A note on $h$-convex functions

TL;DR

This paper explores the properties of $h$-convex functions, including their continuity, geometric interpretation, and conditions under which $h$-midconvexity implies $h$-convexity, along with derivative characterizations.

Contribution

It introduces the concept of $h$-convex curves and provides new insights into the continuity and geometric aspects of $h$-convex functions.

Findings

$h$-convex functions are $h$-continuous

$h$-midconvexity is equivalent to $h$-convexity for $h$-continuous functions

Derivative characterizations of $h$-convexity are discussed

Abstract

In this work, we discuss the continuity of $h$-convex functions by introducing the concepts of $h$-convex curves ($h$-cord). Geometric interpretation of $h$-convexity is given. The fact that for a $h$-continuous function $f$, is being $h$-convex if and only if is $h$-midconvex is proved. Generally, we prove that if $f$ is $h$-convex then $f$ is $h$-continuous. A discussion regarding derivative characterization of $h$-convexity is also proposed.

Equations

$$f\left(t\alpha +\left(1-t\right)\beta \right)\le tf\left(\alpha \right)+\left(1-t\right)f\left(\beta \right),$$

$$f\left(tx+\left(1-t\right)y\right)\le t^{s}f\left(x\right)+{\left(1-t\right)}^{s}f\left(y\right).$$

$$f\left(\alpha x+\beta y\right)\le {\alpha }^{s}f\left(x\right)+{\beta }^{s}f\left(y\right)$$

$$f\left(\lambda x+{\left(1-{\lambda }^s\right)}^{1/s}y\right)\le {\lambda }^{s}f\left(x\right)+\left(1-{\lambda }^s\right)f\left(y\right)$$

$$\underset{J}{\sup }\left(L-f\right)\ge \underset{\partial J}{\sup }\left(L-f\right)$$

$$\underset{J}{\sup }\left(L-f\right)\ge \underset{\partial J}{\sup }\left(L-f\right).$$

$$L\left(\lambda \right)$$

$$f\left(tx+\left(1-t\right)y\right)\le \frac{f\left(x\right)}{t}+\frac{f\left(y\right)}{1-t}.$$

$$f\left(tx+\left(1-t\right)y\right)\le f\left(x\right)+f\left(y\right).$$

$$f\left(t\alpha +\left(1-t\right)\beta \right)\le h\left(t\right)f\left(\alpha \right)+h\left(1-t\right)f\left(\beta \right),$$

$$L\left(t;h\right):=\left[f\left(y\right)-f\left(x\right)\right]h\left(\frac{t-x}{y-x}\right)+f\left(x\right),$$

$$L\left(t;t\right):=\frac{f\left(y\right)-f\left(x\right)}{y-x}\left(t-x\right)+f\left(x\right).$$

$$f\left(t\right)\le \left[f\left(y\right)-f\left(x\right)\right]h\left(\frac{t-x}{y-x}\right)+f\left(x\right)=L\left(t;h\right),$$

$$f\left(\frac{x+y}{2}\right)\le (\ge )h\left(\frac{1}{2}\right)\left[f\left(x\right)+f\left(y\right)\right]$$

$$h\left(\frac{1}{2}\right)\left[f\left(x+p\right)+f\left(x-p\right)\right]-f\left(x\right)\ge 0,$$

$$f\left(\frac{x+y}{2}\right)\le h\left(\frac{1}{2}\right)\left[f\left(x\right)+f\left(y\right)\right],$$

$$f\left(t\right)\ge \left[f\left(y\right)-f\left(x\right)\right]h\left(\frac{t-x}{y-x}\right)+f\left(x\right)=L(t;h),$$

$$g\left(t\right)=f\left(t\right)-\left[f\left(y\right)-f\left(x\right)\right]h\left(\frac{t-x}{y-x}\right)-f\left(x\right),t\in I$$

$$g\left(c-p\right)+g\left(c+p\right)<2g\left(c\right)=\frac{1}{\frac{1}{2}}g\left(c\right)\le \frac{1}{h\left(\frac{1}{2}\right)}g\left(c\right)$$

$${\mathcal{C}}_H^{(\alpha )}\left(h\right)=\left\{h|h\left(\delta \right)=H{\left|\delta \right|}^{\alpha },H>0\right\}$$

$$m_{ϵ}:=\inf \left\{f\left(x\right),x\in \left(a-ϵ,b+ϵ\right)\right\}\text{and}{M}_{ϵ}:=\sup \left\{f\left(x\right),x\in \left(a-ϵ,b+ϵ\right)\right\},$$

$$f\left(y\right)=f\left({\lambda }_{ϵ}z+\left(1-{\lambda }_{ϵ}\right)x\right)$$

$$\le {\lambda }_{ϵ}\left[f\left(z\right)-f\left(x\right)\right]+f\left(x\right)\le h\left({\lambda }_{ϵ}\right)\left[f\left(z\right)-f\left(x\right)\right]+f\left(x\right),$$

$$f\left(y\right)-f\left(x\right)\le h\left({\lambda }_{ϵ}\right)\left[f\left(z\right)-f\left(x\right)\right]$$

$$<h\left(\frac{\left|y-x\right|}{ϵ}\right)\left(M_{ϵ}-m_{ϵ}\right)$$

$$<\frac{h\left(\left|y-x\right|\right)}{h\left(ϵ\right)}\left(M_{ϵ}-m_{ϵ}\right)$$

$$=h\left(\left|y-x\right|\right).$$

$$f\left(y\right)=f\left({\lambda }_{ϵ}z+\left(1-{\lambda }_{ϵ}\right)x\right)$$

$$\le h\left({\lambda }_{ϵ}\right)f\left(z\right)+\left[1-h\left({\lambda }_{ϵ}\right)\right]f\left(x\right)(\mathrm{since}h\left(1-{\lambda }_{ϵ}\right)\le 1-h\left({\lambda }_{ϵ}\right))$$

$$=h\left({\lambda }_{ϵ}\right)\left[f\left(z\right)-f\left(x\right)\right]+f\left(x\right),$$

$$f\left(y\right)-f\left(x\right)\le h\left({\lambda }_{ϵ}\right)\left[f\left(z\right)-f\left(x\right)\right]$$

Full text

1. Introduction

Let $I$ be a real interval. A function $f:I\to\mathbb{R}$ is called convex iff

[TABLE]

for all points $\alpha,\beta\in I$ and all $t\in[0,1]$. If $-f$ is convex then we say that $f$ is concave. Moreover, if $f$ is both convex and concave, then $f$ is said to be affine.

In 1979, Breckner [3] introduced the class of $s$-convex functions (in the second sense), as follows:

Definition 1**.**

Let $I\subseteq\left[0,\infty\right)$ and $s\in(0,1]$, a function $f:I\to\left[0,\infty\right)$ is $s$-convex function or that $f$ belongs to the class $K^{2}_{s}\left(I\right)$ if for all $x,y\in I$ and $t?\in[0,1]$ we have

[TABLE]

In the last years, among others, the notion of $s$-convex functions is discriminated and starred. In literature a few papers devoted to study this type of convexity. The building theories of $s$-convexity as geometric and analytic tools are still under consideration, development and examine. Due to Hudzik and Maligranda (1994), two senses of $s$-convexity $(0<s\leq 1)$ of real-valued functions are known in the literature, and given below:

Definition 2**.**

A function $f:\mathbb{R}_{+}\to\mathbb{R}$, where $\mathbb{R}_{+}=\left[{0,\infty}\right)$, is said to be $s$-convex in the first sense if

[TABLE]

for all $x,y\in\left[{0,\infty}\right)$, $\alpha,\beta\geq 0$ with $\alpha^{s}+\beta^{s}=1$ and for some fixed $s\in\left({0,1}\right]$. This class of functions is denoted by $K_{s}^{1}$.

This definition of $s$-convexity, for so called $\varphi$-functions, was introduced by Orlicz in 1961 and was used in the theory of Orlicz spaces. A function $f:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is said to be a $\varphi$-function if $f(0)=0$ and $f$ is nondecreasing and continuous. The symbol $\varphi$ stands for an Orlicz function, i.e., $\varphi$ is defined on the real line $\mathbb{R}$ with values in $[0,+\infty]$ and is convex, even, vanishing and continuous at zero. For further details see [15], [17] ,[18],[32].

In fact, Breckner [3] walked in the footsteps of Orlicz’s definition (Definition 2) and introduced another type of $s$-convexity or what so called Breckner $s$-convex, as follows:

Definition 3**.**

A function $f:\mathbb{R}^{+}\to\mathbb{R}$, where $\mathbb{R}^{+}=\left[{0,\infty}\right)$, is said to be $s$–convex in the second sense if

[TABLE]

for all $x,y\in\left[{0,\infty}\right)$, $\alpha,\beta\geq 0$ with $\alpha+\beta=1$ and for some fixed $s\in\left({0,1}\right]$. This class of functions is denoted by $K_{s}^{2}$.

Remark 1**.**

We note that, it can be easily seen that for $s=1$, $s$-convexity (in both senses) reduces to the ordinary convexity of functions defined on $\left[{0,\infty}\right)$.

In general, a real-valued function $f$ defined on an open convex subset $C$ of a linear space is called Breckner $s$-convex if \reftagform@1.3 holds for every $x,y\in C$, $\alpha,\beta\in[0,1]$ with $\alpha+\beta=1$, where $s\in(0,1)$ is fixed. More preciously, Breckner considered an open convex subset $\mathbb{M}$ of a linear space $\mathbb{L}$ and defined $f:\mathbb{M}\subseteq\mathbb{L}\to\mathbb{R}$, to be $s$-convex if \reftagform@1.3 holds, for all $x,y\in\mathbb{M}$, $\alpha,\beta\in[0,1]$ with $\alpha+\beta=1$, where $s\in(0,1)$ is fixed. Also, Breckner considered a special case of $s$-convex functions which is so called rational $s$-convex, that is for all rational $\alpha,\beta\in[0,1]$ with $\alpha+\beta=1$ and points $x,y\in\mathbb{M}$, the inequality \reftagform@1.3 holds. Furthermore, Breckner proved that for locally bounded above $s$-convex functions defined on open subsets of linear topological spaces are continuous and nonnegative.

In 1978, Breckner and Orbán [4] considered functions from a convex subset of a real or complex Hausdorff topological linear space of dimension greater than 1 into an ordered topological linear space such that all its order-bounded subsets are bounded, and showed that Breckner $s$-convex functions with $s\in(0,1]$ are continuous on the interior of their domain.

In 1994, Breckner [5] (see also [6]) proved that for a rationally $s$-convex function continuity and local $s$-Hölder continuity are equivalent at each interior point of the domain of definition of the function. Furthermore, it is shown that a rationally $s$-convex function which is bounded on a nonempty open convex set is $s$-Hölder continuous on every compact subset of this set. Indeed, Breckner [2], showed that if a real-valued function defined on a convex subset of a linear space endowed with topology generated by a direct pseudonorm is continuous and rationally Breckner $s$-convex for an $s\in(0,1]$, then it is locally $s$-Hölder.

In 1994, Hudzik and Maligranda [15], realized the importance and undertook a systematic study of $s$-convex functions in both sense. They compared the notion of Breckner $s$-convexity with a similar one of [18]. A function $f$ is Orlicz $s$-convex if the inequality (1.2) is satisfied for all $\alpha,\beta$ such that $\alpha^{s}+\beta^{s}=1$. Hudzik and Maligranda, among others, gave an example of a non-continuous Orlicz $s$-convex function, which is not Breckner $s$-convex.

In 2001, Pycia [24] established a direct proof of Breckner’s result that Breckner $s$-convex real-valued functions on finite dimensional normed spaces are locally $s$-Hölder. The same result was proved in [1] where different context was considered. For the same result regarding convexity see [7] and [8].

In the 2008, Pinheiro [25] studied the class of $K_{s}^{1}$ of $s$-convex functions and explained why the first $s$-convexity sense was abandoned by the literature in the field. In fact, Pinheiro , proposed some criticisms to the current way of presenting the definition of $s$-convex functions. We may summarize Pinheiro criticisms in the following points:

1. (1)

What is the ‘true’ difference between convex and $s$-convex in both senses. 2. (2)

So far, Pinheiro did not find references, in the literature, to the geometry of an $s$-convex function, what, once more, makes it less clear to understand the difference between an $s$-convex and a convex function whilst there are clear references to the geometry of the convex functions.

In the same paper [25], Pinheiro revised the class of $s$–convexity in the first sense. In [26], Pinheiro proposed a geometric interpretation for this type of functions.

Definition 4**.**

Let $U$ be any subset of $[0,\infty)$. A function $f:X\to\mathbb{R}$, is said to be $s$–convex in the first sense if

[TABLE]

for all $x,y\in U$ and $\lambda\in[0,1]$.

The presented reason from Pinheiro to why $s$-convexity in the first sense got abandoned in the literature, is that, if one takes $x=y=\frac{1}{4}$ with $\alpha=\frac{1}{2}$ and $\beta=1$ for example, one gets that $\alpha x+\beta y=0.125+0.25=0.375$. So that, if $s=\frac{1}{2}$, then the value of $\alpha x+\beta y$ would lie outside of the interval $[x,y]$, on the contrary of this, the value of $\alpha x+\beta y$ would lie inside of the interval $[x,y]$ in case of convexity. With this the first sense of $s$-convexity becomes a close to the meaning of convexity and so the geometric explanation of $s$-convex function is easy to be compared with the geometry of convex function if some further restrictions are imposed to it.

The proposed geometric description for $s$-convex curve in the first sense stated by Pinheiro [25]–[30] as follows:

Definition 5**.**

A function $f:X\subset\mathbb{R}_{+}\to\mathbb{R}$ is called $s$-convex in the first sense if and only if one in two situations occur:

• •

$0<s_{1}<1$, $f$ then belonging to $K^{1}_{s}$ , for $0<s\leq s_{1}$: The graph of f lies below (L), which is a convex curve between any two domain points with minimum distance of $(2^{-1}-2^{-1/s})$ (domain points distance), that is, for every compact interval $J\subset I$, where length of J is greater than, or equal to $(2^{-1}-2^{-1/s})$ interval with boundary $\partial J$, it is true that

[TABLE]

and $L$ is such that it is continuous, smooth, and, for each point $x$ of $L$, defined in terms of ninety degrees intercepts with the straight line between the two points of the function, it is true that $1\leq x\leq 2^{-1}+2^{-s}$, where 1 corresponds to the straight line height;

• •

$f$* is convex.*

In general, the class of $s$-convex functions in the second sense would incomplete concept without a geometric interpretations for it is behavior. Recently, Pinheiro devoted her efforts to give a clear geometric definition for $s$-convexity in second sense. In [27] Pinheiro successfully proposed a geometric description for $s$-convex curve, as follows:

Definition 6**.**

$f$* is called $s$-convex in the second sense if and only if one in two situations occur:*

• •

$0<s_{1}<1$, $f$ then belonging to $K^{2}_{s}$ , for $0<s\leq s_{1}$: The graph of f lies below (L), which is a convex curve between any two domain points with minimum distance of $(2^{-s}-2^{-1})$ (domain points distance), that is, for every compact interval $J\subset I$, where length of J is greater than, or equal to $(2^{-s}-2^{-1})$ interval with boundary $\partial J$, it is true that

[TABLE]

and $L$ is such that it is continuous, smooth, and, for each point $x$ of $L$, defined in terms of ninety degrees intercepts with the straight line between the two points of the function, it is true that $1\leq x\leq 2^{1-s}$, where 1 corresponds to the straight line height;

• •

$f$* is convex.*

More geometrically, an interpretation of $s$-convex functions is introduced as follows:

Definition 7**.**

$f$* is called $s$–convex, $0<s<1$, $f\geq 0$, if the graph of $f$ lies below a ‘bent chord’ $L$ between any two points. That is, for every compact interval $J\subset I$, with boundary $\partial J$, it is true that*

[TABLE]

Indeed the geometric view for $s$-convex mapping of second sense is going through which Pinheiro called it ‘limiting curve’, which is going to distinguish curves that are $s$-convex of second sense from those that are not. After that, Pinheiro obtained how the choice of ‘$s$’ affects the limiting curve. In general a ‘limiting curve’ may be described by a bent chord joining $f(x)$ to $f(y)$-corresponding to the verification of the $s$-convexity property of the function $f$ in the interval $[x,y]$-forms representing the limiting height for the curve $f$ to be at, limit included, in case $f$ is $s$-convex. This curve is represented by $\lambda^{s}f\left(x\right)+\left({1-\lambda}\right)^{s}f\left(y\right)$, for each $0<s<1$.

Some properties of the limiting curve such as: maximum height, length, and local inclination are considered in [26]–[29].

• •

Height. The maximum of the limiting $s$-curve is $2^{1-s}$.

• •

Length. Let $f\left(\lambda\right)=\lambda^{s}X+\left({1-\lambda}\right)^{s}Y$, with $X=f\left(x\right)$, and $Y=f\left(y\right)$. The size of the limiting curve from $f\left(x\right)$ to $f\left(y\right)$ is

[TABLE]

which shows that how bent is the limiting curve.

• •

Local inclination. The local inclination of the limiting curve may be founded by means of the first derivative, consider $f\left(\lambda\right)=\lambda^{s}f\left(x\right)+\left({1-\lambda}\right)^{s}f\left(y\right)$, Therefore, the inclination is $f^{\prime}\left(\lambda\right)=s\lambda^{s-1}f\left(x\right)-s\left({1-\lambda}\right)^{s-1}f\left(y\right)$ and varies accordingly to the value of $\lambda$.

In 1985, E. K. Godnova and V. I. Levin (see [13] or [20], pp. 410-433) introduced the following class of functions:

Definition 8**.**

We say that $f:I\to\mathbb{R}$ is a Godunova-Levin function or that $f$ belongs to the class $Q\left(I\right)$ if for all $x,y\in I$ and $t?\in(0,1)$ we have

[TABLE]

In the same work, the authors proved that all nonnegative monotonic and nonnegative convex functions belong to this class. For related works see [12] and [19].

In 1999, Pearce and Rubinov [23], established a new type of convex functions which is called $P$-functions.

Definition 9**.**

We say that $f:I\to\mathbb{R}$ is $P$-function or that $f$ belongs to the class $P\left(I\right)$ if for all $x,y\in I$ and $t?\in[0,1]$ we have

[TABLE]

Indeed, $Q(I)\supseteq P(I)$ and for applications it is important to note that $P(I)$ also consists only of nonnegative monotonic, convex and quasi-convex functions. A related work was considered in [12] and [34].

In 2007, Varošanec [35] introduced the class of $h$-convex functions which generalize convex, $s$-convex, Godunova-Levin functions and $P$-functions. Namely, the $h$-convex function is defined as a non-negative function $f:I\to\mathbb{R}$ which satisfies

[TABLE]

where $h$ is a non-negative function, $t\in(0,1)\subseteq J$ and $x,y\in I$, where $I$ and $J$ are real intervals such that $(0,1)\subseteq J$. Accordingly, some properties of $h$-convex functions were discussed in the same work of Varošanec. For more results; generalization, counterparts and inequalities regarding $h$-convexity see [2], [9]–[11],[14],[16], and [22].

2. On $h$–convex functions

Throughout this work, $I$ and $J$ are two intervals subset of $\left(0,\infty\right)$ such that $\left(0,1\right)\subseteq J$ and $\left[a,b\right]\subseteq I$ with $0<a<b$.

Definition 10**.**

The $h$-cord joining any two points $\left(x,f\left(x\right)\right)$ and $\left(y,f\left(y\right)\right)$ on the graph of $f$ is defined to be

[TABLE]

for all $t\in[x,y]\subseteq\mathcal{I}$. In particular, if $h(t)=t$ then we obtain the well known form of chord, which is

[TABLE]

It’s worth to mention that, if $h\left(0\right)=0$ and $h\left(1\right)=1$, then $L\left(x;h\right)=f\left(x\right)$ and $L\left(y;h\right)=f\left(y\right)$, so that the $h$-cord $L$ agrees with $f$ at endpoints $x,y$, and this true for all such $x,y\in I$.

The $h$-convexity of a function $f:I\to\mathbb{R}$ means geometrically that the points of the graph of $f$ are on or below the $h$-chord joining the endpoints $\left(x,f\left(x\right)\right)$ and $\left(y,f\left(y\right)\right)$ for all $x,y?\in I$, $x<y$. In symbols, we write

[TABLE]

for any $x\leq t\leq y$ and $x,y\in I$.

Hence, \reftagform@1.5 means geometrically that for a given three non-collinear points $P,Q$ and $R$ on the graph of $f$ with $Q$ between $P$ and $R$ (say $P<Q<R$). Let $h$ is super(sub)multiplicative and $h\left(\alpha\right)\geq(\leq)\,\alpha$, for $\alpha\in\left(0,1\right)\subset J$. A function $f$ is $h$–convex (concave) if $Q$ is on or below (above) the $h$-chord $\widehat{PR}$ (see Figure 1).

Caution: In special case, for $h_{k}\left(t\right)=t^{k}$, $t\in\left(0,1\right)$ the proposed geometric interpretation is valid for $k\in(-1,0)\cup(0,\infty)$. In the case that $k\leq-1$ or $k=0$ the geometric meaning is inconclusive so we exclude this case (and (and similar cases) from our proposal above.

Definition 11**.**

Let $h:J\to\left(0,\infty\right)$ be a non-negative function. Let $f:I\to\mathbb{R}$ be any function. We say $f$ is $h$-midconvex ($h$-midconcave) if

[TABLE]

for all $x,y\in I$.

In particular, $f$ is locally $h$-midocnvex if and only if

[TABLE]

for all $x\in\left(x-p,x+p\right)$, $p>0$.

A generalization of Jensen characterization of convex functions could be stated as follows:

Theorem 1**.**

Let $h:J\to\left(0,\infty\right)$ be a non-negative function such that $h\left(\alpha\right)\geq\alpha$, for all $\alpha\in(0,1)$. Let $f:I\to\mathbb{R}_{+}$ be a nonnegative continuous function. $f$ is $h$-convex if and only if it is $h$-midconvex; i.e., the inequality

[TABLE]

holds for all $x,y\in I$.

Proof.

The first direction follows directly by definition of $h$-convexity. To prove the second direction, suppose on the contrary that $f$ is not $h$-convex. Then, there exists a subinterval $[x,y]$ such that the graph of $f$ is not under the chord joining $(x,f(x))$ and $(y,f(y))$; that is,

[TABLE]

for all such $x,y\in I\cap J$. In other words, the function

[TABLE]

satisfies $M=\sup\left\{{g\left(t\right):t\in[x,y]}\right\}>0$. Since $h\left(0\right)=0$ and $h\left(1\right)=1$, then $L\left(x;h\right)=f\left(x\right)$ and $L\left(y;h\right)=f\left(y\right)$, so that the $h$-cord $L$ agrees with $f$ at endpoints $x,y$. Thus, $g$ is continuous and $g(x)=g(y)=0$, direct computation shows that $g$ is also mid $h$-convex. Setting $c=\inf\left\{{t\in[x,y]:g\left(t\right)=M}\right\}$, then necessarily $g(c)=M$ and $c\in(x,y)$. By the definition of $c$, for every $p>0$ for which $c\pm p\in(a,b)$, we have $g\left({c-p}\right)<g\left(c\right)$ and $g\left({c+p}\right)<g\left(c\right)$, so that since $h\left(\alpha\right)\geq\alpha$, for all $\alpha\in(0,1)$ we have

[TABLE]

which contradicts the fact that $g$ is mid $h$-convex. ∎

Corollary 1**.**

Let $h:J\to\left(0,\infty\right)$ be a non-negative function such that $h\left(\alpha\right)\leq\alpha$, for all $\alpha\in(0,1)$. Let $f:I\to\mathbb{R}_{+}$ be a nonnegative continuous function. $f$ is $h$-concave if and only if it is $h$-midconcave.

Sometimes we often need to know how fast limits are converging, and this allows us to control the remainder of a given function in a neighborhood of some point $x_{0}$. So that, we need to extend the concept of continuity. Fortunately, in control theory and numerical analysis, a function $h:J\subseteq[0,\infty)\to[0,\infty]$ is called a control function if

1. (1)

$h$ is nondecreasing, 2. (2)

$\inf_{\delta>0}h\left(\delta\right)=0$.

A function $f:I\to\mathbb{R}$ is $h$-continuous at $x_{0}$ if $\left|f\left(x\right)-f\left(x_{0}\right)\right|\leq h\left(|x-x_{0}|\right)$, for all $x\in I$. Furthermore, a function is continuous in $x_{0}$ if it is $h$-continuous for some control function $h$.

This approach leads us to refining the notion of continuity by restricting the set of admissible control functions.

For a given set of control functions $\mathcal{C}$ a function is $\mathcal{C}$-continuous if it is $h$-continuous for all $h\in\mathcal{C}$. For example the Hölder continuous functions of order $\alpha\in(0,1]$ are defined by the set of control functions

[TABLE]

In case $\alpha=1$, the set $\mathcal{C}^{(1)}_{H}\left(h\right)$ contains all functions satisfying the Lipschitz condition.

Theorem 2**.**

Let $(0,1)\subseteq J$, $h:J\to\left(0,\infty\right)$ be a control function which is supermultiplicative such that $h(\alpha)\geq\alpha$ for each $\alpha\in(0,1)$. Let $I$ be a real interval, $a,b\in\mathbb{R}$ $(a<b)$ with $a,b$ in $I^{\circ}$ (the interior of $I$). If $f:I\to\mathbb{R}$ is non-negative $h$-convex function on $[a,b]$, then $f$ is $h$-continuous on $[a,b]$.

Proof.

Choose $\epsilon>0$ be small enough such that $\left(a-\epsilon,b+\epsilon\right)\subseteq I$ and let

[TABLE]

such that $h\left(\epsilon\right)=M_{\epsilon}-m_{\epsilon}$. If $x,y\in\left[a,b\right]$, such that $x=y+\frac{\epsilon}{\left|y-x\right|}\left(y-x\right)$ and $\lambda_{\epsilon}=\frac{\left|y-x\right|}{\epsilon+\left|y-x\right|}$. Then for $z\in\left[a-\epsilon,b+\epsilon\right]$, $y=\lambda_{\epsilon}z+\left(1-\lambda_{\epsilon}\right)x$, we have

[TABLE]

which implies that $y=\lambda_{\epsilon}z+\left(1-\lambda_{\epsilon}\right)x$, we have

[TABLE]

Since this is true for any $x,y\in\left[a,b\right]$, we conclude that $\left|f\left(y\right)-f\left(x\right)\right|\leq h\left({\left|{y-x}\right|}\right)$, which shows that $f$ is $h$-continuous on $\left[a,b\right]$ as desired. ∎

Another Proof. Alternatively, if one replaces the condition $h(\alpha)+h(1-\alpha)\leq 1$ for each $\alpha\in(0,1)$ instead of $h(\alpha)\geq\alpha$ in Theorem 2. Then by repeating the same steps in the above proof, we have

[TABLE]

which implies that $y=\lambda_{\epsilon}z+\left(1-\lambda_{\epsilon}\right)x$, we have

[TABLE]

Since this is true for any $x,y\in\left[a,b\right]$, we conclude that $\left|f\left(y\right)-f\left(x\right)\right|\leq h\left({\left|{y-x}\right|}\right)$, which shows that $f$ is $h$-continuous on $\left[a,b\right]$. Surely, this is can be considered as an alternative proof of Theorem 2.

It’s well known that if $f$ is twice differentiable then $f$ is convex if and only if $f^{\prime\prime}\geq 0$. In a convenient way Pinheiro in [29] proposed that $f$ is an $s$-convex (in the second sense) if and only if $f^{\prime\prime}\geq 1-2^{1-s}$. Indeed, Pinheiro presented a “proof” to her result, however we can say without doubt that she introduced some good thoughts rather than formal mathematical proof. Following the same way in [29] and in viewing the presented discussion in the introduction we conjecture that:

Conjecture 1**.**

Let $h:J\to\left(0,\infty\right)$ be a non-negative function such that $h\left(\alpha\right)\geq\alpha$, for all $\alpha\in(0,1)$, and consider $f:I\to\mathbb{R}$ be a twice differentiable function. A function $f$ is $h$-convex if and only if $f^{\prime\prime}\left(x\right)\geq 1-2h\left(\frac{1}{2}\right)$.