Ohlin and Levin–Stečkin-Type Results for Strongly Convex Functions

Abstract Counterparts of the Ohlin and Levin–Stečkin theorems for strongly convex functions are proved. An application of these results to obtain some known inequalities related with strongly convex functions in an alternative and unified way is presented.


Introduction
In 1969, J. Ohlin [9] proved the following interesting and very useful result on convex functions in a probabilistic context (as usual, E[X] denotes the expectation of the random variable X): Lemma 1 ( [9]).Let X, Y be two real valued random variables such that E[X] = E[Y ].If the distribution functions F X , F Y crosses one time, i.e. there exists t 0 ∈ R such that for every convex function f : R → R.
For two real valued random variables X, Y with finite expectations, we say that X is dominated by Y in convex stochastic ordering sense, if condition (1) holds for all convex functions f : R → R (see [1]).Thus Ohlin's lemma gives sufficient conditions for X to be dominated by Y in such ordering.It is interesting that earlier, in 1960, V.I.Levin and S.B. Stečkin [4] (see also [6,Theorem 4.2.7 and Lemma 4.2.9])proved a more general result giving a necessary and sufficient condition for convex stochastic ordering.However, their result was clearly unknown for Ohlin.For years the Ohlin lemma also was not well-known in the mathematical community.It has been rediscovered by T. Rajba [12], who found its various applications to the theory of functional inequalities (cf.also [10,13,16,17,18]).In [12], the authoress used the Ohlin lemma to get a very simple proof of known Hermite-Hadamard type inequalities, as well as to obtain new Hermite-Hadamard type inequalities.In the papers [10,13,17,18], furthermore, the Levin-Stečkin theorem [4] is used to examine the Hermite-Hadamard type inequalities.Let us mention also the recent paper by M. Niezgoda [7], in which an extension of the Levin-Stečkin theorem to uniformly convex and superquadratic functions is presented.
In this note we prove counterparts of the Ohlin and Levin-Stečkin theorems for strongly convex functions.We present also applications of these results to obtain some inequalities connected with strongly convex functions.
Let us recall that a function f : for all x, y ∈ I and t ∈ (0, 1).Strongly convex functions have been introduced by Polyak [11] and they play an important role in optimization theory and mathematical economics.Many properties of them can be found, among others, in [2,3,5,8,15].

Main results
Let (Ω, A, P ) be a probability space.Assume that I ⊂ R is an interval and c > 0. Given a random variable X : Ω → R we denote by D 2 [X] the variance of X.The following result is a counterpart of Ohlin's lemma for strongly convex functions.
for every continuous function f : I → R strongly convex with modulus c.
Proof.Let f : I → R be continuous and strongly convex with modulus c.By the characterization of strongly convex functions (see [2,5,15]), the function g : I → R defined by g(x) = f (x) − cx 2 , x ∈ I, is convex.Therefore, by the Ohlin lemma applied for g, we have and hence and finishes the proof.
Remark 3. Note that condition (2) is stronger than (1).Indeed, by the Ohlin lemma applied for the function f Let us recall now the theorem proved by V.I.Levin and S.B. Stečkin [4].
for all continuous convex functions f : [a, b] → R, it is necessary and sufficient that F 1 and F 2 verify the following three conditions: The next result is a version of the above Levin-Stečkin theorem for strongly convex functions (cf.[7] where a similar result for uniformly convex functions is obtained).
for every continuous function f : [a, b] → R strongly convex with modulus c, it is necessary and sufficient that F 1 and F 2 satisfy the following three conditions:  7) is equivalent to conditions (4)-( 6), the proof is finished.
As a consequence of the above theorem, we get the following necessary and sufficient condition for random variables X, Y to satisfy (2).
for every continuous function f : [a, b] → R strongly convex with modulus c, if and only if F X and F Y satisfy the following condition: We have also (10) By the integration by parts formula and ( 9), (10), we obtain Now, using the equalities and and once more the assumption Therefore, by Theorem 4, condition ( 8) is equivalent to because the remaining conditions ( 4), (6) in Theorem 5 are already fulfilled as ( 9) and (11).This finishes the proof.

Applications
In this section we present an application of the Ohlin-type lemma to obtain some known inequalities related with strongly convex functions in an alternative and unified way.The first result is a counterpart of the classical Jensen inequality.
We have also the following converse Jensen inequality for strongly convex functions.
Proof.Take random variables X, Y : Ω → I with the distributions Then the distribution functions F X , F Y satisfy the assumption of Theorem 2, and Therefore, by Theorem 2 we obtain (13).
The next result gives a probabilistic characterization of strong convexity obtained by Rajba and Wąsowicz [14].

Corollary 9 ([14]
).A function f : I → R is strongly convex with modulus c if and only if (14) f for any square integrable random variable Y taking values in I.
Proof.Let f : I → R be strongly convex with modulus c and Y be a random variable with values in I. Take a random variable X : Ω → I with the distributions µ Conversely, assume that f : I → R satisfy (14) for any random variable Y taking values in I. Fix arbitrary x 1 , x 2 ∈ I, t ∈ (0, 1) and take a random variable Y with the distribution µ Thus condition (14) shows that f : I → R is strongly convex with modulus c.
The next corollary is a version of the Hermite-Hadamard inequalities for strongly convex functions.Therefore, by Theorem 2, we obtain (15).

Theorem 5 .
Let a, b ∈ R, a < b and let F 1 , F 2 : [a, b] → R be functions with bounded variation such that F 1 (a) = F 2 (a).Then, in order that

F 2
(t)dt.(6) Proof.By the characterization of strongly convex functions, a function f : [a, b] → R is strongly convex with modulus c if and only if the function g(x) = f (x) − cx 2 , x ∈ [a, b], is convex.Therefore condition (3) holds for all continuous functions f : [a, b] → R strongly convex with modulus c, )dF 2 (t) holds for all continuous convex functions g : [a, b] → R. Since, by the Levin-Stečkin theorem, condition (

Corollary 10 ( 2 for
[5]).If a function f : I → R is strongly convex with modulus c then all a, b ∈ I, a < b.Proof.Let X 1 , X 2 : Ω → I be random variables with the distributionsµ X 1 = δ (a+b)/2 , µ X 2 = 1 2 (δ a +δ b ) and let Y : Ω → I has the uniform distribution on [a, b].Then the pairs X 1 , Y and Y, X 2 satisfy the assumptions of Theorem 2 and for every f : I→ R E[f (X 1 )] = f a + b 2 , E[f (X 2 )] = f (a) + f (b