# On Some New Hermite-Hadamard Type Inequalities fors-Geometrically Convex Functions

Authors: İmdat İşcan
Journal: International Journal of Mathematics and Mathematical Sciences
 (2014)

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## Abstract

Some new integral inequalities of Hermite-Hadamard
type related to thes-geometrically convex functions are established and some
applications to special means of positive real numbers are also given.


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## Body

## 1. Introduction

In this section, we firstly list several definitions and some known results.Definition 1.
LetI be an interval in R. Then f:I→R is said to be convex if
(1)f(λx+(1-λ)y)≤λf(x)+(1-λ)f(y)
for all x,y∈I and λ∈[0,1].Definition 2 (see [1]).
Lets∈(0,1]. A function f:I⊂[0,∞)→[0,∞) is said to be s-convex in the second sense if
(2)f(λx+(1-λ)y)≤λsf(x)+(1-λ)sf(y)
for all x,y∈I and λ∈[0,1].Definition 3 (see [2]).
A functionf:I⊂R+=(0,∞)→R+ is said to be a geometrically convex function if
(3)f(xλy1-λ)≤f(x)λf(y)1-λ
for x,y∈I and λ∈[0,1].Definition 4 (see [2]).
A functionf:I⊂R+=(0,∞)→R+ is said to be an s-geometrically convex function if
(4)f(xλy1-λ)≤f(x)λsf(y)(1-λ)s
for some s∈(0,1], where x,y∈I and λ∈[0,1].Example 5.
(i)f(x)=ex,x>0, is s-geometrically convex for all 0<s≤1.
(ii)f(x)=c,x>0, is s-geometrically convex for all 0<s≤1, where c≥1 is a fixed number.Letf:I⊂R+=(0,∞)→[1,∞). It is easy to show that f is s-geometrically convex function on [a,b],a,b∈I with a<b if and only if ln⁡(f∘exp⁡) is s-convex function on [ln⁡a,ln⁡b].Letf:I⊆R→R be a convex function defined on the interval I of real numbers and a,b∈I with a<b. The following double inequality is well known in the literature as Hermite-Hadamard integral inequality:
(5)f(a+b2)≤1b-a∫abf(x)dx≤f(a)+f(b)2.
Recently, several integral inequalities connected with inequalities (5) for the s-convex functions have been established by many authors; for example, see [1, 3–6]. In [2], the authors have established some integral inequalities connected with inequalities (5) for the s-geometrically convex and monotonically decreasing functions. In [7], Tunç has established inequalities for s-geometrically and geometrically convex functions which are connected with the famous Hermite-Hadamard inequality holding for convex functions. In [7], Tunç also has given the following result for geometrically convex and monotonically decreasing functions.Corollary 6.
Letf:I⊂R+→R+ be geometrically convex and monotonically decreasing on [a,b], a,b∈I with a<b; then one has
(6)f2(ab)≤1ln⁡b-ln⁡a∫abf(x)xf(abx)dx≤f(a)f(b).Note that inequalities (6) are also true without the condition monotonically decreasing and inequalities (6) are sharp. Also, inequality (6) obtained for geometrically convex functions is analogous with Hermite-Hadamard inequality (5).Sincef:I⊂R+→R+ is a geometrically convex function if and only iff is a geometrically convex function, inequality (6) can be given as follows:
(7)f(ab)≤1ln⁡b-ln⁡a∫ab1xf(x)f(abx)dx≤f(a)f(b).In this paper, the author gives new identities for differentiable functions. A consequence of the identities is that the author establishes some new inequalities connected with inequalities (6) for the s-geometrically convex functions.


## 2. Main Results

In order to prove our results, we need the following lemma.Lemma 7.
Letf:I⊆R+→R be differentiable on I∘ and a,b∈I with a<b. If f′∈L[a,b], then
(8)f(a)f(b)-1ln⁡b-ln⁡a∫abf(x)xf(abx)dx=∫01b2ln⁡(ab)(t-1)(ab)t/2f(a)f(b)×f(a1-t(ab)t/2)f′(b1-t(ab)t/2)f(a)f(b)+a2ln⁡(ba)(t-1)(ba)t/2f(a)f(b)×f′(a1-t(ab)t/2)f(b1-t(ab)t/2)dt,(9)f2(ab)-1ln⁡b-ln⁡a∫abf(x)xf(abx)dx=∫01b2ln⁡(ab)t(ab)t/2f2(ab)×f(a1-t(ab)t/2)f′(b1-t(ab)t/2)f(a)f(b)+a2ln⁡(ba)t(ba)t/2f(a)f(b)×f′(a1-t(ab)t/2)f(b1-t(ab)t/2)dt.Proof.
Integrating by part and changing variables of integration yield(10)∫01b2ln⁡(ab)(t-1)(ab)t/2f(a1-t(ab)t/2)f′(b1-t(ab)t/2)+a2ln⁡(ba)(t-1)(ba)t/2×f′(a1-t(ab)t/2)f(b1-t(ab)t/2)dt=∫01(t-1)[f(a1-t(ab)t/2)f(b1-t(ab)t/2)]′dt=(t-1)f(a1-t(ab)t/2)f(b1-t(ab)t/2)|01-∫01f(a1-t(ab)t/2)f(b1-t(ab)t/2)dt=f(a)f(b)-2ln⁡b-ln⁡a∫aabf(x)xf(abx)dx.
By the following equality, we obtain inequality (8):
(11)∫aabf(x)xf(abx)dx=∫abbf(x)xf(abx)dx,∫01b2ln⁡(ab)t(ab)t/2f(a1-t(ab)t/2)f′(b1-t(ab)t/2)+a2ln⁡(ba)t(ba)t/2×f′(a1-t(ab)t/2)f(b1-t(ab)t/2)dt=∫01t[f(a1-t(ab)t/2)f(b1-t(ab)t/2)]′dt=tf(a1-t(ab)t/2)f(b1-t(ab)t/2)|01-∫01f(a1-t(ab)t/2)f(b1-t(ab)t/2)dt=f2(ab)-2ln⁡b-ln⁡a∫aabf(x)xf(abx)dx=f2(ab)-1ln⁡b-ln⁡a∫abf(x)xf(abx)dx.
This completes the proof of Lemma 7.Theorem 8.
Letf:I⊆R+→R+ be differentiable on I∘ and a,b∈I∘ with a<b and f′∈L[a,b]. If |f′|q is s-geometrically convex on [a,b], for q≥1 and s∈(0,1], then
(12)|f(a)f(b)-1ln⁡b-ln⁡a∫abf(x)xf(abx)dx|≤ln⁡(ba)(12)2-(1/q)H1(s,q;h1(θ),h1(ϑ)),(13)|f2(ab)-1ln⁡b-ln⁡a∫abf(x)xf(abx)dx|≤ln⁡(ba)(12)2-(1/q)H2(s,q;h2(θ),h2(ϑ)),
where M1=max⁡|f(x)|x∈[a,ab], M2=max⁡|f(x)|x∈[ab,b],
(14)h1(u)={12,u=1,u-ln⁡u-1(ln⁡u)2,u≠1,h2(u)={12,u=1,uln⁡u-u+1(ln⁡u)2,u≠1,u>0,(15)θ=(a|f′(a)|sb|f′(b)|s)q/2,ϑ=(b|f′(b)|sa|f′(a)|s)q/2,(16)Hi(s,q;hi(θ),hi(ϑ))={b|f′(b)|sM1hi1/q(θ)+a|f′(a)|sM2hi1/q(ϑ),|f′(a)|,|f′(b)|≤1,b|f′(b)||f′(a)|1-sM1hi1/q(θ)+a|f′(a)||f′(b)|1-sM2hi1/q(ϑ),|f′(a)|,|f′(b)|≥1,b|f′(b)|M1hi1/q(θ)+a|f′(a)|s|f′(b)|1-sM2hi1/q(ϑ),|f′(a)|≤1≤|f′(b)|,b|f′(b)|s|f′(a)|1-sM1hi1/q(θ)+a|f′(a)|M2hi1/q(ϑ),|f′(b)|≤1≤|f′(a)|,Hi(s,q;hi(θ),hi(ϑ))Hi(s,q;hi(θ))ni=1,2.Proof.
(1) LetM1=max⁡|f(x)|x∈[a,ab],M2=max⁡|f(x)|x∈[ab,b]. Since |f′|q is s-geometrically convex on [a,b], from Lemma 7 and power mean inequality, we have
(17)|f(a)f(b)-1ln⁡b-ln⁡a∫abf(x)xf(abx)dx|≤∫01b2|ln⁡(ab)||t-1|(ab)t/2mmmmm×|f(a1-t(ab)t/2)||f′(b1-t(ab)t/2)|mmmmm+a2ln⁡(ba)|t-1|(ba)t/2mmmmm×|f′(a1-t(ab)t/2)||f(b1-t(ab)t/2)|dt≤b2|ln⁡(ab)|M1mmim×∫01(1-t)(ab)t/2|f′(b1-t(ab)t/2)|dtmmim+a2ln⁡(ba)M2mmim×∫01(1-t)(ba)t/2|f′(a1-t(ab)t/2)|dt≤b2|ln⁡(ab)|M1(∫01(1-t)dt)1-(1/q)mmim×(∫01(1-t)(ab)qt/2|f′(b1-t(ab)t/2)|qdt)1/qmmim+a2ln⁡(ba)M2(∫01(1-t)dt)1-(1/q)mmim×(∫01(1-t)(ba)qt/2|f′(a1-t(ab)t/2)|qdt)1/q≤bln⁡(b/a)M12(12)1-(1/q)mmim×(∫01(1-t)(ab)qt/2(1-t)vww×|f′(a)|q(t/2)s|f′(b)|q((2-t)/2)sdt(ab)qt/2)1/qmmim+aln⁡(b/a)M22(12)1-(1/q)mmim×(∫01(1-t)(ba)qt/2(1-t)mlww×|f′(b)|q(t/2)s|f′(a)|q((2-t)/2)sdt(ba)qt/2)1/q.
If 0<μ≤1≤η,0<t,s≤1, then
(18)μts≤μts,ηts≤ηts+1-s.(i)
If1≥|f′(a)|,|f′(b)|, by (18), we obtain that
(19)∫01(1-t)(ab)qt/2|f′(a)|q(t/2)s|f′(b)|q((2-t)/2)sdt≤∫01(1-t)(ab)qt/2|f′(a)|qst/2|f′(b)|q(qs(2-t)/2)dt=|f′(b)|qsh1(θ),∫01(1-t)(ba)qt/2|f′(b)|q(t/2)s|f′(a)|q((2-t)/2)sdt≤∫01(1-t)(ba)qt/2|f′(b)|qst/2|f′(a)|qs(2-t)/2dt=|f′(a)|qsh1(ϑ).(ii)
If1≤|f′(a)|,|f′(b)|, by (18), we obtain that
(20)∫01(1-t)(ab)qt/2|f′(a)|q(t/2)s|f′(b)|q((2-t)/2)sdt≤∫01(1-t)(ab)qt/2(1-t)×|f′(a)|q((st/2)+1-s)|f′(b)|q(1-(st/2))dt=(|f′(a)|1-s|f′(b)|)qh1(θ),∫01(1-t)(ba)qt/2|f′(b)|q(t/2)s|f′(a)|q((2-t)/2)sdt≤∫01(1-t)(ba)qt/2(1-t)×|f′(b)|q((st/2)+1-s)|f′(a)|q(1-(st/2))dt=(|f′(b)|1-s|f′(a)|)qh1(ϑ).(iii)
If|f′(a)|≤1≤|f′(b)|, by (18), we obtain that
(21)∫01(1-t)(ab)qt/2|f′(a)|q(t/2)s|f′(b)|q((2-t)/2)sdt≤∫01(1-t)(ab)qt/2|f′(a)|qst/2|f′(b)|q(1-(st/2))dt=|f′(b)|qh1(θ),∫01(1-t)(ba)qt/2|f′(b)|q(t/2)s|f′(a)|q((2-t)/2)sdt≤∫01(1-t)(ba)qt/2|f′(b)|q((st/2)+1-s)|f′(a)|qs(2-t)/2dt=(|f′(a)|s|f′(b)|1-s)qh1(ϑ).(iv)
If|f′(b)|≤1≤|f′(a)|, by (18), we obtain that
(22)∫01(1-t)(ab)qt/2|f′(a)|q(t/2)s|f′(b)|q((2-t)/2)sdt≤∫01(1-t)(ab)qt/2|f′(a)|q((st/2)+1-s)|f′(b)|qs(2-t)/2dt=(|f′(a)|1-s|f′(b)|s)qh1(θ),∫01(1-t)(ba)qt/2|f′(b)|q(t/2)s|f′(a)|q((2-t)/2)sdt≤∫01(1-t)(ba)qt/2|f′(b)|qst/2|f′(a)|q(1-(st/2))dt=|f′(a)|qh1(ϑ).From (17) to (22), (12) holds.
(2) LetM1=max⁡|f(x)|x∈[a,ab],M2=max⁡|f(x)|x∈[ab,b]. Since |f′|q is s-geometrically convex on [a,b], from Lemma 7 and Hölder inequality, we have
(23)|f2(ab)-1ln⁡b-ln⁡a∫abf(x)xf(abx)dx|≤∫01b2|ln⁡(ab)|t(ab)t/2mmimm×|f(a1-t(ab)t/2)||f′(b1-t(ab)t/2)|mmimm+a2ln⁡(ba)t(ba)t/2mmimm×|f′(a1-t(ab)t/2)||f(b1-t(ab)t/2)|dt≤b2ln⁡(ba)M1∫01t(ab)t/2|f′(b1-t(ab)t/2)|dtmmim+a2ln⁡(ba)M2∫01t(ba)t/2|f′(a1-t(ab)t/2)|dt≤b2ln⁡(ba)M1(∫01tdt)1-(1/q)mmim×(∫01t(ab)qt/2|f′(b1-t(ab)t/2)|qdt)1/qmmim+a2ln⁡(ba)M2(∫01tdt)1-(1/q)mmim×(∫01t(ba)qt/2|f′(a1-t(ab)t/2)|qdt)1/q≤bln⁡(b/a)M12(12)1-(1/q)mmim×(∫01t(ab)qt/2|f′(a)|q(t/2)s|f′(b)|q((2-t)/2)sdt)1/qmmim+aln⁡(b/a)M22(12)1-(1/q)mmim×(∫01t(ba)qt/2|f′(b)|q(t/2)s|f′(a)|q((2-t)/2)sdt)1/q.(i)
If1≥|f′(a)|,|f′(b)|, by (18), we obtain that
(24)∫01t(ab)qt/2|f′(a)|q(t/2)s|f′(b)|q((2-t)/2)sdt≤|f′(b)|qsh2(θ),∫01t(ba)qt/2|f′(b)|q(t/2)s|f′(a)|q((2-t)/2)sdt≤|f′(a)|qsh2(ϑ).(ii)
If1≤|f′(a)|,|f′(b)|, by (18), we obtain that
(25)∫01t(ab)qt/2|f′(a)|q(t/2)s|f′(b)|q((2-t)/2)sdt≤(|f′(a)|1-s|f′(b)|)qh2(θ),∫01t(ba)qt/2|f′(b)|q(t/2)s|f′(a)|q((2-t)/2)sdt≤(|f′(b)|1-s|f′(a)|)qh2(ϑ).(iii)
If|f′(a)|≤1≤|f′(b)|, by (18), we obtain that
(26)∫01t(ab)qt/2|f′(a)|q(t/2)s|f′(b)|q((2-t)/2)sdt≤|f′(b)|qh2(θ),∫01t(ba)qt/2|f′(b)|q(t/2)s|f′(a)|q((2-t)/2)sdt≤(|f′(a)|s|f′(b)|1-s)qh2(ϑ).(iv)
If|f′(b)|≤1≤|f′(a)|, by (18), we obtain that
(27)∫01t(ab)qt/2|f′(a)|q(t/2)s|f′(b)|q((2-t)/2)sdt≤(|f′(a)|1-s|f′(b)|s)qh2(θ),∫01t(ba)qt/2|f′(b)|q(t/2)s|f′(a)|q((2-t)/2)sdt≤|f′(a)|qh2(ϑ).From (23) to (27), (13) holds. This completes the required proof.If we takes=1 in Theorem 8, we can derive the following corollary.Corollary 9.
Letf:I⊆R+→R+ be differentiable on I∘ and a,b∈I∘ with a<b and f′∈L[a,b]. If |f′|q is geometrically convex on [a,b], for q≥1, then
(28)|f(a)f(b)-1ln⁡b-ln⁡a∫abf(x)xf(abx)dx|≤ln⁡(ba)(12)2-(1/q)H1(1,q;h1(θ),h1(ϑ)),|f2(ab)-1ln⁡b-ln⁡a∫abf(x)xf(abx)dx|≤ln⁡(ba)(12)2-(1/q)H2(1,q;h2(θ),h2(ϑ)),
where θ,ϑ, H1,H2,h1, and h2 are the same as in Theorem 8.If we takeq=1 in Theorem 8, we can derive the following corollary.Corollary 10.
Letf:I⊆R+→R+ be differentiable on I∘ and a,b∈I∘ with a<b and f′∈L[a,b]. If |f′| is s-geometrically convex on [a,b], for s∈(0,1], then
(29)|f(a)f(b)-1ln⁡b-ln⁡a∫abf(x)xf(abx)dx|≤ln⁡(ba)(12)H1(s,1;h1(θ),h1(ϑ)),|f2(ab)-1ln⁡b-ln⁡a∫abf(x)xf(abx)dx|≤ln⁡(ba)(12)H2(s,1;h2(θ),h2(ϑ)),
where θ,ϑ, H1, H2, h1, and h2 are the same as in Theorem 8.Theorem 11.
Letf:I⊆R+→R+ be differentiable on I∘ and a,b∈I∘ with a<b and f′∈L[a,b]. If |f′|q is s-geometrically convex on [a,b], for q>1 and s∈(0,1], then
(30)|f(a)f(b)-1ln⁡b-ln⁡a∫abf(x)xf(abx)dx|≤ln⁡(b/a)2(q-12q-1)1-(1/q)H3(s,q;h3(θ),h3(ϑ)),(31)|f2(ab)-1ln⁡b-ln⁡a∫abf(x)xf(abx)dx|≤ln⁡(b/a)2(q-12q-1)1-(1/q)H3(s,q;h3(θ),h3(ϑ)),
where M1=max⁡|f(x)|x∈[a,ab], M2=max⁡|f(x)|x∈[ab,b],
(32)h3(u)={1,u=1,u-1ln⁡u,u≠1,u>0,H3(s,q;h3(θ),h3(ϑ))={b|f′(b)|sM1h31/q(θ)+a|f′(a)|sM2h31/q(ϑ),|f′(a)|,|f′(b)|≤1,b|f′(b)||f′(a)|1-sM1h31/q(θ)+a|f′(a)||f′(b)|1-sM2h31/q(ϑ),|f′(a)|,|f′(b)|≥1,b|f′(b)|M1h31/q(θ)+a|f′(a)|s|f′(b)|1-sM2h31/q(ϑ),|f′(a)|≤1≤|f′(b)|,b|f′(b)|sM1h31/q(θ)+a|f′(a)|M2h31/q(ϑ),|f′(b)|≤1≤|f′(a)|,
and θ,ϑ are the same as in (15).Proof.
(1) LetM1=max⁡|f(x)|x∈[a,ab],M2=max⁡|f(x)|x∈[ab,b]. Since |f′|q is s-geometrically convex on [a,b], from Lemma 7 and Hölder inequality, we have
(33)|f(a)f(b)-1ln⁡b-ln⁡a∫abf(x)xf(abx)dx|≤∫01b2|ln⁡(ab)||t-1|(ab)t/2wi(1-t)×|f(a1-t(ab)t/2)||f′(b1-t(ab)t/2)|wwwwwi+a2ln⁡(ba)|t-1|(ba)t/2wwwwwi×|f′(a1-t(ab)t/2)||f(b1-t(ab)t/2)|dt≤b2|ln⁡(ab)|M1(1-t)l×∫01(1-t)(ab)t/2|f′(b1-t(ab)t/2)|dt(1-t)l+a2ln⁡(ba)M2(1-t)l×∫01(1-t)(ba)t/2|f′(a1-t(ab)t/2)|dt≤b2|ln⁡(ab)|M1(∫01(1-t)q/(q-1)dt)1-(1/q)(1-t)×(∫01(ab)qt/2|f′(b1-t(ab)t/2)|qdt)1/q(1-t)+a2ln⁡(ba)M2(∫01(1-t)q/(q-1)dt)1-(1/q)(1-t)×(∫01(ba)qt/2|f′(a1-t(ab)t/2)|qdt)1/q≤bln⁡(b/a)M12(q-12q-1)1-(1/q)(1-t)×(∫01(ab)qt/2|f′(a)|q(t/2)s|f′(b)|q((2-t)/2)sdt)1/q(1-t)+aln⁡(b/a)M22(q-12q-1)1-(1/q)(1-t)×(∫01(ba)qt/2|f′(b)|q(t/2)s|f′(a)|q((2-t)/2)sdt)1/q.(i)
If1≥|f′(a)|,|f′(b)|, by (18), we have
(34)∫01(ab)qt/2|f′(a)|q(t/2)s|f′(b)|q((2-t)/2)sdt=|f′(b)|qsh3(θ),∫01(ba)qt/2|f′(b)|q(t/2)s|f′(a)|q((2-t)/2)sdt=|f′(a)|qsh3(ϑ).(ii)
If1≤|f′(a)|,|f′(b)|, by (18), we have
(35)∫01(ab)qt/2|f′(a)|q(t/2)s|f′(b)|q((2-t)/2)sdt≤(|f′(b)||f′(a)|1-s)qh3(θ),∫01(ba)qt/2|f′(b)|q(t/2)s|f′(a)|q((2-t)/2)sdt≤(|f′(a)||f′(b)|1-s)qh3(ϑ).(iii)
If|f′(a)|≤1≤|f′(b)|, by (18), we obtain that
(36)∫01(ab)qt/2|f′(a)|q(t/2)s|f′(b)|q((2-t)/2)sdt≤|f′(b)|qh3(θ),∫01(ba)qt/2|f′(b)|q(t/2)s|f′(a)|q((2-t)/2)sdt≤(|f′(a)|s|f′(b)|1-s)qh3(ϑ).(iv)
If|f′(b)|≤1≤|f′(a)|, by (18), we obtain that
(37)∫01(ab)qt/2|f′(a)|q(t/2)s|f′(b)|q((2-t)/2)sdt≤(|f′(a)|1-s|f′(b)|s)qh3(θ),∫01(ba)qt/2|f′(b)|q(t/2)s|f′(a)|q((2-t)/2)sdt≤|f′(a)|qh3(ϑ).From (33) to (37), (30) holds.
(2) LetM1=max⁡|f(x)|x∈[a,ab], M2=max⁡|f(x)|x∈[ab,b] Since |f′|q is s-geometrically convex on [a,b], from Lemma 7 and Hölder inequality, we have
(38)|f2(ab)-1ln⁡b-ln⁡a∫abf(x)xf(abx)dx|≤b2ln⁡(ba)M1(q-12q-1)1-(1/q)(1-t)×(∫01(ab)qt/2|f′(a)|q(t/2)s|f′(b)|q((2-t)/2)sdt)1/q(1-t)+a2ln⁡(ba)M2(q-12q-1)1-(1/q)(1-t)×(∫01(ba)qt/2|f′(b)|q(t/2)s|f′(a)|q((2-t)/2)sdt)1/q.
From (38) and (34) to (37), (31) holds.If takings=1 in Theorem 11, we can derive the following corollary.Corollary 12.
Letf:I⊆R+→R+ be differentiable on I∘ and a,b∈I∘ with a<b and f′∈L[a,b]. If |f′|q is geometrically convex on [a,b], for q>1, then
(39)|f(a)f(b)-1ln⁡b-ln⁡a∫abf(x)xf(abx)dx|≤12ln⁡(ba)(q-12q-1)1-(1/q)H3(1,q;h3(θ),h3(ϑ)),|f2(ab)-1ln⁡b-ln⁡a∫abf(x)xf(abx)dx|≤12ln⁡(ba)(q-12q-1)1-(1/q)H3(1,q;h3(θ),h3(ϑ)),
where θ,ϑ, H3, and h3 are the same as in Theorem 11.


## 3. Application to Special Means

Let us recall the following special means of two nonnegative numbersa,b with >a.(1)
The arithmetic mean(40)A=A(a,b):=a+b2.(2)
The geometric mean(41)G=G(a,b):=ab.(3)
The logarithmic mean(42)L=L(a,b):=b-aln⁡b-ln⁡a.(4)
Thep-logarithmic mean
(43)Lp=Lp(a,b):=(bp+1-ap+1(p+1)(b-a))1/p,p∈R∖{-1,0}.Letf(x)=(xs/s)+1,x∈(0,1],0<s<1,q≥1, and then the function |f′(x)|q=x(s-1)q is s-geometrically convex on (0,1] for 0<s<1 (see [2]).Proposition 13.
Let0<a<b≤1,0<s<1, and q≥1. Then
(44)|G2(ass+1,bss+1)-2s2A(G2(as,bs),s2)-2sLs-1s-1(a,b)L(a,b)(ass+1,bss+1)|≤1G(s-1)2(a,b)(1(s2-s+1)q)1/q(b-a4L(a,b))1-(1/q)×[{a(s2-s+1)q/2}1/qM1G(a-(s-1)2,bs)(s2-s+)×{b(s2-s+1)q/2-L(a(s2-s+1)q/2,b(s2-s+1)q/2)}1/q(s2-s+)×M2G(b-(s-1)2,as)(s2-s+)×{L(a(s2-s+1)q/2,b(s2-s+1)q/2)-a(s2-s+1)q/2}1/q],|(Gs(a,b)s+1)2-2ssA(G2(as,bs),s2)-2sLs-1s-1(a,b)L(a,b)(Gs(a,b)s+1)2|≤1G(s-1)2(a,b)(1(s2-s+1)q)1/q(b-a4L(a,b))1-(1/q)×[M1G(a-(s-1)2,bs)(s2-s+)×{L(a(s2-s+1)q/2,b(s2-s+1)q/2)-a(s2-s+1)q/2}1/q(s2-s+)×M2G(b-(s-1)2,as)]×{b(s2-s+1)q/2-L(a(s2-s+1)q/2,b(s2-s+1)q/2)}1/q,
where M1=(abs/s)+1 and M2=(bs/s)+1.Proof.
Letf(x)=(xs/s)+1,x∈(0,1],0<s<1. Then |f′(a)|=as-1>bs-1=|f′(b)|≥1, M1=max⁡|f(x)|x∈[a,ab]=(abs/s)+1, M2=max⁡|f(x)|x∈[ab,b]=(bs/s)+1. Thus, by Theorem 8, Proposition 13 is proved.Proposition 14.
Let0<a<b≤1,0<s<1, and q>1. Then
(45)|G2(ass+1,bss+1)-2s2A(G2(as,bs),s2)-2sLs-1s-1(a,b)L(a,b)(ass+1,bss+1)|≤b-a2L(a,b)(q-12q-1)1-(1/q)×L1/q(a(s2-s+1)q/2,b(s2-s+1)q/2)G(s-1)2(a,b)×{M1G(a-(s-1)2,bs)+M2G(b-(s-1)2,as)},|(Gs(a,b)s+1)2-2ssA(G2(as,bs),s2)-2sLs-1s-1(a,b)L(a,b)(Gs(a,b)s+1)2|≤b-a2L(a,b)(q-12q-1)1-(1/q)×L1/q(a(s2-s+1)q/2,b(s2-s+1)q/2)G(s-1)2(a,b)×{M1G(a-(s-1)2,bs)+M2G(b-(s-1)2,as)},
where M1=(abs/s)+1 and M2=(bs/s)+1.Proof.
Letf(x)=(xs/s)+1,x∈(0,1],0<s<1. Then |f′(a)|=as-1>bs-1=|f′(b)|≥1, M1=max⁡|f(x)|x∈[a,ab]=(abs/s)+1, M2=max⁡|f(x)|x∈[ab,b]=(bs/s)+1. Thus, by Theorem 11, Proposition 14 is proved.



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*Source: 163901-2014-06-19.xml*