In order to prove our main results, we need the following lemma.

Lemma 2.1 (see [4]).

Let

be defined as above, and let

with

. Then,

Now, we are ready to state and prove our results.

Theorem 2.2.

Let

, and

be defined as above. Then

is convex, increasing on

, and for all

, one has the following Fejér-type inequality:

Proof.

It is easily observed from the convexity of

that

is convex on

. Using simple integration techniques and under the hypothesis of

, the following identity holds on

:

Let

in

. By Lemma 2.1, the following inequality holds for all

:

Indeed, it holds when we make the choice

in Lemma 2.1.

Multipling the inequality (2.4) by
, integrating both sides over
on
and using identity (2.3), we derive
. Thus
is increasing on
and then the inequality (2.2) holds. This completes the proof.

Remark 2.3.

Let
in Theorem 2.2. Then
and the inequality (2.2) reduces to the inequality (1.4), where
is defined as in Theorem A.

Theorem 2.4.

Let

be defined as above. Then

is convex, increasing on

, and for all

, one has the following Fejér-type inequality:

Proof.

By using a similar method to that from Theorem 2.2, we can show that

is convex on

, the identity

holds on

, and the inequalities

hold for all
in
and
.

By (2.7)–(2.9) and using a similar method to that from Theorem 2.2, we can show that
is increasing on
and (2.6) holds. This completes the proof.

The following result provides a comparison between the functions
and
.

Theorem 2.5.

Let
,
,
, and
be defined as above. Then
on
.

Proof.

on
, (2.3) and using a similar method to that from Theorem 2.2, we can show that
on
. The details are omited.

Further, the following result incorporates the properties of the function
.

Theorem 2.6.

Let

be defined as above. Then

is convex, increasing on

, and for all

, one has the following Fejér-type inequality:

Proof.

on
. The details are left to the interested reader.

We now present a result concerning the properties of the function
.

Theorem 2.7.

Let

be defined as above. Then

is convex, increasing on

, and for all

, one has the following Fejér-type inequality:

Proof.

on
and using a similar method to that for Theorem 2.2, we can show that
is convex, increasing on
and (2.13) holds.

Remark 2.8.

Let
in Theorem 2.7. Then
and the inequality (2.13) reduces to (1.6), where
is defined as in Theorem B.

Theorem 2.9.

Let
,
,
, and
be defined as above. Then
on
.

Proof.

on
, (2.12) and using a similar method to that for Theorem 2.2, we can show that
on
. This completes the proof.

The following Fejér-type inequality is a natural consequence of Theorems 2.2–2.9.

Corollary 2.10.

Let

be defined as above. Then one has

Remark 2.11.

Let

in Corollary 2.10. Then the inequality (2.16) reduces to

which is a refinement of (1.1).

Remark 2.12.

In Corollary 2.10, the third inequality in (2.16) is the weighted generalization of Bullen's inequality [

5]