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Approximation by modified \((p,q)\)-gamma-type operators
Journal of Inequalities and Applications volume 2024, Article number: 37 (2024)
Abstract
The main object of this paper is to construct a new class of modified \((p,q)\)-Gamma-type operators. For this new class of operators, in section one, the general moments are find; in section two, the Korovkin-type theorem and some direct results are proved by considering the modulus of continuity and modulus of smoothness and their behavior in Lipschitz-type spaces. In section three, some results in the weighted spaces are given, and in the end, some shape-preserving properties are proven.
1 Introduction
One of the central theorems in the approximation theory is a Korovkin-type theorem. It is studied in various function spaces and in the various forms of convergence, starting from standard convergence [1, 12, 18, 27, 29], statistical convergence [3, 9, 10, 16, 23], power summability form of it [4–8, 24], and many other forms. In this paper, we will study the kind of the modified \((p,q)\)-Gamma-type operators, and for these operators, we will prove the Korovkin-type theorem and some direct results by considering the modulus of continuity and modulus of smoothness and their behavior in Lipschitz-type spaces. In Sect. 3, some results in the weighted spaces are given, and in the end, some shape-preserving properties are proven. In [25], the following Gamma-type operators were introduced:
where
Later one, in [29], the above operators have been modified to the following form:
where
Recently, in [21], the above operators have been modified as follows:
where
For any function f, the \((p,q)\)-derivative is given by (for example, see [11, 19])
and in case where f is differentiable at 0, then \(D_{p,q}f(0)=f'(0)\). We know that
for all \(0\leq k\leq n\). In [13], it is proved that (Theorem 1)
Based on this relation, we have
Lemma 1.1
The \((p,q)\)-factorial satisfies the following relation:
Proof
From relation (1.4) and definition of the \((p,q)-\) factorial, for \(k=1\), we get
and we obtain the desired result. □
Some relation related to the p, q-exponential function and p, q-integral are given by the following relations:
\(e_{p,q}(x)E_{p,q} (-x) = 1\).
Further, the p, q-Gamma function is given by
It is known that the following relation is valid (Proposition 3.3, [26]):
for every x.
In this paper, we introduce modified \((p,q)\)-Gamma-type operators:
with
Remark 1.2
Our operators are a generalization of the operators given in [29]; for \(p\to 1\), we obtain their class of operators. For \(p\in (0,1)\) and \(q=0\), we obtain operators defined in [21].
Now, we give some basic results.
Lemma 1.3
For \(p,q\in (0, 1)\) and \(x\in (0,\infty )\), the operators \(G^{(1)}_{n;p,q}\) satisfy
Proof
By setting \(t=x/u\), we have
as required. □
As an application of the above Lemma, we have
Corollary 1.4
For \(p,q\in (0, 1)\) and \(x\in (0,\infty )\), the operators \(G^{(1)}_{n;p,q}\) fulfill
-
(1)
\(G^{(1)}_{n;p,q}(1,x)=1\),
-
(2)
\(G^{(1)}_{n;p,q}(u,x)=\frac{[n]_{pq}x}{[n+2]_{pq}}\),
-
(3)
\(G^{(1)}_{n;p,q}(u^{2},x)= \frac{[n]_{pq}^{2}x^{2}}{[n+1]_{pq}[n+2]_{pq}}\),
-
(4)
\(G^{(1)}_{n;p,q}(u^{3},x)= \frac{[n]_{pq}^{2}x^{3}}{[n+1]_{pq}[n+2]_{pq}}\),
-
(5)
\(G^{(1)}_{n;p,q}(u^{4},x)= \frac{[n]_{pq}^{3}x^{4}}{[n-1]_{pq}[n+1]_{pq}[n+2]_{pq}}\).
Proof
The first one is obvious. For the second, we have:
From relation (1.5), we obtain
Similarly, we obtain
 □
As a result of Lemma 1.3 and the linearity of the operator \(G^{(1)}_{n;p,q}\), we obtain the following:
Lemma 1.5
For \(p,q\in (0, 1)\) and \(x\in (0,\infty )\), the operators \(G^{(1)}_{n;p,q}\) satisfy
Lemma 1.6
For \(p,q\in (0, 1)\) and \(x\in (0,\infty )\), the operators \(G^{(1)}_{n;p,q}\) satisfy
-
(1)
\(G^{(1)}_{n;p,q}((u-x),x)= \frac{[n]_{pq}(1-p^{2})-[2]_{pq}q^{n}}{[n+2]_{pq}}x\),
-
(2)
\(G^{(1)}_{n;p,q}((u-x)^{2},x)= \frac{[n]_{pq}([n]_{pq} +(p^{2}-2)[n+1]_{pq})+[2]_{pq}[n+1]_{pq}q^{n}}{[n+1]_{pq}[n+2]_{pq}}x^{2}\),
-
(3)
\(G^{(1)}_{n;p,q}((u-x)^{3},x)= \frac{[n]_{pq}(-2[n]_{pq}+(3-p^{2}))[n+1]_{pq})-[2]_{pq}[n+1]_{pq}q^{n}}{[n+1]_{pq}[n+2]_{pq}}x^{3}\),
-
(4)
\(G^{(1)}_{n;p,q}((u-x)^{4},x)= \frac{[n]_{pq}([n]_{pq}^{2}+2[n]_{pq}[n-1]_{pq}+(p^{2}-4)[n-1]_{pq}[n+1]_{pq})+[2]_{pq}[n-1]_{pq}[n+1]_{pq}q^{n}}{[n-1]_{pq}[n+1]_{pq}[n+2]_{pq}}x^{4}\).
Proof
Applying Lemma 1.1 and Lemma 1.5 will give:
-
(1)
\(G^{(1)}_{n;p,q}((u-x),x)=G^{(1)}_{n;p,q}(u,x)-x= \frac{[n]_{pq}x\Gamma _{pq}(n+2)}{\Gamma _{pq}(n+3)}-x= \frac{[n]_{pq}-[n+2]_{pq}}{[n+2]_{pq}}x= \frac{[n]_{pq}-(p^{2}[n]_{pq}+[2]_{pq}q^{n})}{[n+2]_{pq}}x= \frac{[n]_{pq}(1-p^{2})-[2]_{pq}q^{n}}{[n+2]_{pq}}x \).
-
(2)
Similarly, we obtain: \(G^{(1)}_{n;p,q}((u-x)^{2},x)= \frac{[n]_{pq}^{2}x^{2} -2x^{2}[n]_{pq}[n+1]_{pq}+x^{2}[n+1]_{pq}[n+2]_{pq} }{[n+1]_{pq}[n+2]_{pq}}= \frac{[n]_{pq}([n]_{pq} +(p^{2}-2)[n+1]_{pq})+[2]_{pq}[n+1]_{pq}q^{n}}{[n+1]_{pq}[n+2]_{pq}}x^{2}\).
-
(3)
\(G^{(1)}_{n;p,q}((u-x)^{4},x)= \frac{[n]_{pq}([n]_{pq}^{2}+2[n]_{pq}[n-1]_{pq}+(p^{2}-4)[n-1]_{pq}[n+1]_{pq})+[2]_{pq}[n-1]_{pq}[n+1]_{pq}q^{n}}{[n-1]_{pq}[n+1]_{pq}[n+2]_{pq}}x^{4}\).
 □
Remark 1.7
Throughout this paper, we assume that \(({p_{n}})_{n\in \mathbb{N}}\) and \(({q_{n}})_{n\in \mathbb{N}}\) are two sequences such that \(0< p_{n},q_{n}<1\), \(p_{n}\neq q_{n}\), satisfying \(\lim_{n\rightarrow \infty}p_{n}=\lim_{n\rightarrow \infty}q_{n}=1\), \(\lim_{n\rightarrow \infty}p_{n}^{n}=\alpha \) and \(\lim_{n\rightarrow \infty}q_{n}^{n}=\beta \), where \(0\leq \alpha ,\beta <1\). Then, from Lemma 1.6, we have
Next results prove the Korovkin-type theorem for the \(G^{(1)}_{n;p,q}\). The Korovkin-type theorem and its versions are widely studied; see, for example, [2–9, 17, 20, 23].
Theorem 1.8
Let \(G^{(1)}_{n;p,q}\) be a sequence of positive linear operators defined on \(C[0,\infty )\), such that for every \(i\in \{0,1,2\}\),
where \(e_{i}=x^{i}\). Then, for every \(f\in C[0,\infty )\),
uniformly for every \(x\in [a,b]\subset [0,\infty )\).
Proof
From Corollary 1.4, we have
and
The proof of theorem follows from the Korovkin theorem [1]. □
2 Some direct results
With \(B[0,\infty )\), \(C[0,\infty )\) and \(C_{B}([0,\infty ))\), we will denote the space of all bounded functions, continuous functions, and continuous, bounded functions defined in the interval \([0,\infty )\). Let be given \(\eta >0\), then the Petree K-functional [28] is defined as follows:
and \(C_{B}^{2}([0,\infty ))=\{r/r^{\prime },r^{\prime \prime }\in C_{B}([0,\infty )) \}\), with the norm
It is proven in [14] and [15] that exists a constant \(C>0\) such that
where
Theorem 2.1
If \(t\in C_{B}[0,\infty )\), then
Proof
From properties of the modulus of continuity and fact that operators \(G^{(1)}_{n;p_{n},q_{n}}\) are positive and linear, for any \(t\in C_{B}[0,\infty )\), we obtain
Let us set
Then, using the Cauchy–Schwarz inequality, we get
Putting \(\eta =\sqrt{n}\), we get the result. □
Next result gives an upper bound for \(G^{(1)}_{n,p_{n},q_{n}}\)-Gamma operators.
Theorem 2.2
For any \(g \in C_{B}[0,\infty )\),
Proof
From the definition of the modified \((p,q)\)-Gamma-type operators in (1.6), we have
 □
Theorem 2.3
For \(y \in (0,\infty )\), \(g\in C_{B}[0,\infty )\), there exists a \(M\in {\mathbb{R}}^{+}\), such that
where \(I(y)= \frac{[n]_{p_{n}q_{n}}(1-p_{n}^{2})-[2]_{p_{n}q_{n}}q_{n}^{n}}{[n+2]_{p_{n}q_{n}}}y \) and
Proof
For any \(y\in (0,\infty )\), we denote by
Then, from Lemma (1.5), we obtain
Let \(y,s\in (0,\infty )\) and \(r(y)\in C_{B}^{2}([0,\infty ))\). Using the Taylor formula, we get:
and it yields
From Theorem 2.2, we have that \(|G^{(1)}_{n,p_{n},q_{n}}(g,y)|\leq \|f\|\), then
Taking infimum for all \(r\in C_{B}^{2}([0,\infty ))\) and relation (2.1), we obtain our result. □
In [15], the following modulus are given:
and
\(\rho (y)=\sqrt{(y-a)(b-y)}\), and K-functional:
where \(\eta >0\).
Theorem 2.4
Let \(\rho =\sqrt{y(1-y)}\), \(g\in C_{B}[0,1]\) and \(y\in [0,1]\), \(n\in \mathbb{N}\). Then,
where \(\alpha _{1}(n,p_{n},q_{n})= \frac{[n]_{p_{n}q_{n}}}{[n+2]_{p_{n}q_{n}}}\).
Proof
Let
where
Then,
Let \(r \in W^{2}(\rho )\). Using the Taylor formula, we obtain
and
Therefore, we have
For \(v=\nu y+(1-\nu )s\) \((\nu \in [0,1])\). Since \(\rho ^{2}\) is concave on \([0,\infty )\), it follows that \(\rho ^{2}(v)\ge \nu \rho ^{2}(y)+(1-\nu )\rho ^{2}(s)\) and hence
Thus, we have
From the above relations, we obtain
On the other hand,
Therefore,
From inequality
-
(1)
$$ \frac{[n]_{p_{n}q_{n}}(1-p_{n}^{2})-[2]_{p_{n}q_{n}}q_{n}^{n}}{[n+2]_{p_{n}q_{n}}}y \leqq \frac{[n]_{p_{n}q_{n}}}{[n+2]_{p_{n}q_{n}}}. $$
It follows from Theorem 2.4
$$\begin{aligned} &K_{2,\rho (y)} \biggl(g, \frac{G^{(1)}_{n,p_{n},q_{n}} ((s-y)^{2};y )+y\beta _{1}(n,p_{n},q_{n},y)}{4\rho ^{2}(y)} \biggr) \\ &\quad \leqq K_{2,\rho (y)} \biggl(g, \frac{G^{(1)}_{n,p_{n},q_{n}} ((s-y)^{2};y )+\alpha _{1}(n,p_{n},q_{n})}{4\rho ^{2}(y)} \biggr), \end{aligned}$$ -
(2)
$$ \omega _{\gamma} \biggl(g; \frac{\beta _{1}(n,p_{n},q_{n},y)}{ \gamma (y)} \biggr)\leqq \omega _{\gamma} \biggl(g; \frac{\alpha _{1}(n,p_{n},q_{n})}{\gamma (y)} \biggr) $$
\(\forall y\in [0,1]\). Finally, we have
as asserted by the theorem. □
Theorem 2.5
Let \(g\in C[0,N]\), N is a finite number. Then,
where
Proof
Let \(g_{S}\) be the Steklov function of the second order for \(g(y)\). We know that
which follows from Corollary (1.4), and
It follows from Lemmas in [30]
As \(g_{S}\in C^{2}[0,N]\), and Lemmas in [17], we get
The following inequality is valid [30]:
In the light of (2.6) and (2.7), we obtain:
From relation (2.7) and the Landau inequality [22], we get
Using relations (2.7) and (2.8) and upon setting
we obtain
The proof of the theorem follows from relation (2.6). □
Theorem 2.6
Let \(g\in C_{B}[0,\infty )\). Then,
for \(y\geqq 0\), where
Proof
From the Taylor formula, it follows
where \(\iota \in (y,s)\). From the above relation, we have
 □
Theorem 2.7
Let \(g\in C[0,\infty )\). Then,
where \(\mathcal{M}>0\) is a constant, and \(D(n,p_{n},q_{n},y)\) is as in Theorem 2.6.
Proof
Let
then
Considering that \(g\in C_{B}^{2}\) and Theorems 2.2 and 2.6, we get
The following relation is valid [15]
for \(\forall \eta >0\), and \(L>0\) is a positive constant. The proof of the theorem follows from the last two relations. □
The next result gives an estimation of \(G^{(1)}_{n,p_{n},q_{n}}\)-operators in Lipschitz space \({\mathrm{Lip}}_{L}{\gamma}\) [27] given by the relation:
\(L>0\) is a constant, \(\gamma \in (0, 1]\).
Theorem 2.8
Let \(g\in {\mathrm{Lip}}_{L}(\gamma )\). Then, \(\forall y, t \in (0,\infty )\), \(n\in \mathbb{N}\) and \(\gamma \in (0, 1]\),
\(T>0\) is a constant.
Proof
Let \(g \in {\mathrm{Lip}}_{L}^{*}(\gamma )\) and \(\gamma \in (0, 1]\). Then,
I. For \(\gamma =1\), we have
for \(T>0\) constant.
Using the Cauchy–Schwarz inequality, we have
II. For \(\gamma \in (0,1)\), we have
From the Hölder inequality under the following conditions
it follows
for \(T>0\) constant. Applying the Cauchy–Schwarz inequality, we have:
 □
3 Weighted approximation
Let \(\zeta (y)=y^{2}+1\) be the weight function. We denote by \(B_{\zeta}[0,\infty )\), \(C_{\zeta}[0,\infty )\) and \(C^{*}_{\zeta}[0,\infty )\) the space of functions g defined on \([0,\infty )\) and satisfying, respectively: \(|g(y)|\leqq T_{g} \zeta (y)\), where \(T_{g}\) is a constant, space of all continuous functions and subspace of \(C_{\zeta}[0,\infty )\) for which \(\frac{g(y)}{\zeta (y)}\) is convergent as \(y\to \infty \).
The space \(B_{\zeta}[0, \infty )\) is a normed linear space defined by the norm as follows:
Next we will consider the weighted modulus of continuity \(\Omega (g;\kappa )\) defined on \([0, \infty )\) as
It is know that for any \(\mu \in [0,\infty )\), the following inequality:
holds true \(\forall g\in C^{*}_{\zeta}[0,\infty )\), and
Theorem 3.1
For \(g \in C^{*}_{\zeta}[0,\infty )\),
Proof
We will achieve our result from the Korovkin-type theorem and relations
which follows from Corollary 1.4.
In what follows, we will prove it for \(i = 1\) and \(i= 2\). Letting \(g \in C^{*}_{\zeta}[0,\infty )\), we get
Using a similar consideration, we have
We thus conclude that
 □
Theorem 3.2
Let \(g \in C^{*}_{\zeta}[0, \infty )\). Then,
for large n, where S is a constant, and \(F>0\) is constants dependent only on n, p, q.
Proof
For \(y \in [0, \infty )\), we have
Then,
Let us define
Then,
and
So, clearly, we get
From Lemma 1.6, it yields
For \(\kappa _{n}=n^{-\frac{1}{4}}\), we get
 □
4 Shape-preserving properties
Next we will prove that modified \((p,q)\)-Gamma-type operators preserve the monotonicity and convexity under certain conditions. We start with
Theorem 4.1
Let \(g \in C[0,\infty )\). If \(g^{\prime } (x)>0 \) and g convex on \([0,\infty )\), then modified \((p_{n},q_{n})\)-Gamma-type operators are increasing.
Proof
We will prove our result in two steps.
Step one. In this case, we will prove the monotonicity of modified \((p_{n},q_{n})\)-Gamma-type operators for the Lagrange interpolation polynomial of function \(g (y)\). Let us suppose that \(y_{0}\), \(y_{1}\) are distinct numbers in the interval \([t,z]\), where \(t< y_{0}< y_{1}< z\). Then, the Lagrangian interpolation polynomial through points \((y_{0},g(y_{0}))\) and \((y_{1},g (y_{1}))\) is:
Based on Corollary 1.4, we have:
which proves that \(G^{(1)}_{n,p_{n},q_{n}}(P(s),y)\) is also increasing.
Step two. From the above condition, it follows
for number \(\xi _{y}\in (\min_{{}}\{y_{0},y_{1}\},\max_{{}}\{y_{0},y_{1}\})\). For \(t< y_{0}< y_{1}< z\) and Corollary 1.4, we have
Therefore, it proves the theorem. □
Question
Prove that the above theorem is valid just only if \(f^{\prime } (x)>0\), on \([0,\infty )\).
Thus, the next results show that modified \((p,q)\)-Gamma-type operators preserve the convexity.
Theorem 4.2
Let \(g \in C[0,\infty )\). If \(g (y)\) is convex on \([0,\infty )\), then \((p_{n},q_{n})\)-Gamma-type operators are also convex.
Proof
Let us consider that \(g ^{{\prime \prime}}(y)>0\). Then,
On the other hand,
From the last relation, it follows
Hence, it proves the theorem. □
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Braha, N.L. Approximation by modified \((p,q)\)-gamma-type operators. J Inequal Appl 2024, 37 (2024). https://doi.org/10.1186/s13660-024-03109-1
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DOI: https://doi.org/10.1186/s13660-024-03109-1