1. Introduction Let a real function f be defined on a nonempty interval I of real line R. The function f is said to be convex on I if inequality(1)ftx+1-ty≤tfx+1-tfyholds for all x,y∈I and t∈0,1.
In [1], Breckner introduced s-convex functions as a generalization of convex functions as follows.
Definition 1. Let s∈(0,1] be a fixed real number. A function f:[0,∞)→[0,∞) is said to be s-convex (in the second sense), or that f belongs to the class Ks2, if (2)ftx+1-ty≤tsfx+1-tsfyfor all x,y∈[0,∞) and t∈[0,1].
Of course, s-convexity means just convexity when s=1.
The following inequalities are well known in the literature as Hermite-Hadamard inequality, Ostrowski inequality, and Simpson inequality, respectively.
Theorem 2. Let f: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 holds:(3)fa+b2≤1b-a∫abfxdx≤fa+fb2.
Theorem 3. Let f:I⊆R→R be a mapping differentiable in I∘, the interior of I, and let a,b∈I∘ with a<b. If f′(x)≤M, x∈a,b, then the following inequality holds: (4)fx-1b-a∫abftdt≤Mb-ax-a2+b-x22for all x∈a,b.
Theorem 4. Let f:a,b→R be a four times’ continuously differentiable mapping on a,b and f(4)∞=supx∈a,bf(4)(x)<∞. Then the following inequality holds: (5)13fa+fb2+2fa+b2-1b-a∫abfxdx≤12880f4∞b-a4.
We will give definitions of the right and left hand side Hadamard fractional integrals which are used throughout this paper.
Definition 5. Let f∈La,b. The right-sided and left-sided Hadamard fractional integrals Ja+αf and Jb-αf of order α>0 with b>a≥0 are defined by(6)Ja+αfx=1Γα∫axlnxtα-1ftdtt, a<x<b,(7)Jb-αfx=1Γα∫xblntxα-1ftdtt, a<x<b,respectively, where Γ(α) is the Gamma function defined by Γ(α)=∫0∞e-ttα-1dt (see [2]).
In recent years, many authors have studied errors estimations for Hermite-Hadamard, Ostrowski, and Simpson inequalities; for refinements, counterparts, and generalization see [3–10].
Definition 6 (see [11, 12]). A function f:I⊆0,∞→R is said to be GA-convex (geometric-arithmetically convex) if (8)fxty1-t≤tfx+1-tfyfor all x,y∈I and t∈0,1.
Definition 7 (see [13]). For s∈0,1, a function f:I⊆0,∞→R is said to be GA-s-convex (geometric-arithmetically s-convex) if (9)fxty1-t≤tsfx+1-tsfyfor all x,y∈I and t∈0,1.
It can be easily seen that if s=1, GA-s-convexity reduces to GA-convexity.
For recent results and generalizations concerning GA-convex and GA-s-convex functions see [13–19].
Lemma 8 (see [20]). For α>0 and μ>0, one has (10)∫01tα-1μtdt=μ∑k=1∞-1k-1lnμk-1αk<∞,where (11)αk=αα+1α+2⋯α+k-1.
Let f:I⊆0,∞→R be a differentiable function on I∘, the interior of I; in sequel of this paper we will take (12)Ifx,λ,α,a,b=1-λlnαxa+lnαbxfx+λfalnαxa+fblnαbx-Γα+1Jx-αfa+Jx+αfb,where a,b∈I with a<b, x∈[a,b], λ∈0,1, α>0, and Γ is Euler Gamma function.
In [21], I˙şcan gave Hermite-Hadamard’s inequalities for GA-convex functions in fractional integral forms as follows.
Theorem 9. Let f:I⊆0,∞→R be a function such that f∈L[a,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: (13)fab≤Γα+12lnb/aαJa+αfb+Jb-αfa≤fa+fb2with α>0.
In [21], I˙şcan obtained some new inequalities for quasi-geometrically convex functions via fractional integrals by using the following lemma.
Lemma 10. Let f:I⊆0,∞→R be a differentiable function on I∘ such that f′∈L[a,b], where a,b∈I with a<b. Then for all x∈[a,b], λ∈0,1, and α>0 one has (14)Ifx,λ,α,a,b=alnxaα+1∫01tα-λxatf′xta1-tdt-blnbxα+1∫01tα-λxbtf′xtb1-tdt.
In this paper, we will use Lemma 10 to obtain some new inequalities on generalization of Hadamard, Ostrowski, and Simpson type inequalities for GA-s-convex functions via Hadamard fractional integral.
2. Generalized Integral Inequalities for Some GA-s-Convex Functions via Fractional Integrals Theorem 11. Let f:I⊂0,∞→R be a differentiable function on I∘ such that f′∈L[a,b], where a,b∈I∘ with a<b. If |f′|q is GA-s-convex on [a,b] in the second sense for some fixed q≥1, x∈[a,b], λ∈0,1, and α>0 then the following inequality for fractional integrals holds:(15)Ifx,λ,α,a,b≤A11-1/qα,λalnxaα+1f′xqA2xaq,α,λ,s+f′aqA3xaq,α,λ,s1/q+blnbxα+1f′xqA2xbq,α,λ,s+f′bqA3xbq,α,λ,s1/q,where (16)A1α,λ=2αλ1+1/α+1α+1-λ,A2xuq,α,λ,s=∫01tα-λxuqttsdt,A3xuq,α,λ,s=∫01tα-λxuqt1-tsdt, u=a,b.
Proof. Using Lemma 10, property of the modulus, and the power-mean inequality, we have(17)Ifx,λ,α,a,b≤alnxaα+1∫01tα-λxatf′xta1-tdt+blnbxα+1∫01tα-λxbtf′xtb1-tdt≤alnxaα+1∫01tα-λdt1-1/q∫01tα-λxaqtf′xta1-tqdt1/q+blnbxα+1∫01tα-λdt1-1/q∫01tα-λxbqtf′xtb1-tqdt1/q. Since f′q is GA-s-convex on [a,b], we get(18)∫01tα-λxaqtf′xta1-tqdt≤∫01tα-λxaqttsf′xq+1-tsf′aqdt=f′xqA2xa,α,λ,s,q+f′aqA3xa,α,λ,s,q,(19)∫01tα-λxbqtf′xtb1-tqdt≤∫01tα-λxbqttsf′xq+1-tsf′bqdt=f′xqA2xb,α,λ,s,q+f′bqA3xb,α,λ,s,q,and by a simple computation, we have(20)∫01tα-λdt=∫0λ1/αλ-tαdt+∫λ1/α1tα-λdt=2αλ1+1/α+1α+1-λ.Hence, If we use (18), (19), and (20) in (17), we obtain the desired result. This completes the proof.
Corollary 12. Under the assumptions of Theorem 11 with s=1, inequality (15) reduces to the following inequality: (21)Ifx,λ,α,a,b≤A11-1/qα,λalnxaα+1f′xqA2xaq,α,λ,1+f′aqA3xaq,α,λ,11/q+blnbxα+1f′xqA2xbq,α,λ,1+f′bqA3xbq,α,λ,11/q.
Corollary 13. Under the assumptions of Theorem 11 with s=1 and α=1, inequality (15) reduces to the following inequality: (22)lnba-1Ifx,λ,1,a,b=1-λfx+λfalnx/a+fblnb/xlnb/a-1lnb/a∫abfuudu≤lnba-1A11-1/q1,λalnxa2f′xqA2μa,1,λ,1+f′aqA3μa,1,λ,11/q+blnbx2f′xqA2μb,1,λ,1+f′bqA3μb,1,λ,11/q,where (23)A11,λ=2λ2-2λ+12,A2μu,1,λ,1=μu-2λ2μuλln2μu+λμuλlnμu-μuλ+1λlnμu+λ+4-λ+2μulnμu-μu+1lnμu3,A3μu,1,λ,1=2μuλ+μulnμu-λ1+μulnμu-μu-1lnμu2-A2μu,1,λ,1,μu=xuq, u=a,b.
Corollary 14. Under the assumptions of Theorem 11 with q=1, inequality (15) reduces to the following inequality: (24)Ifx,λ,α,a,b≤alnxaα+1f′xA2xa,α,λ,s+f′aA3xa,α,λ,s+blnbxα+1f′xA2xb,α,λ,s+f′bA3xb,α,λ,s.
Corollary 15. Under the assumptions of Theorem 11 with x=ab, λ=1/3, from inequality (15), one gets the following Simpson type inequality for fractional integrals: (25)2α-1lnba-αIfab,13,α,a,b=16fa+4fab+fb-2α-1Γα+1lnb/aαJab-αfa+Jab+αfb≤lnb/a4A11-1/qα,13af′abqA2baq/2,α,13,s+f′aqA3baq/2,α,13,s1/q+bf′abqA2abq/2,α,13,s+f′bqA3abq/2,α,13,s1/q.
Corollary 16. Under the assumptions of Theorem 11 with x=ab, λ=0, from inequality (15), one gets (26)2α-1lnba-αIfab,0,α,a,b=fab-2α-1Γα+1lnb/aαJab-αfa+Jab+αfb≤lnb/a41α+11-1/qaf′abqA2baq/2,α,0,s+f′aqA3baq/2,α,0,s1/q+bf′abqA2abq/2,α,0,s+f′bqA3abq/2,α,0,s1/q.
Corollary 17. Under the assumptions of Theorem 11 with x=ab and λ=1, from inequality (15) one gets (27)2α-1lnba-αIfab,1,α,a,b=fa+fb2-2α-1Γα+1lnb/aαJab-αfa+Jab+αfb≤lnb/a4αα+11-1/qaf′abqA2baq/2,α,1,s+f′aqA3baq/2,α,1,s1/q+bf′abqA2abq/2,α,1,s+f′bqA3abq/2,α,1,s1/q.
Corollary 18. Let the assumptions of Theorem 11 hold. If f′(x)≤M for all x∈a,b and λ=0, then from inequality (15), one gets the following Ostrowski type inequality for fractional integrals: (28)lnxaα+lnbxαfx-Γα+1Jx-αfa+Jx+αfb≤M1α+11-1/qalnxaαA2baq/2,α,0,s+A3baq/2,α,0,s1/q+blnbxαA2abq/2,α,0,s+A3abq/2,α,0,s1/qfor all x∈a,b.
Theorem 19. Let f:I⊂0,∞→R be a differentiable function on I∘ such that f′∈L[a,b], where a,b∈I∘ with a<b. If |f′|q is GA-s-convex on [a,b] for some fixed q>1, x∈[a,b], λ∈0,1, and α>0 then the following inequality for fractional integrals holds:(29)Ifx,λ,α,a,b≤C11/pα,λalnxaα+1f′xqC2xaq,s+f′aqC3xaq,s1/q+blnbxα+1f′xqC2xbq,s+f′bqC3xbq,s1/q, where 1/p+1/q=1 and (30)C1α,λ=1αp+1,λ=0λ1+p+1/ααβ1α,p+1+1-λp+1αp+1·F211-1α,1;p+2;1-λ,0<λ≤1,C2xuq,s=xuq∑k=1∞-1k-1lnx/uqk-1s+1k,C3xuq,s=∑k=1∞-1k-1-lnx/uqk-1s+1k, u=a,b.
Proof. Using Lemma 10, property of the modulus, the Hölder inequality, and GA-s-convexity of f′q, we have
(31)
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where μa=x/aq, μb=x/bq and(32)∫01tα-λpdt=∫0λ1/αλ-tαpdt+∫λ1/α1tα-λpdt=1αp+1,λ=0λαp+1/ααβ1α,p+1+1-λp+1αp+1·F211-1α,1;p+2;1-λ,0<λ≤1.Using Lemma 8, we have (33)∫01μuttsdt=μu∑k=1∞-1k-1lnμuk-1s+1k,∫01μut1-tsdt=∫01μu1-ttsdt=∑k=1∞-1k-1-lnμuk-1s+1k, u=a,b.Hence, if we use (32)-(33) in (31) and replacing μa=x/aq, μb=x/bq, we obtain the desired result. This completes the proof.
Corollary 20. Under the assumptions of Theorem 19 with s=1, inequality (29) reduces to the following inequality: (34)Ifx,λ,α,a,b≤C11/pα,λalnxaα+1f′xqC2xaq,1+f′aqC3xaq,11/q+blnbxα+1f′xqC2xbq,1+f′bqC3xbq,11/q.
Corollary 21. Under the assumptions of Theorem 19 with s=1 and α=1, inequality (29) reduces to the following inequality: (35)Ifx,λ,1,a,b=lnba1-λfx+λfalnxa+fblnbx-∫abfuudu≤λp+1+1-λp+1p+11/palnxa2f′xqC2xaq,1+f′aqC3xaq,11/q+blnbx2f′xqC2xbq,1+f′bqC3xbq,11/q.
Corollary 22. Under the assumptions of Theorem 19 with x=ab, λ=1/3, from inequality (29), one gets the following Simpson type inequality for fractional integrals: (36)2α-1lnba-αIfab,13,α,a,b=16fa+4fab+fb-2α-1Γα+1lnb/aαJab-αfa+Jab+αfb≤lnb/a4C11/pα,13af′abqC2baq/2,s+f′aqC3baq/2,s1/q+bf′abqC2abq/2,s+f′bqC3abq/2,s1/q.
Corollary 23. Under the assumptions of Theorem 19 with x=ab, λ=0, from inequality (29), one gets (37)2α-1lnba-αIfab,0,α,a,b=fab-2α-1Γα+1lnb/aαJab-αfa+Jab+αfb≤lnb/a41αp+11/paf′abqC2baq/2,s+f′aqC3baq/2,s1/q+bf′abqC2abq/2,s+f′bqC3abq/2,s1/q.
Corollary 24. Under the assumptions of Theorem 19 with x=ab and λ=1, from inequality (29) one gets (38)2α-1lnba-αIfab,1,α,a,b=fa+fb2-2α-1Γα+1lnb/aαJab-αfa+Jab+αfb≤lnb/a41αβ1α,p+11/paf′abqC2baq/2,s+f′aqC3baq/2,s1/q+bf′abqC2abq/2,s+f′bqC3abq/2,s1/q.
Corollary 25. Let the assumptions of Theorem 19 hold. If f′(x)≤M for all x∈a,b and λ=0, then from inequality (29), one gets the following Ostrowski type inequality for fractional integrals: (39)lnxaα+lnbxαfx-Γα+1Jx-αfa+Jx+αfb≤M1αp+11/palnxaαC2xaq,s+C3xaq,s1/q+blnbxαC2xbq,s+C3xbq,s1/qfor each x∈a,b.