- Open Access
A new version of Jensen’s inequality and related results
© Zabandan and Kılıçman; licensee Springer 2012
- Received: 28 May 2012
- Accepted: 2 October 2012
- Published: 17 October 2012
In this paper we expand Jensen’s inequality to two-variable convex functions and find the lower bound of the Hermite-Hadamard inequality for a convex function on the bounded area from the plane.
- Convex Function
- Reversed Order
- Suitable Condition
- Positive Measure
- Concave Function
The inequality (1) is known as Jensen’s inequality .
holds for all and . A function is called co-ordinated convex on Δ if the partial functions , and , are convex for all and . Note that every convex function is co-ordinated convex, but the converse is not generally true; see . Also note that if F is a convex function on and g, h are real-valued functions such that , then may be not convex on ℝ.
where F is convex on the convex bounded area by , and , .
The inequalities hold in reversed order if f is concave on Δ.
The proof is complete. For the proof of (3), set .
Note the inequalities (2) and (3) are sharp because . □
- (i)for , ,
- (iii)for ,(iv)
- (ii)The function is convex for and is concave for . So, by the inequality (3), we have
The proof of (iii) is similar to that of (ii) and can be omitted. For the proof of (iv), note is convex on . Now, apply the inequality (3). □
In , Dragomir established the following similar inequality (4) for convex functions on the co-ordinates on a rectangle from the plane .
In , Matejíčka proved the left-hand side of the Hermite-Hadamard inequality of several variables for a convex function on certain convex compact sets. In the following theorem, we prove the left-hand side of the Hermite-Hadamard inequality in another way and as a result of Theorem 2.
The proof is complete. □
- Rudin W: Real and Complex Analysis. McGraw-Hill, New York; 1974.Google Scholar
- Dragomir SS: On Hadamard’s inequality for the convex mappings defined on a ball in the space and application. Math. Inequal. Appl. 2000, 3: 177–187.MathSciNetGoogle Scholar
- Mitrinovic DS, Lackoric JB: Hermite and convexity. Aequ. Math. 1985, 28: 229–232. 10.1007/BF02189414View ArticleGoogle Scholar
- Zabandan G: A new refinement of the Hermite-Hadamard inequality for convex functions. JIPAM. J. Inequal. Pure Appl. Math. 2009., 10(2): Article ID 45Google Scholar
- Dragomir SS: On the Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane. Taiwan. J. Math. 2001, 5: 775–788.Google Scholar
- Dragomir SS: On Hadamard’s inequality on a disk. J. Inequal. Pure Appl. Math. 2000., 1: Article ID 2Google Scholar
- Dragomir, SS, Pearce, CEM: Selected Topics on Hermite-Hadamard Inequalities. RGMIA Monographs, Victoria University (2000)Google Scholar
- Matejíčka L: Elementary proof of the left multidimensional Hermite-Hadamard inequality on certain convex sets. J. Math. Inequal. 2010, 4(2):259–270.MathSciNetGoogle Scholar
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