \eta }\right) r^{j}\nonumber \\&\le \frac{2\mu \left( \mu -1\right) \left( \eta +1\right) ^{\mu -2} }{\left( \eta +1\right) ^{\mu }}r^{\mu }=\frac{2\mu \left( \mu -1\right) }{\left( \eta +1\right) ^{2}}r^{\mu },\nonumber \\ \left| I_{13}\right|&\le \frac{\zeta ^{\mu -1}\left( 1-\alpha \right) }{\eta }\left( 1-\frac{\left( \eta -1\right) ^{\mu -1}}{\left( \eta +1\right) ^{\mu -1}}\right) \le \frac{2\left( 1-\alpha \right) \left( \mu -1\right) }{\eta \left( \eta +1\right) }r^{\mu -1},\nonumber \\ \left| I_{15}\right|&\le \frac{\zeta ^{\mu }\left( 1-\alpha \right) }{\eta }\left( 1-\frac{\left( \eta -1\right) ^{\mu -1}}{\left( \eta +1\right) ^{\mu -1}}\right) \le \frac{2\left( 1-\alpha \right) \left( \mu -1\right) }{\eta \left( \eta +1\right) }r^{\mu },\nonumber \\ \left| I_{16}\right|&\le \sum _{j=0}^{\mu -2}\left( \begin{array}{c} \mu -2\\ j \end{array} \right) \frac{\mu -1}{\mu -1-j}\left( \frac{1-\alpha }{\eta }\right) \left( \frac{\eta -1}{\eta }\right) ^{j}\frac{\eta ^{j}}{\left( \eta +1\right) ^{\mu -1}\left( \mu -j\right) }\left\{ B_{\eta -1}\left( e_{j+1}\right) \right. \nonumber \\&\quad \left. -B_{\eta -1}\left( e_{j}\right) \right\} \nonumber \\&\le \frac{\left( 1-\alpha \right) \left( \mu -1\right) }{\eta \left( \eta +1\right) }\left( r^{\mu -1}+r^{\mu -2}\right) . \end{aligned}$$

(15)

Finally we estimate $I_{4},I_{7},I_{10},I_{11},I_{12},I_{14}$. We use [9, Theorem 1.1.2], [1, Theorem 2.4] and Theorem 5(i)

$$\begin{aligned}&\left| I_{4}\right| +\left| I_{7}\right| +\left| I_{10}\right| +\left| I_{11}\right| +\left| I_{12}\right| +\left| I_{14}\right| \nonumber \\&\le \frac{1}{\left( \eta +1\right) }\left| \zeta ^{\mu }-T_{\eta ,\alpha }\left( e_{\mu };\zeta \right) \right| +\frac{1}{2\left( \eta +1\right) }\left| T_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) -\zeta ^{\mu -1}\right| \nonumber \\&\quad +\frac{\alpha \left( 1+r\right) }{\eta }\left| K_{\eta }\left( e_{\mu -1};\zeta \right) -e_{\mu -1}\left( \zeta \right) \right| +\frac{\left( 1+r\right) }{\eta }\left| K_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) -e_{\mu -1}\left( \zeta \right) \right| \nonumber \\&\quad +\frac{1-\alpha }{\eta }\left| B_{\eta -1}\left( e_{\mu -1};\zeta \right) -e_{\mu -1}\left( \zeta \right) \right| +\frac{1-\alpha }{\eta }\left| B_{\eta -1}\left( e_{\mu };\zeta \right) -e_{\mu }\left( \zeta \right) \right| \nonumber \\&\le \frac{1}{\left( \eta +1\right) }\left\{ \left( \frac{3\mu (\mu +1)}{2}+\mu \right) \frac{\left( 1+r\right) }{\eta }r^{\mu -1}+\frac{2\mu (\mu -1)}{\eta }r^{\mu }\right\} \nonumber \\&\quad +\frac{1}{2\left( \eta +1\right) }\left\{ \left( \frac{3\left( \mu -1\right) \mu }{2}+\left( \mu -1\right) \right) \frac{\left( 1+r\right) }{\eta }r^{\mu -2}+\frac{2\left( \mu -1\right) (\mu -2)}{\eta }r^{\mu -1}\right\} \nonumber \\&\quad +\frac{\alpha \left( 1+r\right) }{\eta }\left\{ \frac{2\left( \mu -1\right) \mu }{\eta }r^{\mu -1}\right\} +\frac{\left( 4+\alpha \right) }{\eta }\left\{ \frac{2\left( \mu -1\right) \mu }{\eta }r^{\mu -1}\right\} \nonumber \\&\quad +\frac{1-\alpha }{\eta }\left\{ \frac{3r\left( 1+r\right) \left( \mu -2\right) \left( \mu -1\right) }{2\eta }r^{\mu -3}\right\} +\frac{1-\alpha }{\eta }\left\{ \frac{3r\left( 1+r\right) \left( \mu -1\right) \mu }{2\eta }r^{\mu -2}\right\} . \end{aligned}$$

(16)

Using (12), (14), (15) and (16) in (13) finally we have

$$\begin{aligned}&\left| E_{\eta ,\mu ,\alpha }\left( \zeta \right) \right| \le \frac{r\left( 1+r\right) }{\eta }\left| \left( K_{\eta ,\alpha }\left( e_{\mu -1};\zeta \right) -e_{\mu -1}\left( \zeta \right) \right) ^{^{\prime } }\right| +r\left| E_{\eta ,\mu -1,\alpha }\left( \zeta \right) \right| \\&\quad +\frac{\left( \mu -1\right) }{\eta \left( \eta +1\right) } r^{\mu -1}\left( 1+r\right) \\&\quad +\frac{\left( \mu -1\right) }{\left( \eta +1\right) ^{2}}r^{\mu } +\frac{\mu -1}{2\left( \eta +1\right) ^{2}}r^{\mu -1}+\frac{2\mu \left( \mu -1\right) }{\left( \eta +1\right) ^{2}}r^{\mu }+\frac{\left( 1-\alpha \right) \left( \mu -1\right) }{\eta \left( \eta +1\right) }r^{\mu -1}\\&\quad +\frac{\left( 1-\alpha \right) \left( \mu -1\right) }{\eta \left( \eta +1\right) }r^{\mu }+\frac{\left( 1-\alpha \right) \left( \mu -1\right) }{\eta \left( \eta +1\right) }\left( r^{\mu -1}+r^{\mu -2}\right) \\&\quad +\frac{\left( 1+r\right) }{\eta (\eta +1)}\left( \frac{3\mu (\mu +1)}{2} +\mu \right) r^{\mu -1}+\frac{2\mu (\mu -1)}{\eta (\eta +1)}r^{\mu }\\&\quad +\frac{\left( 1+r\right) }{2\eta (\eta +1)}\left( \frac{3(\mu -1)\mu }{2}+\mu -1\right) r^{\mu -2}+\frac{\left( \mu -1\right) (\mu -2)}{\eta \left( \eta +1\right) }r^{\mu -1}\\&\quad +\frac{2\alpha \mu \left( \mu -1\right) \left( 1+r\right) }{\eta ^{2} }r^{\mu -1}+\frac{\left( 4+\alpha \right) \left( 1+r\right) \left( \mu -1\right) \mu }{\eta ^{2}}r^{\mu -1}\\&\quad +\frac{3\left( 1+r\right) \left( \mu -2\right) \left( \mu -1\right) \left( 1-\alpha \right) }{2\eta ^{2}}r^{\mu -2}+\frac{3\left( 1+r\right) \left( 1-\alpha \right) \mu \left( \mu -1\right) }{2\eta ^{2}}r^{\mu -1}\\&\le \frac{r\left( 1+r\right) }{\eta }\frac{\mu -1}{r}\left\| K_{\eta ,\alpha }\left( e_{\mu -1}\right) -e_{\mu -1}\right\| _{r} +r\left| E_{\eta ,\mu -1,\alpha }\left( \zeta \right) \right| +\frac{23\mu \left( \mu -1\right) }{\eta ^{2}}r^{\mu }\\&\frac{9\mu (\mu -1)\left( 1-\alpha \right) }{\eta ^{2}}r^{\mu }+\frac{3\mu (\mu -1)\alpha }{\eta ^{2}}r^{\mu }+\frac{5\mu (\mu +1)}{\eta ^{2}}r^{\mu }\\&\le +\frac{\left( \mu -1\right) \left( 1+r\right) }{\eta }\frac{(4+\alpha )\left( \mu -1\right) \mu }{\eta }r^{\mu -1}+r\left| E_{\eta ,\mu -1,\alpha }\left( \zeta \right) \right| +\frac{23\mu \left( \mu -1\right) }{\eta ^{2}}r^{\mu }\\&\frac{9\mu (\mu -1)\left( 1-\alpha \right) }{\eta ^{2}}r^{\mu }+\frac{3\mu (\mu -1)\alpha }{\eta ^{2}}r^{\mu }+\frac{5\mu (\mu +1)}{\eta ^{2}}r^{\mu }\\&\le r\left| E_{\eta ,\mu -1,\alpha }\left( \zeta \right) \right| +\frac{2(4+\alpha )\mu \left( \mu -1\right) ^{2}}{\eta ^{2}}r^{\mu }+\frac{23\mu \left( \mu -1\right) }{\eta ^{2}}r^{\mu }\\&\quad +\frac{9\mu (\mu -1)\left( 1-\alpha \right) }{\eta ^{2}}r^{\mu }+\frac{3\mu (\mu -1)\alpha }{\eta ^{2}}r^{\mu }+\frac{5\mu (\mu +1)}{\eta ^{2}}r^{\mu }\\&\le r\left| E_{\eta ,\mu -1,\alpha }\left( \zeta \right) \right| +\frac{\left( 43-7\alpha \right) \mu \left( \mu -1\right) ^{2}}{\eta ^{2} }r^{\mu }+\frac{5\mu (\mu +1)}{\eta ^{2}}r^{\mu } \end{aligned}$$

As a consequence, we get

$$\begin{aligned} \left| E_{\eta ,\mu ,\alpha }\left( \zeta \right) \right| \le \frac{\left( 43-7\alpha \right) \mu \left( \mu -1\right) ^{2}+5\mu (\mu +1)}{\eta ^{2}}r^{\mu } \end{aligned}$$

In conclusion,

$$\begin{aligned}&\left| K_{\eta ,\alpha }\left( \varphi ;\zeta \right) -\varphi \left( \zeta \right) -\frac{1-2\zeta }{2\left( \eta +1\right) }\varphi ^{\prime }\left( \zeta \right) -\frac{\zeta \left( 1-\zeta \right) }{2\left( \eta +1\right) }\varphi ^{\prime \prime }\left( \zeta \right) \right| \\&\quad \le \sum _{\mu =0}^{\infty }\left| a_{\mu }\right| \left| E_{\eta ,\mu ,\alpha }\left( \zeta \right) \right| \\&\quad =\frac{\left( 43-7\alpha \right) }{\eta ^{2}}\sum _{\mu =2}^{\infty }\left| a_{\mu }\right| \mu (\mu -1)^{2}r^{\mu }+\frac{5}{\eta ^{2}} \sum _{\mu =1}^{\infty }\left| a_{\mu }\right| \mu (\mu +1)r^{\mu } \end{aligned}$$

Note that since $\varphi ^{\left( 3\right) }\left( \zeta \right) =\sum _{\mu =3}^{\infty }a_{\mu }\mu \left( \mu -1\right) (\mu -2)\zeta ^{\mu -3}$ and $\varphi ^{\left( 2\right) }\left( \zeta \right) =\sum _{\mu =2}^{\infty }a_{\mu }\mu \left( \mu -1\right) \zeta ^{\mu -2}$, and sumation of these two series is absolutely convergent for all $\left| \zeta \right| <R$, it easily follows that $\left( 43-7\alpha \right) \sum _{\mu =2}^{\infty }\left| a_{\mu }\right| \mu (\mu -1)^{2}r^{\mu }+5\sum _{\mu =1}^{\infty }\left| a_{\mu }\right| \mu (\mu +1)r^{\mu }<\infty .$ $\square$

Remark 7

In hypothesis of in $\varphi$ in Theorem 5 choosing $\alpha \in \left[ 0,1\right]$ as $\eta \rightarrow \infty$, it follows that

$$\begin{aligned} \underset{\eta \rightarrow \infty }{\lim }\left( \eta +1\right) \left[ K_{\eta ,\alpha }\left( \varphi ;\zeta \right) -\varphi \left( \zeta \right) \right] =\frac{1-2\zeta }{2}\varphi ^{\prime }\left( \zeta \right) +\frac{\zeta \left( 1-\zeta \right) }{2}\varphi ^{\prime \prime }\left( \zeta \right) \end{aligned}$$

(17)

Finally, in the next Theorem, we present the order of approximation in Theorem 5 is exactly $\frac{1}{\eta }$.

Theorem 8

Suppose that $0\le \alpha \le 1$ and $\varphi \in H(\mathbb {D)}_{R}.$ (i) If $\varphi$ is not polynomial in $\zeta$ of degree $=0,$ then for all $r\in \left[ 1,R\right) ,$we have

$$\begin{aligned} \left\| K_{\eta ,\alpha }\left( \varphi \right) -\varphi \right\| _{r} \sim \frac{1}{\eta }\text {, \ \ \ }\eta \in {\mathbb {N}} \end{aligned}$$

where the constants in the equivalence depend on r and $\varphi$.(ii) If $1\le r<r_{1}<R$ and $\varphi$ is not a polynomial in $\zeta$ of degree $\le \max \left\{ 1,p-1\right\}$($p\in {\mathbb {N}} ),$ we have

$$\begin{aligned} \left\| K_{\eta ,\alpha }^{\left( p\right) }\left( \varphi \right) -\varphi ^{\left( p\right) }\right\| _{r}\sim \frac{1}{\eta }\text {, \ } \eta \in {\mathbb {N}} \end{aligned}$$

in $\overline{{\mathbb {D}}_{R}}$, where the constant in the equivalence depend on $r,r_{1},\varphi$ and p.

Proof

(i) Taking in to account Remark 7, there exist constants $0<D_{1},D_{2}<\infty$ independent of $\eta$ such that

$$\begin{aligned} D_{1}\le \eta \left\| K_{\eta ,\alpha }\left( \varphi \right) -\varphi \right\| _{r}\le D_{2} \end{aligned}$$

from which it readlily follows that

$$\begin{aligned} \frac{D_{1}}{\eta }\le \left\| K_{\eta ,\alpha }\left( \varphi \right) -\varphi \right\| _{r}\le \frac{D_{2}}{\eta } \end{aligned}$$

in $\overline{{\mathbb {D}}_{r}}.$ Therefore, we get the desired result.

(ii) Denoting by $\rho$ the circle of radius $r<$ $r_{1}$and center 0 (where $r\in \left[ 1,r_{1}\right)$), we have the inequality $r_{1}-r$ $\le \left| v-\zeta \right|$ valid for all $\left| \zeta \right| \le r$ and $v\in \rho$.

Using the Cauchy’s formula, for all $\left| \zeta \right| \le r$ and $\eta ,p\in {\mathbb {N}}$, we obtain

$$\begin{aligned} K_{\eta ,\alpha }^{\left( p\right) }\left( \varphi ;\zeta \right) -\varphi ^{\left( p\right) }(\zeta )=\frac{p!}{2\pi i}\underset{\gamma }{ {\displaystyle \int } }\frac{K_{\eta ,\alpha }\left( \varphi ;v\right) -\varphi (v)}{\left( v-\zeta \right) ^{p+1}}dv. \end{aligned}$$

Thus, from Remark 7 we get

$$\begin{aligned} \underset{\eta \rightarrow \infty }{\lim }\eta \left[ K_{\eta ,\alpha }^{\left( p\right) }\left( \varphi ;\zeta \right) -\varphi ^{\left( p\right) } (\zeta )\right] =\left[ \frac{1-2\zeta }{2}\varphi ^{\prime }\left( \zeta \right) +\frac{\zeta \left( 1-\zeta \right) }{2}\varphi ^{\prime \prime }\left( \zeta \right) \right] ^{(p)} \end{aligned}$$

uniformly in $\overline{{\mathbb {D}}_{r}}.$

It remains to show that $\varphi (\zeta )$ is not constant. Indeed, supposing that $(1-2\zeta )\varphi ^{\prime }(\zeta )+\zeta \left( 1-\zeta \right) \varphi ^{\prime \prime }(\zeta )=0$ for all $|\zeta |\le r$, that is $(\zeta \left( 1-\zeta \right) \varphi ^{\prime }\left( \zeta \right) )^{\prime }=0$ for all $|\zeta |\le r.$ The last equality is equivalent to $\zeta \left( 1-\zeta \right) \varphi ^{\prime }(\zeta )=C$ for all $|\zeta |\le r$ with $\zeta \not =0$. Therefore we get $\varphi ^{\prime }(\zeta )=\frac{2C}{\zeta \left( 1-\zeta \right) }$, for all $|\zeta |\le r$ with $\zeta \not =0$. But since $\varphi$ is analytic in $\overline{D_{r}}$, we necessarily have $C=0,$ which implies $\varphi ^{\prime }(\zeta )=0$ and $\varphi (\zeta )=const$ for all $\zeta \in \overline{D_{r}}.$

Therefore, there exist constant $0<M_{1}<M_{2}<\infty$ independent of $\eta$ such that

$$\begin{aligned} M_{1}\le \eta \left\| K_{\eta ,\alpha }^{\left( p\right) }\left( \varphi \right) -\varphi ^{\left( p\right) }\right\| _{r}\le M_{2}, \end{aligned}$$

which readly follows that

$$\begin{aligned} \frac{M_{1}}{\eta }\le \left\| K_{\eta ,\alpha }^{\left( p\right) }\left( \varphi \right) -\varphi ^{\left( p\right) }\right\| _{r}\le \frac{M_{2} }{\eta } \end{aligned}$$

in $\overline{{\mathbb {D}}_{r}}.$ This completes the proof. $\square$