We start with the following theorem.

Theorem 2.1.

Let

be

-convex functions on

and

with

. Then the following inequality holds:

where
is a logarithmic mean of positive real numbers.

For
a positive
-concave function, the inequality is reversed.

Proof.

Since

are

-convex functions on

, we have

for all

and

Writing (2.2) for

and multiplying the resulting inequalities, it is easy to observe that

for all
and

Integrating inequality (2.3) on

over

, we get

the theorem is proved.

Remark 2.2.

By taking
and
in Theorem 2.1
we obtain (1.7).

Corollary 2.3.

Let

be

-convex functions on

and

with

. Then

If

are positive

-concave functions, then

Proof.

Let

be positive

-convex functions. Then by Theorem 2.1 we have that

for all
, whence (2.7). Similarly we can prove (2.8).

Remark 2.4.

By taking
and
in (2.7) and (2.8)
we obtain the inequalities of Corollary 1.4.

We will now point out some new results of the Hadamard type for log-convex,
-convex, and
-convex functions, respectively.

Theorem 2.5.

Let

be

-convex functions on

and

with

Then the following inequalities hold:

Proof.

Using the elementary inequality

(

reals) and equality (2.11), we have

Since

are

-convex functions, we obtain

for all
and
.

Rewriting (2.12) and (2.13), we have

Integrating both sides of (2.14) and (2.15) on

over

, respectively, we obtain

Combining (2.16), we get the desired inequalities (2.10). The proof is complete.

Theorem 2.6.

Let

be

-convex functions on

and

with

Then the following inequalities hold:

where
is a logarithmic mean of positive real numbers.

Proof.

From inequality (2.14), we have

for all
and

Using the elementary inequality

(

reals) on the right side of the above inequality, we have

Since

are

-convex functions, then we get

Integrating both sides of (2.19) and (2.20) on

over

, respectively, we obtain

Combining (2.21), we get the required inequalities (2.17). The proof is complete.

Theorem 2.7.

Let

be such that

is in

, where

. If

is nonincreasing

-convex function and

is nonincreasing

-convex function on

for some fixed

then the following inequality holds:

Proof.

Since

is

-convex function and

is

-convex function, we have

for all

. It is easy to observe that

Using the elementary inequality

(

reals), (2.25) on the right side of (2.26) and making the charge of variable and since

is nonincreasing, we have

Rewriting (2.27) and (2.28), we get the required inequality in (2.22). The proof is complete.

Theorem 2.8.

Let

be such that

is in

, where

. If

is nonincreasing

-convex function and

is nonincreasing

-convex function on

for some fixed

Then the following inequality holds:

Proof.

Since

is

-convex function and

is

-convex function, then we have

for all

. It is easy to observe that

Using the elementary inequality

(

reals), (2.32) on the right side of (2.33) and making the charge of variable and since

is nonincreasing, we have

Rewriting (2.34) and (2.35), we get the required inequality in (2.29). The proof is complete.

Remark 2.9.

In Theorem 2.8, if we choose
, we obtain the inequality of Theorem 2.7.