Integral Inequalities Involving Strongly Convex Functions

We study the notions of strongly convex function as well as F-strongly convex function. We present here some new integral inequalities of Jensen’s type for these classes of functions. A refinement of companion inequality to Jensen’s inequality established by Matić and Pečarić is shown to be recaptured as a particular instance. Counterpart of the integral Jensen inequality for strongly convex functions is also presented. Furthermore, we present integral Jensen-Steffensen and Slater’s inequality for strongly convex functions.

The main ingredient of our investigation is the strongly convex function [2].Let Ψ be the real function defined on interval  and  be positive number, then we say that the function Ψ is strongly convex with modulus  on  if Every strongly convex function is convex, but the converse is not true in general.Strongly convex functions have been utilized for proving the convergence of a gradient type algorithm for minimizing a function.They play a significant role in mathematical economics, approximation theory, and optimization theory.Many applications and properties of them can be found in [2,9,20].In 2016, Adamek [21] further generalized the notion of strongly convex function.They replaced the nonnegative term ( − ) 2 by a nonnegative real valued function  and defined it as follows: function Ψ is said to be -strongly convex function if for all ,  ∈  and  ∈ [0, 1].From [22], we also have where Ψ is -strongly convex function.
In literature the following inequality is well-known as Jensen inequality.
The following inequality is the integral analogue of another companion inequality to the Jensen inequality.
Matić and Pečarić established a general inequality from which one can directly obtain inequalities (5) and (7).
Merentes and Nikodem improved the Jensen inequality for strongly convex functions as follows.
This paper is organized as follows.In Section 2, we establish general inequalities for -strongly convex function as well as strongly convex functions.As a consequence, we obtain integral Jensen inequality and Slater's inequality for strongly convex functions.Also by the virtue of these general inequalities we deduce converse of Jensen inequality.In Section 3, we give some properties of strongly convex functions.By using these properties of strongly convex functions we prove Jensen-Steffensen and Slater's type inequalities.
By virtue of Theorem 8, we can deduce some new and interesting consequences.Proposition 9. Suppose that all the assumptions of Theorem 8 are satisfied.Then Proof.If we set  =  in (11) and taking integral over Λ and then dividing by (Λ), we have or equivalently Taking the infimum over  ∈ , we obtain (17).
Proposition 10.Suppose that all the assumptions of Theorem 8 are satisfied and  = (1/(Λ)) ∫ Λ Ψ  + (), then Proof.By setting  1 =  and  2 =  ∈  in (10), we have Indeed, the following equivalent form of ( 21) is Taking the infimum over  ∈ , we can easily derive the first and the second inequality in (20).The remaining third inequality in (20) Remark 12.If we put  1 =  in (24) and take probability measure space, then we obtain integral Jensen inequality (9) for strongly convex function.
In the following corollary, we obtain integral Slater's inequality for strongly convex function.
In the following corollary, we obtain a converse of the Jensen inequality for strongly convex function.

Jensen-Steffensen Inequality for Riemann-Stieltjes Integrals
To prove the main results of this section, first we prove the following lemma which will play a key role in the proof of main results.Proof.By definition of strongly convexity we have It means that Δ  () is nonnegative on (, ).
Journal of Function Spaces 5 Also, if  ≤  1 <  2 < , then similarly as above by strongly convexity we have from which it follows that Setting  =  1 and  =  2 in ( 28), then taking the difference we get Hence, Δ  () is increasing on [, ).
In the same manner as above, we also obtain Δ  () is nonnegative on (, ).
The following lemma is given in [35].
In the next result, we prove some general integral inequalities for strongly convex functions.(46) Proof.Under the given conditions in [35], it has been shown that Since  is monotonic and continuous, Ψ() is continuous and (56) Proof.By setting  1 =  2 =  in (46), we obtain (56).
In the following corollary, we obtain integral Slater's inequality for strongly convex functions.Proof.Similar to the proof of Corollary 13, setting  2 =  in the right hand side of (46), we get (57).

Lemma 16 .
Let  : [, ] → R be a nonnegative function and suppose  : [, ] → R is either a bounded variation or continuous.Also assume that the functions  and  have no common discontinuity points.(a) If  is increasing on [, ], then  () inf ≤≤ If  is decreasing on [, ], then  () inf ≤≤