Some new (p, q)-Dragomir–Agarwal and Iyengar type integral inequalities and their applications

1.   Introduction and preliminaries

Quantum calculus which is often known as ${\mathrm{q}}$-calculus is the branch of mathematics in which we obtain $\mathrm{{\mathrm{q}}}$-analogues of the mathematical objects which can be recaptured by taking ${\mathrm{q}}\to 1^{-}$. It is also known as calculus without limits and depends upon finite difference. In recent years it has emerged as a bridge between mathematics and physics. Due to its great many applications in different fields of applied sciences, it has attracted many researchers. Consequently a rapid developments in quantum calculus have been achieved. For example, Tariboon and Ntouyas ^[1] introduced the notions of ${\mathrm{q}}$-derivatives and ${\mathrm{q}}$-integrals on finite intervals. This idea attracted many researchers and resultantly many new ${\mathrm{q}}$-analogues of classical mathematical objects have been obtained using their approach. They themselves have obtained the quantum analogues of Hermite–Hadamard's inequality, Hölders's inequality, trapezoidal inequality and Ostrowski type of inequalities etc. Alp et al. ^[2] obtained a new corrected version of ${\mathrm{q}}$-Hermite–Hadamard's inequality. Noor et al. ^[3] and Sudsutad et al. ^[4] obtained independently some new quantum analogues of trapezoidal like inequalities. Liu and Zhuang ^[5] obtained quantum estimates of Hermite–Hadamard type of inequalities via twice ${\mathrm{q}}$-differentiable convex functions. Noor et al. ^[6] obtained a new ${\mathrm{q}}$-integral identity and obtained associated upper bounds by using the preinvexity property of the functions. Noor et al. ^[7] obtained new ${\mathrm{q}}$-Ostrowski type of inequalities via first order ${\mathrm{q}}$-differentiable convex functions. Deng et al. ^[8] obtained ${\mathrm{q}}$-analogues of Simpson type of inequalities. Zhang et al. ^[9] obtained a very nice generalized ${\mathrm{q}}$-integral identity and obtained associated bounds.

In recent years the classical concepts of quantum calculus have been modified in different directions, see ^{[10,11,12,13,14,15]}. One of the significant generalizations of ${\mathrm{q}}$-calculus is the post quantum often known as $(\mathrm{{\mathrm{p}}}, \mathrm{{\mathrm{q}}})$-calculus. The main idea is that in quantum calculus, we deal with a ${\mathrm{q}}$-number with one base ${\mathrm{q}}$, however, in $(\mathrm{{\mathrm{p}}}, \mathrm{{\mathrm{q}}})$-calculus, we deal with two independent variables ${\mathrm{p}}$ and ${\mathrm{q}}$. This idea was first considered in ^[16]. Tunç and Gov ^[17] recently introduced the concepts of $(\mathrm{{\mathrm{p}}}, \mathrm{{\mathrm{q}}})$-derivatives and $(\mathrm{{\mathrm{p}}}, \mathrm{{\mathrm{q}}})$-integrals on the finite intervals.

Definition 1.1 (^[17]). Let $\Psi:K\to\mathbb{R}$ be a continuous mapping and let $x\in K$ and $0 < {\mathrm{q}} < {\mathrm{p}}\leq1$. Then the $(\mathrm{{\mathrm{p}}}, \mathrm{{\mathrm{q}}})$-derivative on $K$ of mapping $\Psi$ at $x$ is defined as

Definition 1.2 (^[17]). Let $\Psi:K\subset\mathbb{R}\rightarrow \mathbb{R}$ be a continuous mapping. Then $(\mathrm{{\mathrm{p}}}, \mathrm{{\mathrm{q}}})$-integral on $K$ is defined as

for $x\in K$.

For more details, see ^[18,19,20].

Let us recall some basic definitions that will be used in the sequel.

Definition 1.3 (^[21]). A mapping $\Psi:I\rightarrow \mathbb{R}$ is said to be convex, if

holds for all ${\varpi_{1}}, {\varpi_{2}}\in I$ and ${\lambda}\in[0, 1].$

For more details, see ^{[22,23,24,25]}.

Definition 1.4 (^[26]). A set $\mathscr{K}_{\eta}\subset\mathbb{R}$ is said to be invex with respect to $\eta:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$, if

Definition 1.5 (^[9,27]). A mapping $\Psi:\mathscr{K}_{\eta}\to\mathbb{R}$ on the invex set is said to be preinvex, if

holds for all ${\varpi_{1}}, {\varpi_{2}}\in \mathscr{K}_{\eta}$ and ${\lambda}\in[0, 1].$

We now introduce the class of strongly $\varphi$-preinvex mappings.

Definition 1.6. Let $\varphi:(0, 1)\to\mathbb{R}$ be a real mapping. A mapping $\Psi:\mathscr{K}_{\eta}\to\mathbb{R}$ on the invex set is said to be strongly $\varphi$-preinvex, if

holds for all ${\varpi_{1}}, {\varpi_{2}}\in \mathscr{K}_{\eta}$, ${\lambda}\in (0, 1)$ with $\eta({\varpi_{2}}, {\varpi_{1}}) > 0$ and $\sigma > 0$.

Note that, if we take $\eta({\varpi_{2}}, {\varpi_{1}}) = {\varpi_{2}}-{\varpi_{1}}$ in Definition 1.6, then we have the class of strongly $\varphi$-convex functions which was introduced and studied in ^[28].

Remark 1.1. Note that, if we take, respectively $\varphi(\mu) = 1, \; \mu^{-1}, \; \mu^{s-1}$ and $\varphi(\mu) = 1-\mu$ in Definition 1.6, then we recapture the classes of strongly preinvex ^[27], strongly $P$-preinvex ^[29], strongly $s$-preinvex ^[29] and strongly $tgs$-preinvex mappings, respectively. Moreover, if we choose $\sigma\to 0^{+}$, then all of these classes reduce to preinvex ^[26], $P$-preinvex ^[30], $s$-preinvex ^[30] and $tgs$-preinvex mappings ^[31], respectively. This shows that the class of strongly $\varphi$-preinvex mappings is quite unifying one as it relates several other unrelated classes.

For the sake of completeness, let us now recall the Dragomir–Agarwal and Iyengar type of inequalities. Dragomir and Agarwal ^[41] obtained the following new integral identity and obtained associated inequalities essentially using the class of first order differentiable convex functions.

Lemma 1.1. Let $\Psi:I^{\circ}\subseteq\mathbb{R}\to\mathbb{R}$ be a differentiable function, If $\Psi'\in L[\varpi_{1}, \varpi_{2}]$, then

The right side of Hermite–Hadamard inequality can be estimated by the inequality of Iyengar ^[42], which reads as

where by $M$ we denote the Lipschitz constant, that is, $M = \sup\big\{\big|\frac{{\Psi}(x)-{\Psi}(y)}{x-y}\big|; x\neq y\big\}$.

Integral inequalities are important to predict upper and lower bounds in various applied sciences, e.g. in probability theory, functional inequalities, interpolation spaces, Sobolev spaces and information theory. For some recent studies and applications of integral inequalities in these directions, see ^{[32,33,34,35,36,37,38,39,40]}.

The main motivation of this article is to derive a new post-quantum integral identity using $({\mathrm{p}}, {\mathrm{q}})$-differentiable mappings. Using the identity as an auxiliary result, we will obtain some new variants of Dragomir–Agarwal and Iyengar type integral inequalities essentially via the class of strongly $\varphi$-preinvex mappings. We also discuss several new special cases which show that the results obtained are quite unifying. Finally, to support our results, we present some applications to $({\mathrm{p}}, {\mathrm{q}})$-differentiable mappings that are in absolute value bounded. We hope that the ideas and techniques of this paper will inspire interested readers working in this field.

2.   Auxiliary results

In this section, we discuss auxiliary results. These results will be helpful in obtaining the main results of this paper.

Lemma 2.1. Let $\omega\in[0, 1]$ and $\lambda\in[0, \infty)$, then

and

Now we derive a new $(\mathrm{{\mathrm{p}}}, \mathrm{{\mathrm{q}}})$-integral identity which will be used as an auxiliary result for obtaining next results of the paper.

Lemma 2.2. Suppose that $\Psi:[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]\subseteq\mathbb{R}\rightarrow \mathbb{R}$ be a $(\mathrm{{\mathrm{p}}}, \mathrm{{\mathrm{q}}})$-differentiable mapping on $({\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}}))$ with $\eta({\varpi_{2}}, {\varpi_{1}}) > 0$ and $0 < {\mathrm{q}} < {\mathrm{p}}\leq1$. If $\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi$ is a $(\mathrm{{\mathrm{p}}}, \mathrm{{\mathrm{q}}})$-integrable mapping on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$, then

where $[n]_{{\mathrm{p}}, {\mathrm{q}}}$ is the well-known $({\mathrm{p}}, {\mathrm{q}})$-integer expressed as:

Proof. Using Definitions 1 and 2, we have

We can see that

and

Similarly,

and

Substituting equalities (2.3)–(2.6) in (2.2), we get

Multiplying both sides of (2.7) by $\frac{{\mathrm{q}}\eta({\varpi_{2}}, {\varpi_{1}})}{2}$, we obtain the required result.

Remark 2.1. If we take ${\mathrm{p}} = 1$ in Lemma 2.2, then we have the following new equality:

3.   Main results

In this section, we discuss our main results.

3.1.   $(\mathrm{{\mathrm{p}}}, \mathrm{{\mathrm{q}}})$-Dragomir–Agarwal type integral inequalities

We now derive $(\mathrm{{\mathrm{p}}}, \mathrm{{\mathrm{q}}})$-analogues of Dragomir–Agarwal type integral inequalities via strongly $\varphi$-preinvex functions.

Theorem 3.1. For $\eta({\varpi_{2}}, {\varpi_{1}}) > 0$, let $\Psi:[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]\rightarrow \mathbb{R}$ be $(\mathrm{{\mathrm{p}}}, \mathrm{{\mathrm{q}}})$-differentiable mapping on $({\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}}))$ and $\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi$ be integrable on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$ with $0 < {\mathrm{q}} < {\mathrm{p}}\leq1.$ If $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|$ is strongly $\varphi$-preinvex mapping on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$ with modulus $\sigma > 0$, then

where

Proof. Using Lemma 2.2, property of modulus and the strongly $\varphi$-preinvexity of $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|$, we have

This completes the proof. We now discuss some special cases of Theorem 3.1.

Ⅰ. If $\varphi(\mu) = \varphi(\lambda) = 1$, then Theorem 3.1 reduces to the following result for the class of strongly preinvex mapping.

Corollary 3.1. Under the assumptions of Theorem 3.1, if $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|$ is strongly preinvex on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$, then

where

Ⅱ. If $\varphi(\lambda) = \lambda^{-1}, \varphi(\mu) = \mu^{-1}$, then Theorem 3.1 reduces to the following result for the class of strongly $P$-preinvex mapping.

Corollary 3.2. Under the assumptions of Theorem 3.1, if $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|$ is strongly $P$-preinvex on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$, then

where

Ⅲ. If $\varphi(\mu) = \mu^{s-1}, \varphi(\lambda) = \lambda^{s-1}$, then Theorem 3.1 reduces to the following result for the class of strongly $s$-preinvex mapping.

Corollary 3.3. Under the assumptions of Theorem 3.1, if $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|$ is strongly $s$-preinvex on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$, then

where

Ⅳ. If $\varphi(\mu) = 1-\mu, \varphi(\lambda) = 1-\lambda$, then Theorem 3.1 reduces to the following result for the class of strongly $tgs$-preinvex mapping.

Corollary 3.4. Under the assumptions of Theorem 3.1, if $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|$ is strongly $tgs$-preinvex on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$, then

Theorem 3.2. For $\eta({\varpi_{2}}, {\varpi_{1}}) > 0$, let $\Psi:[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]\rightarrow \mathbb{R}$ be $(\mathrm{{\mathrm{p}}}, \mathrm{{\mathrm{q}}})$-differentiable mapping on $({\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}}))$ and $\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi$ be integrable on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$ with $0 < {\mathrm{q}} < {\mathrm{p}}\leq1.$ If $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|^{r_{2}}$ is strongly $\varphi$-preinvex mapping on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$ with modulus $\sigma > 0$ for $r_{1} > 1$ with $r_{1}^{-1}+r_{2}^{-1} = 1$, then

where

Proof. Using Lemma 2.2, Hölder's inequality, property of modulus and the strongly $\varphi$-preinvexity of $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|^{r_{2}}$, we have

This completes the proof.

We now discuss some special cases of Theorem 3.2.

Ⅰ. If $\varphi(\mu) = \varphi(\lambda) = 1$, then Theorem 3.2 reduces to the following result for the class of strongly preinvex mapping.

Corollary 3.5. Under the assumptions of Theorem 3.2, if $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|^{r_{2}}$ is strongly preinvex on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$, then

Ⅱ. If $\varphi(\mu) = \mu^{-1}, \varphi(\lambda) = \lambda^{-1}$, then Theorem 3.2 reduces to the following result for the class of strongly $P$-preinvex mapping.

Corollary 3.6. Under the assumptions of Theorem 3.2, if $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|^{r_{2}}$ is strongly $P$-preinvex on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$, then

Ⅲ. If $\varphi(\mu) = \mu^{s-1}, \varphi(\lambda) = \lambda^{s-1}$, then Theorem 3.2 reduces to the following result for the class of strongly $s$-preinvex mapping.

Corollary 3.7. Under the assumptions of Theorem 3.2, if $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|^{r_{2}}$ is strongly $s$-preinvex on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$, then

Ⅳ. If $\varphi(\mu) = 1-\mu, \varphi(\lambda) = 1-\lambda$, then Theorem 3.2 reduces to the following result for the class of strongly $tgs$-preinvex mapping.

Corollary 3.8. Under the assumptions of Theorem 3.2, if $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|^{r_{2}}$ is strongly $tgs$-preinvex on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$, then

Theorem 3.3. For $\eta({\varpi_{2}}, {\varpi_{1}}) > 0$, let $\Psi:[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]\rightarrow \mathbb{R}$ be $(\mathrm{{\mathrm{p}}}, \mathrm{{\mathrm{q}}})$-differentiable mapping on $({\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}}))$ and $\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi$ be integrable on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$ with $0 < {\mathrm{q}} < {\mathrm{p}}\leq1.$ If $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|^{r_{2}}$ is strongly $\varphi$-preinvex on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$ with modulus $\sigma > 0$ for $r_{2}\geq1$, then

where $\mathbb{M}_{3}$ is given as in Theorem 3.1 and $\mathbb{M}_{5}$ is defined as in Corollary 3.2.

Proof. Using Lemma 2.2, the well-known power mean inequality, property of modulus and the strongly $\varphi$-preinvexity of $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|^{r_{2}}$, we have

This completes the proof.

We now discuss some special cases of Theorem 3.3.

Ⅰ. If $\varphi(\mu) = \varphi(\lambda) = 1$, then Theorem 3.3 reduces to the following result for the class of strongly preinvex mapping.

Corollary 3.9. Under the assumptions of Theorem 3.3, if $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|^{r_{2}}$ is strongly preinvex on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$, then

Ⅱ. If $\varphi(\mu) = \mu^{-1}, \varphi(\lambda) = \lambda^{-1}$, then Theorem 3.3 reduces to the following result for the class of strongly $P$-preinvex mapping.

Corollary 3.10. Under the assumptions of Theorem 3.3, if $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|^{r_{2}}$ is strongly $P$-preinvex on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$, then

Ⅲ. If $\varphi(\mu) = \mu^{s-1}, \varphi(\lambda) = \lambda^{s-1}$, then Theorem 3.3 reduces to the following result for the class of strongly $s$-preinvex mapping.

Corollary 3.11. Under the assumptions of Theorem 3.3, if $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|^{r_{2}}$ is strongly $s$-preinvex on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$, then

Ⅳ. If $\varphi(\mu) = 1-\mu, \varphi(\lambda) = 1-\lambda$, then Theorem 3.3 reduces to the following result for the class of strongly $tgs$-preinvex mapping.

Corollary 3.12. Under the assumptions of Theorem 3.3, if $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|^{r_{2}}$ is strongly $tgs$-preinvex on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$, then

3.2.   $(\mathrm{{\mathrm{p}}}, \mathrm{{\mathrm{q}}})$-Iyengar type integral inequalities

In this section, we derive new $(\mathrm{{\mathrm{p}}}, \mathrm{{\mathrm{q}}})$-Iyengar type integral inequalities essentially by using the strongly quasi-preinvexity property of the mappings. For this, let us recall the following definition.

Definition 3.1. A mapping $\Psi:\mathscr{K}_{\eta}\to\mathbb{R}$ on the invex set is said to be strongly quasi-preinvex, if

holds for all ${\varpi_{1}}, {\varpi_{2}}\in \mathscr{K}_{\eta}$, ${\lambda}\in (0, 1)$ with $\eta({\varpi_{2}}, {\varpi_{1}}) > 0$ and $\sigma > 0$.

Theorem 3.4. For $\eta({\varpi_{2}}, {\varpi_{1}}) > 0$, let $\Psi:[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]\rightarrow \mathbb{R}$ be $(\mathrm{{\mathrm{p}}}, \mathrm{{\mathrm{q}}})$-differentiable mapping on $({\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}}))$ and $\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi$ be integrable on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$ with $0 < {\mathrm{q}} < {\mathrm{p}}\leq1.$ If $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|$ is strongly quasi-preinvex mapping on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$ with modulus $\sigma > 0$, then

where $\mathbb{M}_{3}$ is given as in Theorem 3.1 and $\mathbb{M}_{5}$ is defined as in Corollary 3.2.

Proof. Using Lemma 2.2, property of modulus and the strongly quasi-preinvexity of $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|$, we have

This completes the proof.

Theorem 3.5. For $\eta({\varpi_{2}}, {\varpi_{1}}) > 0$, let $\Psi:[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]\rightarrow \mathbb{R}$ be $(\mathrm{{\mathrm{p}}}, \mathrm{{\mathrm{q}}})$-differentiable mapping on $({\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}}))$ and $\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi$ be integrable on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$ with $0 < {\mathrm{q}} < {\mathrm{p}}\leq1.$ If $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|^{r_{2}}$ is strongly quasi-preinvex mapping on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$ with modulus $\sigma > 0$ for $r_{1} > 1$ with $r_{1}^{-1}+r_{2}^{-1} = 1$, then

where $\mathbb{M}_{7}$ is given as in Theorem 3.2.

Proof. Using Lemma 2.2, Hölder's inequality, property of modulus and the strongly quasi-preinvexity of $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|^{r_{2}}$, we have

This completes the proof.

Theorem 3.6. For $\eta({\varpi_{2}}, {\varpi_{1}}) > 0$, let $\Psi:[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]\rightarrow \mathbb{R}$ be $(\mathrm{{\mathrm{p}}}, \mathrm{{\mathrm{q}}})$-differentiable mapping on $({\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}}))$ and $\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi$ be integrable on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$ with $0 < {\mathrm{q}} < {\mathrm{p}}\leq1.$ If $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|^{r_{2}}$ is strongly quasi-preinvex mapping on $[{\varpi_{1}}, {\varpi_{1}}+\eta({\varpi_{2}}, {\varpi_{1}})]$ for $r_{2}\geq1$, then

where $\mathbb{M}_{3}$ is given as in Theorem 3.1 and $\mathbb{M}_{5}$ is defined as in Corollary 3.2.

Proof. Using Lemma 2.2, the well-known power mean inequality, property of modulus and the strongly quasi-preinvexity of $|\, _{{\varpi_{1}}}\mathrm{D}_{{\mathrm{p}}, {\mathrm{q}}}\Psi|^{r_{2}}$, we have

This completes the proof.

Remark 3.1. If we choose ${\mathrm{p}} = 1$ in our main results, we can get new special cases regarding quantum analogues of Dragomir–Agarwal and Iyengar type of integral inequalities essentially by using the strongly $\varphi$-preinvexity property of the mappings. We omit here their proofs and the details are left to the interested reader.

4.   Application to bounded functions

We suppose that the following condition is satisfied:

which means that the $({\mathrm{p}}, {\mathrm{q}})$-differentiable mapping $\Psi$ is in absolute value bounded from the positive real number $\Omega$. Applying the above condition, we are in a position to derive some new interesting inequalities using our main results.

Proposition 4.1. Under the conditions of Theorem 3.1, the following inequality holds:

Proof. Applying inequality (4.1) in Theorem 3.1, we have the desired result.

Proposition 4.2. Under the conditions of Theorem 3.2, the following inequality holds:

Proof. Using inequality (4.1) in Theorem 3.2, we get the desired result.

Proposition 4.3. Under the conditions of Theorem 3.3, the following inequality holds:

Proof. Applying inequality (4.1) in Theorem 3.3, we obtain the desired result.

Proposition 4.4. Under the conditions of Theorem 3.4, the following inequality holds:

Proof. Using inequality (4.1) in Theorem 3.4, we have the desired result.

Proposition 4.5. Under the conditions of Theorem 3.5, the following inequality holds:

Proof. Applying inequality (4.1) in Theorem 3.5, we get the desired result.

Proposition 4.6. Under the conditions of Theorem 3.6, the following inequality holds:

Proof. Using inequality (4.1) in Theorem 3.6, we obtain the desired result.

5.   Conclusions

In this paper, we have established a new post-quantum integral identity using $({\mathrm{p}}, {\mathrm{q}})$-differentiable mappings. From the applied identity as an auxiliary result, we have obtained some new variants of Dragomir–Agarwal and Iyengar type integral inequalities essentially pertaining to the class of strongly $\varphi$-preinvex and strongly quasi $\varphi$-preinvex mappings, respectively. We also discuss several new special cases which show that the results obtained are quite unifying. In order to illustrate the efficiency of our main results, some applications regarding $({\mathrm{p}}, {\mathrm{q}})$-differentiable mappings that are in absolute value bounded are provided as well. To the best of our knowledge, these results are new in the literature. Since the class of strongly $\varphi$-preinvex mappings have large applications in many mathematical areas, they can be applied to obtain several results in convex analysis, special mappings, quantum mechanics, related optimization theory, and mathematical inequalities and may stimulate further research in different areas of pure and applied sciences. Studies relating convexity, partial convexity, and preinvex mappings (as contractive operators) may have useful applications in complex interdisciplinary studies, such as maximizing the likelihood from multiple linear regressions involving Gauss–Laplace distribution. For more details, please see ^[43,44].

Acknowledgments

This research was supported by Taif University Researchers Supporting Project Number (TURSP-2020/155), Taif University, Taif, Saudi Arabia.

Conflict of interest

The authors declare that they have no conflict of interest.