On the Hermite-Hadamard Inequality and Other Integral Inequalities Involving Two Functions
© Erhan Set et al. 2010
Received: 25 September 2009
Accepted: 31 March 2010
Published: 24 May 2010
We establish some new Hermite-Hadamard-type inequalities involving product of two functions. Other integral inequalities for two functions are obtained as well. The analysis used in the proofs is fairly elementary and based on the use of the Minkowski, Hölder, and Young inequalities.
Integral inequalities have played an important role in the development of all branches of Mathematics.
In [1, 2], Pachpatte established some Hermite-Hadamard-type inequalities involving two convex and log-convex functions, respectively. In , Bakula et al. improved Hermite-Hadamard type inequalities for products of two -convex and -convex functions. In , analogous results for -convex functions were proved by Kirmaci et al.. General companion inequalities related to Jensen's inequality for the classes of -convex and -convex functions were presented by Bakula et al. (see ).
The aim of this paper is to establish several new integral inequalities for nonnegative and integrable functions that are related to the Hermite-Hadamard result. Other integral inequalities for two functions are also established.
In order to prove some inequalities related to the products of two functions we need the following inequalities. One of inequalities of this type is the following one.
To prove our main results we recall some concepts and definitions.
(see, e.g., [10, page 15]).
and is the set of all functions such that .
For several recent results concerning -norms we refer the interested reader to .
Also, we need some important inequalities.
Minkowski Integral Inequality (see page 1 in )
Hermite-Hadamard's Inequality (see page 10 in )
A Reversed Minkowski Integral Inequality (see page 2 in )
One of the most important inequalities of analysis is Hölder's integral inequality which is stated as follows (for its variant see [10, page 106]).
Hölder Integral Inequality
with equality holding if and only if almost everywhere, where and are constants.
For , similarly if is concave on the nonnegative function is concave on .
2. The Results
Here is the Barnes-Gudunova-Levin constant given by (1.1).
Thus, by applying Barnes-Gudunova-Levin inequality to the right-hand side of (2.4) with (2.6), we get (2.1).
Applying the Hölder inequality to the left-hand side of (2.5) with , we get (2.2).
Since and by applying the Minkowski integral inequality to the right hand side of (2.10), we obtain inequality (2.8).
This completes the proof of the inequality in (2.21).
The authors thank the careful referees for some good advices which have improved the final version of this paper.
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