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Chinese Journal of Mathematics
Volume 2013 (2013), Article ID 295146, 3 pages
A Sharp Bound for the Čebyšev Functional of Convex or Concave Functions
Department of Mathematics, Faculty of Science, Jerash University, Jerash 26150, Jordan
Received 15 July 2013; Accepted 9 August 2013
Academic Editors: Y. Fu, Y. He, and Y. Shi
Copyright © 2013 Mohammad W. Alomari. 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.
A sharp bound for the Čebyšev functional of convex or concave functions is proved.
For two Lebesgue functions , the Čebyšev functional defined by has interesting applications in various integral approximations as pointed out in the references below.
The problem of bounding the Čebyšev functional has a long history, starting with Grüss  inequality in 1935, where he had proved that, for two integrable functions , such that and for any , the inequality holds, and the constant is the best possible.
After that, many authors have studied the functional (1), and therefore, several bounds under various assumptions for the functions involved have been obtained.
In 1971, Atkinson  proved that if , are twice differentiable and convex on and then .
This result is implied by that of Lupaş  who proved the following: if are convex functions on the interval , then with equality when at least one of the functions and is a linear function on (see also ).
In recent years, several bounds for the Čebyšev functional in various cases including convexity assumptions for the functions involved are proved. For other results for convex integrands, see [5–9] and [10, page 256], [10, page 262] where further references are given.
In this work, a sharp bound for the Čebyšev functional of convex or concave functions is proved.
2. The Result
We may begin with the following lemma.
Lemma 1. If are both nonnegative, decreasing (increasing), and convex on , then is nonnegative, decreasing (increasing), and convex function on .
Proof. The nonnegativity and monotonicity of follow directly by assumptions. To prove that is convex, by the monotonicity of , and for , we have which implies that Now, by (6) if and , then which proves that is convex and the proof is completely finished.
Throughout this paper, let be any functions. Define the functions , such that
Now, we may state our first result as follows.
Theorem 2. Under the assumptions of Lemma 1, one has The constant is the best possible in the sense that it cannot be replaced by smaller one.
Proof. Firstly, we note that, for any convex function defined on , we have Using the identity [10, page 246], and since and are two nonnegative monotonic convex functions on , then by Lemma 1, we have which gives the desired inequality (6). To prove the sharpness, assume that (6) holds with constant ; that is, Let ; consider that , , so that we have and . Making use of (9), we get which shows that the constant is the best possible in both cases, and thus, the proof is completely finished.
A parallel result for concave functions may be considered as follows.
Lemma 3. If is nonnegative, decreasing (increasing), and concave on , while is nonnegative, increasing (decreasing), and concave on , then is nonnegative, decreasing (increasing), and concave function on .
Proof. The nonnegativity and monotonicity of follow directly by assumptions. To prove that is concave, by the monotonicity of , and for , we have which implies that Now, by (16) if and , then which proves that is concave and the proof is completely finished.
Now, we may state the reverse of (9) as follows.
Theorem 4. Under the assumptions of Lemma 3, one has The constant is the best possible in the sense that it cannot be replaced by smaller one.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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