On a New Generalization of Bernstein-Type Rational Functions and Its Approximation

Abstract

In this study, we introduce a new generalization of a Bernstein-type rational function possessing better estimates than the classical Bernstein-type rational function. We investigate its error of approximation globally and locally in terms of the first and second modulus of continuity and a class of Lipschitz-type functions. We present graphical comparisons of its approximation with illustrative examples.

1. Introduction

Bernstein polynomials [1] are defined to prove the well-known convergence theorem of Weierstreiss for each real-valued function f defined on $\left[0,1\right]$ by

$$B_n\left(f;x\right)=\sum _{k=0}^{n}f\left(k/n\right)\left(\genfrac{}{}{0pt}{}{n}{k}\right)x^k{\left(1-x\right)}^{n-k},n=1,2,....$$

(1)

In 1975, Balázs [2] defined an operator for each real-valued function f defined on $\left[0,\infty \right)$ and appropriately chosen real sequences $\left(a_n\right)$ and $\left(b_n\right)$ such that $a_n=\frac{b_n}{n}$ by

$$R_n\left(f;x\right)=\frac{1}{{\left(1+a_{n}x\right)}^n}\sum _{k=0}^{n}f\left(\frac{k}{b_n}\right)\left(\genfrac{}{}{0pt}{}{n}{k}\right){\left(a_{n}x\right)}^k,n=1,2,....$$

(2)

When $b_n=n$, this operator possesses the following relation with a Bernstein polynomial:

$$R_n\left(f;x\right)=B_n\left(f;\frac{a_{n}x}{1+a_{n}x}\right),$$

which is known as a Bernstein-type rational function. Balàzs estimated its rate of convergence for each continuous function f defined on $\left[0,\infty \right)$ and proved an asymptotic approximation theorem under the condition that $f\left(x\right)=O\left(e^{\tau x}\right),$$x\to \infty$ for some real number $\tau$. In [3], Balàzs and Szabados improved the estimates given in [2] under more restrictive conditions by choosing $a_n=n^{\zeta -1}$ and $b_n=n^{\zeta }$ for $0<\zeta \le \frac{2}{3}$, $n=1,2,...$ by assumming that f is uniformly continuous on $\left[0,\infty \right)$. Additionally, in [4], Balázs presented approximation results for Balázs–Szabados operators on all real axes. Totik investigated in [5] saturation properties of Balázs–Szabados operators, and Abel and Veccia [6] obtained Voronovskaja type asiymtotic results for Balázs–Szabados operators. In [7], Holhoş have presented new approximation results for Balázs–Szabados operators by means of super-exponential functions.

In [8], İspir and Atakut gave a generalization of Bernstein-type rational functions as follows:

$$G_n\left(f;x\right)=\frac{1}{\varphi \left(a_{n}x\right)}\sum _{k=0}^{\infty }\frac{{\varphi }^{\left(k\right)}\left(0\right)}{k!}{\left(a_{n}x\right)}^{k}f\left(\frac{k}{b_n}\right),$$

where $a_n$ and $b_n$ are suitably chosen positive numbers, and ${\varphi }_n$ is a sequence of functions satisfying certain conditions. Recently, Agratini [9] has studied a class of Bernstein-type rational functions by choosing a strictly decreasing positive real sequence $\left({\lambda }_n\right)$ such that ${lim}_{n\to \infty }{\lambda }_n=0$ as follows:

$$L_n\left(f;x\right)=\frac{1}{{\left(1+{\lambda }_{n}x\right)}^n}\sum _{k=0}^{n}f\left(\frac{k}{n{\lambda }_n}\right)\left(\genfrac{}{}{0pt}{}{n}{k}\right){\left({\lambda }_{n}x\right)}^k,n=1,2,...,$$

(3)

where f is continuous on $\left[0,\infty \right)$ satisfying a certain growing condition. Agratini has investigated both a local and global estimation of rate of convergence and has presented a weighted approximation result by using weighted modulus of continuity. Researchers can also find approximation results of some other Bernstein-type rational functions in those references [10,11,12,13,14,15,16,17,18,19,20].

Denoted by $C_B\left(\left[0,\infty \right)\right)$ is the Banach space of all real-valued continuous and bounded functions on $\left[0,\infty \right)$ endowed with the sup-norm ${∥f∥}_{\infty }={sup}_{x\in \left[0,\infty \right)}\left|f\left(x\right)\right|.$

For a compact subinterval $\left[a,b\right]\subset \left[0,\infty \right)$, the same norm is valid and reduced to ${∥f∥}_{\left[a,b\right]}={sup}_{x\in \left[a,b\right]}\left|f\left(x\right)\right|.$

In this paper, we construct a new generalization of Bernstein-type rational function, which is reducible to (2) and (3), and it is a rational function associated with the Bernstein polynomial given in (1). In an effort to define a well-defined Bernstein-type rational function, we choose non-negative real sequences $\left({\alpha }_n\right),\left({\beta }_n\right)$ and $\left({\gamma }_n\right)$ such that ${\gamma }_n=n{\alpha }_n$ satisfying the property

$$\underset{n\to \infty }{lim}{\alpha }_n=0,\underset{n\to \infty }{lim}{\beta }_n=1\mathrm{and}\underset{n\to \infty }{lim}{\gamma }_n=\infty .$$

(4)

We consider a newly defined Bernstein-type rational function as follows:

$$R_n^G\left(f;x\right)=\sum _{k=0}^{n}f\left(\frac{k}{{\gamma }_n}\right)\left(\genfrac{}{}{0pt}{}{n}{k}\right)\frac{{\left({\alpha }_{n}x\right)}^k{\left({\beta }_n\right)}^{n-k}}{{\left({\beta }_n+{\alpha }_{n}x\right)}^n},x\ge 0,n\in \mathbb{N},$$

(5)

where f is a real-valued continuous function on $\left[0,\infty \right)$, $\left({\alpha }_n\right),\left({\beta }_n\right)$ and $\left({\gamma }_n\right)$ are real sequences such that ${\gamma }_n=n{\alpha }_n$ satisfies the property (4). It is clear that $R_n^G$ is a well-defined, linear and positive operator. When ${\beta }_n=1$, ${\alpha }_n=a_n$ and ${\gamma }_n=b_n$, under the condition that $f\left(x\right)=O\left(e^{\tau x}\right),x\to \infty$ for some real number $\tau$, it is reduced to the Bernstein-type rational functions given by (2). When ${\alpha }_n:={\lambda }_n$ is a strictly decreasing positive real sequence, and f is continuous on $\left[0,\infty \right)$ satisfying a certain growing condition, it is induced to Agratini’s modification given by (3). Additionally, since it has the following connection with the Bernstein polynomial given by (1):

$$R_n^G\left(f;x\right)=B_n\left(f;\frac{{\alpha }_{n}x}{{\beta }_n+{\alpha }_{n}x}\right),$$

when ${\gamma }_n=n$ for ${\beta }_n=1$, it can be called a generalized Bernstein-type rational function.

2. Approximation Results

Firstly, we present the following auxiliary result, which will be used throughout the paper:

Lemma 1.

We have the following values of the generalized Bernstein-type rational function at monomials:

$$\begin{array}{ccc}\hfill R_n^G(e_0;x)& =& 1,\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill R_n^G(e_1;x)& =& \frac{x}{{}_{{\beta }_n+{\alpha }_{n}x}},\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill R_n^G(e_2;x)& =& \frac{\left(1-\frac{1}{n}\right)x^2}{{\left({\beta }_n+{\alpha }_{n}x\right)}^2}+\frac{x}{{\gamma }_n\left({\beta }_n+{\alpha }_{n}x\right)},\hfill \end{array}$$

(8)

where $e_i\left(t\right)=t^i$ for $i=0,1,2$, $\left({\alpha }_n\right)$, $\left({\beta }_n\right)$ and $\left({\gamma }_n\right)$ are real sequences such that ${\gamma }_n=n{\alpha }_n$.

Proof. 

By considering

$${\left({\beta }_n+{\alpha }_{n}x\right)}^n=\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right){\left({\alpha }_{n}x\right)}^k{\left({\beta }_n\right)}^{n-k},$$

(9)

We calculate that

$$\begin{array}{ccc}\hfill R_n^G(e_0;x)& =& R_n^G(1;x)=\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)\frac{{\left({\alpha }_{n}x\right)}^k{\left({\beta }_n\right)}^{n-k}}{{\left({\beta }_n+{\alpha }_{n}x\right)}^n}=1,\hfill \\ \hfill R_n^G(e_1;x)& =& R_n^G(t;x)=\sum _{k=0}^n\frac{k}{{\gamma }_n}\left(\genfrac{}{}{0pt}{}{n}{k}\right)\frac{{\left({\alpha }_{n}x\right)}^k{\left({\beta }_n\right)}^{n-k}}{{\left({\beta }_n+{\alpha }_{n}x\right)}^n}\hfill \\ & =& \frac{n{\alpha }_{n}x}{{\gamma }_n\left({\beta }_n+{\alpha }_{n}x\right)}\sum _{k=0}^{n-1}\left(\genfrac{}{}{0pt}{}{n-1}{k}\right){\left(\frac{{\alpha }_{n}x}{{\beta }_n+{\alpha }_{n}x}\right)}^k{\left(\frac{{\beta }_n}{{\beta }_n+{\alpha }_{n}x}\right)}^{n-1-k}\hfill \\ & =& \frac{n{\alpha }_{n}x}{{\gamma }_n\left({\beta }_n+{\alpha }_{n}x\right)}.1\hfill \\ & =& \frac{x}{{\beta }_n+{\alpha }_{n}x},\hfill \\ \hfill R_n^G(e_2;x)& =& R_n^G(t^2;x)=\sum _{k=0}^n\frac{k^2}{{\gamma }_n^2}\left(\genfrac{}{}{0pt}{}{n}{k}\right)\frac{{\left({\alpha }_{n}x\right)}^k{\left({\beta }_n\right)}^{n-k}}{{\left({\beta }_n+{\alpha }_{n}x\right)}^n}\hfill \\ & =& \frac{n(n-1){\alpha }_n^2{x}^2}{{\gamma }_n^2{\left({\beta }_n+{\alpha }_{n}x\right)}^2}\sum _{k=0}^{n-2}\left(\genfrac{}{}{0pt}{}{n-2}{k}\right)\frac{{\left({\alpha }_{n}x\right)}^k{\left({\beta }_n\right)}^{n-2-k}}{{\left({\beta }_n+{\alpha }_{n}x\right)}^{n-2}}\hfill \\ & & +\frac{n{\alpha }_{n}x}{{\gamma }_n^2\left({\beta }_n+{\alpha }_{n}x\right)}\sum _{k=0}^{n-1}\left(\genfrac{}{}{0pt}{}{n-1}{k}\right)\frac{{\left({\alpha }_{n}x\right)}^k{\left({\beta }_n\right)}^{n-1-k}}{{\left({\beta }_n+{\alpha }_{n}x\right)}^{n-1}}\hfill \\ & =& \frac{n(n-1){\alpha }_n^2{x}^2}{{\gamma }_n^2{\left({\beta }_n+{\alpha }_{n}x\right)}^2}+\frac{n{\alpha }_{n}x}{{\gamma }_n^2\left({\beta }_n+{\alpha }_{n}x\right)}\hfill \\ & =& \frac{\left(1-\frac{1}{n}\right)x^2}{{\left({\beta }_n+{\alpha }_{n}x\right)}^2}+\frac{x}{{\gamma }_n\left({\beta }_n+{\alpha }_{n}x\right)}.\hfill \end{array}$$

□

Remark 1.

We have the following first- and second-order central moments by considering Lemma 1:

$$R_n^G(e_1-x;x)=\frac{\left(1-{\beta }_n\right)x}{{\beta }_n+{\alpha }_{n}x}-\frac{{\alpha }_n{x}^2}{{\beta }_n+{\alpha }_{n}x},$$

(10)

$$\begin{array}{ccc}\hfill R_n^G({\left(e_1-x\right)}^2;x)& =& \frac{{\beta }_{n}x}{{\gamma }_n{\left({\beta }_n+{\alpha }_{n}x\right)}^2}+\frac{\left({\alpha }_n+{\left({\beta }_n-1\right)}^2-\frac{1}{n}\right)x^2}{{\left({\beta }_n+{\alpha }_{n}x\right)}^2}\hfill \\ & & +\frac{2{\alpha }_n\left({\beta }_n-1\right)x^3}{{\left({\beta }_n+{\alpha }_{n}x\right)}^2}+\frac{{\alpha }_n^2{x}^4}{{\left({\beta }_n+{\alpha }_{n}x\right)}^2}.\hfill \end{array}$$

(11)

Theorem 1.

Let $R_n^G$, $n\in \mathbb{N}$, be the generalized Bernstein-type rational function defined by (5). If $\left({\alpha }_n\right)$, $\left({\beta }_n\right)$ and $\left({\gamma }_n\right)$ are non-negative real sequences satisfying (4) for each $n\in \mathbb{N}$, then $R_n^G(f;x)$ converges to $f\left(x\right)$ uniformly with respect to x on $\left[0,r\right]\subset \left[0,\infty \right),$$r>0,$ for each $f\in C\left(\left[0,r\right]\right).$

Proof. 

The proof can be fulfilled easily from the well-known Bohman–Korovkin theorem [21]. From (6) of Lemma 1, it is clear that

$$\underset{n\to \infty }{lim}{∥R_n^G\left(e_0;.\right)-e_0(.)∥}_{\left[0,r\right]}=0.$$

(12)

Since $\frac{1}{{\beta }_n+{\alpha }_{n}x}\le \frac{1}{{\beta }_n}$ for each $x\in \left[0,r\right]$, by Remark 1, we can write

$$\begin{array}{ccc}\hfill \left|R_n^G\left(e_1;x\right)-e_1\left(x\right)\right|& =& \left|R_n^G\left(e_1-x;x\right)\right|=\left|\frac{\left(1-{\beta }_n\right)x}{{\beta }_n+{\alpha }_{n}x}-\frac{{\alpha }_n{x}^2}{{\beta }_n+{\alpha }_{n}x}\right|\hfill \\ & \le & \frac{\left|1-{\beta }_n\right|r}{{\beta }_n}+\frac{{\alpha }_n{r}^2}{{\beta }_n}:={\mu }_n^1.\hfill \end{array}$$

(13)

By considering relation (4), in (13), since ${lim}_{n\to \infty }{\mu }_n^1=0,$ we obtain

$$\underset{n\to \infty }{lim}{∥R_n^G\left(e_1;.\right)-e_1\left(.\right)∥}_{\left[0,r\right]}=0.$$

(14)

By considering () of Lemma 1, we can calculate that

$$\begin{array}{ccc}\hfill \left|R_n^G\left(e_2;x\right)-e_2\left(x\right)\right|& =& \left|\frac{x}{{\gamma }_n\left({\beta }_n+{\alpha }_{n}x\right)}+\frac{\left(1-\frac{1}{n}\right)x^2}{{\left({\beta }_n+{\alpha }_{n}x\right)}^2}-x^2\right|\hfill \\ & \le & \frac{x}{{\gamma }_n\left({\beta }_n+{\alpha }_{n}x\right)}+\frac{\left|1-\frac{1}{n}-{\beta }_n^2\right|x^2}{{\left({\beta }_n+{\alpha }_{n}x\right)}^2}\hfill \\ & & +\frac{2{\beta }_n{\alpha }_n{x}^3}{{\left({\beta }_n+{\alpha }_{n}x\right)}^2}+\frac{{\alpha }_n^2{x}^4}{{\left({\beta }_n+{\alpha }_{n}x\right)}^2}\hfill \\ & \le & \frac{r}{{\gamma }_n{\beta }_n}+\frac{\left|1-\frac{1}{n}-{\beta }_n^2\right|r^2}{{\beta }_n^2}+\frac{2{\alpha }_n{r}^3}{{\beta }_n}+\frac{{\alpha }_n^2{r}^4}{{\beta }_n^2}:={\mu }_n^2.\hfill \end{array}$$

(15)

Under conditions of relation (4), from (15), since ${lim}_{n\to \infty }{\mu }_n^2=0,$ we get

$$\underset{n\to \infty }{lim}{∥R_n^G\left(e_2;.\right)-e_2\left(.\right)∥}_{\left[0,r\right]}=0.$$

(16)

From relations (12), (14) and (16), the criterion of the Bohman–Korovkin theorem is satisfied. Therefore, the proof of the theorem is completed. □

3. Local and Global Approximation

In this part, we present local and global results of approximation with the help of the first and second modulus of continuity and a Lipschitz class of functions.

For any $\mu >0$, modulus of continuity of $f\in C_B\left(\left[0,\infty \right)\right)$ is defined as

$$\omega \left(f;\mu \right)=\underset{0<h<\mu }{sup}\underset{x\in \left[0,\infty \right)}{sup}\left|f\left(x+h\right)-f\left(x\right)\right|,$$

(17)

which possesses the following property:

$$\omega \left(f;\kappa \mu \right)\le \left(\kappa +1\right)\omega \left(f;\mu \right),$$

(18)

for $\kappa ,\mu >0$, and ${lim}_{\mu \to 0^{+}}\omega \left(f;\mu \right)=0$, when f is uniformly continuous [22].

Theorem 2.

Let $\left({\alpha }_n\right),\left({\beta }_n\right)$ and $\left({\gamma }_n\right)$ be real sequences such that ${\gamma }_n=n{\alpha }_n$, satisfying the property (4). For any $f\in C_B\left(\left[0,\infty \right)\right)$, we have

$$\left|R_n^G\left(f;x\right)-f\left(x\right)\right|\le 2\omega \left(f;\sqrt{{\mu }_n^x}\right),$$

where

$$\begin{array}{ccc}\hfill {\mu }_n^x& :& =\frac{{\beta }_{n}x}{{\gamma }_n{\left({\beta }_n+{\alpha }_{n}x\right)}^2}+\frac{\left({\alpha }_n+{\left({\beta }_n-1\right)}^2-\frac{1}{n}\right)x^2}{{\left({\beta }_n+{\alpha }_{n}x\right)}^2}\hfill \\ & & +\frac{2{\alpha }_n\left({\beta }_n-1\right)x^3}{{\left({\beta }_n+{\alpha }_{n}x\right)}^2}+\frac{{\alpha }_n^2{x}^4}{{\left({\beta }_n+{\alpha }_{n}x\right)}^2}.\hfill \end{array}$$

(19)

Proof. 

Let $f\in C_B\left(\left[0,\infty \right)\right)$.By (18), we have

$$\left|f\left(t\right)-f\left(x\right)\right|\le \left(1+\frac{\left|t-x\right|}{\mu }\right)\omega \left(f;\mu \right).$$

(20)

By applying the operator $R_n^G$ to (20), by taking linearity and positivity of the operator $R_n^G$ into account and by applying Cauchy–Schwarz inequality, we obtain

$$\begin{array}{ccc}\hfill \left|R_n^G\left(f;x\right)-f\left(x\right)\right|& \le & R_n^G\left(\left|f\left(t\right)-f\left(x\right)\right|;x\right)\hfill \\ & \le & \omega \left(f;\mu \right)\left(1+\frac{1}{\mu }{R}_n^G\left(\left|e_1-x\right|;x\right)\right)\hfill \\ & \le & \omega \left(f;\mu \right)\left(1+\frac{1}{\mu }\sqrt{R_n^G\left({\left(e_1-x\right)}^2;x\right)}\right).\hfill \end{array}$$

(21)

From (11) of Remark 1, by choosing

$$\begin{array}{ccc}\hfill {\mu }_n^x& :& =R_n^G\left({\left(e_1-x\right)}^2;x\right)=\frac{{\beta }_{n}x}{{\gamma }_n{\left({\beta }_n+{\alpha }_{n}x\right)}^2}+\frac{\left({\alpha }_n+{\left({\beta }_n-1\right)}^2-1\right)x^2}{{\left({\beta }_n+{\alpha }_{n}x\right)}^2}\hfill \\ & & +\frac{2{\alpha }_n\left({\beta }_n-1\right)x^3}{{\left({\beta }_n+{\alpha }_{n}x\right)}^2}+\frac{{\alpha }_n^2{x}^4}{{\left({\beta }_n+{\alpha }_{n}x\right)}^2},\hfill \end{array}$$

and by replacing $\mu :=\sqrt{{\mu }_n^x}$, we complete the proof of the theorem. □

Remark 2.

In Theorem 2, ${\mu }_n^x$ is dependent on x and choosing of $\left({\alpha }_n\right),\left({\beta }_n\right)$ and $\left({\gamma }_n\right)$. $\left({\alpha }_n\right),\left({\beta }_n\right)$ and $\left({\gamma }_n\right)$ must be non-negative real sequences satisfying ${\mu }_n^x\ge 0$. Otherwise, Theorem 2 becomes invalid. For example, if ${\beta }_n\ge 1$ and ${\alpha }_n+{\left({\beta }_n-1\right)}^2\ge \frac{1}{n}$ then ${\mu }_n^x\ge 0$. This is not the only possible condition as ${\mu }_n^x\ge 0$.

Moreover, for $f\in C\left(\left[0,r\right]\right),$ Theorem 2 is reduced to the following inequality:

$${∥R_n^G\left(f;.\right)-f∥}_{\left[0,r\right]}\le 2\omega \left(f;\sqrt{{\mu }_n}\right),$$

where

$$\begin{array}{ccc}\hfill {\mu }_n& :& =\frac{r}{{\gamma }_n{\beta }_n}+\frac{\left({\alpha }_n+{\left({\beta }_n-1\right)}^2-\frac{1}{n}\right)r^2}{{\beta }_n^2}\hfill \\ & & +\frac{2{\alpha }_n\left({\beta }_n-1\right)r^3}{{\beta }_n^2}+\frac{{\alpha }_n^2{r}^4}{{\beta }_n^2}.\hfill \end{array}$$

(22)

For $f\in C_B\left(\left[0,\infty \right)\right)$, Petree’s K-functional is defined by

$$K_2\left(f;\mu \right)=\underset{g\in C_B^{\left(2\right)}\left(\left[0,\infty \right)\right)}{inf}\left\{{∥f-g∥}_{\infty }+\mu {∥g^{\prime \prime }∥}_{\infty }\right\},$$

(23)

where

$$C_B^{\left(2\right)}\left(\left[0,\infty \right)\right):=\left\{g\in C_B\left(\left[0,\infty \right)\right):g^{\prime },g^{\prime \prime }\in C_B\left(\left[0,\infty \right)\right)\right\}.$$

We have the following connection (see p. 192 in [23]) between Petree’s K-functional and the second modulus of continuity ${\omega }_2\left(f;.\right)$

$$K_2\left(f;\mu \right)\le C{\omega }_2\left(f;\sqrt{\mu }\right),$$

(24)

where

$${\omega }_2\left(f;\sqrt{\mu }\right)=\underset{0<h<\sqrt{\mu }}{sup}\underset{x\in \left[0,\infty \right)}{sup}\left|f\left(x+2h\right)-2f\left(x+h\right)+f\left(x\right)\right|.$$

(25)

Theorem 3.

Let $\left({\alpha }_n\right),\left({\beta }_n\right)$ and $\left({\gamma }_n\right)$ be real sequences such that ${\gamma }_n=n{\alpha }_n$ satisfying property (4). For each $f\in C_B\left(\left[0,\infty \right)\right)$, then there exists a $C>0$ such that

$$\left|R_n^G\left(f;x\right)-f\left(x\right)\right|\le C\left\{{\omega }_2\left(f;\sqrt{{\mu }_n^x}\right)+\sqrt{{\mu }_n^x}\right\},x\ge 0,$$

where ${\mu }_n^x$ is given as in (19).

Proof. 

We initially define an auxilary operator by

$$E_n^G\left(f;x\right):=R_n^G\left(f;x\right)+f\left(x\right)-f\left({\eta }_n^x\right),$$

(26)

where

$${\eta }_n^x:=R_n^G\left(e_1-x;x\right)=\frac{\left(1-{\beta }_n\right)x-{\alpha }_n{x}^2}{{\beta }_n+{\alpha }_{n}x}.$$

(27)

By (26), we obtain

$$E_n^G\left(e_1-x;x\right)=R_n^G\left(e_1-x;x\right)-x-{\eta }_n^x=0.$$

(28)

For any $g\in C_B^{\left(2\right)}\left(\left[0,\infty \right)\right)$, from Taylor’s formula, we can write

$$g\left(t\right)-g\left(x\right)=g^{\prime }\left(x\right)\left(t-x\right)+{\int }_x^t\left(t-u\right)g^{\prime \prime }\left(u\right)du.$$

(29)

By applying the operator $R_n^G$ to (29), by (28), we obtain

$$\begin{array}{ccc}\hfill \left|R_n^G\left(g;x\right)-g\left(x\right)\right|& \le & \left|g^{\prime }\left(x\right)\right|\left|R_n^G\left(t-x;x\right)\right|+\left|R_n^G\left({\int }_x^t\left(t-u\right)g^{\prime \prime }\left(u\right)du;x\right)\right|\hfill \\ & \le & {∥g^{\prime }∥}_{\infty }\left|R_n^G\left(t-x;x\right)\right|+{∥g^{\prime \prime }∥}_{\infty }{R}_n^G\left({\left(t-x\right)}^2;x\right).\hfill \end{array}$$

(30)

Since $g^{\prime }\in C_B\left(\left[0,\infty \right)\right)$, there exists a $k_0>0$ such that ${∥g^{\prime }∥}_{\infty }=k_0.$ Therefore, by applying Cauchy–Schwarz inequality, we obtain

$$\left|R_n^G\left(g;x\right)-g\left(x\right)\right|\le k_0\sqrt{R_n^G\left({\left(t-x\right)}^2;x\right)}+{∥g^{\prime \prime }∥}_{\infty }{R}_n^G\left({\left(t-x\right)}^2;x\right).$$

(31)

Additionally, for $f\in C_B\left(\left[0,\infty \right)\right)$, we have

$$\begin{array}{ccc}\hfill \left|R_n^G\left(f;x\right)\right|& \le & \sum _{k=0}^n\left|f\left(\frac{k}{{\gamma }_n}\right)\right|\left(\genfrac{}{}{0pt}{}{n}{k}\right)\frac{{\left({\alpha }_{n}x\right)}^k{\beta }_n^{n-k}}{{\left({\beta }_n+{\alpha }_{n}x\right)}^n}\hfill \\ & \le & {∥f∥}_{\infty }{R}_n^G\left(e_0;x\right)={∥f∥}_{\infty }.\hfill \end{array}$$

(32)

By considering (30) and (32), we can write

$$\begin{array}{ccc}\hfill \left|R_n^G\left(f;x\right)-f\left(x\right)\right|& \le & \left|R_n^G\left(\left(f-g\right);x\right)-\left(f-g\right)\left(x\right)\right|+\left|R_n^G\left(g;x\right)-g\left(x\right)\right|\hfill \\ & \le & R_n^G\left(\left|f-g\right|;x\right)+\left|\left(f-g\right)\left(x\right)\right|+\left|R_n^G\left(g;x\right)-g\left(x\right)\right|\hfill \\ & \le & 2{∥f-g∥}_{\infty }+{∥g^{\prime \prime }∥}_{\infty }{R}_n^G\left({\left(e_1-x\right)}^2;x\right)\hfill \\ & & +k_0\sqrt{R_n^G\left({\left(e_1-x\right)}^2;x\right)}.\hfill \end{array}$$

(33)

In (33), considering Remark 1 and choosing ${\mu }_n^x$ as in (19), by taking the infimum right-hand side of the last inequality, for $g\in C_B^{\left(2\right)}\left(\left[0,\infty \right)\right),$ we obtain

$$\left|R_n^G\left(f;x\right)-f\left(x\right)\right|\le 2K\left(f;{\mu }_n^x\right)+k_0\sqrt{{\mu }_n^x}.$$

(34)

Lastly, by applying (24) to (34), we acquire

$$\left|R_n^G\left(f;x\right)-f\left(x\right)\right|\le c_0{\omega }_2\left(f;\sqrt{{\mu }_n^x}\right)+k_0\sqrt{{\mu }_n^x},$$

where $c_0>0$. By choosing $C=max\left\{k_0,c_0\right\}$, we obtain the desired result. □

Remark 3.

For $f\in C\left(\left[0,r\right]\right),$ Theorem 3 is induced to the following result:

$${∥R_n^G\left(f;.\right)-f∥}_{\left[0,r\right]}\le C\left\{{\omega }_2\left(f;\sqrt{{\mu }_n}\right)+\sqrt{{\mu }_n}\right\},C>0,$$

where ${\mu }_n$ is as in (22).

Let E be any subset of $\mathbb{R}$ and $\theta \in \left(0,1\right]$. Let $Li{p}_{M_f}\left(E,\theta \right)$ denote a class of Lipschitz functions in $C_B\left(\left[0,\infty \right)\right)$ satisfying

$$\left|f\left(t\right)-f\left(x\right)\right|\le M_f{\left|t-x\right|}^{\theta },t\in \overline{E},x\ge 0,$$

(35)

where $M_f$ is a constant, and $\overline{E}$ is the closure of E in $\left[0,\infty \right)$.

Theorem 4.

Let $\left({\alpha }_n\right),\left({\beta }_n\right)$ and $\left({\gamma }_n\right)$ be real sequences such that ${\gamma }_n=n{\alpha }_n$, satisfying the property (4). For any $f\in Li{p}_{M_f}\left(E,\theta \right)$, we have

$$\left|R_n^G\left(f;x\right)-f\left(x\right)\right|\le M_f\left\{{\left(\sqrt{{\mu }_n^x}\right)}^{\theta }+2{\left(d\left(x,E\right)\right)}^{\theta }\right\},$$

where ${\mu }_n^x$ is given as in (19), $M_f$ is a constant depending on f, and E is any subset of $\left[0,\infty \right)$, $x\in \left[0,\infty \right)$ and $\theta \in \left(0,1\right]$.

Proof. 

Let $x\in \left[0,\infty \right)$ and $x\in \overline{E}$ such that $d\left(x,x_0\right)=\left|x-x_0\right|$. We can write

$$\left|f-f\left(x\right)\right|\le \left|f-f\left(x_0\right)\right|+\left|f\left(x_0\right)-f\left(x\right)\right|.$$

(36)

By applying $R_n^G$ to (36), and by considering the linearity and positivity of $R_n^G$ and (35), we obtain

$$\begin{array}{ccc}\hfill \left|R_n^G\left(f;x\right)-f\left(x\right)\right|& \le & R_n^G\left(\left|f-f\left(x_0\right)\right|;x\right)+R_n^G\left(\left|f\left(x\right)-f\left(x_0\right)\right|;x\right)\hfill \\ & \le & M_f\left\{R_n^G\left({\left|e_1-x_0\right|}^{\theta }{e}_0;x\right)+R_n^G\left({\left|x-x_0\right|}^{\theta }{e}_0;x\right)\right\}\hfill \\ & =& M_f\left\{R_n^G\left({\left|e_1-x_0\right|}^{\theta }{e}_0;x\right)+{\left|x-x_0\right|}^{\theta }{R}_n^G\left(e_0;x\right)\right\}.\hfill \end{array}$$

(37)

In (37), by using Hölder’s inequality for $p=\frac{2}{\theta }$ and $q=\frac{2}{2-\theta }$ such that $\frac{1}{p}+\frac{1}{q}=1$, by considering Lemma 1 and (11) of Remark 1, we obtain

$$\begin{array}{ccc}\hfill \left|R_n^G\left(f;x\right)-f\left(x\right)\right|& \le & M_f\left\{{\left(R_n^G\left({\left|e_1-x\right|}^{\theta p};x\right)\right)}^{1/p}{\left(R_n^G\left({\left(e_0\right)}^q;x\right)\right)}^{1/q}+2{\left(d\left(x,E\right)\right)}^{\theta }\right\}\hfill \\ & =& M_f\left\{{\left(R_n^G\left({\left(e_1-x\right)}^2;x\right)\right)}^{\theta /2}{\left(1\right)}^{1/q}+2{\left(d\left(x,E\right)\right)}^{\theta }\right\}\hfill \\ & =& M_f\left\{{\left(\sqrt{{\mu }_n^x}\right)}^{\theta }+2{\left(d\left(x,E\right)\right)}^{\theta }\right\},\hfill \end{array}$$

which completes the proof of theorem. □

Remark 4.

When $x\in \left[0,r\right]:=E\subset \left[0,\infty \right)$, it is clear that $d\left(x,E\right)=0$. From Theorem 4, we have the following inequality:

$${∥R_n^G\left(f;.\right)-f∥}_{\left[0,r\right]}\le M_f{\left(\sqrt{{\mu }_n}\right)}^{\theta },$$

where ${\mu }_n$ is as in (22).

4. Graphical Comparison

In this part, we present some graphical results produced in Maple software.

Example 1.

Let us choose $f\left(x\right)=x\left(x+\frac{1}{2}\right)\left(x+\frac{1}{3}\right)$, ${\alpha }_n=\frac{1}{\sqrt{n}}$, ${\gamma }_n=\sqrt{n}$ for $x\ge 0$ and $n\in \mathbb{N}$.

In Figure 1 and Figure 2, by choosing ${\beta }_n=1-\frac{1}{n}$, graphical comparison of approximation of $R_n^G\left(f;x\right)$ to f for $n=50,75$ and 100 is presented on $\left[0,\infty \right)$ and $\left[0,3\right].$ It is clear that approximation of $R_n^G\left(f;x\right)$ to f is better for increasing the value of n.

Figure 1. Approximation by $R_n^G$ on $\left[0,\infty \right)$, for ${\beta }_n=1-\frac{1}{n}$, $n=50,75$ and 100.

Figure 2. Approximation by $R_n^G$ on $\left[0,3\right]$, for ${\beta }_n=1-\frac{1}{n}$, $n=50,75$ and 100.

In Figure 3 and Figure 4, by choosing ${\beta }_n=1-\frac{1}{n}$, $R_n^G\left(f;x\right)$ is compared graphically with the classical Bernstein-type rational function $R_n\left(f;x\right)$ given by (2) to f for $n=25$ on $\left[0,\infty \right)$ and $\left[0,3\right].$ It is obvious that approximation of $R_n^G\left(f;x\right)$ to f is better than approximation by $R_n\left(f;x\right)$ to f on $\left[0,\infty \right)$ and $\left[0,3\right].$

Figure 3. Comparison by $R_n^G$ and $R_n$ on $\left[0,\infty \right)$, for ${\beta }_n=1-\frac{1}{n}$, $n=25$.

Figure 4. Comparison of $R_n^G$ and $R_n$ on $\left[0,3\right]$, for ${\beta }_n=1-\frac{1}{n}$, $n=25$.

In Figure 5 and Figure 6, by denoting $R_n^G\left(f;x\right):=R_n^G\left(f;x,{\beta }_n\right)$ and choosing ${\beta }_n=1-\frac{2}{n},1$ and $1+\frac{2}{n},$$R_n^G\left(f;x,1-\frac{2}{n}\right),$$R_n^G\left(f;x,1\right)$ and $R_n^G\left(f;x,1+\frac{2}{n}\right)$ are graphically compared. Here, it is clear that $R_n^G\left(f;x,1\right)$ is reduced to $R_n\left(f;x\right)$ for ${\beta }_n=1.$ If we choose ${\beta }_n$ as a real sequence such that ${lim}_{n\to \infty }{\beta }_n=1$, and ${\beta }_n$ is not constant, then we see that approximation by $R_n^G\left(f;x,1-\frac{2}{n}\right)$ is better than $R_n^G\left(f;x,1\right)=R_n\left(f;x\right)$ and $R_n^G\left(f;x,1+\frac{2}{n}\right)$ on $\left[0,\infty \right)$ and $\left[0,3\right].$

Figure 5. Approximation by $R_n^G$ on $\left[0,\infty \right)$, for ${\beta }_n=1-\frac{2}{n},1$ and $1+\frac{2}{n}$, $n=25$.

Figure 6. Approximation by $R_n^G$ on $\left[0,3\right]$, for ${\beta }_n=1-\frac{2}{n},1$ and $1+\frac{2}{n}$, $n=25$.

5. Conclusions

In this study, we have introduced a newly defined Bernstein-type rational function $R_n^G$, which is a generalized Bernstein-type rational function in terms of including the classical Bernstein-type rational function defined by (2) and Agratini’s modification, defined by (3). We have estimated the error of its approximation for conveniently chosen non-negative real sequences $\left({\alpha }_n\right)$ and $\left({\beta }_n\right)$. Consequently, the newly defined generalized Bernstein-type rational function possesses better results than the classical Bernstein-type rational function defined by (2) for certain functions.