Research Article Some New Generalized Integral Inequalities for GA-s-Convex Functions via Hadamard Fractional Integrals

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1 Chinese Mathematics Volume 26, Article ID 43686, 8 pages Research Article Some New Generalized Integral Ineualities for GA-s-Convex Functions via Hadamard Fractional Integrals Emdat EGcan and Mustafa Aydin 2 Department of Mathematics, Faculty of Sciences and Arts, Giresun University, 282 Giresun, Turkey 2 Department of Finance-Banking and Insurance, Alucra Turan Barutçu Vocational School, Giresun University, Alucra, 287 Giresun, Turkey Correspondence should be addressed to İmdat İşcan; imdat.iscan@giresun.edu.tr Received 22 April 26; Revised 4 July 26; Accepted August 26 Academic Editor: Chang-Jian Zhao Copyright 26 İ. İşcan and M. Aydin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We prove new generalization of Hadamard, Ostrowski, and Simpson ineualities in the framework of GA-s-convex functions and Hadamard fractional integral.. Introduction Let a real function f be defined on a nonempty interval I of real line R.Thefunctionf is said to be convex on I if ineuality f(tx+( t) y) tf (x) + ( t) f(y) () holds for all x, y I and t [, ]. In [], Breckner introduced s-convex functions as a generalization of convex functions as follows. Definition. Let s (, ] be a fixed real number. A function f:[, ) [, )is said to be s-convex (in the second sense), or that f belongs to the class K 2 s,if f (tx + ( t) y) t s f (x) + ( t) s f (y) (2) for all x, y [, ) and t [,]. Of course, s-convexity means just convexity when s=. The following ineualities are well known in the literature as Hermite-Hadamard ineuality, Ostrowski ineuality, and Simpson ineuality, 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 ineuality holds: f ( a+b 2 ) b b a a f (x) dx f (a) +f(b). (3) 2 Theorem 3. Let f:i R R be a mapping differentiable in I,theinteriorofI,andleta, b I with a<b.if f (x) M, x [a,b], then the following ineuality holds: for all x [a, b]. f (x) b b a f (t) dt a M + (b x) 2 b a [(x a)2 ] 2 Theorem 4. Let f : [a,b] R be a four times continuously differentiable mapping on (a, b) and f (4) = sup x (a,b) f (4) (x) <. Then the following ineuality holds: (a) +f(b) [f 3 2 b b a a +2f( a+b 2 )] f (x) dx 288 f(4) (b a)4. (4) (5)

2 2 Chinese Mathematics We will give definitions of the right and left hand side Hadamard fractional integrals which are used throughout this paper. Definition 5. Let f L[a, b]. The right-sided and left-sided Hadamard fractional integrals J α a+ f and Jα b f of order α> with b>a aredefined by J α a+ f (x) = J α b f (x) = Γ (α) x a Γ (α) b x (ln x t )α f (t) dt, t a<x<b, (6) (ln t α f (t) dt, t a<x<b, (7) respectively, where Γ(α) is the Gamma function defined by Γ(α) = e t t α dt (see [2]). In recent years, many authors have studied errors estimations for Hermite-Hadamard, Ostrowski, and Simpson ineualities; for refinements, counterparts, and generalization see [3 ]. Definition 6 (see [, 2]). A function f:i (, ) R is said to be GA-convex (geometric-arithmetically convex) if f(x t y t )tf(x) + ( t) f(y) (8) for all x, y I and t [,]. Definition 7 (see [3]). For s (, ], a function f : I (, ) R is said to be GA-s-convex (geometricarithmetically s-convex) if f(x t y t )t s f (x) + ( t) s f(y) (9) for all x, y I and t [,]. Itcanbeeasilyseenthatifs=,GA-s-convexity reduces to GA-convexity. For recent results and generalizations concerning GAconvex and GA-s-convex functions see [3 9]. Lemma 8 (see [2]). For α>and μ>,onehas where t α μ t dt = μ k= ( ) k (ln μ) k (α) k <, () (α) k =α(α+)(α+2) (α+k ). () Let f:i (, ) R be a differentiable function on I, the interior of I;inseuelofthispaperwewilltake I f (x,λ,α,a,b) = ( λ) [ln α x a + lnα b x ]f(x) +λ[f(a) ln α x a +f(b) b lnα x ] Γ(α+) [J α x f (a) +Jα x+ f (b)], (2) where a, b I with a<b, x [a,b], λ [,], α>,andγ is Euler Gamma function. In [2], Işcan gave Hermite-Hadamard s ineualities for GA-convex functions in fractional integral forms as follows. Theorem 9. Let f:i (, ) R be a function such that f L[a, b], wherea, b I with a<b.iff is a GA-convex function on [a, b], then the following ineualities for fractional integrals hold: with α>. f( ab) Γ (α+) 2 () α J α a+ f (b) +Jα b f (a) f (a) +f(b) 2 (3) In [2], Işcan obtained some new ineualities for uasigeometrically convex functions via fractional integrals by using the following lemma. Lemma. Let f : I (, ) R be a differentiable function on I such that f L[a,b],wherea, b I with a<b. Then for all x [a,b], λ [,],andα>one has I f (x,λ,α,a,b) =a(ln x a )α+ (t α λ)( x a )t f (x t a t )dt b(ln b α+ (t α λ)( x b )t f (x t b t )dt. (4) In this paper, we will use Lemma to obtain some new ineualities on generalization of Hadamard, Ostrowski, and Simpson type ineualities for GA-s-convex functions via Hadamard fractional integral. 2. Generalized Integral Ineualities for Some GA-s-Convex Functions via Fractional Integrals Theorem. Let f:i (, ) R be a differentiable function on I such that f L[a,b],wherea, b I with a<b. If f is GA-s-convex on [a, b] in the second sense for some fixed, x [a,b], λ [,],andα>then the following ineuality for fractional integrals holds: I f (x,λ,α,a,b) A / (α,λ) a (ln x a )α+ A2 (( x a ),α,λ,s) + f (a) A3 (( x / a ),α,λ,s)) +b(ln b α+

3 Chinese Mathematics 3 A2 (( x b ),α,λ,s) + f (b) A3 (( x / b ),α,λ,s)), (5) tα λ (x b )t f (x t b t ) dt tα λ ( x b )t (t s f (x) + ( t) s f (b) )dt = f (x) A2 ( x b,α,λ,s,)+ f (b) (9) where A (α,λ) = 2αλ+/α + α+ λ, A 2 (( x u ),α,λ,s)= tα λ (x u )t t s dt, A 3 (( x u ),α,λ,s)= tα λ (x u )t ( t) s dt, u = a, b. (6) Proof. Using Lemma, property of the modulus, and the power-mean ineuality, we have I f (x,λ,α,a,b) a(lnx a )α+ tα λ (x a )t f (x t a t ) dt + b (ln b α+ tα λ (x b )t f (x t b t ) dt a (lnx a )α+ / ( tα λ dt) / ( tα λ (x a )t f (x t a t ) dt) +b(ln b α+ / ( tα λ dt) / ( tα λ (x b )t f (x t b t ) dt). Since f is GA-s-convex on [a, b],weget tα λ (x a )t f (x t a t ) dt tα λ ( x a )t (t s f (x) + ( t) s f (a) )dt = f (x) A2 ( x a,α,λ,s,)+ f (a) A 3 ( x a,α,λ,s,), (7) (8) A 3 ( x b,α,λ,s,), and by a simple computation, we have /α λ tα λ dt = = 2αλ+/α + α+ (λ t α )dt+ (t α λ)dt λ /α λ. (2) Hence, If we use (8), (9), and (2) in (7), we obtain the desired result. This completes the proof. Corollary 2. Under the assumptions of Theorem with s=, ineuality (5) reduces to the following ineuality: I f (x, λ, α, a, b) A / (α,λ) a (ln x a )α+ A2 (( x a ),α,λ,) + f (a) A3 (( x / a ),α,λ,)) +b(ln b α+ A2 (( x b ),α,λ,) + f (b) A3 (( x / b ),α,λ,)). (2) Corollary 3. UndertheassumptionsofTheoremwiths= and α=, ineuality (5) reduces to the following ineuality: (ln b a ) I f (x, λ,, a, b) = ( λ) f (x) where +λ[ b a f (a) ln (x/a) +f(b) ln (b/x) ] f (u) u du (ln b a ) A / (, λ) a (ln x a )2 A2 (μ a,,λ,) + f (a) A3 (μ a,,λ,)) / +b(ln b 2 A2 (μ b,,λ,) A3 (μ b,,λ,)) /, (22)

4 4 Chinese Mathematics A (, λ) = (2λ2 2λ+), 2 A 2 (μ u,,λ,)= (μ u 2λ 2 μ λ u ) ln2 μ u +(λμ λ u ln μ u μ λ u +)(λln μ u +λ+4) (λ+2) (μ u ln μ u μ u +) (ln μ u ) 3, A 3 (μ u,,λ,)= [2μλ u +μ u ln μ u λ(+μ u ) ln μ u μ u ] (ln μ u ) 2 A 2 (μ u,,λ,), (23) μ u =( x u ), u = a,b. Corollary 4. UndertheassumptionsofTheoremwith=, ineuality (5) reduces to the following ineuality: Corollary 6. Under the assumptions of Theorem with x= ab, λ =,fromineuality(5),onegets I f (x,λ,α,a,b) a(ln x a )α+ A 2 ( x a,α,λ,s) + f (a) A 3 ( x a,α,λ,s))+b(ln b α+ (24) 2 α (ln b α a ) I f ( ab,,α,a,b) = f( ab) 2α Γ (α+) () α 4 [J α ab f (a) +Jα ab+ f (b)] ( α+ ) / A 2 ( x b,α,λ,s) A 3 ( x b,α,λ,s)). Corollary 5. Under the assumptions of Theorem with x= ab, λ = /3, fromineuality(5),onegetsthefollowing Simpson type ineuality for fractional integrals: a( f ( ab) A2 (( b /2 a ),α,,s) + f (a) A3 (( b /2 / a ),α,,s)) +b( f ( ab) A2 (( a b )/2,α,,s) (26) 2 α (ln b α a ) I f ( ab, 3,α,a,b) = +4f( ab) + f (b)] 2α Γ (α+) () α +J α ab+ f (b)] A / (α, 4 3 ) [f (a) 6 a( f ( ab) A2 (( b /2 a ),α, 3,s) + f (a) A3 (( b /2 a ),α, / 3,s)) +b( f ( ab) A2 (( a b )/2,α, 3,s) + f (b) A3 (( a b )/2,α, / 3, s)). [J α f (a) ab (25) A3 (( a b )/2,α,,s)) /. Corollary 7. UndertheassumptionsofTheoremwith x= ab and λ=,fromineuality(5)onegets 2 α (ln b α a ) f (a) +f(b) I f ( ab,,α,a,b) = 2 2α Γ (α+) () α [J α ab f (a) +Jα ab+ f (b)] 4 ( α α+ ) / a[ f ( ab) A2 (( b /2 a ),α,,s) + f (a) A3 (( b /2 / a ),α,,s)]

5 Chinese Mathematics 5 +b[ f ( ab) A2 (( a b )/2,α,,s) [A 2 (( a b )/2,α,,s) A3 (( a b )/2,α,,s)] /. (27) for all x [a, b]. +A 3 (( a / b )/2,α,,s)] (28) Corollary 8. LettheassumptionsofTheoremhold.If f (x) M for all x [a,b]and λ=,thenfromineuality (5), one gets the following Ostrowski type ineuality for fractional integrals: Theorem 9. Let f:i (, ) R be a differentiable function on I such that f L[a,b],wherea, b I with a<b.if f is GA-s-convex on [a, b] for some fixed >, x [a,b], λ [,],andα>then the following ineuality for fractional integrals holds: [(ln x a )α +(ln b α ]f(x) Γ(α+) [J α x f (a) +J α x+ f (b)] M( / α+ ) a(ln x a )α [A 2 (( b a ) /2,α,,s) +A 3 (( b /2 / a ),α,,s)] +b(ln bx α ) I f (x,λ,α,a,b) C/p (α,λ) a (ln x a )α+ ( f (x) C2 (( x a ),s) + f (a) C3 (( x / a ),s)) +b(ln b α+ C2 (( x b ),s) + f (b) C3 (( x / b ),s)), where /p + / = and (29) (αp + ), λ = C (α,λ) = λ +p+/α β( ( λ)p+,p+)+ α α α(p + ) 2F ( α,;p+2; λ), <λ, C 2 (( x u ),s)=( x u ) k= ( ) k (ln (x/u) ) k (s+) k, (3) C 3 (( x u ),s)= k= ( ) k ( ln (x/u) ) k (s+) k, u = a,b. Proof. Using Lemma, property of the modulus, the Hölder ineuality, and GA-s-convexity of f,wehave I f (x,λ,α,a,b) a(ln α+ a tα λ (x a )t f (x t a t ) dt + b (ln b α+ tα λ ( x b )t f (x t b t ) dt a (ln x a )α+ ( tα λ p /p dt) ( ( x / a )t f (x t a t ) dt) +b(ln b α+ ( tα λ p /p dt) ( ( x / b )t f (x t b t ) dt) ( tα λ p /p dt) a (ln x a )α+ μ t a ts + f (a) / μ t a ( t)s dt)

6 6 Chinese Mathematics +b(ln b α+ ( f (x) μ t b ts / μ t b ( t)s dt), (3) where μ a = (x/a), μ b = (x/b) and tα λ p λ /α dt = (λ t α ) p dt + (t α λ) p dt λ /α (αp + ), λ = = λ (αp+)/α β( ( λ)p+,p+)+ α α α(p + ) 2F ( α,;p+2; λ), <λ. (32) Using Lemma 8, we have Corollary 2. UndertheassumptionsofTheorem9withs= and α=, ineuality (29) reduces to the following ineuality: μ t u ts dt = μ u μ t u ( t)s dt = = k= k= ( ) k (ln μ u ) k (s+) k, μ t u ts dt ( ) k ( ln μ u ) k (s+) k, u = a, b. (33) I f (x, λ,, a, b) = ln b a ( λ) f (x) +λ[f(a) ln x a +f(b) ln b b x ] f (u) a u du ( λp+ + ( λ) p+ /p xa ) a (ln )2 p+ C2 (( x a ),) + f (a) C3 (( x / a ),)) +b(ln b 2 (35) Hence, if we use (32)-(33) in (3) and replacing μ a = (x/a), μ b = (x/b),weobtainthedesiredresult.thiscompletesthe proof. Corollary 2. Under the assumptions of Theorem 9 with s=, ineuality (29) reduces to the following ineuality: C2 (( x b ),) + f (b) C3 (( x / b ),)). Corollary 22. Under the assumptions of Theorem 9 with x= ab, λ = /3, fromineuality(29),onegetsthefollowing Simpson type ineuality for fractional integrals: I f (x,λ,α,a,b) C/p (α,λ) a (ln x a )α+ C2 (( x a ),) + f (a) C3 (( x / a ),)) +b(ln b α+ C2 (( x b ),) + f (b) C3 (( x / b ),)). (34) 2 α (ln b α a ) I f ( ab, 3,α,a,b) = +4f( ab) + f (b)] 2α Γ (α+) () α +J α ab+ f (b)] C /p (α, 4 3 ) a( f ( ab) C2 (( b /2 a ),s) [f (a) 6 [J α f (a) ab

7 Chinese Mathematics 7 + f (a) C3 (( b /2 / a ),s)) +b( f ( ab) C2 (( a b )/2,s) C3 (( a b )/2,s)) /. (36) Corollary 23. UndertheassumptionsofTheorem9withx= ab, λ =,fromineuality(29),onegets 2 α (ln b α a ) I f ( ab,,α,a,b) = f( ab) 2α Γ (α+) () α [J α ab f (a) +Jα ab+ f (b)] 4 ( αp+ ) /p a( f ( ab) C2 (( b /2 a ),s) + f (a) C3 (( b /2 / a ),s)) +b( f ( ab) C2 (( a b )/2,s) / (37) + f (b) C3 (( a b )/2, s)). Corollary 24. Under the assumptions of Theorem 9 with x= ab and λ=,fromineuality(29)onegets 2 α (ln b α a ) f (a) +f(b) I f ( ab,,α,a,b) = 2 2α Γ (α+) () α [J α ab f (a) +Jα ab+ f (b)] 4 ( α β( α,p+)) /p a[ f ( ab) C2 (( b /2 a ),s) + f (a) C3 (( b /2 / a ),s)] +b[ f ( ab) C2 (( a b )/2,s) C3 (( a b )/2, s)] /. (38) Corollary 25. Let the assumptions of Theorem 9 hold. If f (x) M for all x [a,b]and λ=,thenfromineuality (29), one gets the following Ostrowski type ineuality for fractional integrals: [(ln x a )α +(ln b α ]f(x) Γ(α+) [J α x f (a) +Jα x+ f (b)] M( αp+ ) /p a(ln x a )α [C 2 (( x a ),s)+c 3 (( x / a ),s)] +b(ln b α [C 2 (( x b ),s)+c 3 (( x / b ),s)] for each x [a, b]. Competing Interests (39) The authors declare that there are no competing interests regarding the publication of this paper. References [] W. W. Breckner, Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer funktionen in topologischen linearen Räumen, Publications de l Institut Mathématiue, vol. 23, pp. 3 2, 978. [2] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Euations, Elsevier, 26. [3] M. Alomari, M. Darus, S. S. Dragomir, and P. Cerone, Ostrowski type ineualities for functions whose derivatives are s-convex in the second sense, Applied Mathematics Letters,vol. 23,no.9,pp.7 76,2. [4] M. Avci, H. Kavurmaci, and M. E. Özdemir, New ineualities of Hermite-Hadamard type via s-convex functions in the second sense with applications, Applied Mathematics and Computation,vol.27,no.2,pp ,2. [5] S. S. Dragomir and S. Fitzpatrik, The Hadamard s ineuality for s-convex functions in the second sense, Demonstratio Mathematica,vol.32,no.4,pp ,999. [6] İ. İşcan, New estimates on generalization of some integral ineualities for s-convex functions and their applications, International Pure and Applied Mathematics,vol.86, no. 4, pp , 23. [7] J. Park, Generalization of some Simpson-like type ineualities via differentiable s-convex mappings in the second sense, International Mathematics and Mathematical Sciences,vol. 2, Article ID 49353, 3 pages, 2. [8] E.Set, NewineualitiesofOstrowskitypeformappingwhose derivatives are s-convex in the second sense via fractional integrals, Computers & Mathematics with Applications,vol.63, no. 7, pp , 22. [9] M. Z. Sarıkaya, E. Set, and M. E. Özdemir, On new ineualities of Simpson s type for s-convex functions, Computers & Mathematics with Applications,vol.6,no.8,pp ,2.

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