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 [3], Bakula et al. improved Hermite-Hadamard 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 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.

Barnes-Gudunova-Levin Inequality (see [13–15] and references therein)

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 [

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

Hermite-Hadamard's Inequality (see page 10 in [10])

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:

For recent results, refinements, counterparts, generalizations, and new Hermite-Hadamard-type 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,

and

; using the power-mean inequality (1.6), we obtain

For
, similarly if
is concave on
the nonnegative function
is concave on
.