On boundedness of fractional integral operators via several kinds of convex functions

1.   Introduction

Fractional integral operators have great significance in extensive fields of science and engineering. They are widely used to construct and solve fractional order models and fractional dynamical systems. In recent decades fractional integral operators are frequently used to study different types of integral inequalities including well known inequalities of Hadamard ^[1,2,3,4,5], Ostrowski ^[6,7,8,9], Grüss ^[10,11], Opial ^[12], Chebsheve ^[13,14] and Minkowski ^[15,16].

We have motivated by ongoing research in integral inequalities, and interested to establish inequalities for fractional integral operators defined in ^[17]. To attain the desired results, we have used a generalized class of functions called strongly $(\alpha, h-m)$-convex functions. The findings of this paper simultaneously give generalizations as well as refinements of many recently published inequalities.

The unified Mittag-Leffler function is defined as follows:

Definition 1.1. ^[17] For $\underline{a} = (a_{1}, \, a_{2}, ..., \, a_{n})$, $\underline{b} = (b_{1}, b_{2}, ..., b_{n})$, $\underline{c} = (c_{1}, \, c_{2}, \, ..., \, c_{n})$, where $a_{i}$, $b_{i}$, $c_{i}\in\mathbb{C}; \, i = 1, 2, 3, ..., n$ such that $\Re(a_{i}), \Re(b_{i}), \Re(c_{i}) > 0$, $\forall\; i$. Also let $\alpha, \beta, \, \gamma, \, \delta, \, \mu, \, \nu, \, \lambda, \, \rho, \, \theta, t\in\mathbb{C}$, $\min\{{\Re(\alpha), \, \Re(\beta), \, \Re(\gamma), \, \Re(\delta), \, \Re(\theta)}\} > 0$ and $k\in(0, 1)\cup\mathbb{N}$ with $k+\Re(\rho) < \Re(\delta+\nu+\alpha)$, Im$(\rho) =$Im$(\delta+\nu+\alpha)$, then Mittag-Leffler function is defined by

where $\Gamma(\mu)$ is the gamma function, $\Gamma(\mu) = \int_{0}^{\infty}e^{-z}z^{\mu-1}dz$, $(\theta)_{kl}$ is the Pochhammer symbol, $(\theta)_{kl} = \frac{\Gamma(\theta+lk)}{\Gamma(\theta)}$ and $\beta_{\varrho}$ is the extension of beta function and it is defined as follows:

Along with the convergence conditions of the unified Mittag-Leffler function given in Definition 1.1, the unified fractional operators are defined as follows:

Definition 1.2. ^[17] Let $\varPhi\in L_{1}[\xi_{1}, \xi_{2}]$. Then $\forall\; \zeta\in[\xi_{1}, \xi_{2}]$, the fractional integral operator containing the unified Mittag-Leffler function $M^{\lambda, \rho, \theta, k, n}_{\alpha, \beta, \gamma, \delta, \mu, \nu}\left(z; \underline{a}, \underline{b}, \underline{c}, \varrho\right)$ satisfying all the convergence conditions is defined as follows:

By setting $a_{i} = l$, ${\varrho} = 0$ and $\Re(\varrho) > 0$ in (1.3) and (1.4), we get the fractional integral operator associated with generalized Q function as (see ^[18]):

where

is a generalized Q function defined in ^[19].

The more generalized and extended version of integral operator given in Definition 1.2 is defined as follows:

Definition 1.3. ^[20] Let $\varDelta\in L_{1}[\xi_{1}, \xi_{2}],$ $0 < \xi_{1}, \xi_{2} < \infty$ be a positive function and let $\varPsi:[\xi_{1}, \xi_{2}]\rightarrow \mathbb{R}$ be a differentiable and strictly increasing function. Also let $\frac{\Delta}{\zeta}$ be an increasing function on $[\xi_{1}, \infty)$. Then for $\zeta\in[\xi_{1}, \xi_{2}]$ the unified integral operator in its generalized form satisfying all the convergence conditions is defined by:

where

Definition 1.4. ^[18] By setting $a_{i} = l$, ${\varrho} = 0$ and $\Re(\varrho) > 0$ in (1.7) and (1.8), we get the fractional integral operator associated with generalized Q function as follows:

where $\varLambda_{\zeta}^{\vartheta}(Q^{\lambda, \rho, \theta, k, n}_{\alpha, \beta, \gamma, \mu, \nu }\varPsi; \varDelta) =$$\dfrac{\varDelta(\varPsi(\zeta)-\varPsi(\vartheta))}{\varPsi(\zeta)-\varPsi(\vartheta)}Q^{\lambda, \rho, \theta, k, n}_{ \alpha, \beta, \gamma, \delta, \mu, \nu}(\omega(\varPsi(\zeta)-\varPsi(\vartheta))^{\mu}, \underline{a}, \underline{b}, \varrho)$.

One can note that if $\varPsi$ and $\frac{\varDelta}{\zeta}$ are increasing functions, then for $u < \vartheta < v, u, v\in[\xi_{1}, \xi_{2}]$, the kernel $\varLambda_{\zeta}^{\vartheta}(M^{\lambda, \rho, \theta, k, n}_{\alpha, \beta, \gamma, \mu, \nu }\varPsi; \varDelta)$ satisfies the following inequality:

For more details, one can see ^[21]. In the whole paper we will use

Convex functions are extensively utilized in mathematics, physics, mathematical statistics and economics. The geometrical visualization of a convex function can be seen in the Hadamard inequality. Several new classes of functions have been defined to prove generalizations and refinements of many known mathematical inequalities. The definition of strongly $(\alpha, h-m)$-convex function is defined as follows:

Definition 1.5. ^[22] Let $J\subseteq\mathbb{R}$ be an interval containing $(0, 1)$ and let $h:J\rightarrow \mathbb{R}$ be a non-negative function. A function $\varPhi:[0, \xi_{2}]\rightarrow \mathbb{R}$ is called strongly $(\alpha, h-m)$-convex function with modulus $C\geq0$, if $f$ is non-negative and for all $\zeta, y\in[0, \xi_{2}]$, $\vartheta\in(0, 1)$ and $m\in(0, 1],$ one have the inequality

Remark 1. The above definition produces several types of convex functions like $(h-m)$-convex, $(s, m)$-convex, strongly $(\alpha, m)$-convex, $(\alpha, m)$-convex functions etc.

The upcoming section consists of bounds of fractional integral operators given in (1.7) and (1.8). They are constructed by using definition of strongly $(\alpha, h-m)$-convex function. We have established a Hadamard type inequality, by using a lemma for strongly $(\alpha, h-m)$-convex functions. The results of this paper are connected with various inequalities that have been published in recent decades. Finally, by using strongly $(\alpha, h-m)$-convexity of the function $|\varPhi'|$ further bounds are given.

2.   Main results

Theorem 2.1. Let $\varPhi\in L_{1}[\xi_{1}, \xi_{2}]$ be an integrable strongly $(\alpha, h-m)$-convex function. Also let $\frac{\varDelta}{\zeta}$ be an increasing function on $[\xi_{1}, \xi_{2}]$ and, let $\varPsi:[\xi_{1}, \xi_{2}]\longrightarrow \mathbb{R}$ be differentiable and strictly increasing function, also let $\frac{\varDelta}{\zeta}$ be an increasing function on $[\xi_{1}, \xi_{2}]$. Then $\forall\; \zeta\in[\xi_{1}, \xi_{2}]$, we have the following inequality containing the unified Mittag-Leffler function $M^{\lambda, \rho, \theta, k, n}_{\alpha, \beta, \gamma, \delta, \mu, \nu}\left(z; \underline{a}, \underline{b}, \underline{c}, \varrho\right)$ satisfying all the convergence conditions:

where $X_{\zeta}^{\xi_{1}}(r^{\alpha}, h, \varPsi') = \int_{\xi_{1}}^{\zeta}h\left(r^{\alpha}\right)\varPsi'(\zeta-r(\zeta-\xi_{1}))dr$, $X_{\zeta}^{\xi_{1}}(1-r^{\alpha}, h, \varPsi') = \int_{\xi_{1}}^{\zeta}h\left(1-r^{\alpha}\right)\varPsi'(\zeta-r(\zeta-\xi_{1}))dr$.

Proof. Using (1.12), we can write the following inequalities:

Using strongly $(\alpha, h-m)$-convexity of $\varPhi$, we have

From (2.2) and (2.4), one can obtain the following inequality

Using Definition 1.3 and setting $r = \frac{\zeta-\vartheta}{\zeta-\xi_{1}}$, we obtain

The above inequality will become

On the other hand, using the same technique that we did for (2.2) and (2.4), the following inequality from (2.3) and (2.5) can be obtained:

The above inequality can takes the following form

By adding (2.7) and (2.9), (2.1) can be obtained.

Remark 2. (i) If $n = 1,$ $b_{1} = \lambda+lk,$ $a_{1} = \theta-\lambda,$ $c_{1} = \lambda,$ $\rho = \nu = 0$ in (2.1), [22,Theorem 1] is obtained.

(ii) If $h(\zeta) = \zeta$ in the result of (i), then [23,Theorem 7] is obtained.

(iii) If $C = 0,$ $\varDelta(\vartheta) = \vartheta^ {\beta},$ $\varPsi(\zeta) = \zeta,$ $h(\zeta) = \zeta^{s}$ and $(\alpha, m) =$(1, 1) in the result of (i), then [24,Theorem 2.1] is obtained.

(iv) If $\varDelta(\vartheta) = { \vartheta^{\beta}{}}{}$, $h(\zeta) = \varPsi(\zeta) = \zeta$ in the result of (i), then [25,Theorem 4] can be obtained.

(v) If $\varDelta(\vartheta) = { \vartheta^{\beta}{}}{}$, $\varPsi(\zeta) = \zeta$ and $C = 0$ in the result of (i), then [26,Theorem 1] can be obtained.

(vi) If $C = 0$ and $h(\zeta) = \zeta$ in (2.1), then [4,Theorem 1] is obtained.

(vii) If $C = 0$ in the result of (i), [27,Theorem 1] is obtained.

Corollary 1. If $h(\zeta) = \zeta$, then the following inequality holds for strongly $(\alpha, m)$-convex function:

Corollary 2. If $(\alpha, m) = (1, 1)$ and $h(\zeta) = \zeta$, then the following inequality holds for strongly convex function:

where $I_{d}$ is the identity function.

For positive values of $C$ all the results obtained in the aforementioned Remarks/Corollaries get their refinements. The following lemma is required to establish the next result.

Lemma 2.1. ^[22] Let $\varPhi:[\xi_{{1}}, \xi_{2}]\longrightarrow\mathbb{R}$, be a strongly $(\alpha, h-m)$-convex function with modulus $C\geq 0$, $m\in(0, 1]$, $0\leq \xi_{1}\leq m\xi_{2}$. If $\varPhi(\zeta) = \varPhi(\frac{\xi_{1}+m\xi_{2}-\zeta}{m})$, then the following inequality holds:

The following result provides upper and lower bounds of sum of operators (1.7) and (1.8) in the form of a Hadamard type inequality.

Theorem 2.2. Under the assumptions of Theorem 2.1 in addition, if $\varPhi(\zeta) = \varPhi\left(\dfrac{\xi_{1}+m\xi_{2}-\zeta}{m}\right)$, then we have

Proof. Using (1.12), we can write the following inequalities

Using strongly $(\alpha, h-m)$-convexity of $\varPhi$ for $\zeta\in [\xi_{1}, \xi_{2}]$, we have

Multiplying (2.12) and (2.14) and integrating the resulting inequality over $[\xi_{1}, \xi_{2}]$, one can obtain

By using Definition 1.3 and setting $r = \frac{\zeta-\xi_{1}}{\xi_{2}-\xi_{1}}$, the following inequality is obtained:

Using the same technique that we did for (2.12) and (2.14), the following inequality can be observed from (2.14) and (2.13)

By adding (2.15) and (2.16), following inequality can be obtained:

Multiplying both sides of (2.10) by $\varLambda_{\zeta}^{\xi_{1}}(M^{\lambda, \rho, \theta, k, n}_{\alpha, \beta, \gamma, \mu, \nu}\; \varPsi; \varDelta)d(\varPsi(\zeta))$, and integrating over $[\xi_{1}, \xi_{2}]$ we have

From Definition 1.3, the following inequality is obtained:

Similarly multiplying both sides of (2.10) by $\varLambda_{\xi_{2}}^{\zeta}(M^{\lambda, \rho, \theta, k, n}_{\alpha, \beta, \gamma, \mu, \nu}\; \varPsi; \varDelta)\varPsi'(\zeta)$, and integrating over $[\xi_{1}, \xi_{2}]$ we have

From (2.17)–(2.19), inequality (2.11) can be achieved.

Remark 3. (i) If $n = 1$, $b_{1} = \lambda+lk$, $a_{1} = \theta-\lambda$, $c_{1} = \lambda$, $\rho = \nu = 0$, in (2.11), then [22,Theorem 23] is obtained.

(ii) If $h(\zeta) = \zeta$ in the result of (i), then [23,Theorem 11] is obtained.

(iii) If $\omega = \varrho = C = 0$, $(\alpha, m)$ = (1, 1), $\varPhi(\vartheta) = \Gamma(\beta)\vartheta^{\beta+1}$ and $h(\zeta) = \varPsi(\zeta) = \zeta$ in the result of (i), then [28,Theorem 3] is obtained.

(iv) If $\varDelta(\vartheta) = \vartheta^ {\beta+1}$ and $h(\vartheta) = \varPsi(\vartheta) = \vartheta$ in the result of (i), then [25,Theorem 6] can be obtained.

(v) If $\varDelta(\vartheta) = \vartheta^ {\beta+1}$, $\varPsi(\vartheta) = \vartheta$ and $C = 0$ in the result of (i), then [26,Theorem 4] can be obtained.

(vi) If $C = 0$ and $h(\zeta) = \zeta$ in (2.11), then [4,Theorem 2] is obtained.

Corollary 3. If $h(\zeta) = \zeta$ in (2.11), then the following inequality holds for strongly $(\alpha, m)$-convex function:

Corollary 4. If $(\alpha, m) = (1, 1)$ and $h(\zeta) = \zeta$ in (2.11), then the following inequality holds for strongly convex function:

For positive values of $\lambda$ all the results obtained in the aforementioned Remarks/Corollaries got their refinements.

Theorem 2.3. Let $\varPhi:[\xi_{1}, \xi_{2}]\longrightarrow \mathbb{R}$ be a differentiable function. If $|\varPhi'|$ is $(s, m)$-convex and let $\varPsi:[\xi_{1}, \xi_{2}]\longrightarrow \mathbb{R}$ be differentiable and strictly increasing function, also let $\frac{\varDelta}{\zeta}$ be an increasing function on $[\xi_{1}, \xi_{2}]$. Then $\forall\; \zeta\in[\xi_{1}, \xi_{2}]$, we have the following inequality containing the unified Mittag-Leffler function $M^{\lambda, \rho, \theta, k, n}_{\alpha, \beta, \gamma, \delta, \mu, \nu}\left(z; \underline{a}, \underline{b}, \underline{c}, \varrho\right)$ satisfying all the convergence conditions:

where

Proof. Let $\zeta\in [\xi_{1}, \xi_{2}]$ and $\vartheta \in [\xi_{1}, \zeta]$. Then using strongly $(\alpha, h-m)$-convexity of $|\varPhi'|$ we have

The inequality (2.21) can be written as follows:

Let we consider the second inequality of (2.22)

Multiplying (2.2) and (2.23) and integrating over $[\xi_{1}, x]$, we can obtain:

This gives

Considering the left hand side from the inequality (2.22) and adopt the same pattern as did for the right hand side inequality, then

From (2.24) and (2.25), following inequality is observed:

Now using strongly $(\alpha, h-m)$-convexity of $|\varPhi'|$ on $(\zeta, \xi_{2}]$ for $\zeta\in[\xi_{1}, \xi_{2}]$ we have

On the same procedure as we did for (2.2) and (2.21), one can obtain following inequality from (2.3) and (2.27):

By adding (2.26) and (2.28), inequality (2.20) can be achieved.

Remark 4. (i) If $n = 1$, $b_{1} = \lambda+lk$, $a_{1} = \theta-\lambda$, $c_{1} = \lambda$, $\rho = \nu = 0$, in (2.20), [22,Theorem 26].

(ii) If $C = 0$ in the result of (i), [27,Theorem 4] is obtained.

(iii) If $h(\zeta) = \zeta$ in the result of (i), then [23,Theorem 14] is obtained.

(iv) If $\varDelta(\vartheta) = \vartheta^{\beta}$ and $\varPsi(\zeta) = h(\zeta) = \zeta$ in the result of (i), then [25,Theorem 5] is obtained.

(v) If $\varDelta(\vartheta) = \vartheta^{\beta+1}$, $\varPsi(\zeta) = \zeta$ and $C = 0$ in the result of (i), then [26,Theorem 2] is obtained.

(vi) If $C = 0$ and $h(\zeta) = \zeta$ in (2.20), then [4,Theorem 3] is obtained.

Corollary 5. If $h(\zeta) = \zeta$ in (2.20), then the following inequality holds for strongly $(\alpha, m)$-convex function:

Corollary 6. If $(\alpha, m) = (1, 1)$ and $h(\zeta) = \zeta$ in (2.20), then the following inequality holds for strongly convex function:

For positive values of $C$ all the results obtained in the aforementioned Remarks/Corollaries get their refinements.

3.   Conclusions

This article investigates the bounds of fractional integral operators containing unified Mittag-Leffler function via strongly $(\alpha, h-m)$-convex function. The proved results are more generalized and refined as compared to existing inequalities in the literature. Also, they hold implicitly for various classes of functions such as $(\alpha, m)$-convex, $(s, m)$-convex, $(h-m)$-convex, strongly convex, strongly $(\alpha, m)$-convex and strongly $(h-m)$-convex functions.

Acknowledgment

This work was supported by the National Key Research and Development Program under Grant 2018YFB0904205.

Conflict of interest

It is declared that authors have no competing interests.