Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 286845, 12 pages doi:10.1155/2010/286845 Research Article On Hadamard-Type Inequalities Involving Several Kinds of Convexity ă Erhan Set,1 M Emin Ozdemir,1 and Sever S Dragomir2, Department of Mathematics, K.K Education Faculty, Ataturk University, Campus, ă 25240 Erzurum, Turkey Research Group in Mathematical Inequalities & Applications, School of Engineering & Science, Victoria University, P.O Box 14428, Melbourne City, VIC 8001, Australia School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa Correspondence should be addressed to Erhan Set, erhanset@yahoo.com Received 14 May 2010; Accepted 23 August 2010 Academic Editor: Sin E I Takahasi Copyright q 2010 Erhan Set et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We not only give the extensions of the results given by Gill et al 1997 for log-convex functions but also obtain some new Hadamard-type inequalities for log-convex m-convex, and α, m -convex functions Introduction The following inequality is well known in the literature as Hadamard’s inequality: f a b ≤ b−a b a f x dx ≤ f a f b , 1.1 where f : I → R is a convex function on the interval I of real numbers and a, b ∈ I with a < b This inequality is one of the most useful inequalities in mathematical analysis For new proofs, note worthy extension, generalizations, and numerous applications on this inequality; see 1–6 where further references are given 2 Journal of Inequalities and Applications Let I be on interval in R Then f : I → R is said to be convex if, for all x, y ∈ I and λ ∈ 0, , − λ y ≤ λf x f λx 1−λ f y 1.2 see , Page Geometrically, this means that if K, L, and M are three distinct points on the graph of f with L between K and M, then L is on or below chord KM Recall that a function f : I → 0, ∞ is said to be log-convex function if, for all x, y ∈ I and t ∈ 0, , one has the inequality see , Page f tx 1−t y ≤ f x t f y 1−t 1.3 It is said to be log-concave if the inequality in 1.3 is reversed In , Toader defined m-convexity as follows Definition 1.1 The function f : 0, b → R, b > is said to be m-convex, where m ∈ 0, , if one has m − t y ≤ tf x f tx m 1−t f y 1.4 for all x, y ∈ 0, b and t ∈ 0, We say that f is m-concave if −f is m-convex Denote by Km b the class of all m-convex functions on 0, b such that f ≤ if m < Obviously, if we choose m 1, Definition 1.1 recaptures the concept of standard convex functions on 0, b In , Mihesan defined α, m -convexity as in the following: ¸ Definition 1.2 The function f : 0, b → R, b > 0, is said to be α, m -convex, where α, m ∈ 0, , if one has f tx m − t y ≤ tα f x m − tα f y 1.5 for all x, y ∈ 0, b and t ∈ 0, α Denote by Km b the class of all α, m -convex functions on 0, b for which f ≤ It can be easily seen that for α, m 1, m , α, m -convexity reduces to m-convexity and for α, m 1, , α, m -convexity reduces to the concept of usual convexity defined on 0, b , b > For recent results and generalizations concerning m-convex and α, m -convex functions, see 9–12 In the literature, the logarithmic mean of the positive real numbers p, q is defined as the following: L p, q for p q, we put L p, p p p−q ln p − ln q p/q 1.6 Journal of Inequalities and Applications In 13 , Gill et al established the following results Theorem 1.3 Let f be a positive, log-convex function on a, b Then b b−a f t dt ≤ L f a , f b , 1.7 a where L ·, · is a logarithmic mean of the positive real numbers as in 1.6 For f a positive log-concave function, the inequality is reversed Corollary 1.4 Let f be positive log-convex functions on a, b Then b−a b x − a L f a ,f x f t dt ≤ x∈ a,b a b − x L f x ,f b b−a 1.8 If f is a positive log-concave function, then b−a b f x dx ≥ max x − a L f a ,f x x∈ a,b a b − x L f x ,f b b−a 1.9 For some recent results related to the Hadamard’s inequalities involving two logconvex functions, see 14 and the references cited therein The main purpose of this paper is to establish the general version of inequalities 1.7 and new Hadamard-type inequalities involving two log-convex functions, two m-convex functions, or two α, m -convex functions using elementary analysis Main Results We start with the following theorem Theorem 2.1 Let fi : I ⊂ R → 0, ∞ i 1, 2, , n be log-convex functions on I and a, b ∈ I with a < b Then the following inequality holds: b−a b n a i fi x dx ≤ L n n fi a , i fi b i where L is a logarithmic mean of positive real numbers For f a positive log-concave function, the inequality is reversed , 2.1 Journal of Inequalities and Applications Proof Since fi i 1, 2, , n are log-convex functions on I, we have for all a, b ∈ I and t ∈ 0, Writing 2.2 for i inequalities, it is easy to observe that n fi ta t − t b ≤ fi a fi ta t 1−t n fi a i 2.2 1, 2, , n and multiplying the resulting n 1−t b ≤ 1−t fi b fi b i i n n fi b i i 2.3 t fi a fi b for all a, b ∈ I and t ∈ 0, Integrating inequality 2.3 on 0, over t, we get n n n − t b dt ≤ fi ta i fi b i i t fi a fi b 2.4 dt As n − t b dt fi ta i 1 n i fi a fi b t n b n fi x dx, L fi b 2.5 a i n dt b−a n fi a , i fi b , 2.6 i i the theorem is proved Remark 2.2 By taking i and f1 f in Theorem 2.1, we obtain 1.7 Corollary 2.3 Let fi : I ⊂ R → 0, ∞ with a < b Then b−a b i 1, 2, , n be log-convex functions on I and a, b ∈ I n fi x dx a i ≤ x∈ a,b x−a L n i fi a , n i fi x b−x L b−a n i fi x , n i fi 2.7 b Journal of Inequalities and Applications 1, 2, , n are positive log-concave functions, then If fi i b b−a n fi x dx a i 2.8 n i fi x−a L ≥ max n i fi a, x x∈ a,b Proof Let fi i that b n i fi b−x L b−a n i fi x , b 1, 2, , n be positive log-convex functions Then by Theorem 2.1 we have x n b n fi t dt n fi t dt a i fi t dt a i x i 2.9 n ≤ x−a L n fi a , b−x L fi x i n n fi x , i i fi b , i for all x ∈ a, b , whence 2.7 Similarly we can prove 2.8 Remark 2.4 By taking i and f1 f in 2.7 and 2.8 , we obtain the inequalities of Corollary 1.4 We will now point out some new results of the Hadamard type for log-convex, mconvex, and α, m -convex functions, respectively Theorem 2.5 Let f, g : I → 0, ∞ be log-convex functions on I and a, b ∈ I with a < b Then the following inequalities hold: f a b g a b ≤ ≤ b−a b f x f a b−x g x g a a f a f b b − x dx 2.10 g ag b Proof We can write a b ta 1−t b Using the elementary inequality cd ≤ 1/2 c2 have 1−t a d2 tb 2.11 c, d ≥ reals and equality 2.11 , we Journal of Inequalities and Applications a f b a g b 2 a b f 2 ≤ 1−t a 1−t b f ta 1/2 tb tb g 1−t a 1−t b f 1−t a tb 1−t a 1−t b ta g2 f 1−t a 1/2 1−t b g ta f ta b 1−t b 2 ta f ≤ a g2 2.12 1/2 1/2 tb 1−t b g 1−t a g ta tb tb Since f, g are log-convex functions, we obtain f ta 1−t b f 1−t a f a ≤ t f a f b 1−t f b tb f a 1−t b g 1−t a g ta 1−t t f b g a t tb 1−t g b 1−t g a t g b 2.13 g a g b for all a, b ∈ I and t ∈ 0, Rewriting 2.12 and 2.13 , we have f a b g a b ≤ f ta 1−t b f 1−t a tb g ta 1−t b g 1−t a tb , 2.14 f ta 1−t b f 1−t a tb g ta 1−t b g 1−t a tb ≤ f a f b g a g b 2.15 Integrating both sides of 2.14 and 2.15 on 0, over t, respectively, we obtain f a b 1 b−a g a b 1 b−a b f x f a b−x g x g a b − x dx , a 2.16 b f x f a a ≤ b−x g x g a b − x dx ≤ f a f b g a g b Combining 2.16 , we get the desired inequalities 2.10 The proof is complete Journal of Inequalities and Applications Theorem 2.6 Let f, g : I → 0, ∞ be log-convex functions on I and a, b ∈ I with a < b Then the following inequalities hold: a 2f b a g b ≤ ≤ b b−a f2 x g x dx a f a 2.17 f b g a L f a ,f b g b L g a ,g b , where L ·, · is a logarithmic mean of positive real numbers Proof From inequality 2.14 , we have f a b a g b 2.18 ≤ f ta 1−t b f 1−t a tb 1−t b g 1−t a g ta for all a, b ∈ I and t ∈ 0, Using the elementary inequality cd ≤ 1/2 c2 the above inequality, we have f a b g a tb c, d ≥ reals on the right side of d2 b 2.19 ≤ f ta 1−t b f2 − t a 1−t b g ta tb g2 − t a tb Since f, g are log-convex functions, then we get f ta ≤ 1−t b f a f2 b 2t f2 − t a 2−2t f b f a f b 2t f2 a f a f b f a 1−t b g ta tb 2−2t f b 2t g2 b 2t g a g a g b g2 − t a 2t g b 2t g2 a tb 2−2t g b g a g a 2−2t g b 2t 2t 2.20 Journal of Inequalities and Applications Integrating both sides of 2.19 and 2.20 on 0, over t, respectively, we obtain a 2f ≤ b b−a b g f2 x a b ≤ b b−a f2 x g x dx, a g x dx a f2 b f a f b 2t g a g b g2 b ⎛ 1⎝ f b f2 a dt 2t 2t f a g a /g b log g a /g b g b f a f b ⎞ ⎠ 2.21 f2 b − f2 a log f b − log f a g2 b − g2 a log g b − log g a f a f b g a L g a ,g b L f a ,f b 2t f a f b L f a ,f b 2 g b /g a log g b /g a g2 a g2 a − g2 b log g a − log g b g a dt 2t f2 a − f2 b log f a − log f b 2t f b /f a log f b /f a 2t dt g b g a f a /f b log f a /f b g2 b g2 a dt 2t f b f a L f b ,f a g b g a g b L g b ,g a L g a ,g b Combining 2.21 , we get the required inequalities 2.17 The proof is complete Theorem 2.7 Let f, g : 0, ∞ → 0, ∞ be such that fg is in L1 a, b , where ≤ a < b < ∞ If f is nonincreasing m1 -convex function and g is nonincreasing m2 -convex function on a, b for some fixed m1 , m2 ∈ 0, , then the following inequality holds: b−a b a f x g x dx ≤ min{S1 , S2 }, 2.22 Journal of Inequalities and Applications where S1 f2 a g2 a b m1 m1 f a f m2 g a g b m2 m2 f b m1 m2 g 2 b m2 , 2.23 S2 f2 b g2 b a m1 m1 f b f m2 g b g a m2 m2 f a m1 m2 g 2 a m2 2.24 Proof Since f is m1 -convex function and g is m2 -convex function, we have b , m1 − t b ≤ tf a g ta m1 − t f − t b ≤ tg a f ta b m2 − t g m2 2.25 for all t ∈ 0, It is easy to observe that b b−a f x g x dx − t b g ta f ta a − t b dt 2.26 Using the elementary inequality cd ≤ 1/2 c2 d2 c, d ≥ reals , 2.25 on the right side of 2.26 and making the charge of variable and since f, g is nonincreasing, we have b f x g x dx a ≤ ≤ b−a 1−t b f ta g ta 1−t b dt 1 b−a b−a 2 f a m1 − t f tf a b m1 2 b mf m1 tg a b m1 f a f m1 m2 − t g g a b m2 dt 2 b m g m2 b m2 g a g m2 b−a f2 a g2 a m2 g 2 b m2 m1 f a f b m1 m2 g a g b m2 m2 f b m1 2.27 10 Journal of Inequalities and Applications Analogously we obtain b f x g x dx a ≤ b−a f2 b g2 b m1 f b f a m1 m2 g b g a m2 m2 f a m1 m2 g 2 a m2 2.28 Rewriting 2.27 and 2.28 , we get the required inequality in 2.22 The proof is complete Theorem 2.8 Let f, g : 0, ∞ → 0, ∞ be such that fg is in L1 a, b , where ≤ a < b < ∞ If f is nonincreasing α1 , m1 -convex function and g is nonincreasing α2 , m2 -convex function on a, b for some fixed α1 , m1 , α2 , m2 ∈ 0, Then the following inequality holds: b−a b f x g x dx ≤ min{E1 , E2 }, 2.29 a where E1 1 f2 a 2α1 α1 α2 E2 2α1 2α1 2α2 2α2 1 f2 b 2α1 α1 α2 2α1 2α1 2α2 2α2 α1 m2 g 2 α1 1 b m2 2α2 1 2α1 2α2 2α2 α2 2.30 m2 g a g b m2 , a m1 a m1 g2 a 2α2 2α2 m2 f a m2 b m1 α2 m1 f b f m2 g 2 m2 f b m1 m1 f a f 1 2α2 1 2α1 g2 b 2α2 2α2 2.31 m2 g b g a m2 Proof Since f is α1 , m1 -convex function and g is α2 , m2 -convex function, then we have f ta − t b ≤ tα1 f a m1 − tα1 f g ta 1−t b ≤t g a m2 − t α2 α2 b , m1 b g m2 2.32 Journal of Inequalities and Applications 11 for all t ∈ 0, It is easy to observe that b b−a f x g x dx − t b g ta f ta a − t b dt 2.33 c, d ≥ reals , 2.32 on the right side of Using the elementary inequality cd ≤ 1/2 c2 d2 2.33 and making the charge of variable and since f, g is nonincreasing, we have b f x g x dx ≤ a ≤ b−a b−a 1−t b g ta dt 1 m2 − tα2 g 2α1 f2 a 2α1 2α1 α1 α2 2α2 1 2α1 α1 2α2 2α2 m2 g 2 b m2 m1 f a f b m1 m1 − tα1 f tα1 f a tα2 g a b−a 2 1−t b f ta dt b m1 m2 f b m1 b m2 g2 a 2α2 2α2 2α2 α2 m2 g a g b m2 2.34 Analogously we obtain b f x g x dx a ≤ b−a 2α1 α1 α2 f2 b 2α1 2α1 2α2 2α2 α1 2α2 1 2α1 m2 f a m1 f b f m1 m2 g 2 a m2 a m1 2.35 2α2 α2 g b 2α2 2α2 m2 g b g a m2 Rewriting 2.34 and 2.35 , we get the required inequality in 2.29 The proof is complete Remark 2.9 In Theorem 2.8, if we choose α1 α2 1, we obtain the inequality of Theorem 2.7 12 Journal of 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generalization of the convexity,”... m-concave if −f is m-convex Denote by Km b the class of all m-convex functions on 0, b such that f ≤ if m < Obviously, if we choose m 1, Definition 1.1 recaptures the concept of standard convex