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\(M_{\varphi}M_{\psi}\)-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.

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:

  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 [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:

  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.

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. □

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.

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

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

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Correspondence to Mea Bombardelli.

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Bombardelli, M., Varošanec, S. \(M_{\varphi}M_{\psi}\)-convexity and separation theorems. J Inequal Appl 2022, 65 (2022). https://doi.org/10.1186/s13660-022-02797-x

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MSC

  • 26E60
  • 26A51
  • 26D07
  • 39B72

Keywords

  • \(M_{\varphi}M_{\psi}\)-convex function
  • Separation theorem
  • Hyers–Ulam stability
  • Quasi-arithmetic mean