We establish some new HermiteHadamardtype 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.
1. Introduction
Integral inequalities have played an important role in the development of all branches of Mathematics.
In [1, 2], Pachpatte established some HermiteHadamardtype inequalities involving two convex and logconvex functions, respectively. In [3], Bakula et al. improved HermiteHadamard type inequalities for products of two convex and convex functions. In [4], 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 [5]).
For several recent results concerning these types of inequalities, see [6–12] where further references are listed.
The aim of this paper is to establish several new integral inequalities for nonnegative and integrable functions that are related to the HermiteHadamard 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.
BarnesGudunovaLevin Inequality (see [13–15] and references therein)
Let , be nonnegative concave functions on . Then, for we have
where
In the special case we have
with
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 [16] 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 [17].
Also, we need some important inequalities.
Minkowski Integral Inequality (see page 1 in [18])
Let , and . Then
HermiteHadamard's Inequality (see page 10 in [10])
Let be a convex function on interval of real numbers and with . Then the following HermiteHadamard inequality for convex functions holds:
If the function is concave, the inequality (1.10) can be written as follows:
For recent results, refinements, counterparts, generalizations, and new HermiteHadamardtype inequalities, see [19–21].
A Reversed Minkowski Integral Inequality (see page 2 in [18])
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.
Remark 1.1.
Observe that whenever, is concave on the nonnegative function is also concave on . Namely,
that is,
and ; using the powermean inequality (1.6), we obtain
For , similarly if is concave on the nonnegative function is concave on .
2. The Results
Theorem 2.1.
Let and let , , be nonnegative functions such that and are concave on . Then
and if , then one has
Here is the BarnesGudunovaLevin constant given by (1.1).
Proof.
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 BarnesGudunovaLevin inequality to the righthand side of (2.4) with (2.6), we get (2.1).
Applying the Hölder inequality to the lefthand side of (2.5) with , we get (2.2).
Theorem 2.2.
Let , and , and let be positive functions with
Then
where
Proof.
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).
Theorem 2.3.
Let and be as in Theorem 2.1. Then the following inequality holds:
Proof.
If , are concave on , then from (1.11) we get
which imply that
On the other hand, if from (1.6) we get
or
which imply that
Combining (2.13) and (2.16), we obtain the desired inequality as
that is,
To prove the following theorem we need the following Youngtype inequality (see [7, page 117]):
Theorem 2.4.
Let be functions such that , and are in , and
Then
where
and with
Proof.
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).
Acknowledgment
The authors thank the careful referees for some good advices which have improved the final version of this paper.
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