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Certain inequalities via generalized proportional Hadamard fractional integral operators

Abstract

In the article, we introduce the generalized proportional Hadamard fractional integrals and establish several inequalities for convex functions in the framework of the defined class of fractional integrals. The given results are generalizations of some known results.

Introduction

Fractional calculus has gained more attention due to its applications in distinct fields. In the development of fractional calculus, researchers focus on developing several fractional integral operators and their applications in diverse fields. The idea of fractional conformable derivative operators was introduced by Khalil et al. [23] with a deficiency that the new derivative operator does not tend to the original function when the order \(\rho \rightarrow 0\). In [1], Abdeljawad studied studied various properties of the fractional conformable derivative operators and raised the problem of how to use conformable derivative operators to generate more general nonlocal fractional derivative operators; the method was demonstrated in [22]. Later on in [12], Anderson and Unless improved the idea of the fractional conformable derivative by introducing the idea of local proportional derivatives. In [2, 3, 13, 14, 26], some researchers defined new continuous and discrete fractional derivative operators by using the exponential and Mittag-Leffler functions in the kernels. Generalizations of such type promote future research to establish new ideas to unify the fractional derivative and integral operators and obtain fractional integral inequalities via such generalized fractional derivative and integral operators. Integral inequalities and their applications play a vital role in the theory of differential equations and applied mathematics.

Certain weighted Grüss-type inequalities and new inequalities involving Riemann–Liouville fractional integrals are found in [17, 19]. In [29], the authors define several inequalities for the extended gamma and confluent hypergeometric k-functions. In [30], the authors established certain Gronwall inequalities for Riemann–Liouville and Hadamard k-fractional derivatives with applications. Inequalities involving the generalized \((k,\rho )\)-fractional integral operators can be found in [35]. The generalized fractional integral and its applications and Grüss-type inequalities via generalized fractional integrals can be found in [40, 42]. In [39], Sarikaya and Budak have studied the \((k, s)\)-Riemann–Liouville fractional integral and applications. In [41], Set et al. established generalized Hermite–Hadamard-type inequalities via fractional integral operators. In [9], Agarwal et al. introduced Hermite–Hadamard-type inequalities by employing the k-fractional integrals operators. In [18], Dahmani introduced certain classes of fractional integral inequalities by utilizing a family of n positive functions.

In [10], the authors established fractional integral inequalities for a class of family of n (\(n\in \mathbb{N}\)) positive continuous and decreasing functions on \([a,b]\) by employing the \((k,s)\)-fractional integral operators. Recently, the fractional conformable integrals have attracted the attention of many researchers. In particular, many remarkable inequalities, properties, and applications for the fractional conformable integrals can be found in the literature [4,5,6,7,8, 16, 20, 24, 25, 27, 28, 31, 32, 34, 36, 37, 43].

Preliminaries

In [21], Jarad et al. introduced the left and right generalized proportional integral operators which are respectively defined by

$$\begin{aligned} \bigl({}_{a}\mathfrak{I}^{\beta ,\mu }f \bigr) (x)= \frac{1}{\mu ^{ \beta }\varGamma (\beta )} \int _{a}^{x}\exp \biggl[\frac{\mu -1}{\mu }(x-t) \biggr] (x-t)^{ \beta -1}f(t)\,dt,\quad a< x \end{aligned}$$
(2.1)

and

$$\begin{aligned} \bigl(\mathfrak{I}_{b}^{\beta ,\mu }f \bigr) (x)= \frac{1}{\mu ^{ \beta }\varGamma (\beta )} \int _{x}^{b}\exp \biggl[\frac{\mu -1}{\mu }(t-x) \biggr](t-x)^{ \beta -1}f(t)\,dt,\quad x< b, \end{aligned}$$
(2.2)

where the proportionality index \(\mu \in (0,1]\), \(\beta \in \mathbb{C}\), \(\Re (\beta )>0\), and \(\varGamma (x)=\int _{0}^{\infty }t ^{x-1}e^{-t}\,dt\) is the gamma function [44,45,46].

Remark 2.1

If we consider \(\mu =1\) in (2.1) and (2.2), then we get the left and right Riemann–Liouville integrals, which are respectively defined as

$$\begin{aligned} \bigl({}_{a}\mathfrak{I}^{\beta }f \bigr) (x)= \frac{1}{\varGamma ( \beta )} \int _{a}^{x} (x-t)^{\beta -1}f(t)\,dt,\quad a< x, \end{aligned}$$
(2.3)

and

$$\begin{aligned} \bigl(\mathfrak{I}_{b}^{\beta }f \bigr) (x)= \frac{1}{\varGamma (\beta )} \int _{x}^{b}(t-x)^{\beta -1}f(t)\,dt,\quad x< b, \end{aligned}$$
(2.4)

where \(\beta \in \mathbb{C}\) and \(\Re (\beta )>0\).

Recently Alzabut et al. and Rahman et al. [11, 34] studied generalized proportional derivative and integral operators and established certain Gronwall and Minkowski inequalities involving the operators mentioned above.

Next, motivated by the above, we define the following generalized version of Hadamard fractional integrals. Rahman et al. [33] presented certain new classes of integral inequalities for a class of n (\(n\in \mathbb{N}\)) positive continuous and decreasing functions on \([a, b]\) by employing generalized proportional fractional integrals.

Definition 2.1

The left-sided generalized proportional Hadamard fractional integral of order \(\beta >0\) and proportionality index \(\mu \in (0,1]\) is defined by

$$\begin{aligned} \bigl({}_{a}\mathcal{H}^{\beta ,\mu } f \bigr) (x)= \frac{1}{\mu ^{ \beta }\varGamma (\beta )} \int _{a}^{x}\exp \biggl[\frac{\mu -1}{\mu }(\ln x- \ln t)\biggr](\ln x-\ln t)^{\beta -1}\frac{f(t)}{t}\,dt,\quad a< x. \end{aligned}$$
(2.5)

Definition 2.2

The right-sided generalized proportional Hadamard fractional integral of order \(\beta >0\) and proportionality index \(\mu \in (0,1]\) is defined by

$$\begin{aligned} \bigl(\mathcal{H}_{b}^{\beta ,\mu } f \bigr) (x)= \frac{1}{\mu ^{ \beta }\varGamma (\beta )} \int _{x}^{b}\exp \biggl[\frac{\mu -1}{\mu }(\ln t- \ln x)\biggr](\ln t-\ln x)^{\beta -1}\frac{f(t)}{t}\,dt,\quad x< b. \end{aligned}$$
(2.6)

Definition 2.3

The one-sided generalized proportional Hadamard fractional integral of order \(\beta >0\) and proportionality index \(\mu \in (0,1]\) is defined by

$$\begin{aligned} \bigl(\mathcal{H}_{1,x}^{\beta ,\mu } f \bigr) (x)= \frac{1}{\mu ^{ \beta }\varGamma (\beta )} \int _{1}^{x}\exp \biggl[\frac{\mu -1}{\mu }(\ln x- \ln t)\biggr](\ln x-\ln t)^{\beta -1}\frac{f(t)}{t}\,dt,\quad t>1. \end{aligned}$$
(2.7)

Remark 2.2

If we consider \(\mu =1\), then (2.5)–(2.7) will lead to the following well-known Hadamard fractional integrals:

$$\begin{aligned}& \bigl({}_{a}\mathcal{H}^{\beta } f \bigr) (x)=\frac{1}{\varGamma ( \beta )} \int _{a}^{x}(\ln x-\ln t)^{\beta -1} \frac{f(t)}{t}\,dt,\quad a< x, \end{aligned}$$
(2.8)
$$\begin{aligned}& \bigl(\mathcal{H}_{b}^{\beta } f \bigr) (x)= \frac{1}{\varGamma (\beta )} \int _{x}^{b}(\ln t-\ln x)^{\beta -1} \frac{f(t)}{t}\,dt, \quad x< b, \end{aligned}$$
(2.9)

and

$$\begin{aligned} \bigl(\mathcal{H}_{1,x}^{\beta } f \bigr) (x)=\frac{1}{\varGamma ( \beta )} \int _{1}^{x}(\ln x-\ln t)^{\beta -1} \frac{f(t)}{t}\,dt,\quad t>1. \end{aligned}$$
(2.10)

One can easily prove the following results:

Lemma 2.1

$$\begin{aligned} \begin{aligned}[b]& \biggl(\mathcal{H}_{1,x}^{\beta ,\mu }\exp \biggl[ \frac{\mu -1}{\mu }( \ln x)\biggr](\ln x)^{\lambda -1} \biggr) (x)\\&\quad = \frac{\varGamma (\lambda )}{ \mu ^{\beta }\varGamma (\beta +\lambda )}\exp \biggl[\frac{\mu -1}{\mu }(\ln x)\biggr]( \ln x)^{\beta +\lambda -1}\end{aligned} \end{aligned}$$
(2.11)

and the semigroup property holds:

$$\begin{aligned} \bigl(\mathcal{H}_{1,x}^{\beta ,\mu } \bigr) \bigl( \mathcal{H}_{1,x} ^{\lambda ,\mu } \bigr)f(x)= \bigl(\mathcal{H}_{1,x}^{\beta +\lambda ,\mu } \bigr)f(x). \end{aligned}$$
(2.12)

Remark 2.3

If \(\mu =1\), then (2.11) will reduce to the result of [38] as defined by

$$\begin{aligned} \bigl(\mathcal{H}_{1,x}^{\beta }(\ln x)^{\lambda -1} \bigr) (x)= \frac{ \varGamma (\lambda )}{\varGamma (\beta +\lambda )}(\ln x)^{\beta +\lambda -1}. \end{aligned}$$
(2.13)

Main results

In this section, we employ the generalized proportional Hadamard fractional integral operator to establish generalizations of some classical inequalities.

Theorem 3.1

Let f and h be two positive continuous functions on the interval \([1,\infty )\) and \(f\leq h\) on \([1,\infty )\). If \(\frac{f}{h}\) is decreasing and f is increasing on \([1,\infty )\), then for a convex function Φ with \(\varPhi (0)=0\), the generalized proportional Hadamard fractional integral operator given by (2.7) satisfies the inequality

$$\begin{aligned} \frac{\mathcal{H}_{1,x}^{\beta ,\mu } [f(x) ]}{\mathcal{H} _{1,x}^{\beta ,\mu } [h(x) ]}\geq \frac{\mathcal{H}_{1,x} ^{\beta ,\mu } [ \varPhi (f(x)) ]}{\mathcal{H}_{1,x}^{\beta , \mu } [\varPhi (h(x)) ]}, \end{aligned}$$
(3.1)

where \(\mu \in (0,1]\), \(\beta \in \mathbb{C}\), and \(\Re (\beta )>0\).

Proof

Since Φ is convex with \(\varPhi (0)=0\), the function \(\frac{\varPhi (f(x))}{x}\) is increasing. As f is increasing, so is the function \(\frac{\varPhi (f(x))}{f(x)}\). Obviously, the function \(\frac{f(x)}{h(x)}\) is decreasing. Thus for all \(t, \vartheta \in [1, \infty )\), we have

$$\begin{aligned} \biggl(\frac{\varPhi (f(t))}{f(t)}-\frac{\varPhi (f(\vartheta ))}{f(\vartheta )} \biggr) \biggl( \frac{f(\vartheta )}{h(\vartheta )}- \frac{f(t)}{h(t)} \biggr)\geq 0. \end{aligned}$$
(3.2)

It follows that

$$\begin{aligned} \frac{\varPhi (f(t))}{f(t)}\frac{f(\vartheta )}{h(\vartheta )}+\frac{ \varPhi (f(\vartheta ))}{f(\vartheta )} \frac{f(t)}{h(t)}- \frac{\varPhi (f( \vartheta ))}{f(\vartheta )}\frac{f(\vartheta )}{h(\vartheta )}-\frac{ \varPhi (f(t))}{f(t)} \frac{f(t)}{h(t)}\geq 0. \end{aligned}$$
(3.3)

Multiplying (3.3) by \(h(t)h(\vartheta )\), we have

$$\begin{aligned} \frac{\varPhi (f(t))}{f(t)}f(\vartheta )h(t)+\frac{\varPhi (f(\vartheta ))}{f( \vartheta )}f(t)h( \vartheta )- \frac{\varPhi (f(\vartheta ))}{f(\vartheta )}f(\vartheta )h(t)-\frac{\varPhi (f(t))}{f(t)}f(t)h(\vartheta )\geq 0. \end{aligned}$$
(3.4)

Multiplying (3.4) by \(\frac{1}{\mu ^{\beta }\varGamma (\beta )}\frac{ \exp [\frac{\mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{\beta -1}}{t}\), which is positive because \(t\in (1,x)\), \(x>1\) and integrating the resulting identity from 1 to x, we have

$$\begin{aligned} &\frac{1}{\mu ^{\beta }\varGamma (\beta )} \int _{1}^{x}\frac{\exp [\frac{ \mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{\beta -1}}{t} \frac{\varPhi (f(t))}{f(t)}f( \vartheta )h(t)\,dt \\ &\quad{} + \frac{1}{\mu ^{\beta }\varGamma (\beta )} \int _{1}^{x}\frac{\exp [\frac{ \mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{\beta -1}}{t}\frac{\varPhi (f( \vartheta ))}{f(\vartheta )}f(t)h( \vartheta )\,dt \\ &\quad{} - \frac{1}{\mu ^{\beta }\varGamma (\beta )} \int _{1}^{x}\frac{\exp [\frac{ \mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{\beta -1}}{t}\frac{\varPhi (f( \vartheta ))}{f(\vartheta )}f( \vartheta )h(t)\,dt \\ &\quad{} - \frac{1}{\mu ^{\beta }\varGamma (\beta )} \int _{1}^{x}\frac{\exp [\frac{ \mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{\beta -1}}{t} \frac{\varPhi (f(t))}{f(t)}f(t)h(\vartheta )\,dt\geq 0. \end{aligned}$$
(3.5)

This follows that

$$\begin{aligned} \begin{aligned}[b] &f(\vartheta )\mathcal{H}_{1,x}^{\beta ,\mu } \biggl( \frac{\varPhi (f(x))}{f(x)}h(x) \biggr)+ \biggl(\frac{\varPhi (f(\vartheta ))}{f(\vartheta )}h(\vartheta ) \biggr) \mathcal{H}_{1,x}^{\beta ,\mu }\bigl(f(x)\bigr) \\ &\quad{} - \biggl(\frac{\varPhi (f(\vartheta ))}{f(\vartheta )}f(\vartheta ) \biggr) \mathcal{H}_{1,x}^{\beta ,\mu } \bigl(h(x)\bigr)-h(\vartheta ) \mathcal{H}_{1,x} ^{\beta ,\mu } \biggl( \frac{\varPhi (f(x))}{f(x)}f(x) \biggr)\geq 0. \end{aligned} \end{aligned}$$
(3.6)

Again, multiplying both sides of (3.6) by \(\frac{1}{\mu ^{\beta } \varGamma (\beta )}\frac{\exp [\frac{\mu -1}{\mu }(\ln x-\ln \vartheta )]( \ln x-\ln \vartheta )^{\beta -1}}{\vartheta }\), which is positive because \(\vartheta \in (1,x)\), \(x>1\) and integrating the resulting identity from 1 to x, we get

$$\begin{aligned} &\mathcal{H}_{1,x}^{\beta ,\mu }\bigl(f(x)\bigr)\mathcal{H}_{1,x}^{\beta , \mu } \biggl(\frac{\varPhi (f(x))}{f(x)}h(x) \biggr)+ \mathcal{H}_{1,x} ^{\beta ,\mu } \biggl(\frac{\varPhi (f(x))}{f(x)}h(x) \biggr)\mathcal{H} _{1,x}^{\beta ,\mu } \bigl(f(x) \bigr) \\ &\quad \geq \mathcal{H}_{1,x}^{\beta ,\mu } \bigl(h(x) \bigr) \mathcal{H} _{1,x}^{\beta ,\mu } \bigl(\varPhi \bigl(f(x)\bigr) \bigr)+ \mathcal{H}_{1,x}^{ \beta ,\mu } \bigl(\varPhi \bigl(f(x)\bigr) \bigr) \mathcal{H}_{1,x}^{\beta , \mu } \bigl(h(x) \bigr). \end{aligned}$$
(3.7)

It follows that

$$\begin{aligned} \frac{\mathcal{H}_{1,x}^{\beta ,\mu } (f(x) )}{\mathcal{H} _{1,x}^{\beta ,\mu } (h(x) )}\geq \frac{\mathcal{H}_{1,x} ^{\beta ,\mu } (\varPhi (f(x)) )}{\mathcal{H}_{1,x}^{\beta , \mu } (\frac{\varPhi (f(x))}{f(x)}h(x) )}. \end{aligned}$$
(3.8)

Now, since \(f\leq h\) on \([1,\infty )\) and \(\frac{\varPhi (x)}{x}\) is an increasing function, for \(t,\vartheta \in [1,x)\), we have

$$\begin{aligned} \frac{\varPhi (f(t))}{f(t)}\leq \frac{\varPhi (h(t))}{h(t)}. \end{aligned}$$
(3.9)

Multiplying both sides of (3.9) by \(\frac{1}{\mu ^{\beta }\varGamma (\beta )}\frac{\exp [\frac{\mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{ \beta -1}}{t}h(t)\), which is positive because \(t\in (1,x)\), \(x>1\) and integrating the resulting identity from 1 to x, we get

$$\begin{aligned} &\frac{1}{\mu ^{\beta }\varGamma (\beta )} \int _{1}^{x}\frac{\exp [\frac{ \mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{\beta -1}}{t} \frac{\varPhi (f(t))}{f(t)}h(t)\,dt \\ &\quad \leq \frac{1}{\mu ^{\beta }\varGamma (\beta )} \int _{1}^{x}\frac{\exp [\frac{ \mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{\beta -1}}{t} \frac{\varPhi (h(t))}{h(t)}h(t)\,dt, \end{aligned}$$
(3.10)

which, in view of (2.7), can be written as

$$\begin{aligned} \mathcal{H}_{1,x}^{\beta ,\mu } \biggl( \frac{\varPhi (f(x))}{f(x)}h(x) \biggr) \leq \mathcal{H}_{1,x}^{\beta ,\mu } \bigl(\varPhi \bigl(h(x)\bigr) \bigr). \end{aligned}$$
(3.11)

Hence from (3.8) and (3.11), we get (3.1). □

Theorem 3.2

Let f and h be two positive continuous functions on the interval \([1,\infty )\) and \(f\leq h\) on \([1,\infty )\). If \(\frac{f}{h}\) is decreasing and f is increasing on \([1,\infty )\), then for a convex function Φ with \(\varPhi (0)=0\), the generalized proportional Hadamard fractional integral operator given by (2.7) satisfies the inequality

$$\begin{aligned} \frac{\mathcal{H}_{1,x}^{\beta ,\mu } [f(x) ]\mathcal{H} _{1,x}^{\rho ,\mu } [\varPhi (h(x)) ] +\mathcal{H}_{1,x}^{ \rho ,\mu } [f(x) ]\mathcal{H}_{1,x}^{\beta ,\mu } [\varPhi (h(x)) ]}{ \mathcal{H}_{1,x}^{\beta ,\mu } [h(x) ]\mathcal{H}_{1,x} ^{\rho ,\mu } [ \varPhi (f(x)) ]+\mathcal{H}_{1,x}^{\rho , \mu } [h(x) ]\mathcal{H}_{1,x}^{\rho ,\mu } [ \varPhi (f(x)) ]} \geq 1, \end{aligned}$$
(3.12)

where \(\mu \in (0,1]\), \(\beta , \rho \in \mathbb{C}\), \(\Re (\beta )>0\), and \(\Re (\rho )>0\).

Proof

Since Φ is convex with \(\varPhi (0)=0\), the function \(\frac{\varPhi (f(x))}{x}\) is increasing. As f is increasing, so is the function \(\frac{\varPhi (f(x))}{f(x)}\). Obviously, the function \(\frac{f(x)}{h(x)}\) is decreasing for all \(t, \vartheta \in [1,x)\). Multiplying (3.6) by \(\frac{1}{\mu ^{\rho }\varGamma (\rho )}\frac{ \exp [\frac{\mu -1}{\mu }(\ln x-\ln \vartheta )](\ln x-\ln \vartheta )^{\rho -1}}{\vartheta }\), which is positive because \(\vartheta \in (1,x)\), \(\vartheta >1\) and integrating the resulting identity from 1 to x, we get

$$\begin{aligned} \begin{aligned}[b] &\mathcal{H}_{1,x}^{\rho ,\mu }\bigl(f(x)\bigr)\mathcal{H}_{1,x}^{\beta ,\mu } \biggl(\frac{\varPhi (f(x))}{f(x)}h(x) \biggr)+ \mathcal{H}_{1,x}^{ \rho ,\mu } \biggl(\frac{\varPhi (f(x))}{f(x)}h(x) \biggr)\mathcal{H}_{1,x} ^{\beta ,\mu } \bigl(f(x) \bigr) \\ &\quad \geq \mathcal{H}_{1,x}^{\beta ,\mu } \bigl(h(x) \bigr) \mathcal{H} _{1,x}^{\rho ,\mu } \biggl(\frac{\varPhi (f(x)}{f(x)}f(x) \biggr)+ \mathcal{H}_{1,x}^{\beta ,\mu } \biggl(\frac{\varPhi (f(x)}{f(x)}f(x) \biggr) \mathcal{H}_{1,x}^{\rho ,\mu } \bigl(h(x) \bigr). \end{aligned} \end{aligned}$$
(3.13)

Now, since \(f\leq h\) on \([1,\infty )\) and \(\frac{\varPhi (x)}{x}\) is an increasing function, for \(t,\vartheta \in [1,x)\), we have

$$\begin{aligned} \frac{\varPhi (f(t))}{f(t)}\leq \frac{\varPhi (h(t))}{h(t)}. \end{aligned}$$
(3.14)

Multiplying both sides of (3.14) by \(\frac{1}{\mu ^{\beta }\varGamma (\beta )}\frac{\exp [\frac{\mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{ \beta -1}}{t}h(t)\), which is positive because \(t\in (1,x)\), \(x>1\) and integrating the resulting identity with respect to t from 1 to x, we get

$$\begin{aligned} &\frac{1}{\mu ^{\beta }\varGamma (\beta )} \int _{1}^{x}\frac{\exp [\frac{ \mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{\beta -1}}{t} \frac{\varPhi (f(t))}{f(t)}h(t)\,dt \\ &\quad \leq \frac{1}{\mu ^{\beta }\varGamma (\beta )} \int _{1}^{x}\frac{\exp [\frac{ \mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{\beta -1}}{t} \frac{\varPhi (h(t))}{h(t)}h(t)\,dt, \end{aligned}$$
(3.15)

which, in view of (2.7), can be written as

$$\begin{aligned} \mathcal{H}_{1,x}^{\beta ,\mu } \biggl( \frac{\varPhi (f(x))}{f(x)}h(x) \biggr) \leq \mathcal{H}_{1,x}^{\beta ,\mu } \bigl(\varPhi \bigl(h(x)\bigr) \bigr). \end{aligned}$$
(3.16)

Hence, from (3.11), (3.13), and (3.16), we get the desired result. □

Remark 3.1

If we consider \(\beta =\rho \), then Theorem 3.2 will lead to Theorem 3.1.

Theorem 3.3

Let f, h, and g be positive continuous functions on the interval \([1,\infty )\) and \(f\leq h\) on \([1,\infty )\). If \(\frac{f}{h}\) is decreasing and f and g are increasing on \([1,\infty )\), then for a convex function Φ with \(\varPhi (0)=0\), the generalized proportional Hadamard fractional integral operator given by (2.7) satisfies the inequality

$$\begin{aligned} \frac{\mathcal{H}_{1,x}^{\beta ,\mu } [f(x) ]}{\mathcal{H} _{1,x}^{\beta ,\mu } [h(x) ]}\geq \frac{\mathcal{H}_{1,x} ^{\beta ,\mu } [ \varPhi (f(x))g(x) ]}{\mathcal{H}_{1,x}^{ \beta ,\mu } [\varPhi (h(x))g(x) ]}, \end{aligned}$$
(3.17)

where \(\mu \in (0,1]\), \(\beta \in \mathbb{C}\) and \(\Re (\beta )>0\).

Proof

Since \(f\leq h\) on \([1,\infty )\) and \(\frac{\varPhi (x)}{x}\) is increasing, for \(t,\vartheta \in [1,x)\), we have

$$\begin{aligned} \frac{\varPhi (f(t))}{f(t)}\leq \frac{\varPhi (h(t))}{h(t)}. \end{aligned}$$
(3.18)

Multiplying both sides of (3.18) by \(\frac{1}{\mu ^{\beta }\varGamma (\beta )}\frac{\exp [\frac{\mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{ \beta -1}}{t}h(t)g(t)\), which is positive because \(t\in (1,x)\), \(x>1\) and integrating the resulting identity from 1 to x, we get

$$\begin{aligned} \begin{aligned}[b] &\frac{1}{\mu ^{\beta }\varGamma (\beta )} \int _{1}^{x}\frac{\exp [\frac{ \mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{\beta -1}}{t} \frac{\varPhi (f(t))}{f(t)}h(t)g(t)\,dt \\ &\quad \leq \frac{1}{\mu ^{\beta }\varGamma (\beta )} \int _{1}^{x}\frac{\exp [\frac{ \mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{\beta -1}}{t} \frac{\varPhi (h(t))}{h(t)}h(t)g(t)\,dt, \end{aligned} \end{aligned}$$
(3.19)

which, in view of (2.7), can be written as

$$\begin{aligned} \mathcal{H}_{1,x}^{\beta ,\mu } \biggl( \frac{\varPhi (f(x))}{f(x)}h(x)g(x) \biggr) \leq \mathcal{H}_{1,x}^{\beta ,\mu } \bigl(\varPhi \bigl(h(x)\bigr)g(x)\bigr) ). \end{aligned}$$
(3.20)

Also, since the function Φ is convex and such that \(\varPhi (0)=0\), \(\frac{\varPhi (t)}{t}\) is increasing. Since f is increasing, so is \(\frac{\varPhi (f(t))}{f(t)}\). Clearly, the function \(\frac{f(t)}{h(t)}\) is decreasing for all \(t, \vartheta \in [1,x)\), \(x>1 \). Thus,

$$\begin{aligned} \biggl(\frac{\varPhi (f(t))}{f(t)}g(t)-\frac{\varPhi (f(\vartheta ))}{f( \vartheta )}g(\vartheta ) \biggr) \bigl(f(\vartheta )h(t)-f(t)h( \vartheta ) \bigr)\geq 0. \end{aligned}$$
(3.21)

It follows that

$$\begin{aligned} \begin{aligned}[b] & \frac{\varPhi (f(t))g(t)}{f(t)}f(\vartheta )h(t)+\frac{\varPhi (f(\vartheta ))g(\vartheta )}{f(\vartheta )}f(t)h( \vartheta )\\&\quad{} - \frac{\varPhi (f( \vartheta ))g(\vartheta )}{f(\vartheta )}f(\vartheta )h(t)-\frac{ \varPhi (f(t))g(t)}{f(t)}f(t)h(\vartheta )\geq 0.\end{aligned} \end{aligned}$$
(3.22)

Multiplying (3.22) by \(\frac{1}{\mu ^{\beta }\varGamma (\beta )}\frac{ \exp [\frac{\mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{\beta -1}}{t}\), which is positive because \(t\in (1,x)\), \(x>1\) and integrating the resulting identity from 1 to x, we have

$$\begin{aligned} &\frac{1}{\mu ^{\beta }\varGamma (\beta )} \int _{1}^{x}\frac{\exp [\frac{ \mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{\beta -1}}{t} \frac{\varPhi (f(t))}{f(t)}f( \vartheta )h(t)g(t)\,dt \\ &\quad \quad{} + \frac{1}{\mu ^{\beta }\varGamma (\beta )} \int _{1}^{x}\frac{\exp [\frac{ \mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{\beta -1}}{t}\frac{\varPhi (f( \vartheta ))}{f(\vartheta )}f(t)h( \vartheta )g(\vartheta )\,dt \\ &\quad \quad {}- \frac{1}{\mu ^{\beta }\varGamma (\beta )} \int _{1}^{x}\frac{\exp [\frac{ \mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{\beta -1}}{t}\frac{\varPhi (f( \vartheta ))}{f(\vartheta )}f( \vartheta )g(\vartheta )h(t)\,dt \\ &\quad \quad{} - \frac{1}{\mu ^{\beta }\varGamma (\beta )} \int _{1}^{x}\frac{\exp [\frac{ \mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{\beta -1}}{t} \frac{\varPhi (f(t))}{f(t)}f(t)g(t)h(\vartheta )\,dt \\ &\quad \geq 0. \end{aligned}$$
(3.23)

This follows that

$$\begin{aligned} &f(\vartheta )\mathcal{H}_{1,x}^{\beta ,\mu } \biggl( \frac{\varPhi (f(x))}{f(x)}h(x)g(x) \biggr)+ \biggl(\frac{\varPhi (f(\vartheta ))}{f(\vartheta )}h(\vartheta )g( \vartheta ) \biggr)\mathcal{H}_{1,x}^{\beta ,\mu }\bigl(f(x)\bigr) \\ &\quad{} - \biggl(\frac{\varPhi (f(\vartheta ))}{f(\vartheta )}f(\vartheta )g( \vartheta ) \biggr) \mathcal{H}_{1,x}^{\beta ,\mu }\bigl(h(x)\bigr)-h(\vartheta ) \mathcal{H}_{1,x}^{\beta ,\mu } \biggl(\frac{\varPhi (f(x))}{f(x)}f(x)g(x) \biggr) \geq 0. \end{aligned}$$
(3.24)

Again, multiplying both sides of (3.24) by \(\frac{1}{\mu ^{\beta }\varGamma (\beta )}\frac{\exp [\frac{\mu -1}{\mu }(\ln x-\ln \vartheta )](\ln x-\ln \vartheta )^{\beta -1}}{\vartheta }\), which is positive because \(\vartheta \in (1,x)\), \(\vartheta >1\) and integrating the resulting identity from 1 to x, we get

$$\begin{aligned} &\mathcal{H}_{1,x}^{\beta ,\mu }\bigl(f(x)\bigr)\mathcal{H}_{1,x}^{\beta , \mu } \biggl(\frac{\varPhi (f(x))}{f(x)}h(x)g(x) \biggr)+ \mathcal{H}_{1,x} ^{\beta ,\mu } \biggl(\frac{\varPhi (f(x))}{f(x)}h(x)g(x) \biggr) \mathcal{H}_{1,x}^{\beta ,\mu } \bigl(f(x) \bigr) \\ &\quad{} \geq \mathcal{H}_{1,x}^{\beta ,\mu } \bigl(h(x) \bigr) \mathcal{H} _{1,x}^{\beta ,\mu } \bigl(\varPhi \bigl(f(x)\bigr)g(x) \bigr)+\mathcal{H}_{1,x} ^{\beta ,\mu } \bigl(\varPhi \bigl(f(x) \bigr)g(x) \bigr)\mathcal{H}_{1,x}^{\beta ,\mu } \bigl(h(x) \bigr). \end{aligned}$$
(3.25)

It follows that

$$\begin{aligned} \frac{\mathcal{H}_{1,x}^{\beta ,\mu }(f(x))}{\mathcal{H}_{1,x}^{ \beta ,\mu } (h(x) )}\geq \frac{\mathcal{H}_{1,x}^{\beta , \mu } (\varPhi (f(x))g(x) )}{\mathcal{H}_{1,x}^{\beta ,\mu } (\frac{\varPhi (f(x))}{f(x)}h(x)g(x) )}. \end{aligned}$$
(3.26)

Hence, from (3.20) and (3.26), we obtain the required result. □

Theorem 3.4

Let f, h, and g be positive continuous functions on the interval \([1,\infty )\) and \(f\leq h\) on \([1,\infty )\). If \(\frac{f}{h}\) is decreasing and f and g are increasing on \([1,\infty )\), then for a convex function Φ with \(\varPhi (0)=0\), the generalized proportional Hadamard fractional integral operator given by (2.7) satisfies the inequality

$$\begin{aligned} \frac{\mathcal{H}_{1,x}^{\beta ,\mu } [f(x) ]\mathcal{H} _{1,x}^{\rho ,\mu } [ \varPhi (f(x))g(x) ]+\mathcal{H}_{1,x} ^{\rho ,\mu } [f(x) ]\mathcal{H}_{1,x}^{\beta ,\mu } [ \varPhi (f(x))g(x) ]}{\mathcal{H}_{1,x}^{\beta ,\mu } [h(x) ] \mathcal{H}_{1,x}^{\rho ,\mu } [ \varPhi (f(x))g(x) ]+ \mathcal{H}_{1,x}^{\rho ,\mu } [h(x) ]\mathcal{H}_{1,x} ^{\beta ,\mu } [ \varPhi (f(x))g(x) ]}\geq 1, \end{aligned}$$
(3.27)

where \(\mu \in (0,1]\), \(\beta , \rho \in \mathbb{C}\), \(\Re (\beta )>0\), and \(\Re (\rho )>0\).

Proof

Multiplying both sides of (3.27) by \(\frac{1}{\mu ^{\beta }\varGamma (\beta )}\frac{\exp [\frac{\mu -1}{\mu }(\ln x-\ln \vartheta )]( \ln x-\ln \vartheta )^{\rho -1}}{\vartheta }\), which is positive because \(\vartheta \in (1,x)\), \(\vartheta >1\) and integrating the resulting identity from 1 to x, we get

$$\begin{aligned} &\mathcal{H}_{1,x}^{\rho ,\mu }\bigl(f(x)\bigr)\mathcal{H}_{1,x}^{\beta ,\mu } \biggl(\frac{\varPhi (f(x))}{f(x)}h(x)g(x) \biggr)+ \mathcal{H}_{1,x} ^{\rho ,\mu } \biggl(\frac{\varPhi (f(x))}{f(x)}h(x)g(x) \biggr) \mathcal{H}_{1,x}^{\beta ,\mu } \bigl(f(x) \bigr) \\ &\quad \geq \mathcal{H}_{1,x}^{\beta ,\mu } \bigl(h(x) \bigr) \mathcal{H} _{1,x}^{\rho ,\mu } \bigl(\varPhi \bigl(f(x)\bigr)g(x) \bigr)+\mathcal{H}_{1,x} ^{\beta ,\mu } \bigl(\varPhi \bigl(f(x) \bigr)g(x) \bigr)\mathcal{H}_{1,x}^{\rho , \mu } \bigl(h(x) \bigr). \end{aligned}$$
(3.28)

Since \(f\leq h\) on \([1,\infty )\) and \(\frac{\varPhi (x)}{x}\) is increasing, for \(t,\vartheta \in [1,x)\), we have

$$\begin{aligned} \frac{\varPhi (f(t))}{f(t)}\leq \frac{\varPhi (h(t))}{h(t)}. \end{aligned}$$
(3.29)

Multiplying both sides of (3.29) by \(\frac{1}{\mu ^{\beta }\varGamma (\beta )}\frac{\exp [\frac{\mu -1}{\mu }(\ln x-\ln t)](\ln x-\ln t)^{ \beta -1}}{t}h(t)g(t)\), \(t\in (1,x)\), \(x>1\) and integrating the resulting identity from 1 to x, we get

$$\begin{aligned} \mathcal{H}_{1,x}^{\beta ,\mu } \biggl( \frac{\varPhi (f(x))}{f(x)}h(x)g(x) \biggr) \leq \mathcal{H}_{1,x}^{\beta ,\mu } \bigl(\varPhi \bigl(h(x)\bigr)g(x)\bigr) ). \end{aligned}$$
(3.30)

Similarly, multiplying both sides of (3.29) by \(\frac{1}{\mu ^{ \rho }\varGamma (\rho )}\frac{\exp [\frac{\mu -1}{\mu }(\ln x-\ln t)]( \ln x-\ln t)^{\rho -1}}{t}h(t)g(t)\), \(t\in (1,x)\), \(x>1\) and integrating the resulting identity from 1 to x, we get

$$\begin{aligned} \mathcal{H}_{1,x}^{\rho ,\mu } \biggl( \frac{\varPhi (f(x))}{f(x)}h(x)g(x) \biggr) \leq \mathcal{H}_{1,x}^{\rho ,\mu } \bigl(\varPhi \bigl(h(x)\bigr)g(x)\bigr) ). \end{aligned}$$
(3.31)

Hence, from (3.28), (3.30), and (3.31), we obtain the required inequality (3.31). □

Remark 3.2

If we consider \(\beta =\rho \), then Theorem 3.4 will lead to Theorem 3.3.

Concluding remarks

In this paper, first we defined nonlocal generalized proportional Hadamard fractional integral operators and then we established certain inequalities by employing the generalized proportional Hadamard fractional integral operator. The inequalities obtained in this present paper will lead to the classical inequalities which are established earlier by Chinchane and Pachpatte [15]. The results established in this paper give some contribution in the field of fractional calculus and Hadamard fractional integral inequalities. One can establish various integral inequalities by employing the newly defined Hadamard fractional integral operators.

References

  1. Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015). https://doi.org/10.1016/j.cam.2014.10.016

    MathSciNet  Article  MATH  Google Scholar 

  2. Abdeljawad, T., Baleanu, D.: Monotonicity results for fractional difference operators with discrete exponential kernels. Adv. Differ. Equ. 2017, Article ID 78 (2017). https://doi.org/10.1186/s13662-017-1126-1

    MathSciNet  Article  MATH  Google Scholar 

  3. Abdeljawad, T., Baleanu, D.: On fractional derivatives with exponential kernel and their discrete versions. Rep. Math. Phys. 80, 11–27 (2017). https://doi.org/10.1016/S0034-4877(17)30059-9

    MathSciNet  Article  MATH  Google Scholar 

  4. Adil Khan, M., Begum, S., Khurshid, Y., Chu, Y.-M.: Ostrowski type inequalities involving conformable fractional integrals. J. Inequal. Appl. 2018, Article ID 70 (2018)

    MathSciNet  Article  Google Scholar 

  5. Adil Khan, M., Chu, Y.-M., Kashuri, A., Liko, R., Ali, G.: Conformable fractional integrals versions of Hermite–Hadamard inequalities and their generalizations. J. Funct. Spaces 2018, Article ID 6928130 (2018)

    MathSciNet  MATH  Google Scholar 

  6. Adil Khan, M., Chu, Y.-M., Khan, T.U., Khan, J.: Some new inequalities of Hermite–Hadamard type for s-convex functions with applications. Open Math. 15(1), 1414–1430 (2017)

    MathSciNet  MATH  Google Scholar 

  7. Adil Khan, M., Iqbal, A., Suleman, M., Chu, Y.-M.: Hermite–Hadamard type inequalities for fractional integrals via Green’s function. J. Inequal. Appl. 2018, Article ID 161 (2018)

    MathSciNet  Article  Google Scholar 

  8. Adil Khan, M., Khurshid, Y., Du, T.-S., Chu, Y.-M.: Generalization of Hermite–Hadamard type inequalities via conformable fractional integrals. J. Funct. Spaces 2018, Article ID 5357463 (2018)

    MathSciNet  MATH  Google Scholar 

  9. Agarwal, P., Jleli, M., Tomar, M.: Certain Hermite–Hadamard type inequalities via generalized k-fractional integrals. J. Inequal. Appl. 2017, 55 (2017)

    MathSciNet  Article  Google Scholar 

  10. Aldhaifallah, M., Tomar, M., Nisar, K.S., Purohit, S.D.: Some new inequalities for \((k, s)\)-fractional integrals. J. Nonlinear Sci. Appl. 9(9), 5374–5381 (2016)

    MathSciNet  Article  Google Scholar 

  11. Alzabut, J., Abdeljawad, T., Jarad, F., Sudsutad, W.: A Gronwall inequality via the generalized proportional fractional derivative with applications. J. Inequal. Appl. 2019, Article ID 101 (2019)

    MathSciNet  Article  Google Scholar 

  12. Anderson, D.R., Ulness, D.J.: Newly defined conformable derivatives. Adv. Dyn. Syst. Appl. 10, 109–137 (2015)

    MathSciNet  Google Scholar 

  13. Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20, 763–769 (2016). https://doi.org/10.2298/TSCI160111018A

    Article  Google Scholar 

  14. Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 73–85 (2015)

    Google Scholar 

  15. Chinchane, V.L., Pachpatte, D.B.: On new fractional integral inequalities involving convex functions using Hadamard fractional integral. Bull. Allahabad Math. Soc. 31, 183–192 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Chu, Y.-M., Adil Khan, M., Ali, T., Dragomir, S.S.: Inequalities for α fractional differentiable functions. J. Inequal. Appl. 2017, Article ID 93 (2017)

    MathSciNet  Article  Google Scholar 

  17. Dahmani, Z.: New inequalities in fractional integrals. Int. J. Nonlinear Sci. 9, 493–497 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Dahmani, Z.: New classes of integral inequalities of fractional order. Matematiche 69(1), 237–247 (2014). https://doi.org/10.4418/2014.69.1.18

    MathSciNet  Article  MATH  Google Scholar 

  19. Dahmani, Z., Tabharit, L.: On weighted Grüss type inequalities via fractional integration. J. Adv. Res. Pure Math. 2, 31–38 (2010)

    MathSciNet  Article  Google Scholar 

  20. Huang, C.J., Rahman, G., Nisar, K.S., Ghaffar, A., Qi, F.: Some inequalities of Hermite–Hadamard type for k-fractional conformable integrals. Aust. J. Math. Anal. Appl. 16(1), 1–9 (2019)

    MathSciNet  MATH  Google Scholar 

  21. Jarad, F., Abdeljawad, T., Alzabut, J.: Generalized fractional derivatives generated by a class of local proportional derivatives. Eur. Phys. J. Spec. Top. 226, 3457–3471 (2017). https://doi.org/10.1140/epjst/e2018-00021-7

    Article  Google Scholar 

  22. Jarad, F., Ugurlu, E., Abdeljawad, T., Baleanu, D.: On a new class of fractional operators. Adv. Differ. Equ. 2017, Article ID 247 (2017)

    MathSciNet  Article  Google Scholar 

  23. Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)

    MathSciNet  Article  Google Scholar 

  24. Khurshid, Y., Adil Khan, M., Chu, Y.-M.: Conformable integral inequalities of the Hermite–Hadamard type in terms of GG- and GA-2 convexities. J. Funct. Spaces 2019, Article ID 6926107 (2019)

    MathSciNet  MATH  Google Scholar 

  25. Khurshid, Y., Adil Khan, M., Chu, Y.-M., Khan, Z.A.: Hermite–Hadamard–Fejer inequalities for conformable fractional integrals via preinvex functions. J. Funct. Spaces 2019, Article ID 3146210 (2019)

    MathSciNet  MATH  Google Scholar 

  26. Losada, J., Nieto, J.J.: Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 87–92 (2015)

    Google Scholar 

  27. Mubeen, S., Habib, S., Naeem, M.N.: The Minkowski inequality involving generalized k-fractional conformable integral. J. Inequal. Appl. 2019, Article ID 81 (2019). https://doi.org/10.1186/s13660-019-2040-8

    MathSciNet  Article  Google Scholar 

  28. Niasr, K.S., Tassadiq, A., Rahman, G., Khan, A.: Some inequalities via fractional conformable integral operators. J. Inequal. Appl. 2019, 217 (2019). https://doi.org/10.1186/s13660-019-2170-z

    MathSciNet  Article  Google Scholar 

  29. Nisar, K.S., Qi, F., Rahman, G., Mubeen, S., Arshad, M.: Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function. J. Inequal. Appl. 2018, Article ID 135 (2018)

    MathSciNet  Article  Google Scholar 

  30. Nisar, K.S., Rahman, G., Choi, J., Mubeen, S., Arshad, M.: Certain Gronwall type inequalities associated with Riemann–Liouville k- and Hadamard k-fractional derivatives and their applications. East Asian Math. J. 34(3), 249–263 (2018)

    MATH  Google Scholar 

  31. Nisar, K.S., Rahman, G., Mehrez, K.: Chebyshev type inequalities via generalized fractional conformable integrals. J. Inequal. Appl. 2019, 245 (2019). https://doi.org/10.1186/s13660-019-2197-1

    MathSciNet  Article  Google Scholar 

  32. Qi, F., Rahman, G., Hussain, S.M., Du, W.S., Nisar, K.S.: Some inequalities of Čebyšev type for conformable k-fractional integral operators. Symmetry 10, 614 (2018). https://doi.org/10.3390/sym10110614

    Article  Google Scholar 

  33. Rahman, G., Abdeljawad, T., Khan, A., Nisar, K.S.: Some fractional proportional integral inequalities. J. Inequal. Appl. 2019, 244 (2019). https://doi.org/10.1186/s13660-019-2199-z

    MathSciNet  Article  Google Scholar 

  34. Rahman, G., Khan, A., Abdeljawad, T., Nisar, K.S.: The Minkowski inequalities via generalized proportional fractional integral operators. Adv. Differ. Equ. 2019, 287 (2019). https://doi.org/10.1186/s13662-019-2229-7

    MathSciNet  Article  Google Scholar 

  35. Rahman, G., Nisar, K.S., Mubeen, S., Choi, J.: Certain inequalities involving the \((k,\rho )\)-fractional integral operator. Far East J. Math. Sci.: FJMS 103(11), 1879–1888 (2018)

    Google Scholar 

  36. Rahman, G., Nisar, K.S., Qi, F.: Some new inequalities of the Grüss type for conformable fractional integrals. AIMS Math. 3(4), 575–583 (2018)

    Article  Google Scholar 

  37. Rahman, G., Ullah, Z., Khan, A., Set, E., Nisar, K.S.: Certain Chebyshev type inequalities involving fractional conformable integral operators. Mathematics 7, 364 (2019). https://doi.org/10.3390/math7040364

    Article  Google Scholar 

  38. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications. Gordon & Breach, Yverdon (1993) Edited and with a foreword by S.M. Nikol’skǐ. Translated from the 1987 Russian original. Revised by the authors

    MATH  Google Scholar 

  39. Sarikaya, M.Z., Budak, H.: Generalized Ostrowski type inequalities for local fractional integrals. Proc. Am. Math. Soc. 145(4), 1527–1538 (2017)

    MathSciNet  MATH  Google Scholar 

  40. Sarikaya, M.Z., Dahmani, Z., Kiris, M.E., Ahmad, F.: \((k, s)\)-Riemann–Liouville fractional integral and applications. Hacet. J. Math. Stat. 45(1), 77–89 (2016)

    MathSciNet  MATH  Google Scholar 

  41. Set, E., Noor, M.A., Awan, M.U., Gözpinar, A.: Generalized Hermite–Hadamard type inequalities involving fractional integral operators. J. Inequal. Appl. 2017, 169 (2017)

    MathSciNet  Article  Google Scholar 

  42. Set, E., Tomar, M., Sarikaya, M.Z.: On generalized Grüss type inequalities for k-fractional integrals. Appl. Math. Comput. 269, 29–34 (2015)

    MathSciNet  MATH  Google Scholar 

  43. Song, Y.-Q., Adil Khan, M., Zaheer Ullah, S., Chu, Y.-M.: Integral inequalities involving strongly convex functions. J. Funct. Spaces 2018, Article ID 6595921 (2018)

    MathSciNet  MATH  Google Scholar 

  44. Wang, M.-K., Chu, H.-H., Chu, Y.-M.: Precise bounds for the weighted Hölder mean of the complete p-elliptic integrals. J. Math. Anal. Appl. 480(2), 123388 (2019). https://doi.org/10.1016/j.jmaa.2019.123388

    MathSciNet  Article  MATH  Google Scholar 

  45. Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On rational bounds for the gamma function. J. Inequal. Appl. 2017, Article ID 210 (2017)

    MathSciNet  Article  Google Scholar 

  46. Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: Monotonicity rule for the quotient of two functions and its application. J. Inequal. Appl. 2017, Article ID 106 (2017)

    MathSciNet  Article  Google Scholar 

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The second author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

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Rahman, G., Abdeljawad, T., Jarad, F. et al. Certain inequalities via generalized proportional Hadamard fractional integral operators. Adv Differ Equ 2019, 454 (2019). https://doi.org/10.1186/s13662-019-2381-0

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MSC

  • 26A33
  • 26D10
  • 26D53
  • 05A30

Keywords

  • Fractional integrals
  • Generalized proportional Hadamard fractional integrals
  • Inequalities