On new inequalities of Hermite–Hadamard–Fejer type for harmonically convex functions via fractional integrals

SpringerPlus, May 2016

In this paper, firstly, new Hermite–Hadamard type inequalities for harmonically convex functions in fractional integral forms are given. Secondly, Hermite–Hadamard–Fejer inequalities for harmonically convex functions in fractional integral forms are built. Finally, an integral identity and some Hermite–Hadamard–Fejer type integral inequalities for harmonically convex functions in fractional integral forms are obtained. Some results presented here provide extensions of others given in earlier works.

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On new inequalities of Hermite–Hadamard–Fejer type for harmonically convex functions via fractional integrals

Kunt et al. SpringerPlus On new inequalities of Hermite- Hadamard-Fejer type for harmonically convex functions via fractional integrals Mehmet Kunt 0 1 2 4 İmdat İşcan 0 1 2 3 Nazlı Yazıcı 0 1 2 4 Uğur Gözütok 0 1 2 4 0 Technical University , Trabzon 1 of Sciences , Karadeniz 2 of Mathematics , Faculty 3 Department of Mathematics, Faculty of Sciences and Arts, Giresun University , Giresun , Turkey 4 Department of Mathematics, Faculty of Sciences, Karadeniz Technical University , Trabzon , Turkey In this paper, firstly, new Hermite-Hadamard type inequalities for harmonically convex functions in fractional integral forms are given. Secondly, Hermite-Hadamard-Fejer inequalities for harmonically convex functions in fractional integral forms are built. Finally, an integral identity and some Hermite-Hadamard-Fejer type integral inequalities for harmonically convex functions in fractional integral forms are obtained. Some results presented here provide extensions of others given in earlier works. Riemann-Liouville fractional integrals, Harmonically convex functions Mathematics Subject Classification: 26A51, 26A33, 26D10 Hermite-Hadamard inequality; Hermite-Hadamard-Fejer inequality - Turkey article *Correspondence: Background a, b ∈ I. The inequality Let f : I ⊂ R → R be a convex function defined on the interval I of real numbers and f f a + b 2 ≤ 1 b − a a is well known in the literature as Hermite–Hadamard’s inequality (Hadamard 1893; Hermite 1883) . The most well-known inequalities related to the integral mean of a convex function f are the Hermite Hadamard inequalities or their weighted versions, the so-called Hermite–Hadamard-Fejér inequalities. Fejér (1906) established the following Fejér inequality which is the weighted generalization of Hermite–Hadamard inequality (1): Theorem 1 Let f : [a, b]→ R be a convex function. Then the inequality a + b 2 a a b g (x)dx ≤ f (x)g (x)dx ≤ f (a) + f (b) 2 a holds, where g : [a, b]→ R is nonnegative, integrable and symmetric to (a + b)/2. (1) (2) For some results which generalize, improve and extend the inequalities (1) and (2) see Bombardelli and Varošanec (1869) , İşcan (2013a, 2014c) , Minculete and Mitroi (2012) , Sarıkaya (2012) , Tseng et al. (2011 ). We recall the following inequality and special functions which are known as Beta and hypergeometric function respectively: β x, y = 2F1(a, b; c; z) = Γ (x)Γ y = 1 tx−1(1 − t)y−1dt, Γ x + y 0 1 1 β(b, c − b) 0 c > b > 0, |z| < 1 (see Kilbas et al. 2006) . tb−1(1 − t)c−b−1(1 − zt)−adt, x, y > 0, Lemma 1 (Prudnikov et  al. 1981; Wang et  al. 2013) For 0 < α ≤ 1 and 0 ≤ a < b we have aα − bα ≤ (b − a)α. The following definitions and mathematical preliminaries of fractional calculus theory are used further in this paper. Definition 1 (Kilbas et al. 2006) Let f ∈ L[a, b]. The Riemann–Liouville integrals Jaα+f and Jbα−f of oder α > 0 with a ≥ 0 are defined by and Jaα+f (x) = Jbα−f (x) = 1 Γ (α) a 1 Γ (α) x x b (x − t)α−1f (t)dt, x > a (t − x)α−1f (t)dt, x < b respectively, where Γ (α) is the Gamma function defined by Γ (α) = e−t tα−1dt and Ja0+f (x) = Jb0−f (x) = f (x). 0 Because of the wide application of Hermite–Hadamard type inequalities and fractional integrals, many researchers extend their studies to Hermite–Hadamard type inequalities involving fractional integrals not limited to integer integrals. Recently, more and more Hermite–Hadamard inequalities involving fractional integrals have been obtained for different classes of functions; see Dahmani (2010) , İşcan (2013b, 2014a), İşcan and Wu (2014), Mihai and Ion (2014) , Sarıkaya et al. (2013 ), Wang et al. (2012 ), Wang et al. (2013). İşcan (2014b) can defined the so-called harmonically convex functions and established following Hermite–Hadamard type inequality for them as follows: Definition 2 Let I ⊂ R\{0} be a real interval. A function f : I → R is said to be harmonically convex, if ∞ f tx + (x1y− t)y ≤ tf y + (1 − t)f (x) (3) for all x, y ∈ I and t ∈ [0, 1]. If the inequality in (3) is reversed, then f is said to be harmonically concave. Theorem  2 (İşcan 2014b) Let f : I ⊂ R\{0} → R be a harmonically convex function and a, b ∈ I. If f ∈ L[a, b] then the following inequalities holds: f (a) +2 f (b) . Latif et al. (2015 ) gave the following definition: Definition 3 A function g : [a, b] ⊆ R\{0} → R is said to be harmonically symmetric with respect to 2ab/a + b if f f (4) (5) g (x) = g with α > 0 and h(x) = 1/x, x ∈ b1 , a1 . g 1t dt with α > 0 and h(x) = 1/x, x ∈ b1 , a1 . Proof Since f is a harmonically convex function on [a, b], we have for all t ∈ [0, 1] Multiplying both sides of (7) by 2tα−1, then integrating the resulting inequality with respect to t over 0, 12 , we obtain ≤ = 0 Setting x = tb+(1−t)a and dx = ab dx     Then multiplying both sides of (8) by tα−1 and integrating the resulting inequality with respect to t over 0, 21 , we obtain i.e. α ab b − a 21−α f (a) + f (b) ≤ α 2 . The proof (...truncated)


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Mehmet Kunt, İmdat İşcan, Nazlı Yazıcı, Uğur Gözütok. On new inequalities of Hermite–Hadamard–Fejer type for harmonically convex functions via fractional integrals, SpringerPlus, 2016, pp. 635, Volume 5, Issue 1, DOI: 10.1186/s40064-016-2215-4