Hermite-Hadamard Type Inequalities for Fractional Integrals

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1 International Journal of Mathematical Analysis Vol., 27, no. 3, HIKARI Ltd, Hermite-Hadamard Type Inequalities for Fractional Integrals 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 work is properly cited. Abstract Several Hermite-Hadamard type inequalities will be given in this paper for different types of convexity for fractional integrals. Mathematics Subject Classification: 26D2 Keywords: Hermite-Hadamard inequality, convex functions, Holder s inequality, power mean inequality. Introduction The inequality of Hermite-Hadamard type has been considered very useful in mathematical analysis being extended and generalized in many directions by many authors, see [22, 6, 5, 9,, 3, 7, 23, ] 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 kind of integrals, see for example [8, 7, 9,,, 8, 5, 7, 3, 22, 23, 24, 25, 26]. We begin by recalling below the classical definition for the convex functions. Definition. A function f : I R R is said to be convex on an interval I if the inequality () f(tx ( t)y) tf(x) ( t)f(y)

2 626 Loredana Ciurdariu holds for all x, y I and t [, ]. The function f is said to be concave on I if the inequality () takes place in reversed direction. It is necessary to recall below also other kind of convexity and the definition of fractionals integrals, see [8,, 9, 8, 9, 24]. For other type of convexity see also [2, 6]. 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 ( t)y) 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 ( t)y) 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 ( t)y) 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-godunova-levin functions of second kind on an interval I if the inequality f(tx ( t)y) 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 kind are functions P-convex. The classical Hermite-Hadamard s inequality for convex functions is ( ) a b (2) f 2 b a b a f(x)dx f(a) f(b). 2 Moreover, if the function f is concave then the inequality (2) hold in reversed direction. Definition 6. Let f L[a, b]. The Riemann-Liouville integrals J α a f and J α b f of order α > with α are defined by J α a f(x) = x (x t) α f(t)dt, x > a Γ(α) and a

3 On Hermite-Hadamard type inequalities 627 b J α b f(x) = (t x) α f(t)dt, x < b, Γ(α) x respectively, where Γ(α) is the Gamma function defined by Γ(α) = e t t α dt and J a f(x) = J b f(x) = f(x). In this paper, two new identities are established and then by making use of these equalities the author give new estimations of 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 integrals. 2. Main results The following result is a generalization of Lemma from [4] when α > n and n N. Lemma. Let n N and f : I R R be o function such that f (n) exists on the interior I of an interval I and f (n) L[a, b] with a, b I, < a < b. Then for any x [a, b], we have: I(f, x, a, b, α, n) = (xa) = α(α )...(α k 2)f (nk) (x) t α f (n) (tx(t)a)dt(bx) () n Γ(α )[ (x a) J αn α x f(a) where α > n. Proof. By integration by parts and calculus we have, ( () k (x a) k (b x) k (b x) J αn α x t α f (tx ( t)a)dt = f (x) x a αf(x) Γ(α ) (x a) 2 (x a) (t) α f (n) (tb(t)x)dt = f(b)], α J α x and from here by induction we get: I = t α f (n) k α(α )...(α k 2) (tx(t)a)dt = () f (nk) (x) f (n) (x) (x a) k x a n Γ(α ) αn () J (x a) α x f(a). Similarly, for I 2 = ( t)α f (n) (tb ( t)x)dt we will obtain: α(α )...(α k 2) I 2 = f (nk) (x) f (n) (x) Γ(α ) (b x) k b x (b x) ) f(a) αn J α x f(b).

4 628 Loredana Ciurdariu Now, multiplying I by x a and I 2 by b x and adding the resulting identities we have the desired result. Remark. Under conditions of Lemma, for n = 2, we obtain the following equality: I(f, x, a, b, α, 2) = (xa) = Γ(α )[ (x a) J α α x f(a) t α f (tx(t)a)dt(bx) (b x) J α α x f(b)] (t) α f (tb(t)x)dt = b a (x a)(b x) f(x) Theorem. Let n N and f : I R R be o function such that f (n) exists on the interior I of an interval I and t f (n) L[a, b] with a, b I, < a < b. If f (n) q is convex on [a, b] for some fixed q, where = then the p q following inequality takes place: ( ) () I(f, x, a, b, α, n) = α(α)...(αk2)f (nk) k (x) (x a) k (b x) k () n Γ(α )[ (x a) J αn α x f(a) (b x) J αn α x f(b)] 2 {(xa) ( f (n) (x) q f (n) (a) q) q (bx) ( f (n) (b) q f (n) (x) ) q } q (αp ) p Proof. From Lemma using the property of the modulus and the power mean inequality we obtain: ( ) ( I(f, x, a, b, α, n) (x a) t αp p ) dt f (n) (tx ( t)a) q q dt ( ) ( (b x) ( t) αp p ) dt f (n) (tb ( t)x) q q dt. If we take into account that f (n) q is convex we obtain the inequality from below: I(f, x, a, b, α, n) 2 q (αp ) p ( ) {(xa) [t f (n) (x) q ( t) f (n) (a) q q ]dt ( (b x) [t f (n) (b) q ( t) f (n) (x) q ]dt which by calculus leads to desired inequality. ) q }

5 On Hermite-Hadamard type inequalities 629 Theorem 2. Let n N and f : I R R be o function such that f (n) exists on the interior I of an interval I and f (n) L[a, b] with a, b I, < a < b. If f (n) q is quasi-convex on [a, b] for some fixed q, where = then p q the following inequality holds: ( ) () I(f, x, a, b, α, n) = α(α)...(αk2)f (nk) k (x) (x a) k (b x) k () n Γ(α )[ (x a) J αn α x f(a) (b x) J αn α x f(b)] {(x a) sup{ f (n) (x), f (n) (a) } (b x) sup{ f (n) (b), f (n) (x) }} (αp ) p Proof. We will use the property of the modulus, the power mean inequality and then the definition of quasi-convex functions like before. Next result is also a generalization of Lemma 4 from [3]. Lemma 2. Let n N, n 2 and f : I R R be o function such that f (n) exists on the interior I of an interval I and f (n) L[a, b] with a, b I, < a < b, x [a, b], λ (, ). Then the following identity holds: I(f, x, a, b, λ, α, n) = ( λ)(x a) = λ(x a) ( λ)(b x) λ(b x) t α f (n) (t(λa ( λ)x) ( t)a)dt ( t) α f (n) (tx ( t)(λa ( λ)x))dt t α f (n) (t(λx ( λ)b) ( t)x)dt ( t) α f (n) (tb ( t)(λx ( λ)b))dt = α(α )...(α k 2)[( ()k ( λ) k λ k )(f(nk) (λa ( λ)x) (x a) k f (nk) (λx ( λ)b) () n )] Γ(α ){ (b x) k ( λ) α (x a) J αn α (λa(λ)x) f(a) λ α (b x) J αn α (λx(λ)b) f(b) λ α (x a) J αn α (λa(λ)x) f(x) where α > n. () n ( λ) α (b x) J αn α (λx(λ)b) f(x)},

6 63 Loredana Ciurdariu Proof. By integration by parts and then using the substitution u = t(λa( λ)x) ( t)a we get t α f (t(λa(λ)x)(t)a))dt = f (λa ( λ)x) αf(λa ( λ)x) ( λ)(x a) ( λ) 2 (x a) 2 Γ(α ) α J ( λ) α (x a) α Then we check easily by induction that = I = (λa(λ)x) f(a). t α f (n) (t(λa ( λ)x) ( t)a))dt = k α(α )...(α k 2) () f (nk) (λa(λ)x) f (n) (λa ( λ)x) ( λ) k (x a) k ( λ)(x a) ()n Γ(α ) ( λ) α (x a) Analogously we obtain = I 3 = αn J α (λa(λ)x) f(a). t α f (n) (t(λx ( λ)b) ( t)x))dt = k α(α )...(α k 2) () f (nk) (λx(λ)b) f (n) (λx ( λ)b) ( λ) k (b x) k ( λ)(b x) = and = I 2 = ()n Γ(α ) ( λ) α (b x) αn J α (λx(λ)b) f(x)., ( t) α f (n) (tx ( t)(λa ( λ)x))dt = α(α )...(α k 2) f (nk) (λa ( λ)x) f (n) (λa ( λ)x) λ k (x a) k λ(x a) I 4 = Γ(α ) αn λ α J (x a) α (λa(λ)x) f(x). ( t) α f (n) (tb ( t)(λx ( λ)b))dt = α(α )...(α k 2) f (nk) (λx ( λ)b) f (n) (λx ( λ)b) λ k (b x) k λ(b x) Γ(α ) αn λ α J (b x) α (λx(λ)b) f(b). Multiplying now I by ( λ)(x a), I 2 by λ(x a), I 3 by ( λ)(b x) and I 4 by λb x) and summing then these expressions we find by calculus the desired equality.

7 On Hermite-Hadamard type inequalities 63 Theorem 3. Let n N and f : I R R be o function such that f (n) exists on the interior I of an interval I and f (n) L[a, b] with a, b I, < a < b, λ (, ), x [a, b]. If f (n) q is convex on [a, b] for some fixed q, where = then the following inequality holds: p q I(f, x, a, b, λ, α, n) 2 q (αp ) p {(λ)(xa)( f (n) (λa(λ)x) q f (n) (a) q ) q λ(x a)( f (n) (x) q f (n) (λa ( λ)x) q ) q ( λ)(b x)( f (n) (λx ( λ)b) q f (n) (x) q ) q λ(b x)( f (n) (b) q f (n) (λx ( λ)b) q ) q }, where α > n. Proof. We use the power mean inequality and the definition of convex functions as in previous theorem. Theorem 4. Let n N and f : I R R be o function such that f (n) exists on the interior I of an interval I and f (n) L[a, b] with a, b I, < a < b, λ (, ), x [a, b]. If f (n) q is convex on [a, b] then the following inequality holds: I(f, x, a, b, λ, α, n) {(λ)(xa)( f (n) (λa(λ)x) q (α 2) q α f (n) (a) q ) q λ(x a)( α f (n) (x) q f (n) (λa ( λ)x) q ) q ( λ)(b x)( f (n) (λx ( λ)b) q α f (n) (x) q ) q λ(b x)( α f (n) (b) q f (n) (λx ( λ)b) q ) q }, where α > n. Proof. In this case we will use the Holder s inequality and then the definition of convex functions. Theorem 5. Let n N and f : I R R be o function such that f (n) exists on the interior I of an interval I and f (n) L[a, b] with a, b I, < a < b, λ (, ), x [a, b]. If f (n) q is P-convex on [a, b] then the following inequality holds: I(f, x, a, b, λ, α, n) {(λ)(xa)( f (n) (λa(λ)x) q f (n) (a) q ) (αp ) q p λ(x a)( f (n) (x) q f (n) (λa ( λ)x) q ) q

8 632 Loredana Ciurdariu where α > n. ( λ)(b x)( f (n) (λx ( λ)b) q f (n) (x) q ) q λ(b x)( f (n) (b) q f (n) (λx ( λ)b) q ) q }, Proof. In this case we will use the power mean inequality and then the definition of P-convex functions. Theorem 6. Let n N and f : I R R be o function such that f (n) exists on the interior I of an interval I and f (n) L[a, b] with a, b I, < a < b, λ (, ), x [a, b]. If f (n) q is quasi-convex on [a, b] then the following inequality holds: I(f, x, a, b, λ, α, n) where α > n. {(λ)(xa) sup{ f (n) (λa(λ)x), f (n) (a) } (αp ) p λ(x a) sup{ f (n) (x), f (n) (λa ( λ)x) } ( λ)(b x) sup{ f (n) (λx ( λ)b), f (n) (x) } λ(b x) sup{ f (n) (b), f (n) (λx ( λ)b) }}, Proof. In this case we will use the power mean inequality and then the definition of quasi-convex functions. Theorem 7. Let n N and f : I R R be o function such that f (n) exists on the interior I of an interval I and f (n) L[a, b] with a, b I, < a < b, λ (, ), x [a, b]. If f (n) q is s-convex in the first sense on [a, b] and α > n. then the following inequality takes place: I(f, x, a, b, λ, α, n) (αp ) p (s ) q λ(x a)( f (n) (x) q f (n) (λa ( λ)x) q ) q {(λ)(xa)( f (n) (λa(λ)x) q f (n) (a) q ) q ( λ)(b x)( f (n) (λx ( λ)b) q f (n) (x) q ) q λ(b x)( f (n) (b) q f (n) (λx ( λ)b) q ) q }. Proof. In this case we will use the power mean inequality and then the definition of s-convex functions in the first sense.

9 On Hermite-Hadamard type inequalities 633 Theorem 8. Let n N and f : I R R be o nonnegative function such that f (n) exists on the interior I of an interval I and f (n) L[a, b] with a, b I, < a < b, λ (, ), x [a, b]. If f (n) q is s-godunova-levin functiono of second kind on [a, b] and α > n. then the following inequality takes place: I(f, x, a, b, λ, α, n) (αp ) {(λ)(xa)( f (n) (λa(λ)x) q f (n) (a) q ) q p ( s) q λ(x a)( f (n) (x) q f (n) (λa ( λ)x) q ) q ( λ)(b x)( f (n) (λx ( λ)b) q f (n) (x) q ) q λ(b x)( f (n) (b) q f (n) (λx ( λ)b) q ) q }. References [] M. Alomari, M. Darus, U. S. Kirmaci, Some inequalities of Hermite-Hadamard type 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 Mathematics with Applications, 59 (2), [3] L. Ciurdariu, On some Hermite-Hadamard type inequalities for functions whose power of the absolute value of derivatives are (α, m) convex, Int. J. of Math. Anal., 6 (22), no. 48, [4] L. Ciurdariu, A note concerning several Hermite-Hadamard inequalities for different types of convex functions, Int. J. of Math. Anal., 6 (22), no. 33, [5] S. S. Dragomir, C. E. M. Pearce, Selected Topic on Hermite-Hadamard Inequalities and Applications, Melbourne and Adelaide, December, 2. [6] S. S. Dragomir, S. Fitzpatrick, The Hadamard s inequality for s-convex functionsin the second sense, Demonstratio Math., 32 (999), no. 4, [7] 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. [8] Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal., (2), no., [9] 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 values are quasi-convex, Konuralp Journal of Mathematics, (23), no. 2, [] Imdat Iscan, M. Kunt, N. Yazici Gozutok, Tuncay Koroglu, New general integral inequalities for Lipschitzian functions via Riemann-Liouville fractional integrals and applications, Journal of Inequalities and Special Functions, 7 (26), no. 4, -2. [2] H. Kasvurmaci, M. Avci, M. E. Ozdemir, New inequalities of Hermite-Hadamard type for convex functions with applications, arxiv:6.593v[math.ca].

10 634 Loredana Ciurdariu [3] 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., [4] U. S. Kirmaci, M. Klaricic Bakula, M. E. Ozdemir, J. Pecaric, Hadamard-type inequalities for s-convex functions, Appl. Math. Comput., 93 (27), no., [5] M. A. Latif, S. S. Dragomir, New inequalities of Hermite-Hadamard type for functions whose derivatives in absolute value are convex with applications, Acta Univ. Matthiae Belii, Series Math., 2 (23), [6] V. G. Mihesan, A generalization of the convexity, Seminar of Functional Equations, Approx. and Convex, Cluj-Napoca, Romania, (993). [7] E. Set, New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second via fractional integrals, Comput. Math. Appl., 63 (22), [8] M. Z. Sarikaya, E. Set, H. Yildiz, N. Basak, Hermite-Hadamard s inequalities for fractional integrals and related fractional inequalities, Math. and Comput. Model., 57 (2), [9] E. Set, M. Z. Sarikaya, M. E. Ozdemir, Some Ostrowski s type inequalities for functions whose second derivatives are s-convex in the second sense, Demonstratio Mathematica, 47 (24). [2] Gh. Toader, On a generalization of the convexity, Mathematica, 3 (988), no. 53, [2] M. Tunc, On some new inequalities for convex functions, Turk. J. Math., 35 (2), -7. [22] J. Park, New Inequalities of Hermite-Hadamard-like Type for Functions whose Second Derivatives in Absolute Value are Convex, Int. Journal of Math. Analysis, 8, (24), no. 6, [23] J. Park, Hermite-Hadamard-like type inequalities for n-times differentiable functions which are m-convex and s-convex in the second sense, Int. Journal of Math. Analysis, 8 (24), no. 25, [24] J. Park, On some integral inequalities for twice differentiable quasi-convex and convex functions via fractional integrals, Applied Mathematical Sciences, 9 (25), no. 62, [25] J. Park, Inequalities of Hermite-Hadamard-like type for the functions whose second derivatives in absolute value are convex and concave, Applied Mathematical Sciences, 9 (25), no., [26] J. Park, Hermite-Hadamard-like type inequalities for s-convex functions and s- Godunova-Levin functions of two kinds, Applied Mathematical Sciences, 9 (25), no. 69, Received: May 3, 27; Published: June 28, 27