英文文献翻译及原文

原 文

Title: Improved Integral Inequalities for Producets of Convex Functions

A largely applied inequality for convex functions, due to its geometrical significance, is Hadamard’s inequality (see [3] or [2]) which has generated a wide range of directions for

extension and a rich mathematical literature. Below, we recall this inequality, together with its framework.

A function [],f a b R ®：, with [],a b R Ì, is said to be convex if whenever

[],x a b " [],y a b Î,[]0,1t Î the following inequality holds

(1.1) ()()()()()11f tx t y tf x t f y +-?-

This definition has its origins in Jensen’s results from [4] and has opened up the most

extended, useful and multi-disciplinary domain of mathematics, namely, convex analysis. Convex curves and convex bodies have appeared in mathematical literature since antiquity and there are many important results related to them. They were known before the analytical foundation of convexity theory, due to the deep geometrical significance and many

geometrical applications related to the convex shapes (see, for example, [1], [5], [7]). One of these results, known as Hadamard’s inequality, which was first published in [3], states that a convex function f satisfies

(1.2) Recent inequalities derived from Hadamard’s inequality can be found in Pachpatte’s

paper [6] and we recall two of them in the following theorem, because we intend to improve them. Let us suppose that the interval [],a b has the property that 1b a - . Then the

following result holds.

Theorem 1.1. Let f and g be real-valued, nonnegative and convex functions on [],a b . Then

(1.3) ()()()12031(1)(1)2b b a a f tx t y g tx t y dtdydx b a ?-+--蝌

()()()()()

2,,118b a M a b N a b f x g x dx b a b a 轾+犏?犏--犏臌ò and

(1.4) ()()1

03

1122b a a b a b f tx t g tx t dtdx b a 骣骣骣骣++鼢琪琪珑+-+-鼢鼢珑珑鼢鼢珑珑桫桫-桫桫

蝌 ()()()122b a f a f b a b f f x dx b a 骣++÷ç＃÷ç÷ç桫-ò

()()()()111,,4b a b a f x g x dx M a b N a b b a b a +-轾??臌--ò

where

(1.5) ()()()()(),M a b f a g a f b g b =+

and

(1.6) ()()()()(),N a b f a g b f b g a =+

Remark 1.2. The inequalities (1.3) and (1.4) are valid when the length of the interval [],a b does not exceed 1. Unfortunately, this condition is accidentally omitted in [6], but it is implicitly used in the proof of Theorem 1.1.

Of course, there are cases when at least one of the two inequalities from the previous theorem is satisfied for 1b a ->, but it is easy to find counterexamples in this case, as follows.

Example 1.1. Let us take [][],0,2a b =. The functions []:0,2f R ® and []:0,2g R ® are defined by ()f x x = and ()g x x =. Then it is obvious that (),4M a b =, (),0N a b =. Then, the direct calculus of both sides of (1.3) leads to

and, obviously, inequality (1.3) is false.

Remark 1.3. Inequality (1.3) is sharp for linear functions defined on []0,1, while inequality (1.4) does not have the same property.

In this paper we improve the previous theorem, such that the condition

1b a -

is eliminated and the derived inequalities are sharp for the whole class of linear functions.

()()()1203111(1)(1)26b b a a f tx t y g tx t y dtdydx b a ?-+-=-蝌 ()()()()()2,,1135824b a M a b N a b f x g x dx b a b a 轾+犏+=犏--犏臌

ò