^{1}

In this paper generalized Steffensen type inequalities related to the class of functions that are “convex at point

The well-known Steffensen inequality [

Suppose that

Since its appearance in 1918 numerous papers have been devoted to generalizations and refinements of Steffensen’s inequality. In [

Let

By substitutions

Let the conditions of Theorem

In 2000 Mercer [

Motivated by refinement of Steffensen’s inequality given in [

Let

If

Let

If

In this paper we obtain generalized Steffensen type inequalities, related to the aforementioned generalizations and refinements of Steffensen’s inequality, for the class of functions that are convex at point

Let us begin by introducing a class of functions that extends the class of convex functions.

Let

As noted in [

The function

Applying the generalizations of Steffensen’s inequality given in the Introduction to functions that are convex at point

Let

Let

It is similar for

From the proof we deduce that condition (

Moreover, condition (

First, let us show that for

Hence, if the function

Let

Let

It is similar for

For

Additionally, condition (

As a consequence of Theorems

Let

If

Since

Let

If

The proof is similar to that of Corollary

Similarly, as in Remarks

For

In [

Let

Take

Let

Take

In [

Let

Take

Let

Take

Taking

In the following theorems we obtain refined version of results given in Theorems

Let

The proof is similar to that of Theorem

Let

The proof is similar to that of Theorem

Motivated by sharpened and generalized versions of Theorems

Let

The proof is similar to that of Theorem

Let

The proof is similar to that of Theorem

Generalized Steffensen type inequalities obtained in Theorems

Similarly, as in Remarks

Generalizations of Steffensen type inequalities given by (

Under the assumptions of Theorems

Let us begin by showing a Lagrange type mean value theorem for the functional

Let

Since

If

We continue with a Cauchy type mean value theorem for the functional

Let

Define

As in Theorem

We begin by recalling some definitions and results on exponential convexity; see [

A function

A function

It is clear from the definition that

Also,

A function

A function

A function

We also use the following well known results for convex functions.

A function

If

Let

The value

Next, we construct exponentially convex functions using the previously defined functionals

Let

If the mapping

For

If the mapping

If the assumptions of Theorem

Let

If the mapping

Let

If the mapping

If the mapping

(i) By Theorem

(ii) By (i) we have that the mapping

The limit cases

The results stated in Theorem

We continue with an example of a family of functions which satisfies the previous conditions.

Let

For this family of functions

Explicitly for

for

for

for

Theorem

The authors declare that there is no conflict of interests regarding the publication of this paper.

The research of the authors was supported by the Croatian Ministry of Science, Education and Sports under the Research Grant 117-1170889-0888.