logarithmic mean, Resonance 13 (2008), no. 6, 583-594. [5] Carlson B.C., The logarithmic mean, Amer. Math. Monthly 79 (1972), no. 6, 615-618. [6] Cerone P., Dragomir S.S., Mathematical inequalities. A perspective, CRC Press, Boca Raton, 2011. [7] Conde C., A version of the Hermite-Hadamard inequality in a nonpositive curvature space, Banach J. Math. Anal. 6 (2012), no. 2, 159-167. [8] Dragomir S.S., Bounds for the normalised Jensen functional, Bull. Austral. Math. Soc. 74 (2006), no. 3, 471-478. [9] Dragomir S.S., Hermite-Hadamard’s type inequalities for operator convex

References [1] M. Bessenyei, Zs. Páles, Characterization of convexity via Hadamard’s inequality , Math. Inequal. Appl. 9(1) (2006), 53–62. [2] S. S. Dragomir, C. E. M. Pearce, Selected Topics on Hermite–Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2002. (online: http://rgmia.vu.edu.au/monographs/ ). [3] S. Hu, N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol.I: Theory , Kluwer Acad. Publ., Dordrecht, Boston, London, 1997. [4] J. Matkowski, K. Nikodem, An integral Jensen inequality for convex multifunctions

1 Introduction The Hermite–Hadamard inequality plays an important role in convex functions theory, and it is used as a tool to obtain many results in integral inequalities, approximation theory, special mean theory, optimization theory, information theory and numerical analysis. For various results dealing with the Hermite–Hadamard inequality, we refer the reader to a monograph by Dragomir and Pearce [ 5 ], and for the pioneer papers see [ 8 , 9 ]. The discrete fractional Hermite–Hadamard inequality [ 3 ] states that if f : ℤ → ℝ {f:\mathbb{Z}\to\mathbb{R}} is

1 Introduction Throughout the paper, we use the bounded closed interval [ a , b ] {[a,b]} in the line ℝ {\mathbb{R}} with a < b {a<b} , and the rectangle Δ = [ a , b ] × [ c , d ] {\Delta=[a,b]\times[c,d]} in the plane ℝ {\mathbb{R}} with a < b {a<b} and c < d {c<d} . 1.1 Hermite–Hadamard’s inequality The Hermite–Hadamard inequality discovered by C. Hermite and J. Hadamard (see, e.g., [ 6 ], [ 14 , p.137]) is one of the most well established inequalities in the theory of convex functions with a geometrical interpretation and many applications. Over

[1] Bernstein F., Doetsch G., Zur Theorie der konvexen Funktionen, Math. Ann., 1915, 76(4), 514–526 http://dx.doi.org/10.1007/BF01458222 [2] Dragomir S.S., Pearce C.E.M., Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000, preprint available at http://ajmaa.org/RGMIA/papers/monographs/Master.pdf [3] Hadamard J., Étude sur les propriétés des fonctions entières et en particulier d’une fonction considéréé par Riemann, J. Math. Pures Appl., 1893, 58, 171–215 [4] Hyers D.H., Ulam S.M., Approximately convex

4 Hermite-Hadamard inequalities involving Riemann-Liouville fractional integrals 4.1 Inequalities via convex functions The results in this section are taken from [159, 251, 105]. 4.1.1 Main results Theorem 77. Let f : [a, b] → ℝ be a positive function with 0 ≤ a < b and f ∈ L1[a, b]. If f is a convex function on [a, b], then the following inequality for fractional integrals hold f(a + b 2 ) ≤ Γ(α + 1) 2(b − a)α [RLJ α a+ f (b) + RLJαb− f (a)] ≤ f (a) + f (b) 2 . (4.1) Proof. Since f is a convex function on [a, b], we have for x, y ∈ [a, b] with λ = 12 f(x + y 2

5 Hermite-Hadamard inequalities involving Hadamard fractional integrals 5.1 Inequalities via convex functions The results in this section are due to [217, 220, 143, 174]. 5.1.1 Hermite-Hadamard’s inequalities Hermite-Hadamard’s inequalities for Hadamard fractional integrals can be repre- sented as follows. Theorem 252. Let f : [a, b] → ℝ be a positive function with 0 < a < b and f ∈ L1[a, b]. If f is a nondecreasing and convex function on [a, b], then the following inequality for fractional integrals holds: f (√ab) ≤ Γ(α + 1) 2(ln b − ln a)α [H J α a+ f (b) + H

Inequalities in Pure and Applied Mathematics, vol. 5, no. 3, article 74, 2004. [5] Dragomir, S. S., Pearce, C. E. M., Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000. [6] Dragomir, S. S., Agarwal, R.P., Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11(5) (1998), 91-95. [7] Dragomir, S. S., On some new inequalities of Hermite-Hadamard type for m - convex functions, Tamkang J. Math., 3(1) (2002). [8] Dragomir, S. S

Analysis 32, 209–220 (2012) / DOI 10.1524/anly.2012.1161 c Oldenbourg Wissenschaftsverlag, München 2012 Generalizations of Hermite–Hadamard inequality to n-time differentiable functions which are s-convex in the second sense Wei-Dong Jiang, Da-Wei Niu, Yun Hua, Feng Qi Received: April 5, 2012 Summary: In the paper, the famous Hermite–Hadamard integral inequality for convex functions is generalized to and refined as the ones for n-time differentiable functions which are s-convex in the second sense, and some known results are improved. 1 Introduction It is common