Relatively equi-statistical convergence via deferred Nörlund mean based on difference operator of fractional-order and related approximation theorems

1.   Introduction and preliminaries

The rapid growth of sequence spaces has been accompanied by the recent works of many researchers in the field of statistical convergence and it has got tremendous importance over conventional convergence. The basic idea of statistical convergence was initially studied in the year 1951 by Fast ^[18] and Steinhaus ^[52] independently. Recently, the approximation of functions by positive linear operators based on statistical convergence and statistical summability has become a dynamic area of research. Moreover, very recently the introduction of equi-statistical convergence, point-wise statistical convergence and uniform statistical convergence of sequences of functions by Srivastava et al. (see ^[45,51]) has enriched the scope of current research in this direction. As of late, statistical convergence has been very wide and fruitful field for the application in several branches of Analysis; in particular, Approximation Theory, Fourier Analysis, Operator Theory, Theory of Orthogonal series, Number Theory and so on. For more various approaches to statistical convergence and statistical summability, we refer to the current works ^{[22,24,25,30,31,40,44]}.

Let $K\subseteq \mathbb{N}$ and

The asymptotic (or natural) density (see ^[18,52]) of $K$ is given by

presuming that the limit exists, where $|K_{n}|$ be the cardinality of $K_{n}$. A given sequence $(x_{n})$ is statistically convergent to $\ell$ if, for every $\epsilon > 0$,

has asymptotic (or natural) density zero. That is, for every $\epsilon > 0$,

Here, we write

Now, in order to justify the importance of statistical convergence over classical convergence, we present the following example.

Example 1.1. Let $x = (x_{n})$ be a sequence defined by

Here, the sequence $(x_{n})$ is statistically convergent to $0$, even if it is not classically convergent.

The preliminary idea of weighted statistical convergence was introduced in 2009 by Karakaya and Chishti ^[29]. Recently, the concept was modified by Mursaleen et al. (see ^[40]). Suppose that $(s_{k})$ be a sequence of non-negative numbers and such that

Set

then the given sequence $(x_{n})$ is weighted statistically convergent $(\text{or}\quad \text{stat}_{\bar{N}}$-convergent) to a number $\ell$ if, for every $\epsilon > 0$,

has weighted density zero (see ^[40]). That is, for every $\epsilon > 0$,

Here, we write

In the year 2013, Belen and Mohiuddine ^[10] established a new technique for weighted statistical convergence in terms of the de la Vallée Poussin mean and it was subsequently investigated further by Braha et al. ^[11] as the $\Lambda_{n}$-Weighted statistical convergence. Recently, Mohiuddine ^[34] presented another variant of weighted statistical convergence by taking into account non-negative weighted regular matrix. Very recently, a certain class of equi-statistical convergence via deferred Nörlund summability mean and related approximation theorem were studied by Srivastava et al. (see, for details, ^[45]).

Next, the notion of relatively uniform convergence of a sequence of functions was introduced by Moore ^[38]. Later, in the similar lines it was modified by Chittenden ^[13] and was defined over a closed interval, $I = [a, b]\subseteq\mathbb{R}$. We recall here, the definition of uniform convergence relative to a scale function as follows. A sequence of functions $(f_{n})$, defined over $[a, b]$, converges relatively uniformly to a limit function $f(x)$ if, there exists a scale function $\sigma(x)$ ($|\sigma(x)| > 0$) defined over $[a, b]$, such that for every $\epsilon > 0$, there exists an integer $n_{\epsilon}$ and for every $n > n_{\epsilon}$,

holds true uniformly for all $x \in [a, b]\subseteq\mathbb{R}$.

Now, to see the importance of relatively uniform convergence over classical uniform convergence, we present the following example (see ^[15,37]).

Example 1.2. For each $n\in\mathbb{N}$, define $f_{n}:[0, 1]\rightarrow\mathbb{R}$ by

The sequence $(f_{n})$ of functions is not classically uniformly convergent on $[0, 1]$; however, it is converging to $f = 0$ uniformly, relative to a scale function

on $[0, 1]$. Here, we write

Subsequently, uniform convergence can be viewed as a special case of uniform convergence relative to the non-zero scale function (that is, relatively uniform convergence). Recently, based on the natural density of a set, Demirci and Orhan ^[15] introduced the notion of relatively uniform statistical convergence of a sequence of functions. Later, they have also established the definitions of relative modular convergence and relative modular statistical convergence for double sequences of measurable real-valued functions in the same year (see ^[16]). Furthermore, very recently, Kadak at al. ^[28] studied the theory of relatively uniform weighted $\alpha\beta$-statistical convergence and relatively uniform statistical $\Phi$-summability based on linear difference operators of fractional-order.

Let $E\subset \mathbb{R}$ be compact and $(f_{n})$ be a sequence of functions defined on $E$. The sequence $(f_{n})$ is relatively uniform statistically converges to a limit function $f$ defined on $E$, if there exists a scale function $\sigma(x)$ $(|\sigma(x)| > 0)$ on $E$ such that, for each $\epsilon > 0$,

has zero natural density, that is, for every $\epsilon > 0$, we have

Here, we write

The notion of fractional-order difference operator was firstly introduced by Chapman ^[12] and later on, it has been used by many researchers. For more details, see the current works ^[8,9,26]. In the year 2016, some new classes of fractional-order difference sequence spaces were studied by Baliarsingh in ^[7]. Very recently, Kadak ^[25] has developed the concept of weighted statistical convergence based on generalized difference sequence including $(p, q)$-integers. Again, Kadak ^[24] extended it for $(p, q)$-gamma function.

Balcerzak et al. ^[6] introduced the concept of equi-statistical convergence in the year 2007 and later by Aktuǧlu and Gezer^[4] and Mohiuddine and Alamri ^[35] by using the idea of lacunary. Subsequently, it was extended to the $\lambda_{n}$-equi-statistical convergence and accordingly some associated approximation theorems were proved by Srivastava et al. ^[51]. Furthermore, based on the equi-statistical convergence via deferred Nörlund summability mean, Srivastava et al. ^[45] has also proved certain Korovkin type approximation theorems.

Motivated essentially by the above-mentioned works, here in this paper, we introduce the concept of relatively $\psi_{n}^{p, q}$-equi-statistical convergence, point-wise relatively $\psi_{n}^{p, q}$-statistical convergence and relatively uniform $\psi_{n}^{p, q}$-statistical convergence for a sequence of real valued functions and show that the relatively $\psi_{n}^{p, q}$-equi-statistical convergence lies between the point-wise relatively $\psi_{n}^{p, q}$-statistical convergence and relatively uniform $\psi_{n}^{p, q}$-statistical convergence. Moreover, as an application point of view, we have proved a Korovkin type approximation theorem by using the relatively deferred Nörlund equi-statistical convergence of difference sequences of functions including $(p, q)$-gamma function and exhibited that our theorem is a generalization of some well-established approximation theorems of Korovkin type which were presented in earlier works. Also, we estimate the rate of the relatively deferred Nörlund equi-statistical convergence involving a non-zero scale function. Furthermore, we use the modulus of continuity to estimate the rate of convergence of approximating linear positive operators. Finally, we set up various remarks and observations in connection with our results and definitions.

2.   Deferred Nörlund-type mean

We now recall and introduce some definitions which are required for the present investigation. Let $\omega$ be the set of all real valued sequences and suppose any subspace of $\omega$ be the sequence space. Suppose $\ell_{\infty}$ be the class of all bounded linear spaces and let $c$, $c_{0}$ be the respective classes for convergent and null sequences with real and complex terms. Let $(x_{k})$ be any sequence in $\omega$ and $h$ be a positive constant. For real numbers $a, b$ and $c$. Recently, a new class of difference sequence space of fractional order has been introduced by Baliarsingh ^[7] and is as follows:

where $(\alpha)_{k}$ denote the Pochhammer symbol or shifted factorial of the real number $\alpha$ and moreover, it is defined by using the familiar Euler's gamma function as

Note that the series defined in (2.1) converges for all $c > a+b$ (see ^[19]).

In the similar lines, he has also introduced the difference sequence space of fractional order for differentiable functions.

Let $h\rightarrow0$ and $f(x)$ be a differentiable function with fractional order. Associated with this function $f(x)$, define the sequence

and the sequence spaces $\Delta_{h, x}^{a, b, c}(f_{h}(x))$ via the difference operator $(\Delta_{h, x}^{a, b, c})$ as

Subsequently, with the development of $q$-calculus, various researchers worked on certain new generalizations of positive linear operators based on $q$-integers (see ^[32]). Recently, Mursaleen et al. ^[39] developed the $(p, q)$-analogue of Bernstein operators in connection with $(p, q)$-integers and later on, some approximation results for Baskakov operators and Bernstein-Schurer operators are studied for $(p, q)$-integers by ^[1,41].

We, next recall some definitions and basic notations on $(p, q)$-integers for our present study. For any $(n\in \mathbb{N})$ the $(p, q)$-integer $[n]_{p, q}$ is defined by,

where $0 < q < p\leqq1$. The $(p, q)$-factorial is defined by

The $(p, q)$-binomial coefficient is defined by

The formula for $(p, q)$-binomial expansion is as follows:

In the year 1738, Euler first studied the gamma function in order to extend the factorial function to the real and complex numbers. The gamma function $\Gamma(x)$ is defined by an improper integral

Integration by parts gives the important functional relation of the gamma function,

If $x$ is positive say $k$, then (2.3) gives

In the year 1904, Jackson ^[20] studied the notion of $q$-gamma function, as an extension of ordinary gamma function. It is defined by

where the symbol $(.; .)$ is called the infinite $q$-Pochhammer and also called the $q$-shifted factorial, it is defined by

If limit $q\rightarrow1$, $\Gamma_{q}(x)$ reduces to the general gamma function $\Gamma(x)$ and it also fulfils the property that

and for any positive integers $k$ such that $\Gamma_{q}(k) = [k-1]!_{q}$, where the symbol $[.]!_{q}$ is the $q$-factorial function.

Subsequently, the $(p, q)$-gamma function has been introduced as an extension of $q$-gamma function (see ^[43]). For $0 < q < p\leqq1$, the $(p, q)$-gamma function is defined by

where

and the symbol $(.; .)$ is called the $(p, q)$-Pochhammer or $(p, q)$-shifted factorial. Here if $p = 1$, then the $\Gamma_{p, q}(x)$ reduces to the $\Gamma_{q}(x)$.

Definition 2.1. Let the sequence $(f_{h}(x))$ in $\omega$, $0 < q < p\leqq1$ and $a, b, c$ be a non-negative integer, and let $h$ is positive constant. We define a generalized difference sequence under the difference operator

by

where $(s)_{p, q}^{i}$ is $(p, q)$-shifted factorial of non-negative integers $s$ which is being defined $(p, q)$-gamma function as

That is,

Throughout the paper, we assume that the series defined in (2.4) converges for all $c > a+b$ (see ^[19]).

Example 2.2. Let $f(x) = x, \; \; (\forall\; x\in\mathbb{N}\subset\mathbb{R})$, and we choose $a = 2$, $b = c$, we have

Clearly, depending on the choice of the values of $p$ and $q$, the difference sequence $\Delta_{1, p, q}^{2, b, b}(f(x))$ of third order has different limits. This situation is due to the definition of $(p, q)$-integers. However, in order to obtain a convergence criterion for all values of $p$ and $q$, belonging to the operator $\Delta_{1, p, q}^{2, b, b}$, we must have to overcome this difficulty. This type of difficulties can be avoided in the following two ways. The first one is taking $p = q = 1$ and thus the operator reduces to the usual difference sequence. Next, the second way is to replace $p = p_{n}$ and $q = q_{n}$ under the limits, $\lim_{n}q_{n} = \alpha$ and $\lim_{n}p_{n} = \beta$ $(0\leq \alpha, \beta \leq1)$ where $0 < q_{n} < p_{n}\leqq1$, for all $(n\in\mathbb{N})$. Afterwards, the difference sequence $\Delta_{1, p, q}^{a, b, c}(f(x))$ of third order $2$, converges to the value $3-(\beta^{2}+\alpha\beta+\alpha^{2})$. Thus, if we take $q_{n} = \left(\frac{n}{n+s}\right) < \left(\frac{n}{n+t}\right) = p_{n}$ such that $0 < q_{n} < p_{n}\leqq1$ $(s > t > 0)$, then $\lim_{n}q_{n} = 1 = \lim_{n}p_{n}$ and hence $\Delta_{1, p, q}^{2, b.b}(f(x))\rightarrow0$.

Remark 2.3. If $\lim_{n}q_{n} = 1$ and $\lim_{n}p_{n} = 1$, then the difference operator $\Delta_{h, p, q}^{a, b, c}$ reduces to the $\Delta_{h}^{a, b, c}$ (see ^[28]).

Next, before presenting the notion of the deferred Nörlund mean under the generalized difference operator of fractional order, including $(p, q)$-gamma function, we recall the regularity condition as follows: Let $(a_{n})$ and $(b_{n})$ be sequences of non-negative integers fulfilling the conditions: (a) $a_{n} < b_{n}\; \; (n\in\mathbb{N})$ and (b) $\mathop {\lim }\limits_{n \to \infty }b_{n} = \infty.$ Note that, the conditions (a) and (b) are regularity conditions for the proposed deferred Nörlund mean (see Agnew ^[3]).

We, now assume that $(s_{n})$ and $(t_{n})$ be the sequences of non-negative real numbers such that

The convolution of the above sequences can be presented as (see ^[45]),

Now, for defining the sequence $(f_{n})$ of function as deferred Nörlund $(D_{a}^{b}(N, s, t)$ mean by the fractional difference operator $(\Delta^{a, b, c}_{h, p, q})$ including $(p, q)$-gamma function, we first set,

We say that the sequence $(f_{n})$ of functions is deferred Nörlund summable to a number $\ell$ by the operator $\Delta^{a, b, c}_{h, p, q}$ if,

3.   Definitions and associated inclusion relations

In the earlier works of Srivastava et al. ^[45] the definitions of point-wise statistical convergence, equi-statistical convergence and uniform statistical convergence via the deferred Nörlund mean were introduced. In the similar lines, here we introduce the definitions of point-wise relatively statistical convergence, relatively equi-statistical convergence and relatively uniform statistical convergence via the deferred Nörlund mean. Let $\mathcal{C}(E)$ is the space of continuous real-valued functions defined over $E$. Also, it is a Banach space.

We, now introduce the following new definitions which are required for the present investigation.

Definition 3.1. Let $0 < q_{n} < p_{n}\leq1$ such that $\lim_{n}q_{n} = \alpha$ and $\lim_{n}p_{n} = \beta$ $(0 < \alpha, \beta\leq1)$ and let $r$ is a non-negative integer. Also, let $(a_{n})$ and $(b_{n})$ be sequences of non-negative integers. A sequence $(f_{n})$ of functions is point-wise relatively $\psi_{n}^{p, q}$-statistical convergent to a function $f$ on $E$ if, there exists a scale function $\sigma(x)$ ($|\sigma(x)| > 0$) on $E$ such that, for every $\epsilon > 0$ and for every $x\in E$,

where

Here, we can write

Definition 3.2. Let $0 < q_{n} < p_{n}\leq1$ such that $\lim_{n}q_{n} = \alpha$ and $\lim_{n}p_{n} = \beta$ $(0 < \alpha, \beta\leq1)$ and let $r$ is a non-negative integer. Also, let $(a_{n})$ and $(b_{n})$ be sequences of non-negative integers. A sequence $(f_{n})$ of functions is relatively $\psi_{n}^{p, q}$-equi-statistical convergent to a function $f$ defined on $E$ if, there exists a scale function $\sigma(x)$ ($|\sigma(x)| > 0$) on $E$ such that, for every $\epsilon > 0$,

uniformly relatively with respect to $x\in E$, that is,

where

Here, we write

Definition 3.3. Let $0 < q_{n} < p_{n}\leq1$ such that $\lim_{n}q_{n} = \alpha$ and $\lim_{n}p_{n} = \beta$ $(0 < \alpha, \beta\leq1)$ and let $r$ is a non-negative integer. Also, let $(a_{n})$ and $(b_{n})$ be sequences of non-negative integers. A sequence $(f_{n})$ of functions is relatively uniform $\psi_{n}^{p, q}$-statistical convergent to a function $f$ on $E$ if, there exists a scale function $\sigma(x)$ ($|\sigma(x)| > 0$) on $E$ such that, for every $\epsilon > 0$,

where

Here, we write

As a consequence of the above definitions, we present the following inclusion relations (without proof in the form of a Lemma) in the similar lines of our earlier work (see ^[4,35,45]).

Lemma 3.4. The implications,

are fairly true.

Moreover, reverse the implications of (3.1) are not necessarily true, that is, the above inclusions are strict.

The following examples justifies the strictness of the implications of our Lemma 3.4.

Example 3.5. The present example shows the strictness of the first inclusion under (3.1). We have, for

let $f_{n}:E\rightarrow\mathbb{R}, \; n\in \mathbb{N}$ be the real sequence of continuous functions defined on $E = [0, 1]$ (see ^[4]) is given as,

Then, for every $\epsilon > 0$,

uniformly on $E$. This implies

But since,

we see that

does not fairly true.

Example 3.6. The present example shows the strictness of the first inclusion under (3.1). We have, for

let $f_{n}:E\rightarrow\mathbb{R}, \; n\in \mathbb{N}$ be the real sequence of continuous functions defined over $E = [0, 1]$ is given as

We can see that $(f_{n})$ converges relatively point-wise to $f$, where

Hence

If we take $\epsilon = \frac{1}{3}$ then $\forall\; n\in \mathbb{N}$, there exists $r > n$ such that, for every $m\in[3^{r-1}, 3^{r})$ and $x\in \left(\sqrt[3r]{\frac{1}{3}}, 1\right)$, we have

Thus, clearly we have the following condition,

does not fairly true.

4.   A Korovkin-type theorem for relatively equi-statistical convergence via $\psi_{n}^{p, q}$-mean

Recently, a few researchers worked on extending the approximation theorems of Korovkin-type in several ways on the basis of various aspects, involving (for instance) function spaces, abstract Banach lattices, and so on. This well established theory is highly valuable in analysis and many other fields. The main concern of this paper is to introduce the notion of point-wise relatively $\psi_{n}^{p, q}$-statistical convergence, relatively $\psi_{n}^{p, q}$-equi-statistical convergence and relatively uniform $\psi_{n}^{p, q}$-statistical convergence with respect to the difference operator of fractional order, including $(p, q)$-gamma function, and then to establish some associated approximation type results in relevance to our presumably defined new concept of relatively $\psi_{n}^{p, q}$-equi-statistical convergence, that will effectively extend and improve most (if not all) of the existing results depending on the choice of deferred Nörlund $\psi_{n}^{p, q}$-mean. Furthermore, based upon the proposed methods, we wish to approximate the order of convergence and to investigate a Korovkin type approximation result for a sequence of functions. In fact, we extend here the result of Srivastava et al. ^[45] by using the notion of the relatively $\psi_{n}^{p, q}$-equi-statistical convergence for the generalized difference operator of fractional order, including $(p, q)$-gamma function and prove the following theorem. For more various approaches in this direction, we refer to the current works ^{[2,11,14,15,16,17,21,23,24,25,27,30,33,36,42,44,45,46,47,48,49]}.

Let $\mathfrak{L}:\mathcal{C}[a, b]\rightarrow\mathcal{C}[a, b]$ be a positive linear operator, where $\mathcal{C}[a, b]$ is the linear space of all continuous real-valued functions defined over $[a, b]$. Also, $\mathcal{C}[a, b]$ is a Banach space and for $f\in\mathcal{C}[a, b],$ the sup norm of $f$, is given by

In the similar lines of the well established approximation theorems of Korovkin-type (see ^[31,45]), here we present the following theorem for relatively equi-statical convergence via our proposed mean.

Theorem 4.1. Let $\mathfrak{L}_{m}:\mathcal{C}[a, b]\rightarrow\mathcal{C}[a, b]\; \; (m\in\mathbb{N})$ is a sequence of positive linear operators and let $f\in \mathcal{C}[E]\; \; (E = [a, b])$, then

if and only if

where

and

Proof. Since each of the functions given by

are continuous, the implication,

holds true. Now for the competition of the proof, we first assume that (4.2) is true. Further, suppose $f\in \mathcal{C}[E]$, then there exists a constant $\mathcal{K} > 0$ such that $|f(x)|\leq \mathcal{K}\quad(\forall\; x\in E)$, for which

Certainly, for given $\epsilon > 0$, there exists $\delta > 0$, such that

for all $x, y\in X$.

Let us choose

Next, if

then, we clearly get

From equation (4.3) and (4.4), we have

Now, since the operator $\mathfrak{L}_{m}(1, x)$ is linear and monotone, by using this in (4.5), we obtain

We note here that $x$ is fixed and so $f(x)$ is a constant number. Therefore, we get

and moreover in association with the following identity:

yields

Further, computing $\mathfrak{L}_{m}(\phi, x)$ as,

and using (4.9), we get

Then,

so that, by writing $\Delta_{h, p, q}^{a, b, c}\mathfrak{L}_{m}(., x)$ instead of $\mathfrak{L}_{m}(., x)$,

where

Now, for given $r' > 0$, choose $\epsilon > 0$ such that, $\frac{\epsilon}{\sigma(x)} < r'$. Moreover, for

and

we easily obtain from (4.11) that

Thus, we fairly have

Consequently, by Definition 3.2 and under the above assumption for the implication in (4.2), the right-hand side of (4.12) tends to zero $(n\rightarrow\infty)$. We, thus get

Hence, the implication in (4.1) is true. Which completes the proof of the Theorem.

Example 4.2. Suppose $E = [0, 1]$ and let the $(p, q)$-analogue of Bernstein operators $\mathfrak{B}_{n, p, q}(f; x)$ on $\mathcal{C}[E]$ is considered as (see ^[39]),

Also, observe that

Further, we consider $\mathfrak{L}_{m}:\mathcal{C}[E]\rightarrow \mathcal{C}[E]$ as the sequence of operators (positive linear) of the form:

Also, earlier a part of our operator, that is $x\left(1+x\frac{d}{dx}\right)$ was used by Al-Salam ^[5]. For several families of operators of this type we refer ^[53] and ^[50]. Next, on choosing the sequence $f_{n}(x)$ of functions as in Example 3.5, we can have

and

Further, since

so in similar lines as in Example 3.5, we can conclude that

for each $i = 0, 1, 2$. Thus, by Theorem 4.1, we see that

for all $f\in \mathcal{C}[E]$

Furthermore, as the sequence of functions $(f_{n})$ is not relatively uniform $\psi_{n}^{p, q}$-statistical convergent to $f = 0$ on $E$, we can certainly say that the outcomes of Srivastava et al. (^[45], p. 10, Theorem 1) and also the outcomes in ^[15,16] are not fairly true under the operators defined by us in (4.13). Also, the sequence of functions $(f_{n})$ is not being converge uniformly to the function $f = 0$ on $E$, the traditional Korovkin type theorem (see ^[31]) here also does not work. Moreover, for the operators defined under (4.13), our Theorem 4.1 fairly holds true.

5.   Rate of the relatively $\psi_{n}^{p, q}$-eqiu-statistical convergence

We intend here to investigate the order of the relatively $\psi_{n}^{p, q}$-equi-statistical convergence of a sequence of positive linear operators defined on $\mathcal{C}(E)$ into itself under the modulus of continuity.

Definition 5.1. Let $0 < q_{n} < p_{n}\leq1$ such that $\lim_{n}q_{n} = \alpha$ and $\lim_{n}p_{n} = \beta$ $(0 < \alpha, \beta\leq1)$, $r$ is a non-negative integer and let $(a_{n})$ and $(b_{n})$ be sequences of non-negative integers, and also let $(u_{n})$ be a positive non-increasing sequence. A sequence $(f_{n})$ of functions is relatively $\psi_{n}^{p, q}$-equi-statistical convergent to a function $f$ on $E$ with rate $o(u_{n})$ if, for every $\epsilon > 0$,

uniformly relatively with respect to $x\in E$ or, otherwise, if

where

We, write here

We now need to prove the following Lemma.

Lemma 5.2. Suppose the sequences $(f_{n})$ and $(g_{n})\in \mathcal{C}[E]$ satisfy the conditions:

and

where $\sigma_{0} > 0$ and $\sigma_{1} > 0$; then all the following assertions are true:

(i) $(f_{n}(x)+g_{n}(x))-(f(x)+g(x)) = o(w_{n})\; \; (\psi_{n}^{p, q} {-equi-stat}) \; {on}\; (E; \max\{\sigma_{0}, \sigma_{1}\});$

(ii) $(f_{n}(x)-f(x))(g_{n}(x)-g(x)) = o(u_{n}v_{n})\; \; (\psi_{n}^{p, q} {-equi-stat}) \; {on}\; (E; \{\sigma_{0}, \sigma_{1}\})$;

(iii) $\gamma(f_{n}(x)-f(x)) = o(u_{n})\; \; (\psi_{n}^{p, q} {-equi-stat})\; {on}\; (E; \sigma_{0}), \; \; {for\ any\ scalar}\; \gamma;$

(iv) $\left\{|f_{n}(x)-f(x)|\right\}^{\frac{1}{2}} = o(u_{n})\; \; (\psi_{n}^{p, q} {-equi-stat})\; {on}\; (E; \sqrt{|\sigma_{0}(x)|}),$ where $w_{n} = \max\{u_{n}, v_{n}\}$.

Proof. For proving the assertion (ⅰ) of Lemma 5.2, we consider the following sets for $\epsilon > 0$ and $x\in E$:

and

where

Clearly, we have

Moreover, since

by using the assertion (4.1) of Theorem 4.1, we obtain

Also, by using the assertion (4.2) of Theorem 4.1, we obtain

Thus, the assertion (ⅰ) of Lemma 5.2 is proved.

Next, since all other assertions ((ⅱ) - (ⅳ)) of Lemma 5.2 are similar as in the assertion (ⅰ), so these can be proved along similar lines to complete the proof Lemma 5.2.

We now recall the modulus of continuity of a function $f\in \mathcal{C}[E]$ as,

We now introduce a theorem to obtain the rates of relatively deferred Nörlund equi-statistical convergence based on difference operator of fractional order, including $(p, q)$-gamma function under the support of the modulus of continuity in (5.4).

Theorem 5.3. Let $E\subset\mathbb{R}$ be compact and let $\mathfrak{L}_{m}:\mathcal{C}(E)\rightarrow \mathcal{C}(E)$ be a sequence of positive linear operators. Suppose that the following conditions hold true:

(a) $\mathfrak{L}_{m}(1, x)-1 = o(u_{n})\; \; (\psi_{n}^{p, q} {-equi-stat})\; {on}\; (E; \sigma_{0})$;

(b) $\omega(f, \delta_{n}) = o(v_{n})\; \; (\psi_{n}^{p, q} {-equi-stat})\; {on}\; (E; \sigma_{1})$,

where

then, for each $f\in \mathcal{C}(E)$, the assertion as below holds true:

where $w_{n}$ is given by (5.1) and

Proof. Suppose $E\subset\mathbb{R}$ be compact and let $f\in \mathcal{C}[E]$, $x\in E$. Then, it is obvious that,

where

Which yields

Finally, for the conditions (a) and (b) (of Theorem 5.3) along with Lemma 5.2, the last inequality (5.6) helps us to the achieve the assertion (5.5). Therefore, proof of Theorem 5.3 is completed.

6.   Observations and concluding remarks

Here, in the last section of our study, we put forth some further concluding remarks and observations connecting to different outcomes which we have demonstrated here.

Remark 6.1. If we consider the sequence of functions $(f_{n})_{n\in \mathbb{N}}$ as given in Example 3.5. Then, since

we, fairly have

Thus, by using Theorem 4.1, we can write

Moreover, since $(f_{n})$ is not relatively uniform $\psi_{n}^{p, q}$-statistically convergent to $f = 0$ on $E$, and also since $(f_{n})$ is not relatively convergent uniformly to the function $f = 0$ on $E$, the well established theorem of Korovkin type fairly does not hold under the operators defined in (4.13). Thus, clearly this outcome indicates that our Theorem 1 is a generalization of traditional and statistical versions of Korovkin-type theorems (see ^[31], ^[45]).

Remark 6.2. Suppose in Theorem 5.3, we substitute the conditions (a) and (b) by the following condition:

Then, since

we can write

where

Clearly, from (6.3), (6.4) and Lemma 5.2, it follows that

where

Thus, definitely, we obtain

By applying (6.5) in Theorem 5.3, we instantly see that, $\forall\; f\in \mathcal{C}[E]$,

Therefore, instead of conditions (a) and (b) of Theorem 5.3, if we use the condition (6.3), then we certainly find the rates of the relatively $\psi_{n}^{p, q}$-equi-statistical convergence for the sequence $(\mathfrak{L}_{m})$ of positive linear operators in our Theorem 4.1.

Remark 6.3. In our present investigation, we have considered a number of fascinating special cases and illustrative examples in relevance to our definitions and also of the outcomes which have been established here.

Acknowledgments

The authors would like to express their heart-felt thanks to the editors and anonymous referees for their most valuable comments and constructive suggestions which leads to the significant improvement of the earlier version of the manuscript.

Conflict of interest

The authors declare no conflict of interest.