# Hermite-Hadamard-Fejér Type Inequalities for Quasi-Geometrically Convex Functions via Fractional Integrals

Authors: İmdat İşcan; Mehmet Kunt
Journal: Journal of Mathematics
 (2016)

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## Abstract

Some Hermite-Hadamard-Fejér type integral inequalities for quasi-geometrically convex functions in fractional integral forms have been obtained.


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## Body

## 1. Introduction

Letf:I⊆R→R be a convex function defined on interval I of real numbers and a,b∈I with a<b. The inequality(1)fa+b2≤1b-a∫abfxdx≤fa+fb2is well known in the literature as Hermite-Hadamard’s inequality [1].The most well-known inequalities related to the integral mean of a convex functionf are the Hermite-Hadamard inequalities or its weighted versions, the so-called Hermite-Hadamard-Fejér inequalities.In [2], Fejér established the following Fejér inequality which is the weighted generalization of Hermite-Hadamard inequality (1).Theorem 1.
Letf:a,b→R be convex function. Then, the inequality(2)fa+b2∫abgxdx≤∫abfxgxdx≤fa+fb2∫abgxdxholds, where g:a,b→R is nonnegative, integrable, and symmetric to (a+b)/2.For some results which generalize, improve, and extend inequalities (1) and (2), see [3–6].Definition 2 (see [7, 8]).
A functionf:I⊆0,∞→R is said to be GA-convex (geometric-arithmetically convex) if (3)fxty1-t≤tfx+1-tfyfor all x,y∈I and t∈0,1.Definition 3 (see [9]).
A functionf:I⊆0,∞→R is said to be quasi-geometrically convex on I if (4)fxty1-t≤sup⁡fx,fyfor all x,y∈I and t∈0,1.In [10], Latif et al. established the following inequality which is the weighted generalization of Hermite-Hadamard inequality for GA-convex functions as follows.Theorem 4.
Letf:I⊆0,∞→R be a GA-convex function and a,b∈I with a<b. Let g:a,b→0,∞ be continuous positive mapping and geometrically symmetric to ab. Then,(5)fab∫abgxxdx≤∫abfxgxxdx≤fa+fb2∫abgxxdx.We will now give definitions of the right-hand side and left-hand side Hadamard fractional integrals which are used throughout this paper.Definition 5 (see [11]).
Letf∈La,b. The right-hand side and left-hand side Hadamard fractional integrals Ja+αf and Jb-αf of order α>0 with b>a≥0 are defined by (6)Ja+αfx=1Γα∫axln⁡xtα-1ftdtt,x>a,Jb-αfx=1Γα∫xbln⁡txα-1ftdtt,x<b,respectively, where Γ(α) is the Gamma function defined by Γ(α)=∫0∞e-ttα-1dt.Because of the wide application of Hermite-Hadamard type inequalities and fractional integrals, many researchers extend their studies to Hermite-Hadamard type inequalities involving fractional integrals not limited to integer integrals. Recently, more and more Hermite-Hadamard inequalities involving fractional integrals have been obtained for different classes of functions; see [9, 12–15].In [9], İşcan represented Hermite-Hadamard’s inequalities for GA-convex functions in fractional integral forms as follows.Theorem 6.
Letf:I⊆0,∞→R be a function such that f∈La,b, where a,b∈I with a<b. If f is a GA-convex function on a,b, then the following inequalities for fractional integrals hold:(7)fab≤Γα+12ln⁡b/aαJa+αfb+Jb-αfa≤fa+fb2with α>0.In [16], the authors represented Hermite-Hadamard-Fejér inequalities for GA-convex functions in fractional integral forms as follows.Theorem 7.
Letf:a,b⊆R+→R be a GA-convex function with a<b and f∈La,b. If g:a,b→R is nonnegative, integrable, and geometrically symmetric with respect to ab, then the following inequalities for fractional integrals hold:(8)fabJa+αgb+Jb-αga≤Ja+αfgb+Jb-αfga≤fa+fb2Ja+αgb+Jb-αgawith α>0.Lemma 8 (see [16]).
Letf:a,b⊆R+→R be a differentiable mapping on a,b with a<b and f′∈La,b. If g:a,b→R is integrable and geometrically symmetric with respect to ab, then the following equality for fractional integrals holds:(9)fa+fb2Ja+αgb+Jb-αga-Ja+αfgb+Jb-αfga=1Γα∫ab∫atln⁡bsα-1gsdss-∫tbln⁡saα-1gsdssf′tdtwith α>0.Lemma 9 (see [15, 17]).
For0<α≤1 and 0≤a<b, one has (10)aα-bα≤b-aα.In [9], İşcan introduced the quasi-geometrically convex functions and presented some Hermite-Hadamard type integral inequalities for quasi-geometrically convex functions via fractional integrals. In [18], İşcan obtained generalized Hermite-Hadamard type integral inequalities for quasi-geometrically convex functions. In [19], İşcan et al. showed some Hermite-Hadamard and Simpson type inequalities for differentiable quasi-geometrically convex functions.In this paper, it is the first time Hermite-Hadamard-Fejér type integral inequality for quasi-geometrically convex function has been studied. Obtained results are new and generalize the Hermite-Hadamard type integral inequality for quasi-geometrically convex functions.


## 2. Main Results

Throughout this section, letg∞=supt∈a,bg(x), for the continuous function g:a,b→R.Theorem 10.
Letf:I⊆R+→R be a differentiable mapping on I∘ and f′∈La,b with a<b. If f′ is quasi-geometrically convex on a,b and g:a,b→R is continuous and geometrically symmetric with respect to ab, then the following inequality for fractional integrals holds:(11)fa+fb2Ja+αgb+Jb-αga-Ja+αfgb+Jb-αfga≤g∞lnα+1⁡b/aΓα+1C1αsup⁡f′a,f′b,where(12)C1α=∫01/21-uα-uαa1-ubu+aub1-uduwith α>0.Proof.
From Lemma8 we have(13)fa+fb2Ja+αgb+Jb-αga-Ja+αfgb+Jb-αfga≤1Γα∫ab∫atln⁡bsα-1gsdss-∫tbln⁡saα-1gsdssf′tdt.Setting t=a1-ubu and dt=a1-ubuln⁡b/adu gives(14)fa+fb2Ja+αgb+Jb-αga-Ja+αfgb+Jb-αfga≤1Γα∫01∫aa1-ubuln⁡bsα-1gsdss-∫a1-ububln⁡saα-1gsdssf′a1-ubua1-ubuln⁡badu.
Sinceg:a,b→R is geometrically symmetric with respect to ab, we write(15)∫a1-ububln⁡saα-1gsdss=∫aaub1-uln⁡bsα-1gabsdss=∫aaub1-uln⁡bsα-1gsdssand then we have(16)∫aa1-ubuln⁡bsα-1gsdss-∫a1-ububln⁡saα-1gsdss=∫a1-ubuaub1-uln⁡bsα-1gsdss≤∫a1-ubuaub1-uln⁡bsα-1gsdssu∈0,12∫aub1-ua1-ubuln⁡bsα-1gsdssu∈12,1≤g∞∫a1-ubuaub1-uln⁡bsα-1dssu∈0,12∫aub1-ua1-ubuln⁡bsα-1dssu∈12,1=g∞ln⁡b/aαα1-uα-uαu∈0,12uα-1-uαu∈12,1.Since f′ is quasi-geometrically convex on a,b, we know that, for u∈0,1,(17)f′a1-ubu≤sup⁡f′a,f′b,and a combination of (14), (16), and (17) gives (18)fa+fb2Ja+αgb+Jb-αga-Ja+αfgb+Jb-αfga≤1Γα∫01∫aa1-ubuln⁡bsα-1gsdss-∫a1-ububln⁡saα-1gsdssf′a1-ubua1-ubuln⁡badu≤1Γα∫01/2g∞ln⁡b/aαα1-uα-uαsupf′a,f′ba1-ubulnbadu+1Γα∫1/21g∞ln⁡b/aααuα-1-uαsup⁡f′a,f′ba1-ubuln⁡badu≤lnα+1⁡b/ag∞Γα+1∫01/21-uα-uαa1-ubudu+∫1/21uα-1-uαa1-ubudusupf′a,f′b=lnα+1⁡b/ag∞Γα+1∫01/21-uα-uαa1-ubu+aub1-udusup⁡f′a,f′b.
This completes the proof.Corollary 11.
In Theorem10, one can see the following.
(1) If one takesα=1, one has the following Hermite-Hadamard-Fejér type inequality for quasi-geometrically convex function which is related to the right-hand side of (5): (19)fa+fb2∫abgxxdx-∫abfxgxxdx≤g∞ln2⁡b/a2C11sup⁡f′a,f′b.
(2) If one takesgx=1, one has the following Hermite-Hadamard type inequality for quasi-geometrically convex function in fractional integral forms which is related to the right-hand side of (7):(20)fa+fb2-Γα+12ln⁡b/aαJa+αfb+Jb-αfa≤ln⁡b/a2C1αsup⁡f′a,f′b.
(3) If one takesα=1 and gx=1, one has the following Hermite-Hadamard type inequality for quasi-geometrically convex function: (21)fa+fb2-1ln⁡b/a∫abfxxdx≤ln⁡b/a2C11sup⁡f′a,f′b.Theorem 12.
Letf:I⊆R+→R be a differentiable mapping on I∘ and f′∈La,b with a<b. If f′q, q≥1, is quasi-geometrically convex on a,b and g:a,b→R is continuous and geometrically symmetric with respect to ab, then the following inequalities for fractional integrals hold: (22)fa+fb2Ja+αgb+Jb-αga-Ja+αfgb+Jb-αfga≤g∞lnα+1⁡b/aΓα+11-12α2α+11-1/qC21/qαsup⁡f′aq,f′bq1/q,where(23)C2α=∫01/21-uα-uαa1-ubuq+aub1-uqdu,with α>0.Proof.
Similar to the proof of Theorem10, using Lemma 8, (14), (16), power mean inequality, and quasi-geometrically convexity of f′q, we have (24)fa+fb2Ja+αgb+Jb-αga-Ja+αfgb+Jb-αfga≤1Γα∫01∫a1-ubuaub1-uln⁡bsα-1gsdssf′a1-ubua1-ubuln⁡badu≤ln⁡b/aΓα∫01∫a1-ubuaub1-uln⁡bsα-1gsdssdu1-1/q∫01∫a1-ubuaub1-uln⁡bsα-1gsdssf′a1-ubuqa1-ubuqdu1/q≤g∞1-1/qlnα1-1/q+1⁡b/aα1-1/qΓα∫01/21-uα-uαdu+∫1/21uα-1-uαdu1-1/q∫01∫a1-ubuaub1-ulnbsα-1dsssup⁡f′aq,f′bqa1-ubuqdu1/q=g∞1-1/qlnα1-1/q+1⁡b/aα1-1/qΓα1-12α2α+11-1/qsup⁡f′aq,f′bq1/q∫01∫a1-ubuaub1-uln⁡bsα-1dssa1-ubuqdu1/q=g∞lnα+1⁡b/aΓα+11-12α2α+11-1/qsup⁡f′aq,f′bq1/q∫011-uα-uαa1-ubuqdu1/q=g∞lnα+1⁡b/aΓα+11-12α2α+11-1/qsup⁡f′aq,f′bq1/q∫01/21-uα-uαa1-ubuqdu+∫1/21uα-1-uαa1-ubuqdu1/q=g∞lnα+1⁡b/aΓα+11-12α2α+11-1/qsup⁡f′aq,f′bq1/q∫01/21-uα-uαa1-ubuq+aub1-uqdu1/q.
This completes the proof.Corollary 13.
In Theorem12, one has the following.
(1) If one takesα=1, one has the following Hermite-Hadamard-Fejér type inequality for quasi-geometrically convex function which is related to the right-hand side of (5): (25)fa+fb2∫abgxxdx-∫abfxgxxdx≤g∞ln2⁡ba122-1/qC21/q1sup⁡f′aq,f′bq1/q.
(2) If one takesgx=1, one has the following Hermite-Hadamard type inequality for quasi-geometrically convex function in fractional integral forms which is related to the right-hand side of (7): (26)fa+fb2-Γα+12ln⁡b/aαJa+αfb+Jb-αfa≤ln⁡b/a21-12α2α+11-1/qC21/qαsup⁡f′aq,f′bq1/q.
(3) If one takesα=1 and gx=1, one has the following Hermite-Hadamard type inequality for quasi-geometrically convex function: (27)fa+fb2-1ln⁡b/a∫abfxxdx≤ln⁡ba122-1/qC21/q1sup⁡f′aq,f′bq1/q.Theorem 14.
Letf:I⊆R+→R be a differentiable mapping on I∘ and f′∈La,b with a<b. If f′q, q>1, is quasi-geometrically convex on a,b and g:a,b→R is continuous and geometrically symmetric with respect to ab, then the following inequalities for fractional integrals hold:
(i)(28)fa+fb2Ja+αgb+Jb-αga-Ja+αfgb+Jb-αfga≤g∞lnα+1-1/q⁡b/aq1/qΓα+12αp+11-12αp1/pbq-aq1/qsup⁡f′aq,f′bq1/qwith α>0;
(ii)(29)fa+fb2Ja+αgb+Jb-αga-Ja+αfgb+Jb-αfga≤g∞lnα+1-1/q⁡b/aq1/qΓα+11αp+11/pbq-aq1/qsup⁡f′aq,f′bq1/qfor 0<α≤1, where 1/p+1/q=1.Proof.
(i) Using Lemma8, (14), (16), Hölder’s inequality, and quasi-geometrically convexity of f′q, we have(30)fa+fb2Ja+αgb+Jb-αga-Ja+αfgb+Jb-αfga≤1Γα∫01∫a1-ubuaub1-uln⁡bsα-1gsdssf′a1-ubua1-ubuln⁡badu≤ln⁡b/aΓα∫01∫a1-ubuaub1-uln⁡bsα-1gsdsspdu1/p∫01f′a1-ubuqa1-ubuqdu1/q≤g∞lnα+1⁡b/aΓα+1∫01/21-uα-uαpdu+∫1/21uα-1-uαpdu1/p∫01sup⁡f′aq,f′bqa1-ubuqdu1/q=g∞lnα+1-1/q⁡b/aq1/qΓα+1∫01/21-uα-uαpdu+∫1/21uα-1-uαpdu1/pbq-aq1/qsup⁡f′aq,f′bq1/q≤g∞lnα+1-1/q⁡b/aq1/qΓα+1∫01/21-uαp-uαpdu+∫1/21uαp-1-uαpdu1/pbq-aq1/qsup⁡f′aq,f′bq1/q=g∞lnα+1-1/q⁡b/aq1/qΓα+12αp+11-12αp1/pbq-aq1/qsup⁡f′aq,f′bq1/q.
Here, we use(31)1-tα-tαp≤1-tαp-tαpfor t∈0,1/2 and (32)tα-1-tαp≤tαp-1-tαpfor t∈1/2,1, which follows from (33)A-Bq≤Aq-Bq,for any A≥B≥0 and q≥1. Hence, inequality (28) is proved.
(ii) Inequality (29) is easily proved using (30) and Lemma 9.Corollary 15.
In Theorem14, one can see the following.
(i) In (28), consider the following:(1)
If one takesα=1, one has the following Hermite-Hadamard-Fejér type inequality for quasi-geometrically convex function which is related to the right-hand side of (5): (34)fa+fb2∫abgxxdx-∫abfxgxxdx≤g∞ln2-1/q⁡b/a2q1/q2p+11-12p1/pbq-aq1/qsup⁡f′aq,f′bq1/q.(2)
If one takesgx=1, one has the following Hermite-Hadamard type inequality for quasi-geometrically convex function in fractional integral forms which is related to the right-hand side of (7): (35)fa+fb2-Γα+12ln⁡b/aαJa+αfb+Jb-αfa≤ln1-1/q⁡b/a2q1/q2αp+11-12αp1/pbq-aq1/qsup⁡f′aq,f′bq1/q.(3)
If one takesα=1 and gx=1, one has the following Hermite-Hadamard type inequality for quasi-geometrically convex function: (36)fa+fb2-1ln⁡b/a∫abfxxdx≤ln1-1/q⁡b/a2q1/q2p+11-12p1/pbq-aq1/qsup⁡f′aq,f′bq1/q.
(ii) In (29), consider the following:(1)
If one takesα=1, one has the following Hermite-Hadamard-Fejér type inequality for quasi-geometrically convex function which is related to the right-hand side of (5): (37)fa+fb2∫abgxxdx-∫abfxgxxdx≤g∞ln2-1/q⁡b/a2q1/q1p+11/pbq-aq1/qsup⁡f′aq,f′bq1/q.(2)
If one takesgx=1, one has the following Hermite-Hadamard type inequality for quasi-geometrically convex function in fractional integral forms which is related to the right-hand side of (7): (38)fa+fb2-Γα+12ln⁡b/aαJa+αfb+Jb-αfa≤ln1-1/q⁡b/a2q1/q1αp+11/pbq-aq1/qsup⁡f′aq,f′bq1/q.(3)
If one takesα=1 and gx=1, one has the following Hermite-Hadamard type inequality for quasi-geometrically convex function: (39)fa+fb2-1ln⁡b/a∫abfxxdx≤ln1-1/q⁡b/a2q1/q1p+11/pbq-aq1/qsup⁡f′aq,f′bq1/q.



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