# Hermite-Hadamard Type Inequalities Obtained via Fractional Integral for Differentiablem-Convex and α,m-Convex Functions

Authors: Erhan Set; Süleyman Sami Karataş; Muhammad Adil Khan
Journal: International Journal of Analysis
 (2016)

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## Abstract

Some Hermite-Hadamard type inequalities involving fractional integrals form-convex and (α,m)-convex functions are obtained.


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## Body

## 1. Introduction and Preliminaries

The following definition is well known in the literature.Definition 1.
A functionf:I→R,∅≠I⊆R, is said to be convex on the interval I if the inequality (1)ftx+1-ty≤tfx+1-tfyholds for all x,y∈I and t∈[0,1].Geometrically, this means that ifP,Q, and R are three points on the graph of f with Q between P and R, then Q is on or below the chord PR.Theorem 2 (Hermite-Hadamard inequality).
Letf:I⊂R→R be a convex function and a,b∈I with a<b. Then(2)fa+b2≤1b-a∫abfxdx≤fa+fb2.m-convexity was defined by Toader as follows.Definition 3 (see [1]).
The functionf:[0,b]→R,b>0, is said to be m-convex, where m∈[0,1], if one has (3)ftx+m1-ty≤tfx+m1-tfy for all x,y∈[0,b] and t∈[0,1]. One says that f is m-concave if -f is m-convex. Denote by Km(b) the class of all m-convex functions on [0,b] for which f(0)≤0.Obviously, form=1, Definition 3 recaptures the concept of standard convex functions on [a,b] and for m=0 the concept of starshaped functions. The notion of m-convexity has been further generalized in [2] as it is stated in the following definition.Definition 4 (see [2]).
The functionf:[0,b]→R,b>0, is said to be (α,m)-convex, where (α,m)∈0,12, if one has (4)ftx+m1-ty≤tαfx+m1-tαfy for all x,y∈[0,b] and t∈[0,1].
Denote byKmα(b) the class of all (α,m)-convex functions on [0,b] for which f(0)≤0.It can be easily seen that when(α,m)∈{(1,1),(1,m)} one obtains the following classes of functions: convex and m-convex, respectively. Note that K11(b) is a proper subclass of m-convex and (α,m)-functions [0,b]. The interested reader can find more about partial ordering of convexity in [3].Definition 5 (see [4]).
Letf∈L1[a,b]. Then Riemann-Liouville integrals Ja+αf and Jb-αf of order α>0 with a≥0 are defined by (5)Ja+αfx=1Γα∫axx-tα-1ftdt,x>a,Jb-αfx=1Γα∫xbt-xα-1ftdt,x<b,where(6)Γα=∫0∞e-xxα-1dx is the Gamma function.We now give the definition of the hypergeometric series which will be used in obtaining some integrals.Definition 6 (see [4]).
The integral representation of the hypergeometric functions is as follows:(7)F12a,b,c;z=1Bb,c-b∫01tb-11-tc-b-11-zt-adt,where z<1,c>b>0, and(8)Bx,y=∫01tx-11-ty-1dtis Beta function with (9)Bx,y=ΓxΓyΓx+y.In the present paper, we establish some new Hermite-Hadamard’s type inequalities for the classes ofm-convex and (α,m)-convex functions via Riemann-Liouville fractional integrals.To prove our main results, we consider the following lemma.Lemma 7 (see [5]).
Letf:[a,b]⊂R→R be a differentiable function such that f′∈L[a,b]. Then, for n∈N, k>0, and x∈[a,b], one has(10)Gk;n;a,x,bf=x-ak+1b-a∫01tk2f′n+tn+1x+1-tn+1adt-∫01tk2f′1-tn+1x+n+tn+1adt-b-xk+1b-a∫01tk2f′n+tn+1x+1-tn+1bdt-∫01tk2f′1-tn+1x+n+tn+1bdt,where(11)Gk;n;a,x,bf=n+12x-ak+b-xkb-afx+x-akfa+b-xkfbb-a-n+1k+1Γk+12b-aJx-kfnn+1x+1n+1a+Ja+kf1n+1x+nn+1a+Jx+kfnn+1x+1n+1b+Jb-kf1n+1x+nn+1a.


## 2. Generalized Inequalities form-Convex Functions

Theorem 8.
LetI be on open real interval such that [0,∞)⊂I. Let f:I→R be a differentiable function on I such that n∈N, k>0 and f′∈L[a,b], where 0≤a<b<∞. If f′ is an m-convex function on [a,b] for some fixed m∈(0,1], then(12)Gk;n;a,x,bf≤x-ak+12b-ak+1f′x+mf′am+b-xk+12b-ak+1f′x+mf′bm,where x∈[a,b].Proof.
Using Lemma7, taking modulus and the fact that f′ is an m-convex function, we have(13)Gk;n;a,x,bf≤x-ak+1b-a∫01tk2f′n+tn+1x+1-tn+1adt+∫01tk2f′1-tn+1x+n+tn+1adt+b-xk+1b-a∫01tk2f′n+tn+1x+1-tn+1bdt+∫01tk2f′1-tn+1x+n+tn+1bdt=x-ak+1b-a∫01tk2f′n+tn+1x+m1-tn+1amdt+∫01tk2f′1-tn+1x+mn+tn+1amdt+b-xk+1b-a∫01tk2f′n+tn+1x+m1-tn+1bmdt+∫01tk2f′1-tn+1x+mn+tn+1bmdt≤x-ak+12b-ak+1f′x+mf′am+b-xk+12b-ak+1f′x+mf′bm.
This completes the proof.Remark 9.
Observe that if in Theorem8 we have m=n=1, the statement of Theorem 8 becomes the statement of Theorem  1 in [6].Theorem 10.
LetI be on open real interval such that [0,∞)⊂I. Let f:I→R be a differentiable function on I such that n∈N, k>0 and f′∈L[a,b], where 0≤a<b<∞. If f′q is an m-convex function on [a,b] for some fixed m∈(0,1] and 1/p+1/q=1, q>1, then(14)Gk;n;a,x,bf≤x-ak+12b-a1kp+11/p1n+11/q×2n+1f′xq+mf′a/mq21/q+f′xq+m2n+1f′a/mq21/q+b-xk+12b-a1kp+11/p1n+11/q×2n+1f′xq+mf′b/mq21/q+f′xq+m2n+1f′b/mq21/q,where x∈[a,b].Proof.
Using Lemma7, Hölder’s inequality, and the fact that f′q is an m-convex function, (15)Gk;n;a,x,bf≤x-ak+1b-a∫01tk2f′n+tn+1x+1-tn+1adt+∫01tk2f′1-tn+1x+n+tn+1adt+b-xk+1b-a∫01tk2f′n+tn+1x+1-tn+1bdt+∫01tk2f′1-tn+1x+n+tn+1bdt≤x-ak+1b-a∫01tk2pdt1/p∫01f′n+tn+1x+1-tn+1aqdt1/q+∫01f′1-tn+1x+n+1n+1aqdt1/q+b-xk+1b-a∫01tk2pdt1/p∫01f′n+tn+1x+1-tn+1bqdt1/q+∫01f′1-tn+1x+n+1n+1bqdt1/q≤x-ak+12b-a1kp+11/p1n+11/q∫01n+tf′xq+m1-tf′amqdt1/q+∫011-tf′xq+mn+tf′amqdt1/q+b-xk+12b-a1kp+11/p1n+11/q∫01n+tf′xq+m1-tf′bmqdt1/q+∫011-tf′xq+mn+tf′bmqdt1/q=x-ak+12b-a1kp+11/p1n+11/q×2n+1f′xq+mf′a/mq21/q+f′xq+m2n+1f′a/mq21/q+b-xk+12b-a1kp+11/p1n+11/q×2n+1f′xq+mf′b/mq21/q+f′xq+m2n+1f′b/mq21/q.This completes the proof.Remark 11.
Observe that if in Theorem10 we have m=n=1, the statement of Theorem 10 becomes the statement of Theorem  2 in [6].Theorem 12.
LetI be on open real interval such that [0,∞)⊂I. Let f:I→R be a differentiable function on I such that n∈N, k>0 and f′∈L[a,b], where 0≤a<b<∞. If f′q is an m-convex function on [a,b] for some fixed m∈(0,1] and q≥1, then(16)Gk;n;a,x,bf≤x-ak+12b-ak+11k+2n+11/qnk+2+k+1f′xq+mf′amq1/q+f′xq+mnk+2+k+1f′amq1/q+b-xk+12b-ak+11k+2n+11/qnk+2+k+1f′xq+mf′bmq1/q+f′xq+mnk+2+k+1f′bmq1/q,where x∈[a,b].Proof.
Using Lemma7, Power’s mean inequality, and the fact that f′q is an m-convex function,(17)Gk;n;a,x,bf≤x-ak+1b-a∫01tk2f′n+tn+1x+1-tn+1adt+∫01tk2f′1-tn+1x+n+tn+1adt+b-xk+1b-a∫01tk2f′n+tn+1x+1-tn+1bdt+∫01tk2f′1-tn+1x+n+tn+1bdt≤x-ak+1b-a∫01tk2dt1-1/q∫01tk2f′n+tn+1x+1-tn+1aqdt1/q+∫01tk2f′1-tn+1x+n+1n+1aqdt1/q+b-xk+1b-a∫01tk2dt1-1/q∫01tk2f′n+tn+1x+1-tn+1bqdt1/q+∫01tk2f′1-tn+1x+n+1n+1bqdt1/q≤x-ak+12b-a1k+11-1/q1n+11/q∫01tkn+tf′xq+m1-tf′amqdt1/q+∫01tk1-tf′xq+mn+tf′amqdt1/q+b-xk+12b-a1k+11-1/q1n+11/q∫01tkn+tf′xq+m1-tf′bmqdt1/q+∫01tk1-tf′xq+mn+tf′bmqdt1/q=x-ak+12b-ak+11k+2n+11/qnk+2+k+1f′xq+mf′amq1/q+f′xq+mnk+2+k+1f′amq1/q+b-xk+12b-ak+11k+2n+11/qnk+2+k+1f′xq+mf′bmq1/q+f′xq+mnk+2+k+1f′bmq1/q.This completes the proof.Remark 13.
Observe that if in Theorem12 we have m=n=1, the statement of Theorem 12 becomes the statement of Theorem  3 in [6].


## 3. Generalized Inequalities forα,m-Convex Functions

Theorem 14.
LetI be on open real interval such that [0,∞)⊂I. Let f:I→R be a differentiable function on I such that n∈N, k>0, and f′∈L[a,b], where 0≤a<b<∞. If f′ is (α,m)-convex function on [a,b] for some fixed (α,m)∈(0,1]2, then(18)Gk;n;a,x,bf≤x-ak+12b-an+1αAf′x+mB-Af′am+b-xk+12b-an+1αAf′x+mB-Af′bm,where x∈[a,b] and(19)A=nαk+1F21-α,k+1,k+2;-1n+Γα+1Γk+1Γα+k+2B=2n+1αk+1.Proof.
Using Lemma7 and taking modulus and the fact that f′ is α,m-convex function, we have(20)Gk;n;a,x,bf≤x-ak+1b-a∫01tk2f′n+tn+1x+1-tn+1adt+∫01tk2f′1-tn+1x+n+tn+1adt+b-xk+1b-a∫01tk2f′n+tn+1x+1-tn+1bdt+∫01tk2f′1-tn+1x+n+tn+1bdt≤x-ak+1b-a∫01tk2n+tn+1αf′x+m1-n+tn+1αf′amdt+∫01tk21-tn+1αf′x+m1-1-tn+1αf′amdt+b-xk+1b-a∫01tk2n+tn+1αf′x+m1-n+tn+1αf′bmdt+∫01tk21-tn+1αf′x+m1-1-tn+1αf′bmdt=x-ak+12b-an+1αAf′x+mB-Af′am+b-xk+12b-an+1αAf′x+mB-Af′bm.This completes the proof.Remark 15.
Observe that if in Theorem14 we have α=1, the statement of Theorem 14 becomes the statement of Theorem 8.Theorem 16.
LetI be on open real interval such that [0,∞)⊂I. Let f:I→R be a differentiable function on I such that n∈N, k>0, and f′∈L[a,b], where 0≤a<b<∞. If f′q is (α,m)-convex function on [a,b] for some fixed (α,m)∈(0,1]2 and 1/p+1/q=1, q>1, then(21)Gk;n;a,x,bf≤x-ak+1b-a1pk+11/p1n+1α/qCf′xq+mD-Cf′amq1/q+D-Cf′xq+mCf′amq1/q+b-xk+1b-a1pk+11/p1n+1α/qCf′xq+mD-Cf′bmq1/q+D-Cf′xq+mCf′bmq1/q,where x∈[a,b] and(22)C=n+1α-nαα+1D=n+1α.Proof.
Using Lemma7, Hölder’s inequality, and the fact that f′q is α,m-convex function, (23)Gk;n;a,x,bf≤x-ak+1b-a∫01tk2f′n+tn+1x+1-tn+1adt+∫01tk2f′1-tn+1x+n+tn+1adt+b-xk+1b-a∫01tk2f′n+tn+1x+1-tn+1bdt+∫01tk2f′1-tn+1x+n+tn+1bdt≤x-ak+1b-a∫01tk2pdt1/p∫01f′n+tn+1x+1-tn+1aqdt1/q+∫01f′1-tn+1x+n+1n+1aqdt1/q+b-xk+1b-a∫01tk2pdt1/p∫01f′n+tn+1x+1-tn+1bqdt1/q+∫01f′1-tn+1x+n+1n+1bqdt1/q≤x-ak+1b-a1pk+11/p∫01n+tn+1αf′xq+m1-n+tn+1αf′amqdt1/q+∫011-tn+1αf′xq+m1-1-tn+1αf′amqdt1/q+b-xk+1b-a1pk+11/p∫01n+tn+1αf′xq+m1-n+tn+1αf′bmqdt1/q+∫011-tn+1αf′xq+m1-1-tn+1αf′bmqdt1/q=x-ak+1b-a1pk+11/p1n+1α/qCf′xq+mD-Cf′amq1/q+D-Cf′xq+mCf′amq1/q+b-xk+1b-a1pk+11/p1n+1α/qCf′xq+mD-Cf′bmq1/q+D-Cf′xq+mCf′bmq1/q.This completes the proof.Remark 17.
Observe that if in Theorem16 we have α=1, the statement of Theorem 16 becomes the statement of Theorem 10.Theorem 18.
LetI be on open real interval such that [0,∞)⊂I. Let f:I→R be a differentiable function on I such that n∈N, k>0, and f′∈L[a,b], where 0≤a<b<∞. If f′q is (α,m)-convex function on [a,b] for some fixed (α,m)∈(0,1]2 and q≥1, then(24)Gk;n;a,x,bf=x-ak+12b-a1k+11-1/q1n+1α/qA-Γα+1Γk+1Γα+k+2f′xq+mB2-A+Γα+1Γk+1Γα+k+2f′amq1/q+Γα+1Γk+1Γα+k+2f′xq+mB2-Γα+1Γk+1Γα+k+2f′amq1/q+b-xk+12b-a1k+11-1/q1n+1α/qA-Γα+1Γk+1Γα+k+2f′xq+mB2-A+Γα+1Γk+1Γα+k+2f′bmq1/q+Γα+1Γk+1Γα+k+2f′xq+mB2-Γα+1Γk+1Γα+k+2f′bmq1/q,where A and B are given by (19) and x∈[a,b].Proof.
Using Lemma7, Power’s mean inequality, and the fact that f′q is α,m-convex function,(25)Gk;n;a,x,bf≤x-ak+1b-a∫01tk2f′n+tn+1x+1-tn+1adt+∫01tk2f′1-tn+1x+n+tn+1adt+b-xk+1b-a∫01tk2f′n+tn+1x+1-tn+1bdt+∫01tk2f′1-tn+1x+n+tn+1bdt≤x-ak+1b-a∫01tk2dt1-1/q∫01tk2f′n+tn+1x+1-tn+1aqdt1/q+∫01tk2f′1-tn+1x+n+1n+1aqdt1/q+b-xk+1b-a∫01tk2dt1-1/q∫01tk2f′n+tn+1x+1-tn+1bqdt1/q+∫01tk2f′1-tn+1x+n+1n+1bqdt1/q≤x-ak+1b-a12k+11-1/q∫01tk2n+tn+1αf′xq+m1-n+tn+1αf′amqdt1/q+∫01tk21-tn+1αf′xq+m1-1-tn+1αf′amqdt1/q≤b-xk+1b-a12k+11-1/q∫01tk2n+tn+1αf′xq+m1-n+tn+1αf′bmqdt1/q+∫01tk21-tn+1αf′xq+m1-1-tn+1αf′bmqdt1/q=x-ak+12b-a1k+11-1/q1n+1α/qA-Γα+1Γk+1Γα+k+2f′xq+mB2-A+Γα+1Γk+1Γα+k+2f′amq1/q+Γα+1Γk+1Γα+k+2f′xq+mB2-Γα+1Γk+1Γα+k+2f′amq1/q+b-xk+12b-a1k+11-1/q1n+1α/qA-Γα+1Γk+1Γα+k+2f′xq+mB2-A+Γα+1Γk+1Γα+k+2f′bmq1/q+Γα+1Γk+1Γα+k+2f′xq+mB2-Γα+1Γk+1Γα+k+2f′bmq1/q.
This completes the proof.Remark 19.
Observe that if in Theorem18 we have α=1, the statement of Theorem 18 becomes the statement of Theorem 12.



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