On some fractional integral inequalities for generalized strongly modified $h$-convex functions

1.   Introduction

Linear functions are considered as simplest functions in linear spaces. The class of functions and sets that are just a step more complicated then linear ones namely convex functions and convex sets.

The subset $C$ of $\mathbb{R}^{n}$ is said to be convex if

$\forall x, y\in C$, $p\in(0, 1)$ and $q = 1-p$. The function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is said to be convex if its epigraph is convex subset of $\mathbb{R}$.The convexity of sets and functions are the the objects of many studies during the past few decades. The convexity of a function and set make it so special because of its interesting properties like convex function has global minima, it has non-empty relative interior and convex set is connected having feasible directions at any point.

Some of early contributions to convex analysis were made by Holder, Jensen and Minkowski. The importance of convex analysis is well known in optimization theory ^[2,3], inspite of many applications, many recent problems in economics and engineering the notion of convexity does not longer suffices. Hence it is always necessary to extend the notion of convexity to some general form to meet recent problems see ^{[4,5,6,7,8,9,10]}, for further reading on fractional integral inequalities we refer ^{[11,12,13,14,15,16,17]}. Moreover, the new inequalities in analysis is always appreciable. The present paper is organized as follow: in the second section, we give some preliminary material. In the third section, we derive some fractional integral inequalities for generalized strongly modified $h$-convex function, whereas in the fourth section. we present applications of results to the mean. Finally, we conclude our results.

2.   Preliminaries

We start from some preliminaries material and basic definitions.

Definition 2.1. ^[18] Let $f:\varphi\rightarrow\mathbb{R}$ be an extended-real-valued function define on a convex set $\varphi\subset\mathbb{R}^{n}$. Then the function $f$ is convex on $\varphi$ if

for all $b_1, b_{2}\in\varphi$ and $t\in(0, 1)$.

Definition 2.2. ^[19] Choose the functions $f, h:J\subset\mathbb{R}\rightarrow\mathbb{R}$ are non-negative. Then $f$ is called $h$-convex function if

for all $b_1, b_{2}\in J$ and $t\in[0, 1]$.

Definition 2.3. ^[20] Choose the functions $f, h:J\subset\mathbb{R}\rightarrow\mathbb{R}$ are non-negative. Then $f$ is called modified $h$-convex function if

for all $b_1, b_{2}\in J$ and $t\in[0, 1]$.

Definition 2.4. ^[21] Let $\varphi$ be an interval in real line $\mathbb{R}$. A function $f:\varphi = [b_{1}, b_{2}]\rightarrow\mathbb{R}$ is said to be generalized convex with respect to an arbitrary bifunction $\eta(b_{1}, b_{2}):\mathbb{E}\times\mathbb{E}\rightarrow\mathbb{F}$ where $E, F\in\mathbb{R}$ if

for all $b_1, b_{2}\in \varphi, t\in[0, 1]$.

Definition 2.5. A function $f:\varphi = [b_1, b_{2}]\rightarrow\mathbb{R}$ is called $\eta_{h}$ convex function if

for all $b_1, b_{2}\in \varphi, t\in[0, 1]$ and $h:J\rightarrow\mathbb{R}$ is a non-negative function.

Definition 2.6. ^[22] A function $f:\varphi = [b_1, b_{2}]\rightarrow\mathbb{R}$ is called strongly convex function with modulus $\mu$ on $\varphi$, where $\mu\geq0$ if

for all $b_1, b_{2}\in\varphi$ and $t\in[0, 1]$.

Definition 2.7. ^[23] A function $f:J \subset\mathbb{R}\rightarrow\mathbb{R}$ is said to be strongly $\eta$-convex function with respect to $\eta:\mathbb{E}\times\mathbb{E}\rightarrow \mathbb{F}$ where $E, F\in\mathbb{R}$ and modulus $\mu\geq 0$, if

for all $b_1, b_{2}\in J, t\in[0, 1]$.

Definition 2.8. ^[24] Choose the functions $f, h:J\subset\mathbb{R}\rightarrow\mathbb{R}$ are non-negative. Then $f$ is called generalized strongly modified $h$-convex function if

for all $b_1, b_{2}\in J$ and $t\in[0, 1]$.

Definition 2.9. ^[12] Let $0 < s\leq1$. A function $f:J\subset\mathbb{R}\rightarrow\mathbb{R}$ is called $s$-$\phi$ -$convex$ with respect to bifunction $\phi:\mathbb{E}\times\mathbb{E}\rightarrow\mathbb{F}$ where $E, F\in\mathbb{R}$ (briefly $\phi$-$convex$) if

The next remark provides the relations among the convexities.

Remark 1. 1. If $\eta(b_1, b_{2}) = b_1-b_{2}$ then, (2.4) reduces to (2.1);

2. If $h(t) = t$ then, (2.5) reduces to (2.4);

3. If $h(t) = t$ and $\eta(b_1, b_{2}) = b_1-b_{2}$ then, (2.5) reduces to(2.1);

4. If $\eta(b_1, b_{2}) = b_1-b_{2}$ then, (2.5) reduces to (2.3);

5. If $\mu = 0$ and $\eta(b_1, b_{2}) = b_1-b_{2}$ then, (2.8) reduces to (2.3);

6. If $\mu = 0$, $\eta(b_1, b_{2}) = b_1-b_{2}$ and $h(t) = t$ then, (2.8) reduces to(2.1);

7. If $\mu = 0$ then, (2.8) reduces to (2.5);

8. If $h(t) = t$ then, (2.8) reduces to (2.7);

9. If $\mu = 0$ and $h(t) = t^{s}$ then, (2.8) reduces to (2.9).

Utilization of more complicated convex functions

Most of the modern problems in engineering and other applied sciences are non-convex in nature. So it is difficult to reach at favorite results by only the classical convexity. That's why the convexity is generalized in many directions. To understand the generalization of convexity it may categorize as:

Some generalization are made to change the form of defining e.g. quasi convex ^[26], pseudo convex ^[27] and strongly convex ^[28].

Some generalizations are made by expanding the domain e.g. ^[29] and some generalization are made by changing the range set of convex functions e.g. ^[30]. So generalizations the convex is always appreciable.

The next lemmas are useful in proving the main results.

Lemma 2.10. ^[31] Let $f:J\subseteq\mathbb{R}\rightarrow\mathbb{R}$, be a differentiable mapping on $\mathring{J}$ such that $f' \in L^{1}[b_{1}, b_{2}]$, where $b_1, b_{2} \in J$ with $b_1 < b_{2}.$ If $\alpha, \beta \in \mathbb{R}$, then

Lemma 2.11. ^[31] For $s > 0$ and $0\leq\varepsilon\leq1$, we have

Lemma 2.12. ^[32] Let $f:J\rightarrow \mathbb{R}$, $J \subseteq\mathbb{R}$ be a differentiable mapping on $\mathring{J}$ with $f'' \in L^{1}[b_1, b_{2}],$ where $b_1, b_{2} \in J, b_1 < b_{2}$, then

Lemma 2.13. ^[23] If $f^{n}$ for $n\epsilon\mathbb{N}$ exists and is integrable on $[b_{1}, b_{2}]$, then

Lemma 2.14. ^[12] Suppose that $f:[b_{1}, b_{2}]\rightarrow\mathbb{R}$ is a differentiable function, $g:[b_{1}, b_{2}]\rightarrow\mathbb{R}^{+}$ is a continuous function and symmetric about $\frac{b_{1}+b_{2}}{2}$ and $f'$ is an integrable function on $[b_{1}, b_{2}]$. Then

3.   Fractional integral inequalities

Theorem 3.1. Let $f:J\rightarrow \mathbb{R}$, $J\subseteq\mathbb{R}$ be a differentiable mapping on $\mathring{J}$ with $f' \in L^{1}[b_1, b_{2}],$ where $b_1, b_{2} \in J, b_1 < b_{2}$. If $|f'(x)|^{q}$ for $q\geq 1$ and $0\leq \alpha, \beta\leq 1$, is generalized strongly modified $h$-convex function on $[b_1, b_{2}]$, then

Proof. The proof begins with $f'(x)\in [b_{1}, b_{2}]$, then using Lemma (2.10), and power mean inequality we have for $q > 1$

Using Lemma (2.11), we have

And

Substituting values from Eqs (3.3), (3.4) in inequality (3.2), we obtain

For $q = 1$, using Lemma (2.10) and Lemma (2.11), we have

This completes the proof.

Remark 2. If we take $h(t) = t$ and $\mu = 0$ then inequality (3.1) reduces to inequality (13) in ^[33].

Taking $\alpha = \beta$ in Theorem (3.1), we have following corollary.

Corollary 1. Let $f:J\rightarrow \mathbb{R}$, $J\subseteq\mathbb{R}$ be a differentiable mapping on $\mathring{J}$ with $f' \in L^{1}[b_1, b_{2}],$ where $b_1, b_{2} \in J, b_1 < b_{2}$. If $|f'(x)|^{q}$ for $q\geq 1$ is generalized strongly modified $h$-convex function on $[b_1, b_{2}]$ and $0\leq \alpha, \beta\leq 1$, then

Remark 3. If we take $h(t) = t$ and $\mu = 0$ then inequality (3.6) reduces to inequality (16) in ^[33].

By choosing $\alpha = \beta = \frac{1}{2}, \frac{1}{3}$ in Theorem (3.1) respectively, we obtain following corollary.

Corollary 2. Let $f:J\rightarrow \mathbb{R}$, $J\subseteq\mathbb{R}$ be a differentiable mapping on $\mathring{J}$ with $f' \in L^{1}[b_1, b_{2}],$ where $b_1, b_{2} \in J, b_1 < b_{2}$. If $|f'(x)|^{q}$ for $q\geq 1$ is generalized strongly modified $h$-convex function on $[b_1, b_{2}]$ and $0\leq \alpha, \beta\leq 1$, then

and

Remark 4. Setting $q = 1$ in Corollary (2), we have the following result.

Corollary 3. Let $f:J\rightarrow \mathbb{R}$, $J\subseteq\mathbb{R}$ be a differentiable mapping on $\mathring{J}$ with $f' \in L^{1}[b_1, b_{2}],$ where $b_1, b_{2} \in I, b_1 < b_{2}$. If $|f'(x)|^{q}$ for $q\geq 1$ is generalized strongly modified $h$-convex function on $[b_1, b_{2}]$ and $0\leq \alpha, \beta\leq 1$, then

and

Remark 5. If we take $h(t) = t$ and $\mu = 0$ then inequalities (3.7)–(3.10) reduce to inequalities (17) and (18) in ^[33].

Theorem 3.2. Let $f:J\subset[0, 1)\rightarrow \mathbb{R}$ be a differentiable mapping on $\mathring{J}$ with $f'' \in L^{1}[b_1, b_{2}]$, where $b_1, b_{2} \in J$ and $b_1 < b_{2}$. If $\left|f''\right|$ is genarilized strongly modified $h$-convex on $[b_1, b_{2}]$, then

Proof. From Lemma (2.12), we have

Since $\left|f''\right|$ is generalized strongly modified $h$ convex function, so

And

This completes the proof.

Remark 6. If we take $h(t) = t$ and $\mu = 0$ then inequality (3.11) reduces to inequality (24) in ^[33].

Theorem 3.3. Let $f:J\subset[0, 1)\rightarrow \mathbb{R}$ be a differentiable mapping on $\mathring{J}$ with $f'' \in L^{1}[b_1, b_{2}]$, where $b_1, b_{2} \in J$ and $b_1 < b_{2}$.If $\left|f''\right|^{q}$ for $q\geq 1$ with $\frac{1}{p}+\frac{1}{q} = 1$ is generelized strongly modified $h$-convx on $[b_1, b_{2}]$, then

Proof. Suppose that p$\geq1$, using Lemma (2.12) and power mean inequality, we have

Since $\left|f''\right|^{q}$ is generalized strongly modified $h$-convex, then we have

And

After simplification, we have

Which completes the proof.

Remark 7. If we take $h(t) = t$ and $\mu = 0$, then inequality (3.13) reduces to inequality (25) in ^[33].

Theorem 3.4. Let $f:\mathring{J} \subset R\rightarrow R$ be a n-times differentiable generalized strongly modified $h$-convex, function on $\mathring{J}$ where $b_{1}, b_{2}\in\mathring{J}$ with $b_{1} < b_{2}$ and $f^{'}\in L^{1}[b_{1}, b_{2}]$. If $\left|f^{'}\right|^{p}$ is generalized strongly modified $h$-convex, function with $\mu\geq1$, then for $n\geq2$ and $p\geq1$, we have

Proof. Case-i: Since it is known that $\left|f^{'}\right|$ is generalized strongly modified $h$-convex function, then using the property of modules, and Lemma (2.13), we have following inequality for $p = 1$

Using the definition of generalized strongly modified $h$-convex function, we have

As

Substituting (3.17) and (3.18) in (3.16), we have

Case-ii For $p > 1$ applying Holder inequality, we have

Using definition of generalized strongly modified $h$-convex function, we have

Which completes the proof.

Remark 8. If we take $h(t) = t$ then Theorem (3.13) reduces to Theorem (2.5) in ^[23].

Theorem 3.5. Suppose that $f:[b_{1}, b_{2}]\rightarrow\mathbb{R}$ is a differentiable function, $g:[b_{1}, b_{2}]\rightarrow\mathbb{R}^{+}$ is a continuous function and symmetric about $\frac{b_{1}+b_{2}}{2}$ and $\left|f'\right|$ is a generalized strongly modified $h$-convex function. Then

where $k = max_{t\in[0, 1]}\left|g(t)\right|$ and $g(t) = h\left(\frac{1-t}{2}\right)+h\left(\frac{1+t}{2}\right).$

Proof. From Lemma (2.14) and the fact that $\left| f'\right|$ is generalized strongly modified $h$-convex, we have

where $k = max_{t\in[0, 1]}\left|g(t)\right|$ and $g(t) = h\left(\frac{1-t}{2}\right)+h\left(\frac{1+t}{2}\right)$.

Remark 9. If we take $\mu = 0, h(t) = t^{s}$ then Theorem (3.15) reduces to Theorem (3) in ^[12].

Corollary 4. In theorem (3.15) if we choose $h(t) = t$ then $k = 1$ and $\mu = 0$, we have the inequality of the theorem (2) (Gordji, Dragomir and Delaver).

Corollary 5. In corollary (3.17) if we choose $g = 1, \eta(x, y) = x-y$, we have the following inequality

for convex functions that is equivalent to Theorem (1.2) in ^[1].

4.   Application to means

For two positive numbers $b_1 > 0$ and $b_2 > 0$, define

for $0 \leq w \leq \infty$. These means are respectively called the arithematic, geometric, harmonic, generalized logarithmic, identric and Heronian means of two positive numbers $b_1$ and $b_2$.

Applying Theorem (3.1) to $f(x) = x^s$ for $s \neq 0$ and $x > 0$ result in the following inequalities for means.

Theorem 4.1. Let $b_1 > 0$, $b_2 > 0$, $q\geq 1$, either $s > 1$ and $(s -1)q \geq 1$ or $s < 0$. Then

Taking $f(x) = lnx$ for $x > 0$ in Theorem (3.1) results in the following inequality for means.

Theorem 4.2. For $b_{1} > 0$, $b_{2} > 0$, $b_{1}\neq b_{2}$ and $q\geq1$, we have

Finally, we can establish an inequality for the Heronian mean as follows.

Theorem 4.3. For $b_2 > b_1 > 0$, $b_{1}\neq b_{2}$ $w \geq 0$ and $s \geq 4$ or $0 \neq s < 1$, we have

Proof. Let $f(x) = \frac{x^s + wx^{\frac{s}{2}} + 1}{w + 2}$ for $x > 0$ and $s \notin (1, 4)$. Then

By corollary (3.6) it follows that

On the other hand, we have

Obviously (4.5) and (4.6) yield (4.4).

5.   Conclusions

Fractional differential and integral equations play increasingly important roles in the modeling of engineering and science problems. It has been established fact that, in many situations, these models provide more suitable results than analogous models with integer derivatives. Fractional integral inequality results when $0 < q < 1$ can be developed when the nonlinear term is increasing and satisfies a one sided Lipschitz condition. Using the integral inequality result and the computation of the solution of the linear fractional equation of variable coefficients, Gronwall inequality results can be established. In the present report, we developed the fractional integral inequalities for more broader class of convex functions named as generalized strongly modified $h$-convex functions, we also established some applications of derived inequalities to means. Our results extend and generalize many existing results, for example ^{[1,23,33,34,35]}.

Acknowledgment

1. This work is supported by shandong Provincial Education science "12th Five-Year Plan" project (code: CBS15007), Shandong Provincial Humanities and Social Science Research Project (code: J16WB01) and Shandong Huayu university of Technology achievement Cultivation project (Practical Research on the teaching Reform of advanced Mathematics course in application-oriented undergraduate course of "curriculum thought and politics + Mixed learning").

2. The research was supported by the National Natural Science Foundation of China (Grant Nos. 11971142, 11871202, 61673169, 11701176, 11626101, 11601485).

3. Authors are thankful to both reviewers and editor for their valuable time and constructive comments.

Conflict of interest

The authors declare that no competing interests exist.