Hermite-Hadamard Type Inequalities for Fractional Integrals Operators

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1 Applied Mathematical Sciences, Vol., 27, no. 35, HIKARI Ltd, Hermite-Hadamard Type Inequalities for ractional Integrals Operators Loredana Ciurdariu Department of Mathematics Politehnica University of Timisoara P-ta. Victoriei, No.2, 36-Timisoara, Romania Copyright c 27 Loredana Ciurdariu. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. Abstract Several Hermite-Hadamard type inequalities will be given in this paper for n-time differentiable functions whose n-time derivative in absolute value satisfy different ind of convexities via Riemann-Liouville fractional integral operators. Mathematics Subject Classification: 26A33, 26D, 26D5 Keywords: Hermite-Hadamard inequality, convex functions, Holder inequality, Riemann-Liouville fractional integral, fractional integral operator, power mean inequality. Introduction The inequality of Hermite-Hadamard type has been considered very useful in mathematical analysis being very intensely studied, extended and generalized in many directions by many authors, see [24, 7, 6,,, 4, 8, 25, 2] and the references therein. Many papers study the Riemann-Liouville fractionals integrals and give new and interesant generalizations of Hermite-Hadamard type inequalities using these ind of integrals, see for instance [9, 8,,, 2, 9, 6, 8, 4, 24, 25, 26, 27, 28, 2, 3]. We will begin now by recalling the classical definition for the convex functions and then the definitions for other ind of convexities.

2 746 Loredana Ciurdariu Definition. A function f : I R R is said to be convex on an interval I if the inequality ( f(tx ( ty tf(x ( tf(y holds for all x, y I and t [, ]. The function f is said to be concave on I if the inequality ( taes place in reversed direction. It is necessary to recall below also the definition of fractionals integrals, see [9,,, 9, 2, 26] and then the definition of fractional integral operators. or other type of convexity see also [22, 7]. Definition 2. A function f : [a, b] R is said to be quasi-convex onl [a, b] if holds for all x, y [a, b] and t [, ]. f(tx ( ty sup{f(x, f(y} Definition 3. A function f : I R is said to be P-convex on [a, b] if it is nonnegative and for all x, y I and λ [9, ] f(tx ( ty f(x f(y. Definition 4. A function f : I R R is said to be s-convex in the first sense on an interval I if the inequality f(tx ( ty t s f(x ( t s f(y holds for all x, y I, t [, ] and for some fixed s (, ]. Definition 5. A function f : I R R is said to be s-convex in the second sense on an interval I if the inequality f(tx ( ty t s f(x ( t s f(y holds for all x, y I, t [, ] and for some fixed s (, ]. Definition 6. A function f : I R R is said to be s-godunova-levin functions of second ind on an interval I if the inequality f(tx ( ty t s f(x ( t s f(y holds for all x, y I, t (, and for some fixed s [, ]. It is easy to see that for s = s-godunova-levin functions of second ind are functions P-convex. The classical Hermite-Hadamard s inequality for convex functions is

3 On Hermite-Hadamard type inequalities 747 ( a b (2 f 2 b a b a f(xdx f(a f(b. 2 Moreover, if the function f is concave then the inequality (2 hold in reversed direction. Definition 7. Let f L[a, b]. The Riemann-Liouville integrals α a f and α b f of order α > with α are defined by and α a f(x = Γ(α x a b (x t α f(tdt, x > a α b f(x = (t x α f(tdt, x < b, Γ(α x respectively, where Γ(α is the Gamma function defined by Γ(α = e t t α dt and a f(x = b f(x = f(x. It is well-nown that the beta function is defined when a, b > by R(a, b = Γ(aΓ(b Γ(a b = t a ( t b dt. The following class of functions defined formally by ρ,λ(x ( = Γ(ρ λ x (ρ, λ > ; x < R, = where the coefficients (, ( N = N {} is a bounded sequence of positive real numbers and R is the set of real numbers, as in [2], was introduced in [29] and was used for giving in [3] the following left-sided and right-sided fractional integral operators from below: and x ( ρ,λ,a ;w ϕ(x = (x t λ ρ,λ[w(x t ρ ]ϕ(tdt, (x > a >, a b ( ρ,λ,b ;w ϕ(x = (t x λ ρ,λ[w(t x ρ ]ϕ(tdt, ( < x < b, x where ρ, λ >, w R and ϕ(t is such that the integral on the right side exists. There are new integral inequalities for this operator, seet [2, 3, 3] and references therein. It is important to mention that for example the classical Riemann-Liouville fractional integrals α a and α b of order α were obtained by setting λ = α, sigma( = and w = in previous integrals.

4 748 Loredana Ciurdariu In this paper, two new identities are given and then some applications, lie Hermite-Hadamard type inequalities for functions whose the n-time derivative iin absolute value of certain powers satisfies different type of convexities via Riemann-Liouville fractional integral operators are established. 2. Main results The following result is a generalization of Lemma from [5] for fractional integral operators for functions n-time differentiable. Lemma. Let f : [a, b] R be an n-time differentiable mapping on (a, b with < a < b, λ > n, x (a, b and t [, ]. If f (n L[a, b] then the following equality for generalized fractional integrals holds: = t λ ρ,λ[(x a ρ t ρ ]f (n (tx ( tadt ( t λ ρ,λ[(b x ρ ( t ρ ]f (n (tb ( txdt = { ( (x a ρ,λ 2[(x a ρ ] (b x ρ,λ 2[(b x ρ ]}f (n (x ( n ( (x a λ ρ,λ n,x ;w f ( (a (b x λ ρ,λ n,x ;w f (b. Proof. As in [2], we compute first t λ ρ,λ[(x a ρ t ρ ]f (tx ( tadt and then we will prove by induction that = I = t λ ρ,λ[(x a ρ t ρ ]f (n (tx ( tadt = ( (x a f (n (x ρ,λ 2[(x a ρ ] ( n (x a λ ( ρ,λ n,x ;w f (a. Integrating by parts and then changing variables with u = tx ( ta we get t λ ρ,λ[(x a ρ t ρ ]f (tx ( tadt = = ρ,λ[(x a ρ ] f (x x a f(x (x a ρ,λ[(x a ρ ] 2 t λ 2 (x a ρ,λ [(x a ρ t ρ ]f(tx ( tadt 2 or t λ ρ,λ[(x a ρ t ρ ]f (tx ( tadt =

5 On Hermite-Hadamard type inequalities 749 = ρ,λ[(x a ρ ] f (x x a f(x (x a ρ,λ[(x a ρ ] 2 ( (x a λ ρ,λ,x ;w f (a. Analogously, by using the same method we get: ( t λ ρ,λ[(b x ρ ( t ρ ]f (tb ( txdt = = ρ,λ[(b x ρ ] f (x b x f(x (b x ρ,λ[(b x ρ ] 2 ( t λ 2 (b x ρ,λ [(b x ρ ( t ρ ]f(tb ( txdt. 2 or by substitution u = tb ( tx, ( t λ ρ,λ[(b x ρ ( t ρ ]f (tb ( txdt = = ρ,λ[(b x ρ ] f (x b x f(x (b x ρ,λ[(b x ρ ] 2 ( (b x λ ρ,λ,x ;w f (b. Therefore by induction we have, I 2 = f (n (x (b x ρ,λ 2[(b x ρ ( ] (b x λ ρ,λ n,x ;w f (b. Now summing I and I 2 we obtain the desired equality. Using this lemma we obtain the following result for n-time differentiable functions whose absolute value is convex via fractional integral operator. Theorem. Let f : [a, b] R be an n-time differentiable mapping on (a, b with < a < b, λ > n, x (a, b and t [, ]. If f (n L[a, b] and f (n is convex on (a, b then the following inequality for generalized fractional integral operators taes place: ( f (n (x{ ρ,λ 2 [(x aρ ] (x a ρ,λ 2 [(b xρ ] (b x } ( n ( (x a λ ρ,λ n,x ;w f (a ρ,λ[w(x a ρ ] ( (b x λ ρ,λ n,x ;w f (b ( f (n (x λ 2 f (n (a (λ (λ 2

6 75 Loredana Ciurdariu ( f ρ,λ[w(b x ρ (n (x ] λ 2 f (n (b. (λ (λ 2 Proof. Using the properties of modulus, Lemma and that f (n is convex function we get: ( f (n (x{ ρ,λ 2 [(x aρ ] ρ,λ 2 [(b xρ ] } (x a (b x ( n (x a λ ( = I I 2 ρ,λ n,x ;w f (a (b x λ ( ρ,λ n,x ;w f (b = t λ ρ,λ[(x a ρ t ρ ]f (n (tx ( ta dt ( t λ ρ,λ[(b x ρ ( t ρ ]f (n (tb ( tx dt ( w (x a ρ Γ(ρ λ = = ( w (b x ρ Γ(ρ λ ( f (n (x ( f (n (b t λ dt f (n (a ( t λ tdt f (n (x rom here by easily calculus we get the desired inequality. t λ ( tdt ( t λ dt. BY this lemma we also obtain the following result for n-time differentiable functions whose absolute value is s-convex in the second sense via fractional integral operator. Theorem 2. Let f : [a, b] R be an n-time differentiable mapping on (a, b with < a < b, λ > n, x (a, b, s (, ] and t [, ]. If f (n L[a, b] and f (n is s-convex in the second sense on (a, b then the following inequality for generalized fractional integral operators taes place: ( f (n (x{ ρ,λ 2 [(x aρ ] (x a ρ,λ 2 [(b xρ ] (b x } ( n ( (x a λ ρ,λ n,x ;w f ( (a (b x λ ρ,λ n,x ;w f (b ( f ρ,λ[w(x a ρ (n (x ] λ s f (n (a B(λ, s ( f ρ,λ[w(b x ρ (n (x ] λ s f (n (b B(λ, s. Proof. We use the same method as in Theorem, but this time we apply the definition of s-convex function in the second sense.

7 On Hermite-Hadamard type inequalities 75 Next result is a generalization of Lemma 4 from [4] for fractional integral operators for functions n-time differentiable. Lemma 2. Let f : [a, b] R be an n-time differentiable mapping on (a, b with < a < b, λ > n, x (a, b and t, r [, ]. If f (n L[a, b] then the following equality for generalized fractional integrals holds: t λ ρ,λ[( r ρ (x a ρ t ρ ]f (n (t(ra ( rx ( tadt ( t λ ρ,λ[r ρ (x a ρ ( t ρ ]f (n (tx ( t(ra ( rxdt t λ ρ,λ[( r ρ (b x ρ t ρ ]f (n (tb ( t(rx ( rbdt ( t λ ρ,λ[r ρ (b x ρ ( t ρ ]f (n (tb ( t(rx ( rbdt = = ( ( r {f(n (ra ( rx (x a ρ,λ 2[( r ρ (x a ρ ] f (n (rx ( rb (b x ρ,λ 2[( r ρ (b x ρ ]} {f(n (ra ( rx r (x a ρ,λ 2[r ρ (x a ρ ] Proof. We denote I 2 = I = I 3 = f (n (rx ( rb (b x ρ,λ 2[r ρ (b x ρ ]} ( n ( ( r λ (x a λ ρ,λ n,(ra( rx ;w f(a ( r λ (x a λ ρ,λ n,(ra( rx ;w f(x ( n ( ( r λ (b x λ ρ,λ n,(rx( rb ;w f(x ( r λ (b x λ ρ,λ n,(rx( rb ;w f(b. t λ ρ,λ[( r ρ (x a ρ t ρ ]f (n (t(ra ( rx ( tadt, ( t λ ρ,λ[r ρ (x a ρ ( t ρ ]f (n (tx ( t(ra ( rxdt, t λ ρ,λ[( r ρ (b x ρ t ρ ]f (n (tb ( t(rx ( rbdt

8 752 Loredana Ciurdariu and I 4 = ( t λ ρ,λ[r ρ (b x ρ ( t ρ ]f (n (tb ( t(rx ( rbdt. As in Lemma we prove by induction that ( I = ( r (x a f (n (ra ( rxρ,λ 2[( r ρ (x a ρ ] ( n ( ( r λ (x a λ ρ,λ n,(ra( rx ;w f(a and then similarly we can find I 2, I 3 and I 4. Therefore we have: I 2 = r (x a f (n (ra ( rxρ,λ 2[r ρ (x a ρ ] ( n ( r λ (x a λ ρ,λ n,(ra( rx ;w f(x Summing now I, I 2 I 3 and I 4 we find the desired equality. Theorem 3. Let f : [a, b] R be an n-time differentiable mapping on (a, b with < a < b, λ > n, x (a, b and t, r [, ]. If f (n L[a, b] and f (n is convex on (a, b then the following inequality for generalized fractional integral operators taes place: ( ( r {f(n (ra ( rx (x a ρ,λ 2[( r ρ (x a ρ ] f (n (rx ( rb (b x ρ,λ 2[( r ρ (b x ρ ]} {f(n (ra ( rx r (x a ρ,λ 2[r ρ (x a ρ ] f (n (rx ( rb (b x ρ,λ 2[r ρ (b x ρ ]} ( n ( ( r λ (x a λ ρ,λ n,(ra( rx ;w f(a ( r λ (x a λ ρ,λ n,(ra( rx ;w f(x ( n ( ( r λ (b x λ ρ,λ n,(rx( rb ;w f(x ( r λ (b x λ ρ,λ n,(rx( rb ;w f(b

9 On Hermite-Hadamard type inequalities 753 ( f ρ,λ[( r ρ (x a ρ (n (ra ( rx f (n (a w] λ 2 (λ (λ 2 ( f ρ,λ[r ρ (x a ρ (n (x w] (λ 2(λ f (n (ra ( rx λ 2 ( f ρ,λ[( r ρ (b x ρ (n (rx ( rb f (n (x w] λ 2 (λ (λ 2 ( f ρ,λ[r ρ (b x ρ (n (b w] (λ 2(λ f (n (rx ( rb λ 2. Proof. We use the same method as in Theorem, we shall apply Lemma 2 and the definition of the convex functions. References [] M. Alomari, M. Darus, U. S. Kirmaci, Some inequalities of Hermite-Hadamard tyepe for s-convex functions, Acta Mathematica Scientia, 3 (2, no. 4, [2] M. Alomari, M. Darus, U. S. Kirmaci, Refinements of Hadamard-type inequalities for quasi-convex functions with applications to trapezoidal formula and to specail means, Computers and Mathematica with Applications, 59 (2, [3] R. P. Agarwal, M.-. Luo, R. K. Raina, On Ostrowsi type inequalities, asciculi Mathematici, 56 (26, [4] L. Ciurdariu, On some Hermite-Hadamard type inequalities for functions whose power of absolute value of derivatives are (α, m convex, Int.. of Math. Anal., 48 (22, no. 6, [5] L. Ciurdariu, A note concerning several Hermite-Hadamard inequalities for different types of convex functions, Int.. of Math. Anal., 33 (22, no. 6, [6] S. S. Dragomir, C. E. M. Pearce, Selected Topic on Hermite-Hadamard Inequalities and Applications, Melbourne and Adelaide, December, 2. [7] S. S. Dragomir, S. itzpatric, The Hadamard s inequality for s-convex functionsin the second sense, Demonstratio Math., 32 (999, no. 4, [8] M. A. Latif, S. S. Dragomir, New inequalities of Hermite-Hadamard type for n-times differentiable convex and concave functions with applications, Res. Rep. Coll., (24, 7. [9] Z. Dahmani, On Minowsi and Hermite-Hadamard integral inequalities via fractional integration, Ann. unct. Anal., (2, no., [] Imdat Iscan, Generalizations of different type integral inequalities for s-convex functions via fractional integrals, Appl. Anal., 93 (23, [] Imdat Iscan, Generalization of different type integral inequalities via fractional integrals for functions whose second derivatives absolute value are quasi-convex, Konuralp ournal of Mathematics, (23, no. 2, [2] Imdat Iscan M. Kunt, N. Yazici, Gozuto, K. Tuncay, New general integral inequalities for Lipschitzian functions via Riemann-Liouville fractional integrals and applications, oirnal of Inequalities and Special unctions, 7 (26, no. 4, -2.

10 754 Loredana Ciurdariu [3] H. Kasvurmaci, M. Avci, M. E. Ozdemir, New inequalities of Hermite-Hadamard type for convex functions with applications, arxiv:6.593v[math.ca]. [4] U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput., 47 (24, no., [5] U. S. Kirmaci, K. Klaricic, Baula, M. E. Ozdemir,. Pecaric, Hadamard-type inequalities for s-convex functions, Appl. Math. Comput., 93 (27, no., [6] M. A. Latif, S. S. Dragomir, New inequalities of Hermite-Hadanard type for functions whose derivatives in absolute value are convex with applications, Acta Univ. Matthiae Belii, Series Math., (23, [7] V. G. Mihesan, A Generalization of the Convexity, Seminar of unctional Equations, Approx. and Convex, Cluj-Napoca, Romania, (993. [8] E. Set, New inequalities of Ostrowsi type for mappings whose derivatives are s-convex in the second via fractional integrals, Comput. Math. Appl., 63 (22, [9] M. Z. Sariaya, E. Set, H. Yildiz, N. Basa, Hermite-Hadamard s inequalities for fractional integrals and related fractional inequalities, Math. and Comput. Model., 57 (23, [2] E. Set, M. Z. Sariaya, M. E. Ozdemir, Some Ostrowsi s type inequalities for functions whose second derivatives are s-convex in the second sense, Demonstratio Mathematica, 47 (24. [2] E. Set, S. S. Dragomir, A. Gozpinar, Some generalized Hermite-Hadamard type inequalities involving fractional integral operator for functions whose second derivatives in absolute value are s-convex, Res. Rep. Coll., 2 (27, Art. 4. [22] Gh. Toader, On a generalization of the convexity, Mathematica, 3 (988, no. 53, [23] M. Tunc, On some new inequalities for convex functions, Tur.. Math., 35 (2, -7. [24]. Par, New Inequalities of Hermite-Hadamard-lie Type for unctions whose Second Derivatives in Absolute Value are Convex, Int. ournal of Math. Analysis, 8 (24, no. 6, [25]. Par, Hermite-Hadamard-lie type inequalities for n-times differentiable functions which are m-convex and s-convex in the second sense, Int. ournal of Math. Analysis, 8 (24, no. 25, [26]. Par, On some integral inequalities for twice differentiable quasi-convex and convex functions via fractional integrals, Applied Mathematical Sciences, 9 (25, no. 62, [27]. Par, Inequalities of Hermite-Hadamard-lie type for the functions whose second derivatives in absolute value are convex and concave, Applied Mathematical Sciences, 9 (25, no., [28]. Par, Hermite-Hadamard-lie type inequalities for s-convex functions and s- Godunova-Levin functions of two inds, Applied Mathematical Sciences, 9 (25, no. 69, [29] R. K. Raina, On generalized Wright s hypergeometric functions and fractional calculus operators, East Asian Math.., 2 (25, no. 2, [3] H. Yaldiz, M. Z. Sariaya, On the Hermite-Hadamard type inequalities for fractional integral operator, Submitted. Received: May 3, 27; Published: une 28, 27