$M_{\varphi}M_{\psi}$-convexity and separation theorems

Abstract

A characterization of pairs of functions that can be separated by an $M_{\varphi}M_{\psi}$-convex function and related results are obtained. Also, a Hyers–Ulam stability result for $M_{\varphi}M_{\psi}$-convex functions is given.

1 Preliminaries

The concept of classical convexity has been generalized in various ways. Among numerous generalizations, we pay attention to the $M_{\varphi}M_{\psi}$-convexity described in [11].

Let φ and ψ be two continuous, strictly monotone functions defined on intervals I and J respectively. By $M_{\varphi}$ we denote a quasi-arithmetic mean:

$$ M_{\varphi}(x,y;t) := \varphi ^{-1} \bigl( t \varphi (x)+ (1-t) \varphi (y)\bigr), \quad x,y \in I, t\in [0,1]. $$

It is obvious that the power mean $M_{p}$ corresponds to $\varphi (x)=x^{p}$ if $p\neq 0$ and to $\varphi (x)=\log x$ if $p=0$. If it is clear from the text that the weight next to $\varphi (x)$ equals t, then we omit parameter t and simply write $M_{\varphi}(x,y)$.

We say that a function $f: I \rightarrow J$ is $M_{\varphi}M_{\psi}$-convex if

$$ f\bigl(M_{\varphi}(x,y)\bigr) \leq M_{\psi}\bigl(f(x), f(y)\bigr) $$

for all $x,y\in I$ and $t\in [0,1]$. The $M_{\varphi}M_{\psi}$-concavity and $M_{\varphi}M_{\psi}$-affinity are defined in a natural way. If ψ is strictly increasing (strictly decreasing), then f is $M_{\varphi}M_{\psi}$-convex if and only if $\psi \circ f \circ \varphi ^{-1}$ is convex (concave) in the usual sense [11, p. 68].

The most known examples are classes of $M_{\varphi}M_{\psi}$-convex functions where $M_{\varphi}$ and $M_{\psi}$ belong to $\{ A,G,H\}$, where A, G, and H are weighted arithmetic, geometric, and harmonic mean, respectively. Some of them are known under specific names. For example, AG-convex function is usually known as log-convex function, GG-convex function is called multiplicatively convex function, HA-convex function is named harmonically convex function. Of course, AA-convex function is the usual convex function.

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 $f(x)=\sum a_{n}x^{n}$ with $a_{n} \geq 0$ is GG-convex on $[0,R\rangle$ where R is the radius of convergence [11, Chap. 2]. Particularly, functions exp, sinh, cosh on $\langle 0,\infty \rangle$, arcsin on $\langle 0,1]$ 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 $V_{n}^{-1}(p)=2^{-n} \frac{\Gamma (1+{n}/p)}{\Gamma (1+1/p)^{n}}$ which is connected with the volume of the ellipsoid $\{ x\in \mathbb{R}^{n} : \| x \|_{L^{p}} \leq 1 \}$ is HG-convex on $(0,\infty )$. Also, it is AG-convex.

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 $f,g:I\to \mathbb{R}$ be two functions. The following statements are equivalent:

• (i) For all $x,y \in I$ and $t\in [0,1]$,

  $$ f\bigl(tx+(1-t)y\bigr) \leq t g(x) + (1-t) g(y). $$

• (ii) There exists a convex function $h:I\to \mathbb{R}$ such that

  $$ f \leq h\leq g. $$

As a consequence of the above-mentioned theorem, the Hyers–Ulam stability result for convex functions is obtained also in [2]. Namely, if $\varepsilon >0$ and $f:I\to \mathbb{R}$ is a function such that

$$ f\bigl(tx+(1-t)y\bigr) \leq tf(x)+(1-t)f(y)+\varepsilon , \quad x,y\in I, t\in [0,1], $$

then there exists a convex function $h:I\to \mathbb{R}$ such that

$$ \bigl\vert f(x)-h(x) \bigr\vert \leq \frac{\varepsilon}{2}, \quad x\in I. $$

Finally, we mention a sandwich theorem involving affine functions which are considered in [12].

Theorem 1.2

Let $I \subseteq \mathbb{R}$ be an interval and f and g be real functions defined on I. The following conditions are equivalent:

• (i) There exists an affine function $h : I \to \mathbb{R}$ such that $f \leq h \leq g$ on I.

• (ii) There exist a convex function $h_{1} : I \to \mathbb{R}$ and a concave function $h_{2} : I \to \mathbb{R}$ such that $f \leq h_{1} \leq g$ and $f \leq h_{2} \leq g$ on I.

• (iii) The following inequalities hold:

  $$\begin{aligned}& f\bigl(tx+(1-t)y\bigr) \leq tg(x)+(1-t)g(y), \\& g\bigl(tx+(1-t)y\bigr) \geq tf(x)+(1-t)f(y) \end{aligned}$$

  for all $x, y \in I$ and $t\in [0,1]$.

In this paper we show that the above-mentioned theorems have their counterparts in the setting of $M_{\varphi}M_{\psi}$-convex functions. We prove that two functions f, g can be separated by an $M_{\varphi}M_{\psi}$-convex function h if and only if

$$ f\bigl(M_{\varphi}(x,y)\bigr) \leq M_{\psi}\bigl(g(x), g(y)\bigr) $$

for all $x,y\in I$ and $t\in [0,1]$. In the same section we give a result for an $M_{\varphi}M_{\psi}$-affine function which is a generalization of Theorem 1.2. The last section is devoted to the counterpart of the Hyers–Ulam stability theorem.

2 Separation theorems

Theorem 2.1

Let φ and ψ be two continuous, strictly monotone functions defined on intervals I and J respectively. Let $f,g:I \to J$ be real functions.

The following statements are equivalent:

• (i) There exists an $M_{\varphi}M_{\psi}$-convex function $h:I\to J$ such that

  $$ f\leq h \leq g.$$

• (ii) The following inequality holds:

  $$ f\bigl(M_{\varphi}(x,y;t)\bigr) \leq M_{\psi}\bigl(g(x), g(y);t\bigr) $$

  (1)

  for all $x,y\in I$, $t\in [0,1]$.

Proof

Assume that ψ is an increasing function. Then $\psi ^{-1}$ is also increasing.

First we prove that (i) implies (ii).

Since $h\leq g$,

$$ t \psi \bigl(h(x)\bigr) + (1-t) \psi \bigl(h(y)\bigr) \leq t\psi \bigl(g(x) \bigr)+(1-t)\psi \bigl(g(y)\bigr)$$

and then

$$ \psi ^{-1} \bigl(t \psi \bigl(h(x)\bigr) + (1-t) \psi \bigl(h(y)\bigr) \bigr) \leq \psi ^{-1} \bigl(t\psi \bigl(g(x)\bigr)+(1-t)\psi \bigl(g(y) \bigr) \bigr), $$

i.e.,

$$ M_{\psi}\bigl(h(x),h(y)\bigr) \leq M_{\psi}\bigl(g(x),g(y)\bigr). $$

(2)

Using the fact that $f\leq h$, h is $M_{\varphi}M_{\psi}$-convex and inequality (2)

$$\begin{aligned} f\bigl(M_{\varphi}(x,y)\bigr) &\leq h\bigl(M_{\varphi}(x,y)\bigr) \\ & \leq M_{\psi}\bigl(h(x),h(y)\bigr) \leq M_{\psi}\bigl(g(x),g(y)\bigr). \end{aligned}$$

Now assume that (ii) holds. For any $u,v \in \operatorname{Im}\varphi$, there exist $x,y\in I$ such that $u=\varphi (x)$, $v=\varphi (y)$. From (1) it follows

$$ \psi \bigl( f \bigl( \varphi ^{-1}\bigl(t u +(1-t) v\bigr) \bigr) \bigr) \leq t\psi \bigl(g\bigl( \varphi ^{-1}(u)\bigr)\bigr)+(1-t)\psi \bigl(g\bigl(\varphi ^{-1}(v)\bigr)\bigr). $$

This can be written as

$$ F\bigl(tu+(1-t)v\bigr) \leq t G(u) + (1-t) G(v), $$

(3)

where $F=\psi \circ f \circ \varphi ^{-1}$ and $G=\psi \circ g \circ \varphi ^{-1}$, $F , G : \operatorname{Im}\varphi \to \mathbb{R}$. Inequality (3) holds for all $u,v\in \operatorname{Im}\varphi$ and for all $t\in [0,1]$.

Now we may apply Theorem 1.1 to conclude that there exists a convex function $H : \operatorname{Im}\varphi \to \mathbb{R}$ such that

$$ F\leq H\leq G. $$

(4)

Then $H\circ \varphi$ is well defined.

Since $F(u)\leq H(u)\leq G(u)$, i.e., $\psi (f (x)) \leq (H\circ \varphi )(x) \leq \psi ( g (x))$, and since ψ is a continuous, strictly increasing function defined on the interval J, the value $(H\circ \varphi )(x)$ is in the domain of ψ. This allows us to define $h=\psi ^{-1}\circ H\circ \varphi$, $h:I\to J$. As H is convex, it follows that h is $M_{\varphi}M_{\psi}$-convex, and from (4) it follows that $f\leq h\leq g$, 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 $f,g:I \rightarrow J$ be real functions.

The following statements are equivalent:

• (i) There exists an $M_{\varphi}M_{\psi}$-affine function h such that

  $$ f\leq h \leq g. $$

• (ii) The following inequalities:

  $$\begin{aligned}& f\bigl(M_{\varphi}(x,y;t)\bigr) \leq M_{\psi}\bigl(g(x), g(y);t \bigr), \\& g\bigl(M_{\varphi}(x,y;t)\bigr) \geq M_{\psi}\bigl(f(x), f(y);t \bigr) \end{aligned}$$

  (5)

  hold for all $x,y\in I$ and $t\in [0,1]$.

Proof

Let h be an $M_{\varphi}M_{\psi}$-affine function such that $f\leq h \leq g$. This means that

$$ h\bigl(M_{\varphi}(x,y)\bigr)=M_{\psi}\bigl(h(x),h(y)\bigr),\quad \forall x,y \in I.$$

Let $F=\psi \circ f\circ \varphi ^{-1}$, $G=\psi \circ g\circ \varphi ^{-1}$, $H=\psi \circ h\circ \varphi ^{-1}$. It is easy to show that H is an affine function.

Let ψ be an increasing function. Then $F\leq H\leq G$ on Imφ. (If ψ is decreasing, then $G\leq H\leq F$, and the proof is similar.)

Applying Theorem 1.2 ((i) implies (iii)), we obtain

$$\begin{aligned}& F\bigl(tu+(1-t)v\bigr) \leq t G(u)+(1-t) G(v), \end{aligned}$$

(6)

$$\begin{aligned}& G\bigl(tu+(1-t)v\bigr) \geq t F(u)+(1-t) F(v) \end{aligned}$$

(7)

for all $u,v \in \operatorname{Im}\varphi$ and $t\in [0,1]$.

From (6), for all $x,y \in I$ and $t\in [0,1]$, it follows

$$\begin{aligned}& \bigl(\psi \circ f\circ \varphi ^{-1}\bigr) \bigl(t\varphi (x)+(1-t) \varphi (y)\bigr) \leq t (\psi \circ g) (x)+(1-t) (\psi \circ g) (y), \\& f \bigl(\varphi ^{-1}\bigl(t\varphi (x)+(1-t)\varphi (y)\bigr) \bigr) \leq \psi ^{-1} \bigl( t \psi \bigl(g(x)\bigr)+(1-t) \psi \bigl(g(y) \bigr) \bigr) , \end{aligned}$$

i.e., $f(M_{\varphi}(x,y;t))\leq M_{\psi}(g(x),g(y);t)$.

In the same way, $g(M_{\varphi}(x,y;t))\geq M_{\psi}(f(x),f(y);t)$.

Now assume (ii).

From (5) it follows

$$\begin{aligned}& F\bigl(tu+(1-t)v\bigr) \leq t G(u)+(1-t) G(v), \\& G\bigl(tu+(1-t)v\bigr) \geq t F(u)+(1-t) F(v), \quad \forall u,v \in \operatorname{Im}\varphi , \forall t\in [0,1], \end{aligned}$$

where $F=\psi \circ f\circ \varphi ^{-1}$ and $G=\psi \circ g\circ \varphi ^{-1}$, $F,G : \operatorname{Im}\varphi \to \mathbb{R}$.

From Theorem 1.2 ((iii) implies (i)) we conclude that there exists an affine function $H: \operatorname{Im}\varphi \to \mathbb{R}$ such that $F(w)\leq H(w) \leq G(w)$ for all $w\in \operatorname{Im}\varphi$.

Then, as in the proof of the previous theorem, $h=\psi ^{-1}\circ H\circ \varphi$, $h:I \to \mathbb{R}$ is well defined, and $f\leq h\leq g$. It is easy to verify that h is an $M_{\varphi}M_{\psi}$-affine function. □

3 Hyers–Ulam stability

Theorem 3.1

Let φ be a continuous strictly monotone function on an interval I. Let $\varepsilon >0$ be a fixed number. A function $f:I \to \mathbb{R}$ satisfies

$$ f\bigl(M_{\varphi}(x,y)\bigr) \leq tf(x)+(1-t) f(y) + \varepsilon $$

(8)

for all $x,y\in I$, $t\in [0,1]$, if and only if there exists an $M_{\varphi}A$-convex function $h: I \to \mathbb{R}$ such that

$$ \bigl\vert f(x) - h(x) \bigr\vert \leq \frac{1}{2} \varepsilon , \quad \forall x \in I. $$

(9)

Proof

Assume that f satisfies (8). For $g= f+\varepsilon$, we have

$$ A\bigl(f(x),f(y)\bigr)+\varepsilon = A\bigl(g(x),g(y) \bigr).$$

Therefore, from (8) it follows

$$ f\bigl(M_{\varphi}(x,y)\bigr) \leq A\bigl(g(x),g(y)\bigr), $$

which is a form of condition (ii) from Theorem 2.1.

We conclude that there exists an $M_{\varphi}A$-convex function $h_{1}:I\to \mathbb{R}$ such that $f\leq h_{1} \leq g$, i.e., $f\leq h_{1}\leq f+\varepsilon$.

Let $h=h_{1} - \frac{1}{2}\varepsilon$. Then $-\frac{1}{2}\varepsilon \leq f(x)-h(x)\leq \frac{1}{2}\varepsilon$ for all $x\in I$, so (9) holds.

Since

$$ h(M_{\varphi}(x,y)= h_{1}(M_{\varphi}(x,y) - \frac{1}{2}\varepsilon \leq A\bigl(h_{1}(x),h_{1}(y) \bigr)- \frac{1}{2}\varepsilon = A\bigl(h(x),h(y)\bigr),$$

h is also $M_{\varphi}A$-convex, which completes the proof.

Now let $h: I \to \mathbb{R}$ be an $M_{\varphi}A$-convex function such that (9) holds. This condition can be written in the form

$$ f(x) - \frac{1}{2} \varepsilon \leq h(x) \leq f(x)+\frac{1}{2} \varepsilon . $$

Using Theorem 2.1 we can conclude that functions $f_{1}=f-\frac{1}{2} \varepsilon$ and $f_{2}=f+\frac{1}{2} \varepsilon$ satisfy

$$ f_{1}\bigl(M_{\varphi}(x,y)\bigr) \leq A\bigl(f_{2}(x), f_{2}(y)\bigr). $$

This is equivalent to

$$ f\bigl(M_{\varphi}(x,y)\bigr) -\frac{1}{2}\varepsilon \leq A \bigl(f(x),f(y)\bigr) +\frac{1}{2} \varepsilon , $$

which proves (8). □

As we mentioned in the first section, the corresponding results for convex functions, i.e., for AA-convex functions, are given in [2] and [12]. A special case of Theorem 2.2, where $\psi =\varphi$, is given in [8]. Particular cases of Theorem 2.1 and Theorem 3.1 for HA-convex functions are given in [4].

Results about the separation problem for some other classes of functions which are not particular cases of the class of $M_{\varphi}M_{\psi}$-convex functions, i.e., for strongly convex functions, m-convex and h-convex functions, set-valued functions, and convex functions with control function, are given in [7, 9, 10, 13], and [1] respectively.