# Applying GG-Convex Function to Hermite-Hadamard Inequalities Involving Hadamard Fractional Integrals

Authors: Zhi Zhang; JinRong Wang; JianHua Deng
Journal: International Journal of Mathematics and Mathematical Sciences
 (2014)

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## Abstract

By virtue of fractional integral identities, incomplete beta function, useful series, and inequalities, we apply the concept of GG-convex function to derive new type Hermite-Hadamard inequalities involving Hadamard fractional integrals. Finally, some applications to special means of real numbers are demonstrated.


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## Body

## 1. Introduction

Fractional calculus played an important role in various fields such as electricity, biology, economics, and signal and image processing [1–8]. The fractional Hermite-Hadamard inequality gives a lower and an upper estimation for both right-hand and left-hand integrals average of any convex function defined on a compact interval, involving the midpoint and the endpoints of the domain.As we know, Set [9] firstly studied fractional Ostrowski inequalities involving Riemann-Liouville fractional integrals. Then, Sarikaya et al. [10] studied Hermite-Hadamard type inequalities involving Riemann-Liouville fractional integrals. Further, our group go on studying fractional version Hermite-Hadamard inequality involving Riemann-Liouville and Hadamard fractional integrals for all kinds of functions [11–19].Recently, Wang et al. [16, 17] established the following two powerful fractional integral identities involving Hadamard fractional integrals.Lemma 1 (see [12, Lemma 3.1]).
Letf:[a,b]→R be a differentiable mapping on (a,b). If f′∈L[a,b], then the following equality for fractional integrals holds:(1)f(a)+f(b)2-Γ(α+1)2(ln⁡b-ln⁡a)α[JHa+αf(b)+JHb-αf(a)]=ln⁡b-ln⁡a2∫01‍[(1-t)α-tα]atb1-tf′(atb1-t)dt,
where the symbols JHa+αf and JHb-αf are defined by
(2)(JHa+αf)(x)=1Γ(α)∫ax‍(ln⁡xt)α-1f(t)dtt,(Ja+αHf)(x)=1Γ(α)lllllllll(0<a<x≤b),(JHb-αf)(x)=1Γ(α)∫xb‍(ln⁡tx)α-1f(t)dtt,(Ja+αHf)(x)=1Γ(α)lllllllll(0<a≤x<b),
where Γ(·) is the Gamma function.Lemma 2 (see [11, Lemma 2.1]).
Letf:[a,b]→R be a differentiable mapping on (a,b) with 0<a<b. If f′∈L[a,b], then the following equality for fractional integrals holds:(3)Γ(α+1)2(ln⁡b-ln⁡a)α[JHa+αf(b)+JHb-αf(a)]-f(a+b2)=b-a2∫01kf′(ta+(1-t)b)dt‍-ln⁡b-ln⁡a2∫01‍[(1-t)α-tα]atb1-tf′(atb1-t)dt,
where
(4)k={1,0≤t<12,-1,12≤t<1.Remark 3.
It is remarkable that Professor Srivastava et al. [20] give some further refinements and extensions of the Hermite-Hadamard inequalities in n variables. In the forthcoming works, we will try to extend to study fractional version Hermite-Hadamard inequalities in n variables based on such fundamental results.Next, we recall the following basic concepts and results in our previous papers.Definition 4 (see [21, 22]).
Letf:I⊆R+→R+. A function f is said to be GG-convex on I if, for every x,y∈I and λ∈[0,1], one has
(5)f(xλy1-λ)≤[f(x)]λ[f(y)]1-λ.Remark 5.
By the arithmetic-geometric mean inequality, we have(6)[f(x)]λ[f(y)]1-λ≤λf(x)+(1-λ)f(y).
Linking (5) and (6), we obtain
(7)f(xλy1-λ)≤λf(x)+(1-λ)f(y),
which appears in the standard definition of GA-convex function [21]. So GG-convex function is GA-convex function.Lemma 6 (see [19, Lemma 2.5]).
Fort∈[0,1],x,y>0, one has
(8)tx+(1-t)y≥y1-txt.Lemma 7 (see [13, Lemma 2.1]).
Forα>0 and k>0, one has
(9)I(α)=∫01‍tα-1ktdt=k∑i=1∞‍(-1)i-1(ln⁡k)i-1(α)i<+∞,
where (α)i=α(α+1)(α+2)⋯(α+i-1).Lemma 8 (see [13, Lemma 2.2]).
Forα>0 and k>0,z>0, one has
(10)J(α,k)=∫01(1-t)α-1ktdt‍=∑i=1∞‍(ln⁡k)i-1(α)i<+∞,H(α,k,z)=∫0z‍tα-1ktdt=zαkz∑i=1∞(-zln⁡k)i-1(α)i‍<+∞.Lemma 9 (see [13, Lemma 2.3]).
Forα>0 and k>0,1>z>0, one has
(11)R(α,k,z)=∫0z‍(1-t)α-1ktdt=∑i=0∞‍(ln⁡k)i-1(α)i(1-kz(1-z)α+i-1).In the present paper, we will use the above concepts and lemmas to derive some new fractional Hermite-Hadamard inequalities involving Hadamard fractional integrals.


## 2. Main Results Based on Lemma1

Now we are ready to state the following main results in this section.Theorem 10.
Letf:[0,b]→R+ be a differentiable mapping. If |f′| is measurable and |f′| is GG-convex on [a,b] for some fixed α∈(0,∞) and t∈[0,1], 0≤a<b, then the following integrals hold:
(12)|f(a)+f(b)2-Γ(α+1)2(ln⁡b-ln⁡a)α[JHa+αf(b)+JHb-αf(a)]|≤ln⁡b-ln⁡a2×{a|f′(a)|(α+1)2α+2(1-12α+1)lllll×[b|f′(b)|α+1+b|f′(a)|+a|f′(b)|-2b|f′(b)|α+2]≤ll+a|f′(a)|-a|f′(b)|-b|f′(a)|α+3≤ll-a|f′(a)|-b|f′(a)|(α+3)2α+3+2a|f′(a)|(α+1)(α+2)(α+3)≤ll+b|f′(a)|+a|f′(b)|-2a|f′(a)|(α+1)(α+3)≤ll+2a(|f′(a)|-|f′(b)|)-2b|f′(a)|2α+3(α+2)(α+3)≤ll+b|f′(a)|+a|f′(b)|2α+2(α+2)≤ll+a|f′(a)|(α+1)2α+2-a|f′(a)|(α+2)22α+2}.Proof.
Noting Definition4 and Lemmas 1 and 6, we have
(13)|f(a)+f(b)2-Γ(α+1)2(ln⁡b-ln⁡a)α[JHa+αf(b)+JHb-αf(a)]|=|ln⁡b-ln⁡a2∫01‍atb1-t[(1-t)α-tα]f′(atb1-t)dt|≤ln⁡b-ln⁡a2∫01‍atb1-t|(1-t)α-tα||f′(atb1-t)|dt≤ln⁡b-ln⁡a2∫1/21atb1-t[tα-(1-t)α]|f′(atb1-t)|dt‍+ln⁡b-ln⁡a2∫01/2‍atb1-t[(1-t)α-tα]+ln⁡b-ln⁡a2lllllll×|f′(atb1-t)|dt≤ln⁡b-ln⁡a2∫1/21‍[at+b(1-t)][tα-(1-t)α]+ln⁡b-ln⁡a2lllll×|f′(a)|t|f′(b)|1-tdt+ln⁡b-ln⁡a2∫01/2‍[at+b(1-t)][(1-t)α-tα]+ln⁡b-ln⁡a2lllllll×|f′(a)|t|f′(b)|1-tdt≤ln⁡b-ln⁡a2∫1/21‍[at+b(1-t)][tα-(1-t)α]+ln⁡b-ln⁡a2lllll×[|f′(a)|t+|f′(b)|(1-t)]dt+ln⁡b-ln⁡a2∫01/2‍[at+b(1-t)][(1-t)α-tα]+ln⁡b-ln⁡a2lllllll×[|f′(a)|t+|f′(b)|(1-t)]dt≤ln⁡b-ln⁡a2{a|f′(a)|α+3∫1/21‍(α+3)tα+2dtlllllllllllllllllll+b|f′(a)|α+2∫1/21(α+2)tα+1dt‍lllllllllllllllllll-b|f′(a)|α+3∫1/21‍(α+3)tα+2dtlllllllllllllllllll+a|f′(b)|α+2∫1/21‍(α+2)tα+1dtlllllllllllllllllll-a|f′(b)|α+3∫1/21‍(α+3)tα+2dtlllllllllllllllllll+b|f′(b)|α+1∫1/21‍(α+1)tαdtlllllllllllllllllll-2b|f′(b)|α+2∫1/21‍(α+2)tα+1dtlllllllllllllllllll+b|f′(b)|α+3∫1/21(α+3)tα+2dt‍lllllllllllllllllll-a|f′(a)|∫1/21‍(1-t)αt2dtlllllllllllllllllll-b|f′(a)|∫1/21‍t(1-t)α+1dtlllllllllllllllllll-a|f′(b)|∫1/21‍t(1-t)α+1dtlllllllllllllllllll-b|f′(b)|α+3∫1/21‍(α+3)(1-t)α+2dt}+ln⁡b-ln⁡a2{b|f′(b)|α+3∫01/2‍(α+3)tα+2dta|f′(a)|∫01/2‍t2(1-t)αdtllllllllllllllllllllll+b|f′(a)|∫01/2‍t(1-t)α+1dtllllllllllllllllllllll+a|f′(b)|∫01/2‍t(1-t)α+1dtllllllllllllllllllllll+b|f′(b)|α+3∫01/2‍(α+3)(1-t)α+2dtllllllllllllllllllllll-a|f′(a)|α+3∫01/2‍(α+3)tα+2dtllllllllllllllllllllll-b|f′(a)|α+2∫01/2(α+2)tα+1dt‍llllllllllllllllllllll+b|f′(a)|α+3∫01/2‍(α+3)tα+2dtllllllllllllllllllllll-a|f′(b)|α+2∫01/2‍(α+2)tα+1dtllllllllllllllllllllll+a|f′(b)|α+3∫01/2‍(α+3)tα+2dtlllllllllllllllllllllll-b|f′(b)|α+1∫01/2‍(α+1)tαdtllllllllllllllllllllll+2b|f′(b)|α+2∫01/2‍(α+2)tα+1dtllllllllllllllllllllll-b|f′(b)|α+3∫01/2‍(α+3)tα+2dt}≤ln⁡b-ln⁡a2×{a|f′(a)|α+3(1-12α+3)≤ll+b|f′(a)|α+2(1-12α+2)-b|f′(a)|α+3(1-12α+3)≤ll+a|f′(b)|α+2(1-12α+2)-a|f′(b)|α+3(1-12α+3)≤ll+b|f′(b)|α+1(1-12α+1)-2b|f′(b)|α+2(1-12α+2)≤ll+b|f′(b)|α+3(1-12α+3)-a|f′(a)|∫1/21‍(1-t)αt2dt≤ll-b|f′(a)|∫1/21‍t(1-t)α+1dt≤ll-a|f′(b)|∫1/21‍t(1-t)α+1dt≤ll-b|f′(b)|α+3∫1/21‍(α+3)(1-t)α+2dt}+ln⁡b-ln⁡a2×{b|f′(a)|α+312α+3a|f′(a)|∫01/2‍t2(1-t)αdt≤lll+b|f′(a)|∫01/2‍t(1-t)α+1dt≤lll+a|f′(b)|∫01/2‍t(1-t)α+1dt≤lll+b|f′(b)|α+3(12α+3-1)-a|f′(a)|α+312α+3≤lll-b|f′(a)|α+212α+2+b|f′(a)|α+312α+3≤lll-a|f′(b)|α+212α+2+a|f′(b)|α+312α+3≤lll-b|f′(b)|α+112α+1+2b|f′(b)|α+212α+2≤lll-b|f′(a)|α+312α+3}≤ln⁡b-ln⁡a2{(a-b)[|f′(a)|-|f′(b)|]α+3(1-12α+3)≤lllllllllllllllllll+b|f′(b)|α+1(1-12α+1)≤lllllllllllllllllll+b[|f′(a)|-|f′(b)|]+(a-b)|f′(b)|α+2≤lllllllllllllllllll×(1-12α+2)-a|f′(a)|∫01(1-t)αt2dt‍≤lllllllllllllllllll+a|f′(a)|∫01/2(1-t)αt2dt‍≤lllllllllllllllllll-b|f′(a)|∫01‍t(1-t)α+1dt≤lllllllllllllllllll+b|f′(a)|∫01/2‍t(1-t)α+1dt≤lllllllllllllllllll-a|f′(b)|∫01‍t(1-t)α+1dt≤lllllllllllllllllll+a|f′(b)|∫01/2‍t(1-t)α+1dt(a-b)[|f′(a)|-|f′(b)|]α+3(1-12α+3)}+ln⁡b-ln⁡a2{b|f′(a)|α+312α+3a|f′(a)|∫01/2‍t2(1-t)αdt+ln⁡b-ln⁡a2ll+b|f′(a)|∫01/2‍t(1-t)α+1dt+ln⁡b-ln⁡a2ll+a|f′(b)|∫01/2t(1-t)α+1dt‍+ln⁡b-ln⁡a2ll+b|f′(b)|α+3(12α+3-1)-a|f′(b)|(α+3)2α+3+ln⁡b-ln⁡a2ll-b|f′(a)|+a|f′(b)|-2b|f′(b)|(α+2)2α+2+ln⁡b-ln⁡a2l-b|f′(b)|(α+1)2α+1b|f′(a)|α+312α+3}≤ln⁡b-ln⁡a2×{a|f′(a)|(α+2)22α+2(1-12α+1)lllllllll×[b|f′(b)|α+1llllllllllllll+b|f′(a)|+a|f′(b)|-2b|f′(b)|α+2]lllllllll+a|f′(a)|-a|f′(b)|-b|f′(a)|α+3lllllllll-a|f′(a)|-b|f′(a)|(α+3)2α+3lllllllll+2a|f′(a)|(α+1)(α+2)(α+3)lllllllll+b|f′(a)|+a|f′(b)|-2a|f′(a)|(α+1)(α+3)lllllllll+2a(|f′(a)|-|f′(b)|)-2b|f′(a)|2α+3(α+2)(α+3)lllllllll+b|f′(a)|+a|f′(b)|2α+2(α+2)+a|f′(a)|(α+1)2α+2lllllllll-a|f′(a)|(α+2)22α+2}.
The proof is done.Theorem 11.
Letf:[0,b]→R+ be a differentiable mapping. If |f′|q is measurable and |f′|q, (q>1) is GG-convex on [a,b] for some fired α∈(0,∞) and t∈[0,1], 0≤a<b, then the following integrals hold:(14)|f(a)+f(b)2-Γ(α+1)2(ln⁡b-ln⁡a)α[JHa+αf(b)+JHb-αf(a)]|≤ln⁡b-ln⁡a2(|f′(a)|q+2|f′(b)|q-|f′(b)|q2)1/q×(ap-2bppα+2+bppα+1-2bp+ap2pα+2(pα+1)mm+2bp-ap2pα+2(pα+2)-appα(pα+1)mm-ap+bp2pα+2(pα+1)(pα+2)mm+2ap(pα+1)(pα+2))1/p,
where 1/p+1/q=1.Proof.
By using Definition4 and Lemmas 1 and 6, we have
(15)|f(a)+f(b)2-Γ(α+1)2(ln⁡b-ln⁡a)α[JHa+αf(b)+JHb-αf(a)]|≤ln⁡b-ln⁡a2(∫01‍aptbp(1-t)|(1-t)α-tα|pdt)1/p×(∫01‍|f′(a)|qt|f′(b)|q(1-t)dt)1/q≤ln⁡b-ln⁡a2×(∫01‍[tap+(1-t)bp]|(1-t)α-tα|pdt)1/p×(∫01‍[t|f′(a)|q+(1-t)|f′(b)|q]dt)1/q≤ln⁡b-ln⁡a2×(∫1/21‍[tap+(1-t)bp](tpa-(1-t)pa)dtll≤+∫01/2‍[tap+(1-t)bp][(1-t)pα-tpα]dt)1/p×(|f′(a)|q2+|f′(b)|q-|f′(b)|q2)1/q≤ln⁡b-ln⁡a2(|f′(a)|q+2|f′(b)|q-|f′(b)|q2)1/q×(ap∫1/21‍tpα+1dt-ap∫1/21‍t(1-t)pαdt×m+bp∫1/21‍tpα(1-t)dt-bp∫1/21‍(1-t)pα+1dt×m+ap∫01/2‍t(1-t)pαdt-ap∫01/2‍tpα+1dt×m+bp∫01/2‍(1-t)pα+1dt×m-bp∫01/2‍tpα(1-t)dt)1/p≤ln⁡b-ln⁡a2(|f′(a)|q+2|f′(b)|q-|f′(b)|q2)1/q×(appα+2-ap∫01‍t(1-t)pα-1dt×m+ap∫01/2‍t(1-t)pαdt+bppα+1×m-12pα+1bppα+1-bppα+2+12pα+2bppα+2×m+ap∫01/2‍t(1-t)pαdt-12pα+2appα+2×m+12pα+2bppα+2-bppα+2×m-bp∫01/2‍tpα(1-t)dt)1/p≤ln⁡b-ln⁡a2(|f′(a)|q+2|f′(b)|q-|f′(b)|q2)1/q×(ap-2bppα+2+bppα+1-2bp+ap2pα+2(pα+1)×m+2bp-ap2pα+2(pα+2)-appα(pα+1)×m-ap+bp2pα+2(pα+1)(pα+2)×m+2ap(pα+1)(pα+2))1/p.
The proof is done.Theorem 12.
Letf:[0,b]→R+ be a differentiable mapping. If |f′| is measurable and |f′| is GG-convex on [a,b] for some fired α∈(0,∞), t∈[0,1] and k>0, 0≤a<b, then the following integrals hold:
(16)|f(a)+f(b)2-Γ(α+1)2(ln⁡b-ln⁡a)α[JHa+αf(b)+JHb-αf(a)]|≤b|f′(b)|ln⁡b-ln⁡a2×{1-(1/2)α(a|f′(a)|/b|f′(b)|)1/2ln⁡(a|f′(a)|/b|f′(b)|)a|f′(a)|b|f′(b)|≤m×∑i=1∞‍(-1)i-1[ln⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)i≤m+∑i=1∞‍[ln⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)i≤m-(1/2)α-1(a|f′(a)|/b|f′(b)|)1/2α+1≤m+21-(1/2)α(a|f′(a)|/b|f′(b)|)1/2ln⁡(a|f′(a)|/b|f′(b)|)∑i=1∞‍}.Proof.
By using Definition4 and Lemmas 1, 7, 8, and 9, we have
(17)|f(a)+f(b)2-Γ(α+1)2(ln⁡b-ln⁡a)α[JHa+αf(b)+JHb-αf(a)]|≤ln⁡b-ln⁡a2∫1/21‍atb1-t[tα-(1-t)α]|f′(atb1-t)|dt+ln⁡b-ln⁡a2∫01/2‍atb1-t[(1-t)α-tα]+ln⁡b-ln⁡a2lllll×|f′(atb1-t)|dt≤ln⁡b-ln⁡a2∫1/21‍atb1-t[tα-(1-t)α]+ln⁡b-ln⁡a2ll×|f′(a)|t|f′(b)|1-tdt+ln⁡b-ln⁡a2∫01/2‍atb1-t[(1-t)α-tα]+ln⁡b-ln⁡a2lll×|f′(a)|t|f′(b)|1-tdt≤b|f′(b)|ln⁡b-ln⁡a2×∫1/21‍(a|f′(a)|b|f′(b)|)t[tα-(1-t)α]dt+b|f′(b)|ln⁡b-ln⁡a2×∫01/2‍(a|f′(a)|b|f′(b)|)t[(1-t)α-tα]dt≤b|f′(b)|ln⁡b-ln⁡a2×{∫01‍(a|f′(a)|b|f′(b)|)ttαdt≤ll-2∫01/2‍(a|f′(a)|b|f′(b)|)ttαdt≤ll-∫01‍(a|f′(a)|b|f′(b)|)t(1-t)αdt≤ll+2∫01/2‍(a|f′(a)|b|f′(b)|)t(1-t)αdt}≤b|f′(b)|ln⁡b-ln⁡a2×{(a|f′(a)|b|f′(b)|)1/2a|f′(a)|b|f′(b)|llllll×∑i=1∞‍(-1)i-1[ln⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)illllll-∑i=1∞‍[ln⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)illllll-(12)α(a|f′(a)|b|f′(b)|)1/2llllll×∑i=1∞‍[-(1/2)ln⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)illllll+2∑i=0∞‍[ln⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)illllll×[1-(12)α+i(a|f′(a)|b|f′(b)|)1/2]}≤b|f′(b)|ln⁡b-ln⁡a2×{∑i=1∞‍∑i=1∞‍[(1/2)ln⁡⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)ia|f′(a)|b|f′(b)|llllll×∑i=1∞‍(-1)i-1[ln⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)illllll-∑i=1∞‍[ln⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)illllll-(12)α(a|f′(a)|b|f′(b)|)1/2llllll×∑i=1∞‍[-(1/2)ln⁡⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)illllll+21-(1/2)α(a|f′(a)|/b|f′(b)|)1/2ln⁡(a|f′(a)|/b|f′(b)|)llllll+2∑i=1∞‍[ln⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)illllll-(12)α(a|f′(a)|b|f′(b)|)1/2llllll×∑i=1∞‍[(1/2)ln⁡⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)i}≤b|f′(b)|ln⁡b-ln⁡a2×{(a|f′(a)|/b|f′(b)|)1/2ln⁡(a|f′(a)|/b|f′(b)|)a|f′(a)|b|f′(b)|llllll×∑i=1∞‍(-1)i-1[ln⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)illllll+∑i=1∞‍[ln⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)illllll-(1/2)α-1(a|f′(a)|/b|f′(b)|)1/2α+1llllll+21-(1/2)α(a|f′(a)|/b|f′(b)|)1/2ln⁡(a|f′(a)|/b|f′(b)|)}.
The proof is done.Theorem 13.
Letf:[0,b]→R+ be a differentiable mapping. If |f′|q is measurable and |f′|q, (q>1) is GG-convex on [a,b], for some fired α∈(0,∞), t∈[0,1], and k>0, 0≤a<b, then the following integrals hold:
(18)|f(a)+f(b)2-Γ(α+1)2(ln⁡b-ln⁡a)α[JHa+αf(b)+JHb-αf(a)]|≤ln⁡b-ln⁡a2×(ap∑i=1∞‍(-1)i-1[(1/2)ln⁡(a/b)p]i-1(pα+1)i≤llll+bp∑i=1∞‍[ln⁡(a/b)p]i-1(pα+1)i-(1/2)pα-1(abp/2)pα+1≤llll-[ln⁡⁡(ab)p]-1[(12)pα-1(ab)p/2+2bp]∑i=1∞‍)1/p×(|f′(b)|qln⁡(|f′(a)|/|f′(b)|)q[|f′(a)f′(b)|q-1])1/q,
where 1/p+1/q=1.Proof.
By using Definition4 and Lemmas 1, 7, 8, and 9, we have
(19)|f(a)+f(b)2-Γ(α+1)2(ln⁡b-ln⁡a)α[JHa+αf(b)+JHb-αf(a)]|≤ln⁡b-ln⁡a2(∫01‍(atb1-t|(1-t)α-tα|)pdt)1/p×(∫01‍|f′(atb1-t)|qdt)1/q≤ln⁡b-ln⁡a2(bp∫1/21‍(ab)pttpαdt≤ln⁡b-ln⁡a2lll-bp∫1/21‍(ab)pt(1-t)pαdt≤ln⁡b-ln⁡a2lll+bp∫01/2‍(ab)pt(1-t)pαdt≤ln⁡b-ln⁡a2lll-bp∫01/2‍(ab)pttpαdt)1/p×(|f′(b)|qln⁡(|f′(a)|/|f′(b)|)qllllllll×∫01‍ln⁡(|f′(a)||f′(b)|)q[|f′(a)||f′(b)|]qtdt)1/q≤ln⁡b-ln⁡a2(bp(ab)p∑i=1∞‍(-1)i-1[ln⁡⁡(a/b)p]i-1(pα+1)i≤ln⁡b-ln⁡a2-bp∑i=0∞‍[ln⁡⁡(a/b)p]i-1(pα+1)i≤ln⁡b-ln⁡a2-2bp∫01/2‍(ab)pttpαdt≤ln⁡b-ln⁡a2+2bp∫01/2‍(ab)pt(1-t)pαdt∑i=1∞‍)1/p×(|f′(b)|qln⁡(|f′(a)|/|f′(b)|)q[|f′(a)f′(b)|q-1])1/q≤ln⁡b-ln⁡a2×(bp(ab)p∑i=1∞‍(-1)i-1[ln⁡(a/b)p]i-1(pα+1)i≤lll-bp∑i=1∞‍[ln⁡(a/b)p]i-1(pα+1)i≤lll-(12)pαbp(ab)p/2∑i=1∞‍[-(1/2)ln⁡⁡(a/b)p]i-1(pα+1)i≤lll+2bp∑i=0∞‍[ln⁡(a/b)p]i-1(pα+1)i≤+2bp+2lll×[1-(ab)p/2(1-12)pα+i]∑i=0∞‍)1/p×(|f′(b)|qln⁡⁡(|f′(a)|/|f′(b)|)q[|f′(a)f′(b)|q-1])1/q≤ln⁡b-ln⁡a2×(ap∑i=1∞‍(-1)i-1[ln⁡(a/b)p]i-1(pα+1)illlllll-bp∑i=1∞‍[ln⁡(a/b)p]i-1(pα+1)illlllll-(12)pα(ab)p/2∑i=1∞‍[-(1/2)ln⁡(a/b)p]i-1(pα+1)illlllll+2bp∑i=0∞‍[ln⁡(a/b)p]i-1(pα+1)illlllll-2bp(ab)p/2∑i=0∞‍(1-12)pα+i[ln⁡(a/b)p]i-1(pα+1)i)1/p×(|f′(b)|qln⁡(|f′(a)|/|f′(b)|)q[|f′(a)f′(b)|q-1])1/q≤ln⁡b-ln⁡a2×(ap∑i=1∞‍(-1)i-1[ln⁡(a/b)p]i-1(pα+1)i-bp∑i=1∞‍[ln⁡(a/b)p]i-1(pα+1)illllll-(12)pα(ab)p/2∑i=1∞‍[-(1/2)ln⁡(a/b)p]i-1(pα+1)illllll+2bp∑i=1∞‍[ln⁡⁡(a/b)p]i-1(pα+1)i+2bp[ln⁡⁡(a/b)p]-1llllll-2bp(ab)p/2∑i=0∞‍(1-12)pα+i[ln⁡⁡(a/b)p]i-1(pα+1)i)1/p×(|f′(b)|qln⁡(|f′(a)|/|f′(b)|)q[|f′(a)f′(b)|q-1])1/q≤ln⁡b-ln⁡a2×(ap∑i=1∞‍(-1)i-1[ln⁡(a/b)p]i-1(pα+1)i+bp∑i=1∞‍[ln⁡⁡(a/b)p]i-1(pα+1)illllll-(12)pα(ab)p/2∑i=1∞‍[-(1/2)ln⁡(a/b)p]i-1(pα+1)illllll-(12)pα(ab)p/2∑i=0∞‍(12)i-1[ln⁡(a/b)p]i-1(pα+1)illllll+2bp[ln⁡(ab)p]-1)1/p×(|f′(b)|qln⁡(|f′(a)|/|f′(b)|)q[|f′(a)f′(b)|q-1])1/q≤ln⁡b-ln⁡a2×(ap∑i=1∞‍(-1)i-1[ln⁡(a/b)p]i-1(pα+1)i+bp∑i=1∞‍[ln⁡⁡(a/b)p]i-1(pα+1)illllll-(12)pα(ab)p/2∑i=1∞‍[-(1/2)ln⁡(a/b)p]i-1(pα+1)illllll-(12)pα(ab)p/2∑i=1∞‍[(1/2)ln⁡(a/b)p]i-1(pα+1)illllll-(12)pα-1(ab)p/2[ln⁡(ab)p]-1llllll+2bp[ln⁡(ab)p]-1)1/p×(|f′(b)|qln⁡(|f′(a)|/|f′(b)|)q[|f′(a)f′(b)|q-1])1/q≤ln⁡b-ln⁡a2×(ap∑i=1∞‍(-1)i-1[ln⁡(a/b)p]i-1(pα+1)i+bp∑i=1∞‍[ln⁡⁡(a/b)p]i-1(pα+1)illllll-2(12)pα(ab)p/21pα+1llllll-[ln⁡(ab)p]-1[(12)pα-1(ab)p/2+2bp]∑i=1∞‍)1/p×(|f′(b)|qln⁡(|f′(a)|/|f′(b)|)q[|f′(a)f′(b)|q-1])1/q≤ln⁡b-ln⁡a2×(ap∑i=1∞‍(-1)i-1[(1/2)ln⁡(a/b)p]i-1(pα+1)illllll+bp∑i=1∞‍[ln⁡(a/b)p]i-1(pα+1)i-(1/2)pα-1(abp/2)pα+1llllll-[ln⁡(ab)p]-1[(12)pα-1(ab)p/2+2bp]∑i=1∞‍)1/p×(|f′(b)|qln⁡(|f′(a)|/|f′(b)|)q[|f′(a)f′(b)|q-1])1/q.
The proof is done.


## 3. Main Results Based on Lemma2

Theorem 14.
Letf:[0,b]→R+ be a differentiable mapping. If |f′| is measurable and |f′| is GG-convex on [a,b], for some fired α∈(0,∞) and t∈[0,1], 0≤a<b, then the following integrals hold:
(20)|Γ(α+1)2(ln⁡b-ln⁡a)α[JHa+αf(b)+JHb-αf(a)]-f(a+b2)|≤|f′(b)|-|f′(a)|2+ln⁡b-ln⁡a2×{a|f′(a)|(α+2)22α+2(1-12α+1)LL×[b|f′(b)|α+1+b|f′(a)|+a|f′(b)|-2b|f′(b)|α+2]LL+a|f′(a)|-a|f′(b)|-b|f′(a)|α+3LL-a|f′(a)|-b|f′(a)|(α+3)2α+3+2a|f′(a)|(α+1)(α+2)(α+3)LL+b|f′(a)|+a|f′(b)|-2a|f′(a)|(α+1)(α+3)LL+2a(|f′(a)|-|f′(b)|)-2b|f′(a)|2α+3(α+2)(α+3)LL+b|f′(a)|+a|f′(b)|2α+2(α+2)+a|f′(a)|(α+1)2α+2LL-a|f′(a)|(α+2)22α+2}.Proof.
By using Definition4 and Lemmas 2 and 6, we have
(21)|Γ(α+1)2(ln⁡b-ln⁡a)α[JHa+αf(b)+JHb-αf(a)]-f(a+b2)|≤b-a2∫01‍|kf′(ta+(1-t)b)|dt+ln⁡b-ln⁡a2∫01‍atb1-t|(1-t)α-tα|+ln⁡b-ln⁡a2llll×|f′(atb1-t)|dt≤-12(|f(a)|-|f(b)|)+ln⁡b-ln⁡a2×{∫1/21‍atb1-t[tα-(1-t)α]|f′(atb1-t)|dtMM+∫01/2‍atb1-t[(1-t)α-tα]|f′(atb1-t)|dt}≤-12(|f(a)|-|f(b)|)+ln⁡b-ln⁡a2×{∫1/21‍tα[at+b(1-t)]MMML×[t|f′(a)|+(1-t)|f′(b)|]dtMM-∫1/21‍(1-t)α[at+b(1-t)]MMMMM×[t|f′(a)|+(1-t)|f′(b)|]dt∫01/2‍}+ln⁡b-ln⁡a2×{∫01/2‍(1-t)α[at+b(1-t)]MMM×[t|f′(a)|+(1-t)|f′(b)|]dtMM-∫01/2‍tα[at+b(1-t)]MMMMM×[t|f′(a)|+(1-t)|f′(b)|]dt∫01/2‍}≤-12(|f(a)|-|f(b)|)+ln⁡b-ln⁡a2×{|f′(a)|∫1/21‍tα+1[at+b(1-t)]dtMM+|f′(b)|∫1/21‍(1-t)tα[at+b(1-t)]dtMM-|f′(a)|∫1/21‍(1-t)αt[at+b(1-t)]dtMM-|f′(b)|∫1/21‍(1-t)α+1[at+b(1-t)]dt}+ln⁡b-ln⁡a2×{|f′(a)|∫01/2‍t(1-t)α[at+b(1-t)]dtMM+|f′(b)|∫01/2‍(1-t)α+1[at+b(1-t)]dtMM-|f′(a)|∫01/2‍tα+1[at+b(1-t)]dtMM-|f′(b)|∫01/2‍(1-t)tα[at+b(1-t)]dt}≤|f′(b)|-|f′(a)|2+ln⁡b-ln⁡a2×{a|f′(a)|α+3(1-12α+3)+b|f′(a)|α+2(1-12α+2)MM-b|f′(a)|α+3(1-12α+3)+a|f′(b)|α+2(1-12α+2)MM-a|f′(b)|α+3(1-12α+3)+b|f′(b)|α+1(1-12α+1)MM-2b|f′(b)|α+2(1-12α+2)+b|f′(b)|α+3(1-12α+3)MM-a|f′(a)|∫1/21‍(1-t)αt2dtMM-b|f′(a)|∫1/21‍t(1-t)α+1dtMM-a|f′(b)|∫1/21‍t(1-t)α+1dtMM-b|f′(b)|α+3∫1/21‍(α+3)(1-t)α+2dt}+ln⁡b-ln⁡a2×{a|f′(a)|∫01/2t2(1-t)αdt‍MM+b|f′(a)|∫01/2‍t(1-t)α+1dtMM+a|f′(b)|∫01/2‍t(1-t)α+1dtMM+b|f′(b)|α+3(12α+3-1)-a|f′(a)|α+312α+3MM-b|f′(a)|α+212α+2+b|f′(a)|α+312α+3MM-a|f′(b)|α+212α+2+a|f′(b)|α+312α+3-b|f′(b)|α+1MM×12α+1+2b|f′(b)|α+212α+2-b|f′(a)|α+312α+3}≤|f′(b)|-|f′(a)|2+ln⁡b-ln⁡a2×{(a-b)[|f′(a)|-|f′(b)|]α+3(1-12α+3)MM+b|f′(b)|α+1(1-12α+1)MM+b[|f′(a)|-|f′(b)|]+(a-b)|f′(b)|α+2MM×(1-12α+2)-a|f′(a)|∫01‍(1-t)αt2dtMM+a|f′(a)|∫01/2‍(1-t)αt2dtMM-b|f′(a)|∫01‍t(1-t)α+1dtMM+b|f′(a)|∫01/2‍t(1-t)α+1dtMM-a|f′(b)|∫01‍t(1-t)α+1dtMM+a|f′(b)|∫01/2‍t(1-t)α+1(a-b)[|f′(a)|-|f′(b)|]α+3(1-12α+3)}+ln⁡b-ln⁡a2×{b|f′(b)|(α+1)2α+1a|f′(a)|∫01/2‍t2(1-t)αdtMM+b|f′(a)|∫01/2‍t(1-t)α+1dtMM+a|f′(b)|∫01/2‍t(1-t)α+1dtMM+b|f′(b)|α+3(12α+3-1)-a|f′(b)|(α+3)2α+3MM-b|f′(a)|+a|f′(b)|-2b|f′(b)|(α+2)2α+2MM-b|f′(b)|(α+1)2α+1}≤|f′(b)|-|f′(a)|2+ln⁡b-ln⁡a2×{(a-b)[|f′(a)|-|f′(b)|]α+3(1-12α+3)LLL+b|f′(b)|α+1(1-12α+1)LLL+b[|f′(a)|-|f′(b)|]+(a-b)|f′(b)|α+2LLL×(1-12α+2)-a|f′(a)|∫01‍t2(1-t)αdtLLL+a|f′(a)|∫01/2‍t2(1-t)αdtLLL-b|f′(a)|∫01‍t(1-t)α+1dtLLL+b|f′(a)|∫01/2‍t(1-t)α+1dtLLL-a|f′(b)|∫01‍t(1-t)α+1dtLLL+a|f′(b)|∫01/2‍t(1-t)α+1dt(a-b)[|f′(a)|-|f′(b)|]α+3(1-12α+3)}+ln⁡b-ln⁡a2{b|f′(b)|(α+1)2α+1a|f′(a)|∫01/2‍t2(1-t)αdtL+ln⁡b-ln⁡a2+b|f′(a)|∫01/2‍t(1-t)α+1dtL+ln⁡b-ln⁡a2+a|f′(b)|∫01/2‍t(1-t)α+1dtL+ln⁡b-ln⁡a2+b|f′(b)|α+3(12α+3-1)-a|f′(b)|(α+3)2α+3L+ln⁡b-ln⁡a2-b|f′(a)|+a|f′(b)|-2b|f′(b)|(α+2)2α+2L+ln⁡b-ln⁡a2-b|f′(b)|(α+1)2α+1}≤|f′(b)|-|f′(a)|2+ln⁡b-ln⁡a2×{a|f′(a)|(α+2)22α+2(1-12α+1)LLl×[b|f′(b)|α+1+b|f′(a)|+a|f′(b)|-2b|f′(b)|α+2]LLl+a|f′(a)|-a|f′(b)|-b|f′(a)|α+3LLl-a|f′(a)|-b|f′(a)|(α+3)2α+3+2a|f′(a)|(α+1)(α+2)(α+3)LLl+b|f′(a)|+a|f′(b)|-2a|f′(a)|(α+1)(α+3)LLl+2a(|f′(a)|-|f′(b)|)-2b|f′(a)|2α+3(α+2)(α+3)LLl+b|f′(a)|+a|f′(b)|2α+2(α+2)LLl+a|f′(a)|(α+1)2α+2-a|f′(a)|(α+2)22α+2}.
The proof is done.Theorem 15.
Letf:[0,b]→R+ be a differentiable mapping. If |f′|q is measurable and |f′|q, (q>1) is GG-convex on [a,b], for some fired α∈(0,∞) and t∈[0,1], 0≤a<b, then the following integrals hold:
(22)|Γ(α+1)2(ln⁡b-ln⁡a)α[JHa+αf(b)+JHb-αf(a)]-f(a+b2)|≤|f′(b)|-|f′(a)|2+ln⁡b-ln⁡a2×(|f′(a)|q+2|f′(a)|q-|f′(a)|q2)1/q×(ap-2bppα+2+bppα+1-2bp+ap2pα+2(pα+1)mm+2bp-ap2pα+2(pα+2)-appα(pα+1)mm-ap+bp2pα+2(pα+1)(pα+2)mm+2ap(pα+1)(pα+2))1/p,
where 1/p+1/q=1.Proof.
By using Definition4 and Lemmas 2 and 6, we have
(23)|Γ(α+1)2(ln⁡b-ln⁡a)α[JHa+αf(b)+JHb-αf(a)]-f(a+b2)|≤b-a2∫01‍|kf′(ta+(1-t)b)|dt+ln⁡b-ln⁡a2∫01‍atb1-t|(1-t)α-tα||f′(atb1-t)|dt≤b-a2∫01‍|k||f′(at+b(1-t))|dt+ln⁡b-ln⁡a2∫01atb1-t|(1-t)α-tα||f′(atb1-t)|dt‍≤(b-a)2∫01‍|f′(at+b(1-t))|dt+ln⁡b-ln⁡a2∫01‍atb1-t|(1-t)α-tα||f′(atb1-t)|dt≤-12(|f′(a)|-|f′(b)|)+ln⁡b-ln⁡a2(∫01‍(atb1-t|(1-t)α-tα|)pdt)1/p×(∫01‍|f′(atb1-t)|qdt)1/q≤-12(|f′(a)|-|f′(b)|)+ln⁡b-ln⁡a2(∫01‍aptbp(1-t)|(1-t)α-tα|pdt)1/p×(∫01‍|f′(a)|qt|f′(b)|q(1-t)dt)1/q≤-12(|f′(a)|-|f′(b)|)+ln⁡b-ln⁡a2×(∫01‍[tap+(1-t)bp]|(1-t)α-tα|pdt)1/p×(∫01‍[t|f′(a)|q+(1-t)|f′(b)|q]dt)1/q≤-12(|f′(a)|-|f′(b)|)+ln⁡b-ln⁡a2×(∫1/21‍[tap+(1-t)bp](tpa-(1-t)pa)dtLLL+∫01/2‍[tap+(1-t)bp][(1-t)pα-tpα]dt)1/p×(|f′(a)|q2+|f′(b)|q-|f′(b)|q2)1/q≤-12(|f′(a)|-|f′(b)|)+ln⁡b-ln⁡a2(|f′(a)|q+2|f′(b)|q-|f′(b)|q2)1/q×(ap∫1/21‍tpα+1dt-ap∫1/21‍t(1-t)pαdtMM+bp∫1/21‍tpα(1-t)dt-bp∫1/21‍(1-t)pα+1dtMM+ap∫01/2‍t(1-t)pαdt-ap∫01/2‍tpα+1dtMM+bp∫01/2‍(1-t)pα+1-bp∫01/2‍tpα(1-t)dt)1/p≤-12(|f′(a)|-|f′(b)|)+ln⁡b-ln⁡a2(|f′(a)|q+2|f′(b)|q-|f′(b)|q2)1/q×(appα+2-ap∫01‍t(1-t)pα-1dtMM+ap∫01/2‍t(1-t)pαdt+bppα+1-12pα+1bppα+1MM-bppα+2+12pα+2bppα+2+ap∫01/2‍t(1-t)pαdtMM-12pα+2appα+2+12pα+2bppα+2-bppα+2MM-bp∫01/2‍tpα(1-t)dt)1/p≤|f′(b)|-|f′(a)|2+ln⁡b-ln⁡a2(|f′(a)|q+2|f′(b)|q-|f′(b)|q2)1/q×(ap-2bppα+2+bppα+1-2bp+ap2pα+2(pα+1)MM+2bp-ap2pα+2(pα+2)-appα(pα+1)MM-ap+bp2pα+2(pα+1)(pα+2)MM+2ap(pα+1)(pα+2))1/p.
The proof is done.Theorem 16.
Letf:[0,b]→R+ be a differentiable mapping. If |f′| is measurable and |f′| is GG-convex on [a,b], for some fired α∈(0,∞) and t∈[0,1], 0≤a<b, then the following integrals hold:
(24)|Γ(α+1)2(ln⁡b-ln⁡a)α[JHa+αf(b)+JHb-αf(a)]-f(a+b2)|≤12(|f′(b)|-|f′(a)|)+b|f′(b)|ln⁡b-ln⁡a2×{a|f′(a)|b|f′(b)|∑i=1∞‍(-1)i-1[ln⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)iMM+∑i=1∞‍[ln⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)iMM-(1/2)α-1(a|f′(a)|/b|f′(b)|)1/2α+1MM+21-(1/2)α(a|f′(a)|/b|f′(b)|)1/2ln⁡(a|f′(a)|/b|f′(b)|)∑i=1∞‍}.Proof.
By using Definition4 and Lemmas 2, 7, 8, and 9, we have
(25)|Γ(α+1)2(ln⁡b-ln⁡a)α[JHa+αf(b)+JHb-αf(a)]-f(a+b2)|≤|b-a2∫01‍kf′(ta+(1-t)b)dt|+ln⁡b-ln⁡a2∫01‍atb1-t|(1-t)α-tα|+ln⁡b-ln⁡a2lll×|f′(atb1-t)|dt≤-12(|f(a)|-|f(b)|)+ln⁡b-ln⁡a2×{∫1/21‍atb1-t[tα-(1-t)α]|f′(atb1-t)|dtLLL+∫01/2‍atb1-t[(1-t)α-tα]|f′(atb1-t)|dt}≤-12(|f(a)|-|f(b)|)+ln⁡b-ln⁡a2×∫1/21‍atb1-t[tα-(1-t)α]|f′(a)|t|f′(b)|1-tdt+ln⁡b-ln⁡a2×∫01/2‍atb1-t[(1-t)α-tα]|f′(a)|t|f′(b)|1-tdt≤-12(|f(a)|-|f(b)|)+b|f′(b)|ln⁡b-ln⁡a2×∫1/21(a|f′(a)|b|f′(b)|)t[tα-(1-t)α]dt‍+b|f′(b)|ln⁡b-ln⁡a2×∫01/2‍(a|f′(a)|b|f′(b)|)t[(1-t)α-tα]dt≤12(|f(b)|-|f(a)|)+b|f′(b)|ln⁡b-ln⁡a2×{∫01‍(a|f′(a)|b|f′(b)|)ttαdt00000000-∫01‍(a|f′(a)|b|f′(b)|)t(1-t)αdt00000000-2∫01/2‍(a|f′(a)|b|f′(b)|)ttαdt00000000+2∫01/2‍(a|f′(a)|b|f′(b)|)t(1-t)αdt}≤12(|f(b)|-|f(a)|)+b|f′(b)|ln⁡b-ln⁡a2×{a|f′(a)|b|f′(b)|∑i=1∞(-1)i-1[ln⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)i‍m-∑i=1∞‍[ln⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)im-(12)α-1(a|f′(a)|b|f′(b)|)1/2m×∑i=1∞‍[-(1/2)ln⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)im+2∑i=0∞‍[ln⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)im×[1-(12)α+i(a|f′(a)|b|f′(b)|)1/2]∑i=0∞‍}≤12(|f′(b)|-|f′(a)|)+b|f′(b)|ln⁡b-ln⁡a2×{a|f′(a)|b|f′(b)|∑i=1∞‍(-1)i-1[ln⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)im+∑i=1∞‍[ln⁡(a|f′(a)|/b|f′(b)|)]i-1(α+1)im-(1/2)α-1(a|f′(a)|/b|f′(b)|)1/2α+1m+21-(1/2)α(a|f′(a)|/b|f′(b)|)1/2ln⁡(a|f′(a)|/b|f′(b)|)∑i=1∞‍}.
The proof is done.Theorem 17.
Letf:[0,b]→R+ be a differentiable mapping. If |f′|q is measurable and |f′|q, (q>1) is GG-convex on [a,b], for some fired α∈(0,∞) and t∈[0,1], 0≤a<b, then the following integrals hold:
(26)|Γ(α+1)2(ln⁡b-ln⁡a)α[JHa+αf(b)+JHb-αf(a)]-f(a+b2)|≤12(|f(b)|-|f(a)|)+ln⁡b-ln⁡a2×(ap∑i=1∞‍(-1)i-1[(1/2)ln⁡(a/b)p]i-1(pα+1)imm+bp∑i=1∞‍[ln⁡⁡(a/b)p]i-1(pα+1)i-(1/2)pα-1(abp/2)pα+1mm-[ln⁡⁡(ab)p]-1[(12)pα-1(ab)p/2+2bp])1/p×(|f′(b)|qln⁡(|f′(a)|/|f′(b)|)q[|f′(a)f′(b)|q-1])1/q,
where 1/p+1/q=1.Proof.
By using Definition4 and Lemmas 2, 7, 8, and 9, we have
(27)|Γ(α+1)2(ln⁡b-ln⁡a)α[JHa+αf(b)+JHb-αf(a)]-f(a+b2)|≤b-a2∫01‍|kf′(ta+(1-t)b)|dt+ln⁡b-ln⁡a2∫01‍atb1-t|(1-t)α-tα||f′(atb1-t)|dt≤b-a2∫01‍|k||f′(ta+(1-t)b)|dt+ln⁡b-ln⁡a2∫01‍atb1-t|(1-t)α-tα||f′(atb1-t)|dt≤b-a2∫01‍|k||f′(ta+(1-t)b)|dt+ln⁡b-ln⁡a2(∫01‍(atb1-t|(1-t)α-tα|)pdt)1/p×(∫01‍|f′(atb1-t)|qdt)1/q≤-12(|f(a)|-|f(b)|)+ln⁡b-ln⁡a2×(bp∫1/21‍(ab)pttpαdt-bp∫1/21‍(ab)pt(1-t)pαdtllll+bp∫01/2‍(ab)pt(1-t)pαdt-bp∫01/2(ab)pttpαdt)1/p×(|f′(b)|q∫01‍[|f′(a)||f′(b)|]qtdt)1/q≤-12(|f(a)|-|f(b)|)+ln⁡b-ln⁡a2×(bp∫01‍(ab)pttpαdt-bp∫01/2‍(ab)pttpαdtllll-bp∫01‍(ab)pt(1-t)pαdt+∫01/2‍(ab)pt(1-t)pαdtllll+bp∫01/2‍(ab)pt(1-t)pαdt-bp∫01/2‍(ab)pttpαdt)1/p×(|f′(b)|qln⁡(|f′(a)|/|f′(b)|)q[|f′(a)f′(b)|q-1])1/q≤-12(|f(a)|-|f(b)|)+ln⁡b-ln⁡a2×(bp(ab)p∑i=1∞‍(-1)i-1[ln⁡⁡(a/b)p]i-1(pα+1)im-bp∑i=0∞‍[ln⁡⁡(a/b)p]i-1(pα+1)i-2bp∫01/2‍(ab)pttpαdtm+2bp∫01/2‍(ab)pt(1-t)pαdt)1/p×(|f′(b)|qln⁡(|f′(a)|/|f′(b)|)q[|f′(a)f′(b)|q-1])1/q≤-12(|f(a)|-|f(b)|)+ln⁡b-ln⁡a2×(bp(ab)p∑i=1∞‍(-1)i-1[ln⁡⁡(a/b)p]i-1(pα+1)im-bp∑i=1∞‍[ln⁡(a/b)p]i-1(pα+1)im-(12)pαbp(ab)p/2∑i=1∞‍[-(1/2)ln⁡⁡(a/b)p]i-1(pα+1)im+2bp∑i=0∞‍[ln⁡(a/b)p]i-1(pα+1)i[1-(ab)p/2(1-12)pα+i])1/p×(|f′(b)|qln⁡(|f′(a)|/|f′(b)|)q[|f′(a)f′(b)|q-1])1/q≤-12(|f(a)|-|f(b)|)+ln⁡b-ln⁡a2×(ap∑i=1∞‍(-1)i-1[ln⁡⁡(a/b)p]i-1(pα+1)i-bp∑i=1∞‍[ln⁡⁡(a/b)p]i-1(pα+1)imm-(12)pα(ab)p/2∑i=1∞‍[-(1/2)ln⁡(a/b)p]i-1(pα+1)imm+2bp∑i=0∞‍[ln⁡⁡(a/b)p]i-1(pα+1)imm-2bp(ab)p/2∑i=0∞‍(1-12)pα+i[ln⁡⁡(a/b)p]i-1(pα+1)i)1/p×(|f′(b)|qln⁡(|f′(a)|/|f′(b)|)q[|f′(a)f′(b)|q-1])1/q≤-12(|f(a)|-|f(b)|)+ln⁡b-ln⁡a2×(ap∑i=1∞‍(-1)i-1[ln⁡⁡(a/b)p]i-1(pα+1)i-bp∑i=1∞‍[ln⁡⁡(a/b)p]i-1(pα+1)imm-(12)pα(ab)p/2∑i=1∞‍[-(1/2)ln⁡(a/b)p]i-1(pα+1)imm+2bp∑i=1∞‍[ln⁡⁡(a/b)p]i-1(pα+1)i+2bp[ln⁡⁡(ab)p]-1mm-2bp(ab)p/2∑i=0∞‍(1-12)pα+i[ln⁡⁡(a/b)p]i-1(pα+1)i)1/p×(|f′(b)|qln⁡(|f′(a)|/|f′(b)|)q[|f′(a)/f′(b)|q-1])1/q≤12(|f(b)|-|f(a)|)+ln⁡b-ln⁡a2×(ap∑i=1∞‍(-1)i-1[ln⁡⁡(a/b)p]i-1(pα+1)i+bp∑i=1∞‍[ln⁡⁡(a/b)p]i-1(pα+1)im-(12)pα(ab)p/2∑i=1∞‍[-(1/2)ln⁡(a/b)p]i-1(pα+1)im-(12)pα(ab)p/2∑i=1∞‍[(1/2)ln⁡(a/b)p]i-1(pα+1)im-(12)pα-1(ab)p/2[ln⁡(ab)p]-1m+2bp[ln⁡(ab)p]-1∑i=1∞‍)1/p×(|f′(b)|qln⁡(|f′(a)|/|f′(b)|)q[|f′(a)f′(b)|q-1])1/q≤12(|f(b)|-|f(a)|)+ln⁡b-ln⁡a2×(ap∑i=1∞‍(-1)i-1[(1/2)ln⁡(a/b)p]i-1(pα+1)im+bp∑i=1∞‍[ln⁡⁡(a/b)p]i-1(pα+1)i-(1/2)pα-1(abp/2)pα+1m-[ln⁡⁡(ab)p]-1[(12)pα-1(ab)p/2+2bp]∑i=1∞‍)1/p×(|f′(b)|qln⁡(|f′(a)|/|f′(b)|)q[|f′(a)f′(b)|q-1])1/q.
The proof is done.


## 4. Applications to Special Means

Consider the following special means (see [23]) for arbitrary real numbers x,y,x≠y as follows:(M1)
A(x,y)=(x+y)/2, x,y∈R;(M2)
H(x,y)=2/(1/x+1/y), x,y∈R∖{0};(M3)
G(x,y)=xy;(M4)
L(x,y)=(y-x)/(ln⁡|y|-ln⁡|x|), |x|≠|y|, xy≠0;(M5)
Ln(x,y)=[(yn+1-xn+1)/(n+1)(y-x)]1/n, n∈Z∖{-1,0}, x,y∈R, x≠y.We give some applications to special means of real numbers.Proposition 18.
Leta,b∈R+∖{0}, 0≤a<b, x∈[0,b]. Then,
(28)|A(a,b)-L(a,b)|≤ln⁡b-ln⁡a263a-b192;|A(a,b)-L(a,b)|≤ln⁡b-ln⁡a2(12)1/q|A(a,b)-L(a,b)|×(ap-2bpp+2+bpp+1-2bp+ap2p+1(p+1)|A(a,b)-L(a,b)|-bp-ap2p+2(p+2)+ap(p+1)(p+2)|A(a,b)-L(a,b)|-ap2p+1(p+1)(p+2))1/p;|L(a,b)-A(a,b)|≤ln⁡b-ln⁡a263a-b192;|L(a,b)-A(a,b)|≤ln⁡b-ln⁡a2(12)1/q|A(a,b)-L(a,b)|×(ap-2bpp+2+bpp+1-2bp+ap2p+1(p+1)|A(a,b)-L(a,b)|-bp-ap2p+2(p+2)+ap(p+1)(p+2)|A(a,b)-L(a,b)|-ap2p+1(p+1)(p+2))1/p;|A(a,b)-L(a,b)|≤bln⁡b-ln⁡a2|A(a,b)-L(a,b)|×[a/b-1ln⁡(a/b)|A(a,b)-L(a,b)|≤llll+2(a/b)1/2-a/b+(a/b)2-1(ln⁡(a/b))2].
By using Lemmas 7, 8, and 9, we have
(29)|A(a,b)-L(a,b)|≤bln⁡b-ln⁡a2|A(a,b)-L(a,b)|≤×(I[p+1]-H(p+1,(ab)p,12)|A(a,b)-L(a,b)|≤××-2J(p+1,(ab)p)|A(a,b)-L(a,b)|≤××+2bR(p+1,(ab)p,12))1/p;|L(a,b)-A(a,b)|≤bln⁡b-ln⁡a2|L(a,b)-A(a,b)|≤×[a/b-1ln⁡(a/b)|L(a,b)-A(a,b)|≤lllll+2(a/b)1/2-a/b+(a/b)2-1(ln⁡(a/b))2];|L(a,b)-A(a,b)|≤bln⁡b-ln⁡a2|L(a,b)-A(a,b)|≤×(I[p+1]-H(p+1,(ab)p,12)|L(a,b)-A(a,b)|≤lllll-2J(p+1,(ab)p)|L(a,b)-A(a,b)|≤lllll+2bR(p+1,(ab)p,12))1/p.Proof.
Applying Theorems10, 11, 14, 15, 12, 13, 16, and 17, for f(x)=x and α=1, one can obtain the results immediately.Proposition 19.
Leta,b∈R+∖{0}, 0≤a<b, x∈[0,b], n≥2. Then,
(30)|A(an,bn)-Ln-1n-1(a,b)L(a,b)|≤ln⁡b-ln⁡a222nan+12nban-1+9nabn-1-12nbn96;|A(an,bn)-Ln-1n-1(a,b)L(a,b)|≤ln⁡b-ln⁡a2(nqaq(n-1)+nqbq(n-1)2)1/q×(ap-2bpp+2+bpp+1l××-12p+1bpp+1+12p+2bp-app+2-12p+31p+3l××-app(p+1)+2ap(p+1)(p+2))1/p;|Ln-1n-1(a,b)L(a,b)-An(x,y)|≤nbn-1-nan-12+ln⁡b-ln⁡a222nan+12nban-1+9nabn-1-12nbn96;|Ln-1n-1(a,b)L(a,b)-An(x,y)|≤nbn-1-nan-12+ln⁡b-ln⁡a2(nqaq(n-1)+nqbq(n-1)2)1/q×(ap-2bpp+2+bpp+1-12p+1bpp+1lllllll+12p+2bp-app+2-12p+31p+3lllllll-app(p+1)+2ap(p+1)(p+2))1/p;|A(an,bn)-Ln-1n-1(a,b)L(a,b)|≤nbnln⁡b-ln⁡a2×(an/bn-1ln⁡(an/bn)+4(an/bn)-2(an/bn)-2(ln⁡(an/bn))2).
By using Lemmas 7, 8, and 9, we have
(31)|A(an,bn)-Ln-1n-1(a,b)L(a,b)|≤bln⁡b-ln⁡a2(I[p+1]-H(p+1,(ab)p,12)≤bln⁡b-ln⁡a2llll-2J(p+1,(ab)p)≤bln⁡b-ln⁡a2llll+2bR(p+1,(ab)p,12))1/p×(|nbn-1|qln⁡(an-1/bn-1)q[(an-1bn-1)q-1])1/q;|Ln-1n-1(a,b)L(a,b)-An(x,y)|≤nbn-1-nan-12+nbnln⁡b-ln⁡a2×(an/bn-1ln⁡(an/bn)+4(an/bn)-2(an/bn)-2(ln⁡(an/bn))2);|Ln-1n-1(a,b)L(a,b)-An(x,y)|≤nbn-1-nan-12+bln⁡b-ln⁡a2(I[p+1]-H(p+1,(ab)p,12)+bln⁡b-ln⁡a2lllll-2J(p+1,(ab)p)+bln⁡b-ln⁡a2lllll+2bR(p+1,(ab)p,12))1/p×(|nbn-1|qln⁡(an-1/bn-1)q[(an-1bn-1)q-1])1/q.Proof.
Applying Theorems10, 11, 14, 15, 12, 13, 16, and 17 for f(x)=xn and α=1, one can obtain the results immediately.Proposition 20.
Leta,b∈R+∖{0}, 0≤a<b, x∈[0,b], n≥2. Then,
(32)|H-1(a,b)-G-2(a,b)L(a,b)|≤ln⁡b-ln⁡a2×(450(1/a)+60(a/b2)-30(b/a2)llllll-32(1/a)+42(1/a)(a/b2))(1920)-1;|H-1(a,b)-G-2(a,b)L(a,b)|≤ln⁡b-ln⁡a2(a-2q+b-2q2)1/q×(ap-2bpp+2+bpp+1-12p+22ap+3bpp+1llllll+12p+22bp-app+2-1p(p+1)llllll-2ap+bp2p+2(p+1)(p+2))1/p;|G-2(a,b)L(a,b)-A-1(a,b)|≤b2-a22a2b2+ln⁡b-ln⁡a2×(450(1/a)+60(a/b2)-30(b/a2)llllll-32(1/a)+42(1/a)(a/b2))(1920)-1;|G-2(a,b)L(a,b)-A-1(a,b)|≤b2-a22a2b2+ln⁡b-ln⁡a2(a-2q+b-2q2)1/q×(ap-2bpp+2+bpp+1-12p+22ap+3bpp+1lllllll+12p+22bp-app+2-1p(p+1)lllllll-2ap+bp2p+2(p+1)(p+2))1/p;|H-1(a,b)-G-2(a,b)L(a,b)|≤ln⁡b-ln⁡a2b(b/a-1ln⁡(b/a)+4(b/a)1/2-2(b/a)-2(ln⁡(b/a))2).
By using Lemmas 7, 8, and 9, we have
(33)|H-1(a,b)-G-2(a,b)L(a,b)|≤bln⁡b-ln⁡a2(I[p+1]-H(p+1,(ab)p,12)≤bln⁡b-ln⁡a2llll-2J(p+1,(ab)p)≤bln⁡b-ln⁡a2llll+2bR(p+1,(ab)p,12))1/p×(|1/b2|qln⁡|b2/a2|q[|b2a2|q-1])1/q;|G-2(a,b)L(a,b)-A-1(a,b)|≤b2-a22a2b2+ln⁡b-ln⁡a2b(b/a-1ln⁡(b/a)+4(b/a)1/2-2(b/a)-2(ln⁡(b/a))2);|G-2(a,b)L(a,b)-A-1(a,b)|≤b2-a22a2b2+bln⁡b-ln⁡a2(I[p+1]-H(p+1,(ab)p,12)+bln⁡b-ln⁡a2lllll-2J(p+1,(ab)p)+bln⁡b-ln⁡a2lllll+2bR(p+1,(ab)p,12))1/p×(|1/b2|qln⁡|b2/a2|q[|b2/a2|q-1])1/q.Proof.
Applying Theorems10, 11, 14, 15, 12, 13, 16, and 17 for f(x)=1/x and α=1, one can obtain the results immediately.



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*Source: 136035-2014-07-14.xml*