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.
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 , .
2 Main results
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). □
3 Hermite-Hadamard inequality
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. □
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