# $$M_{\varphi}M_{\psi}$$-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 .

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  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 , 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 , and it is stated as follows.

### Theorem 1.1

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

1. (i)

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

$$f\bigl(tx+(1-t)y\bigr) \leq t g(x) + (1-t) g(y).$$
2. (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 . 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 .

### 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:

1. (i)

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

2. (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.

3. (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:

1. (i)

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

$$f\leq h \leq g.$$
2. (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:

1. (i)

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

$$f\leq h \leq g.$$
2. (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  and . A special case of Theorem 2.2, where $$\psi =\varphi$$, is given in . Particular cases of Theorem 2.1 and Theorem 3.1 for HA-convex functions are given in .

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  respectively.

Not applicable.

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Not applicable.

## Funding

The publication charges for this manuscript are supported by the University of Zagreb, Faculty of Science, Department of Mathematics.

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Authors

### Contributions

SV: conceptualization, writing the original draft, computation. MB: computation, analyzing the results, writing and editing. Both authors read and approved the final manuscript.

### Corresponding author

Correspondence to Mea Bombardelli.

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