Abstract
We present some new Hermite-Hadamard-type inequalities and other integral inequalities involving several functions.
1. Introduction and Preliminaries
In 2003, Pachpatte [1] gave some Hermite-Hadamard-type inequalities involving two convex functions, and then Pachpatte [2] also gave, in 2004, some Hermite-Hadamard-type inequalities involving two log-convex functions. In 2007, Kirmaci et al. [3] gave some Hadamard-type inequalities involving -convex functions. In 2008, Bakula et al. [4] presented some Hadamard-type inequalities involving -convex functions and -convex functions. In 2010, Set et al. [5] gave some new Hermite-Hadamard-type inequalities and other integral inequalities involving two functions. More details about results proved in [5] will be given in Section 2. In this paper, we present more general Hermite-Hadamard-type inequalities and some integral inequalities involving several functions.
Let and . The -norm of the function on is defined by
Below we recall few well-known inequalities that will be useful in the proofs of our results.
Hermite-Hadamard’s Inequality (see [6–8]). If is a convex function on , then If is a concave function on , then
Barnes-Gugunova-Levin Inequality (see [9–11]). If and are nonnegative concave functions on , and if , then where
The Power-Mean Inequality (see [12]). Let and let . Then Notice that if then
A Generalization of Hölder Integral Inequality. For any , , if are nonnegative functions on and if are integrable functions on , then
A Generalization of Minkowski Integral Inequality. If and if are non-negative functions on such that for all , then
A Generalization of Young Inequality. If and , , then
To prove results, we refer to the following lemma.
Lemma 1 (see [13]). If and if and are positive functions on such that for all , then
Proof. Let . Assume that and are positive functions on such that for all . Then Then Thus,
2. Main Results
We start this section with the following.
Theorem 2. Let be a positive even integer and and let be non-negative functions on such that are concave on . Then where for all . Moreover, if , then
Proof. Applying the inequality (3) with , for any , we get
and, consequently,
By the Barnes-Gudunova-Levin inequality (4), it follows that
where
for all .
By the power-mean inequality (7), we have
for all .
This implies the inequality (15).
Next, we assume that . By the inequality (19) and the generalized Hölder inequality, we obtain that
This proof is completed.
It is easy to notice that if we put in Theorem 2 then we get the following.
Corollary 3 (see [5]). Let and let be non-negative functions on such that are concave on . Then where Moreover, if , then
Theorem 4. Let and be a positive integer such that and let be positive functions on such that the functions are integrable functions on , for all , and for all and for all . Then where for all .
Proof. By Lemma 1, we have
for all .
Then
for all .
Let , , and for all .
It follows that
for all .
By multiplying the above inequalities and the generalized Minkowski inequality, we obtain that
Then
This implies the inequality (28).
Notice that from above theorem one can easily get the following.
Corollary 5 (see [5]). Let and let be positive functions on such that , , and for all . Then where .
Proof. By Theorem 4 where , we have
where
Let . Then
This implies the inequality (36).
Theorem 6. Let be a positive integer such that and and let be non-negative functions on such that are concave on . Then
Proof. Using the inequality (3) with , for any , we obtain
Then
By the power-mean inequality (7), we have
so
for all .
Then
This implies the inequality (40).
One can easily check that if we put in Theorem 6 then we get the following.
Corollary 7 (see [5]). Let and let be non-negative functions on such that are concave on . Then
Theorem 8. Let be a positive integer such that and , , and let be positive functions on such that the function is integrable on , for all , and for all and . Then where for all .
Proof. By Lemma 1, we have
for all .
Using the elementary inequality where and , we get
for all .
Then
for all .
Let , , and
for all .
It follows that
for all .
By the generalized Young inequality, we obtain that
This proof is completed.
Applying Theorem 8 with and putting there , , and , we get the following.
Corollary 9 (see [5]). Let , , , and let and be positive functions on such that , , and for all . Then where
Acknowledgment
The author would like to thank the referees for their useful comments and suggestions.