MφMψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{\varphi}M_{\psi}$\end{document}-convexity and separation theorems

A characterization of pairs of functions that can be separated by an MφMψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{\varphi}M_{\psi}$\end{document}-convex function and related results are obtained. Also, a Hyers–Ulam stability result for MφMψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{\varphi}M_{\psi}$\end{document}-convex functions is given.

A lot of examples of AG-convex or log-convex functions connected with various functionals, which have appeared in the investigation of n-convexity, are given in [5] and [6, pp. 105, 155-160, 177]. Every polynomial with nonnegative coefficients is GG-convex or multiplicatively convex function, every real analytic function f (x) = a n x n with a n ≥ 0 is GG-convex on [0, R where R is the radius of convergence [11,Chap. 2]. Particularly, functions exp, sinh, cosh on 0, ∞ , arcsin on 0, 1] are GG-convex. Examples of special functions which are GG-convex are the following: the gamma function, the Lobacevski function, and the integral sine. In [3], an example of HG-convex function is given. Namely, the function V -1 n (p) = 2 -n (1+n/p) (1+1/p) n which is connected with the volume of the ellipsoid The aim of this paper is to give a separation (sandwich) theorem in this settings. A characterization of pairs of functions that can be separated by a convex function is given in [2], and it is stated as follows. Theorem 1.1 Let f , g : I → R be two functions. The following statements are equivalent: (i) For all x, y ∈ I and t ∈ [0, 1], (ii) There exists a convex function h : As a consequence of the above-mentioned theorem, the Hyers-Ulam stability result for convex functions is obtained also in [2]. Namely, if ε > 0 and f : I → R is a function such that then there exists a convex function h : I → R such that Finally, we mention a sandwich theorem involving affine functions which are considered in [12].

Theorem 1.2 Let I ⊆ R be an interval and f and g be real functions defined on I. The following conditions are equivalent:
(i) There exists an affine function h : (ii) There exist a convex function h 1 : I → R and a concave function h 2 : The following inequalities hold: for all x, y ∈ I and t ∈ [0, 1].
In this paper we show that the above-mentioned theorems have their counterparts in the setting of M ϕ M ψ -convex functions. We prove that two functions f , g can be separated by an for all x, y ∈ I and t ∈ [0, 1]. In the same section we give a result for an M ϕ M ψ -affine function which is a generalization of Theorem 1.2. The last section is devoted to the counterpart of the Hyers-Ulam stability theorem.

Separation theorems
Theorem 2.1 Let ϕ and ψ be two continuous, strictly monotone functions defined on intervals I and J respectively. Let f , g : I → J be real functions.
The following statements are equivalent: (ii) The following inequality holds: for all x, y ∈ I, t ∈ [0, 1].
Proof Assume that ψ is an increasing function. Then ψ -1 is also increasing. First we prove that (i) implies (ii).
Since h ≤ g, and then i.e., Using the fact that f ≤ h, h is M ϕ M ψ -convex and inequality (2) Now assume that (ii) holds. For any u, v ∈ Im ϕ, there exist x, y ∈ I such that u = ϕ(x), v = ϕ(y). From (1) it follows This can be written as where F = ψ • f • ϕ -1 and G = ψ • g • ϕ -1 , F, G : Im ϕ → R. Inequality (3) holds for all u, v ∈ Im ϕ and for all t ∈ [0, 1]. Now we may apply Theorem 1.1 to conclude that there exists a convex function H : Then H • ϕ is well defined. ψ(g(x)), and since ψ is a continuous, strictly increasing function defined on the interval J, the value (H • ϕ)(x) is in the domain of ψ. This allows us to define h = ψ -1 • H • ϕ, h : I → J. As H is convex, it follows that h is M ϕ M ψ -convex, and from (4) it follows that f ≤ h ≤ g, i.e., (i) holds.
If ψ is decreasing, the proof is analogous.

Theorem 2.2 Let ϕ and ψ be two continuous, strictly monotone functions defined on intervals I and J
respectively. Let f , g : I → J be real functions.
The following statements are equivalent: (ii) The following inequalities: hold for all x, y ∈ I and t ∈ [0, 1].
Proof Let h be an M ϕ M ψ -affine function such that f ≤ h ≤ g. This means that It is easy to show that H is an affine function.
Let ψ be an increasing function. Then F ≤ H ≤ G on Im ϕ. (If ψ is decreasing, then G ≤ H ≤ F, and the proof is similar.) Applying Theorem 1.2 ((i) implies (iii)), we obtain for all u, v ∈ Im ϕ and t ∈ [0, 1].
In the same way, g(M ϕ (x, y; t)) ≥ M ψ (f (x), f (y); t). Now assume (ii). From (5) it follows From Theorem 1.2 ((iii) implies (i)) we conclude that there exists an affine function H : Then, as in the proof of the previous theorem, h = ψ -1 • H • ϕ, h : I → R is well defined, and f ≤ h ≤ g. It is easy to verify that h is an M ϕ M ψ -affine function.
Therefore, from (8) it follows f M ϕ (x, y) ≤ A g(x), g(y) , which is a form of condition (ii) from Theorem 2.1. We conclude that there exists an M ϕ A-convex function h 1 : I → R such that f ≤ h 1 ≤ g, i.e., f ≤ h 1 ≤ f + ε.