On a generalization of sandwich type theorems

We introduce affine and convex functions with a control function and present some sandwich type theorems for them. Also, Hyers–Ulam stability type results for affine and convex functions with a control function are given.


Introduction
Let I ⊂ R be an open interval. In paper [3] the authors proved that two functions f, g : I → R can be separated by a convex function if and only if f (tx + (1 − t)y) ≤ tg(x) + (1 − t)g(y), x,y ∈ I, t ∈ [0, 1].
A counterpart of this result for strongly convex functions, i.e. functions satisfying the inequality where c is a fixed positive number, is presented in [7] and it appears that the necessary and sufficient condition for the separation of two functions f, g : I → R by strongly convex function is the following In [1] the author introduced a concept of strong convexity in a more general case, i.e. functions satisfying the inequality where F is a fixed positive function are considered, and called them F -strongly convex. It is a natural question in the context of the aforementioned separation results and F -strong convexity, whether the inequality guaranties the separation of the functions f and g by an F -strongly convex function. In general, the answer to this question is that it does not guarantee such a separation. It can be verified that for constant functions f ≡ 0, g ≡ 1 and F ≡ 1 the above inequality holds true, but we cannot separate them by a 1-strongly convex function, because a 1-strongly convex function does not exist. The aim of this paper is to present a condition under which a separation result holds true and to show it in a more general case than F -strong convexity.

Main result
We start with the following two definitions.
for all t ∈ [0, 1] and x, y ∈ I. Definition 2. Let G : [0, 1] × I 2 → R be a given function. A function f : I → R we will call an affine function with a control function G if for all t ∈ [0, 1] and x, y ∈ I.
Of course, if we take a function G(t, x, y) = −ct(1 − t) |x − y| 2 we will obtain a well known strong convexity case, and if we take a function G(t, x, y) = −t(1−t)F (x−y) then we will get F -strong convexity. It appears, that between convex functions and strongly convex functions we have some connections (see [2,6,9,[12][13][14]). In particular, under some assumptions, a function f is strongly convex if and only if the function f − |·| 2 is convex and also a function f is F -strongly convex if and only if the function f − F is convex (see [1]). Now we present an obvious counterpart of these results. At this moment we will focus our attention on affine functions with a control function G and we will try to describe a structure of the family F := {φ : I → R | φ is affine with a control function G} .
Taking into consideration the definitions of affine functions with a control function G and affine functions, we have the next lemma.

Lemma 2. φ ∈ F if and only if φ + a ∈ F and a is an affine function on I.
And finally Theorem 1 gives a full description of the family F. Proof. If φ = φ 0 + a then from Lemma 2 also φ ∈ F. Assume now that φ ∈ F. Let's fix different points x 1 , x 2 ∈ I and adjust an affine function a such that the function μ = φ 0 + a satisfies the conditions μ(x 1 ) = φ(x 1 ) and μ(x 2 ) = φ(x 2 ).
In view of Lemma 1 and Lemma 2 we get that φ = φ 0 + a. The proof ends.
From the above theorem we immediately have the following corollary. Proof. The "'only if"' part is evident. To prove the "'if"' assume that functions f , g satisfy the inequality G(t, x, y) and φ is a member of the family F, i.e. G(t, x, y), for all t ∈ [0, 1] and x, y ∈ I. Subtracting from the inequality this equation side by side we get for all t ∈ [0, 1] and x, y ∈ I. It means that the functions f −φ and g −φ satisfy the sufficient conditions of the Baron-Matkowski-Nikodem theorem [3]. Thus, there exists a convex function h * : I → R such that It means that the function h := h * + φ is between f and g and from the aforementioned observation it is convex with a control function G. The proof is complete.
Taking a function g := f + , where is a fixed positive number, and substituting a function h by a function h + 2 in the sandwich theorem above we get the following Hyers-Ulam type stability result for convex functions with a control function G (the classical Hyers-Ulam theorem we can find in [5]). Remark. Taking a control function G(t, x, y) = −ct(1 − t) 2 it is easy to check that the function φ 0 (x) = cx 2 belongs to the family F. Thus we get the results presented in [7].