In the present paper we establish some new integral inequalities analogous to the well known Hadamard’s inequality by using a fairly elementary analysis. | Turk J Math 36 (2012) , 245 – 251. ¨ ITAK ˙ c TUB doi: On some new inequalities for convex functions Mevl¨ ut Tun¸c Abstract In the present paper we establish some new integral inequalities analogous to the well known Hadamard’s inequality by using a fairly elementary analysis. Key Words: Hadamard’s inequality, convex function, means 1. Introduction The inequality (see [1]) f a+b 2 1 ≤ b−a b f (x) dx ≤ a f (a) + f (b) , 2 () which holds for all convex functions f : [a, b] → R, is known in the literature as Hadamard’s inequality. Since its discovery in 1893, Hadamard’s inequality [2] has been proven to be one of the most useful inequalities in mathematical analysis. A number of papers have been written on this inequality providing new proofs, noteworthy extensions, generalizations and numerous applications; see [1, 8] and the references cited therein. The main purpose of this paper is to establish some new integral inequalities analogous to Hadamard’s inequality given in () involving two convex functions. The analysis used in the proof is elementary and we believe that the inequalities established here are of independent interest. 2. Main results We need the following Lemma proved in [6] which deals with the simple characterization of convex functions. Lemma 1 The following statements are equivalent for a mapping: f : [a, b] → R; i) f is convex on [a, b] , ii) for all x, y in [a, b] the mapping g : [0, 1] → R, defined by g(t) = f(tx + (1 − t)y) is convex on [0, 1] . For the proof of this Lemma, see [6]. Our main result is given in the following theorem. 2000 AMS Mathematics Subject Classification: 26A51, 26D15. 245 TUNC ¸ Theorem 2 Let f, g : [a, b] → R be two convex functions and fg ∈ L1 ([a, b]) . Then, 1 (b − x) (f (a) g (x) + g (a) f (x)) dx 2 (b − a) a 1 + b 2 (b − a) b (x − a) (f (b) g (x) + g (b) f (x)) dx M (a, b) N (a, b) 1 + + 3 6 b−a ≤ () a b f (x) g (x) dx, a where M (a, b) =
