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.
Let , be nonnegative concave functions on . Then, for we have
In the special case we have
To prove our main results we recall some concepts and definitions.
Let and be two positive -tuples, and let . Then, on putting the th power mean of with weights is defined  by
Note that if , then
(see, e.g., [10, page 15]).
Let and . The -norm of the function on is defined by
and is the set of all functions such that .
One can rewrite the inequality (1.1) as follows:
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 )
Let , and . Then
Hermite-Hadamard's Inequality (see page 10 in )
Let be a convex function on interval of real numbers and with . Then the following Hermite-Hadamard inequality for convex functions holds:
If the function is concave, the inequality (1.10) can be written as follows:
A Reversed Minkowski Integral Inequality (see page 2 in )
Let and be positive functions satisfying
Then, putting , we have
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
Let and If and are real functions defined on and if and are integrable functions on , then
with equality holding if and only if almost everywhere, where and are constants.
Observe that whenever, is concave on the nonnegative function is also concave on . Namely,
and ; using the power-mean inequality (1.6), we obtain
For , similarly if is concave on the nonnegative function is concave on .
2. The Results
Let and let , , be nonnegative functions such that and are concave on . Then
and if , then one has
Here is the Barnes-Gudunova-Levin constant given by (1.1).
Since are concave functions on , then from (1.11) and Remark 1.1 we get
By multiplying the above inequalities, we obtain (2.4) and (2.5)
If , then it easy to show that
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).
Let , and , and let be positive functions with
Since are positive, as in the proof of the inequality (1.13) (see [18, page 2]), we have that
By multiplying the above inequalities, we get
Since and by applying the Minkowski integral inequality to the right hand side of (2.10), we obtain inequality (2.8).
Let and be as in Theorem 2.1. Then the following inequality holds:
If , are concave on , then from (1.11) we get
which imply that
On the other hand, if from (1.6) we get
which imply that
Combining (2.13) and (2.16), we obtain the desired inequality as
To prove the following theorem we need the following Young-type inequality (see [7, page 117]):
Let be functions such that , and are in , and
From we have
From (2.19) with (2.23) we obtain
Using the elementary inequality , ( and ) in (2.24), we get
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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