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\(M_{\varphi}M_{\psi}\)convexity and separation theorems
Journal of Inequalities and Applications volume 2022, Article number: 65 (2022)
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 quasiarithmetic mean:
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
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, AGconvex function is usually known as logconvex function, GGconvex function is called multiplicatively convex function, HAconvex function is named harmonically convex function. Of course, AAconvex function is the usual convex function.
A lot of examples of AGconvex or logconvex functions connected with various functionals, which have appeared in the investigation of nconvexity, are given in [5] and [6, pp. 105, 155160, 177]. Every polynomial with nonnegative coefficients is GGconvex or multiplicatively convex function, every real analytic function \(f(x)=\sum a_{n}x^{n}\) with \(a_{n} \geq 0\) is GGconvex 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 GGconvex. Examples of special functions which are GGconvex are the following: the gamma function, the Lobacevski function, and the integral sine. In [3], an example of HGconvex 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 HGconvex on \((0,\infty )\). Also, it is AGconvex.
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 \(t\in [0,1]\),
$$ f\bigl(tx+(1t)y\bigr) \leq t g(x) + (1t) g(y). $$ 
(ii)
There exists a convex function \(h:I\to \mathbb{R}\) such that
$$ f \leq h\leq g. $$
As a consequence of the abovementioned 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
then there exists a convex function \(h:I\to \mathbb{R}\) such that
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:
$$\begin{aligned}& f\bigl(tx+(1t)y\bigr) \leq tg(x)+(1t)g(y), \\& g\bigl(tx+(1t)y\bigr) \geq tf(x)+(1t)f(y) \end{aligned}$$for all \(x, y \in I\) and \(t\in [0,1]\).
In this paper we show that the abovementioned 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
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:

(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 \(h\leq g\),
and then
i.e.,
Using the fact that \(f\leq h\), h is \(M_{\varphi}M_{\psi}\)convex and inequality (2)
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
This can be written as
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
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
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
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
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
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
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
Proof
Assume that f satisfies (8). For \(g= f+\varepsilon \), we have
Therefore, from (8) it follows
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 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
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
This is equivalent to
which proves (8). □
As we mentioned in the first section, the corresponding results for convex functions, i.e., for AAconvex 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 HAconvex 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, mconvex and hconvex functions, setvalued 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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Bombardelli, M., Varošanec, S. \(M_{\varphi}M_{\psi}\)convexity and separation theorems. J Inequal Appl 2022, 65 (2022). https://doi.org/10.1186/s1366002202797x
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DOI: https://doi.org/10.1186/s1366002202797x
MSC
 26E60
 26A51
 26D07
 39B72
Keywords
 \(M_{\varphi}M_{\psi}\)convex function
 Separation theorem
 Hyers–Ulam stability
 Quasiarithmetic mean