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Half convex functions
Journal of Inequalities and Applications volume 2014, Article number: 13 (2014)
Abstract
The paper provides the characteristic properties of half convex functions. The analytic and geometric image of half convex functions is presented using convex combinations and support lines. The results relating to convex combinations are applied to quasiarithmetic means.
MSC:26A51, 26D15.
1 Introduction
Through the paper we will use an interval $\mathcal{I}\subseteq \mathbb{R}$ with the nonempty interior ${\mathcal{I}}^{0}$. Two subintervals of ℐ specified by the point $c\in {\mathcal{I}}^{0}$ will be denoted by
We say that a function $f:\mathcal{I}\to \mathbb{R}$ is right convex if it is convex on ${\mathcal{I}}_{x\ge c}$ for some point $c\in {\mathcal{I}}^{0}$. A left convex function is similarly defined on the interval ${\mathcal{I}}_{x\le c}$. A function is half convex if it is right or left convex.
A combination
of points ${x}_{i}$ and real coefficients ${p}_{i}$ is affine if the coefficient sum ${\sum}_{i=1}^{n}{p}_{i}=1$. The above combination is convex if all coefficients ${p}_{i}$ are nonnegative and the coefficient sum is equal to 1. The point c itself is called the combination center, and it is important for mathematical inequalities.
Every affine function $f:\mathbb{R}\to \mathbb{R}$ satisfies the equality
for all affine combinations from ℝ. Every convex function $f:\mathcal{I}\to \mathbb{R}$ satisfies the Jensen inequality
for all convex combinations from ℐ.
If $a,b\in \mathbb{R}$ are different numbers, say $a<b$, then every number $x\in \mathbb{R}$ can be uniquely presented as the affine combination
The above binomial combination of a and b is convex if, and only if, the number x belongs to the interval $[a,b]$. Given the function $f:\mathbb{R}\to \mathbb{R}$, let ${f}_{\{a,b\}}^{\mathrm{line}}:\mathbb{R}\to \mathbb{R}$ be the function of the line passing through the points $(a,f(a))$ and $(b,f(b))$ of the graph of f. Using the affinity of ${f}_{\{a,b\}}^{\mathrm{line}}$, we get the equation
Assuming that f is convex, we get the inequality
and the reverse inequality
A general overview of convexity, convex functions, and applications can be found in [1], and many details of this branch are in [2]. The different forms of the famous Jensen’s inequality (discrete form in [3] and integral form in [4]) can be found in [5] and [6]. Global bounds for Jensen’s functional were investigated in [7].
2 Recent results
The following results on half convex functions have been presented in [8].
WRCFTheorem (Weighted right convex function theorem)
Let $f(x)$ be a function defined on ℐ and convex for $x\ge c\in \mathcal{I}$, and let ${p}_{1},\dots ,{p}_{n}$ be positive real numbers such that
The inequality
holds for all ${x}_{1},\dots ,{x}_{n}\in \mathcal{I}$ satisfying ${\sum}_{i=1}^{n}{p}_{i}{x}_{i}\ge c$ if and only if the inequality
holds for all $x,y\in \mathcal{I}$ such that $x\le c\le y$ and $px+(1p)y=c$.
The WLCFTheorem (weighted left convex function theorem) is presented in a similar way. The final common theorem on half convexity reads as follows.
WHCFTheorem (Weighted half convex function theorem)
Let $f(x)$ be a function defined on ℐ and convex for $x\ge c$ or $x\le c$, where $c\in \mathcal{I}$, and let ${p}_{1},\dots ,{p}_{n}$ be positive real numbers such that
The inequality
holds for all ${x}_{1},\dots ,{x}_{n}\in \mathcal{I}$ satisfying ${\sum}_{i=1}^{n}{p}_{i}{x}_{i}=c$ if and only if the inequality
holds for all $x,y\in \mathcal{I}$ such that $px+(1p)y=c$.
3 Main results
3.1 Half convexity with convex combinations
The main result is Lemma 3.1, which extends the right convexity of WRCFTheorem to all convex combinations, without preselected coefficients ${p}_{1},\dots ,{p}_{n}$ and without emphasizing the coefficient $p=min\{{p}_{1},\dots ,{p}_{n}\}$. Therefore, Jensen’s inequality for right convex functions follows (using as usual nonnegative coefficients in convex combinations).
Lemma 3.1 Let $f:\mathcal{I}\to \mathbb{R}$ be a function that is convex on ${\mathcal{I}}_{x\ge c}$ for some point $c\in {\mathcal{I}}^{0}$. Then the inequality
holds for all convex combinations from ℐ satisfying ${\sum}_{i=1}^{n}{p}_{i}{x}_{i}\ge c$ if, and only if, the inequality
holds for all binomial convex combinations from ℐ satisfying $px+qy=c$.
Proof Let us prove the sufficiency using the induction on the integer $n\ge 2$.
The base of induction. Take a binomial convex combination from ℐ satisfying $px+qy\ge c$ with $x\le y$ and $0<p<1$. If $x\ge c$, we apply the Jensen inequality to get the inequality in equation (3.1) for $n=2$. If $x<c$, we have $x<c\le px+qy<y$. Since $px+qy$ lies between c and y, we have some convex combination ${p}_{0}c+{q}_{0}y=px+qy$ with ${p}_{0}>0$. Then the affine combination
is convex since c lies between x and y, and hence
by assumption in equation (3.2). Using the convexity of f on ${\mathcal{I}}_{x\ge c}$ and the above inequality, it follows that
The step of induction. Suppose that the inequality in equation (3.1) is true for all corresponding nmembered convex combinations with $n\ge 2$. Take an $(n+1)$membered convex combination from ℐ satisfying ${\sum}_{i=1}^{n+1}{p}_{i}{x}_{i}\ge c$ with ${x}_{1}=min\{{x}_{1},\dots ,{x}_{n}\}$ and $0<{p}_{1}<1$. If ${x}_{1}\ge c$, we can use the Jensen inequality to get the inequality in equation (3.1) for $n+1$. If ${x}_{1}<c$, relying on the representation
we can apply the induction base with the points $x={x}_{1}$ and $y={\sum}_{i=2}^{n+1}{p}_{i}/(1{p}_{1}){x}_{i}$, and the coefficients $p={p}_{1}$ and $q=1{p}_{1}$. Using the inequality in equation (3.5) and the induction premise
we get the inequality in equation (3.1) for the observed $(n+1)$membered combination. □
Remark 3.2 The formula in equation (3.5) for the case $x<c$ in the proof of Lemma 3.1 can be derived with exactly calculated convex combination coefficients. If $d=px+qy$, then we have the order $x<c\le d<y$. Using the presentation formula in equation (1.4), we determine the convex combinations
Applying the right convex function f, it follows that
since
Following Lemma 3.1 we can state a similar lemma for left convex functions. So, if a function $f(x)$ is convex on ${\mathcal{I}}_{x\ge c}$, and if the interval ℐ is symmetric respecting the point c, then the function $g(x)=f(2cx)$ is convex on ${\mathcal{I}}_{x\le c}$. The function g verifies the same inequalities as the function f in Lemma 3.1, wherein the condition ${\sum}_{i=1}^{n}{p}_{i}(2c{x}_{i})\ge c$ becomes ${\sum}_{i=1}^{n}{p}_{i}{x}_{i}\le c$.
Connecting the lemmas on right and left convex functions, we have the following result for half convex functions.
Theorem 3.3 Let $f:\mathcal{I}\to \mathbb{R}$ be a function that is convex on ${\mathcal{I}}_{x\ge c}$ or ${\mathcal{I}}_{x\le c}$ for some point $c\in {\mathcal{I}}^{0}$. Then the inequality
holds for all convex combinations from ℐ satisfying ${\sum}_{i=1}^{n}{p}_{i}{x}_{i}=c$ if, and only if, the inequality
holds for all binomial convex combinations from ℐ satisfying $px+qy=c$.
If f is concave on ${\mathcal{I}}_{x\ge c}$ or ${\mathcal{I}}_{x\le c}$ for some point $c\in {\mathcal{I}}^{0}$, then the reverse inequalities are valid in equations (3.8) and (3.9).
3.2 Half convexity with support lines
The main result is Lemma 3.5 complementing a geometric image of a right convex function satisfying Jensen’s inequality for all binomial convex combinations with the center at the convexity edgepoint c. The graph of such a function is located above the right tangent line at c.
Regardless of convexity, we start with a trivial lemma which can be taken as Jensen’s inequality for functions supported with the line passing through a point of its graph.
Lemma 3.4 Let $f:\mathcal{I}\to \mathbb{R}$ be a function, and let $C(c,f(c))$ be a point of the graph of f with $c\in {\mathcal{I}}^{0}$.
If there exists a line that passes through the point C below the graph of f, then the function f satisfies the inequality
for all convex combinations from ℐ satisfying ${\sum}_{i=1}^{n}{p}_{i}{x}_{i}=c$.
If there exists a line that passes through the point C above the graph of f, then the reverse inequality is valid in equation (3.10).
Proof Let ${f}_{\{c\}}^{\mathrm{line}}$ be the function of the line passing through the point C. Assume the case ${f}_{\{c\}}^{\mathrm{line}}\le f$. If ${\sum}_{i=1}^{n}{p}_{i}{x}_{i}=c$ is any convex combination from ℐ with the center at c, then using the affinity of ${f}_{\{c\}}^{\mathrm{line}}$, it follows that
concluding the proof in the case that the line is below the curve. □
The function f with the line passing through the point C below its graph is shown in Figure 1. In what follows, f will be convex to the right or left of c, and the right or left tangent line will be used as the support line at c.
If f is convex on ${\mathcal{I}}_{x\ge c}$, then the slope $(f(x)f(c))/(xc)$ approaches the real number ${k}_{r}={f}^{\prime}(c+)$ or to the negative infinity −∞ as x approaches c+.
Lemma 3.5 Let $f:\mathcal{I}\to \mathbb{R}$ be a function that is convex on ${\mathcal{I}}_{x\ge c}$ for some point $c\in {\mathcal{I}}^{0}$. Let ${k}_{r}\in \mathbb{R}$ be the slope of the right tangent line of the function f at the point c.
Then the inequality
holds for all binomial convex combinations from ℐ satisfying $px+qy=c$ if, and only if, the right tangent line inequality
holds for all points $x\in \mathcal{I}$.
Proof The proof of necessity. Since the tangent line of a convex function is the support line, it is sufficient to prove the inequality in equation (3.12) for all x in ℐ which are less than c. Take any such x. Then the ccentered convex combinations $px+qy=c$ from ℐ include only $y>c$. The inequality in equation (3.11) with these combinations can be adapted to the form
Letting y go to c and p to 0 so that x stays the same, it follows that
Multiplying the above inequality with $xc<0$, we get the inequality in equation (3.12).
The proof of sufficiency. Using the convex function $g:\mathcal{I}\to \mathbb{R}$ defined by
we see that the inequality
holds for all binomial convex combinations from ℐ satisfying $px+qy=c$. □
The graph of the right convex function f with the right tangent line at the point c is presented in Figure 2.
Based on Lemma 3.5 and its analogy for left convex functions, and also on Theorem 3.3, we have the following characterization of half convex and half concave functions.
Theorem 3.6 Let $f:\mathcal{I}\to \mathbb{R}$ be a function that is convex on ${\mathcal{I}}_{x\ge c}$ or ${\mathcal{I}}_{x\le c}$ for some point $c\in {\mathcal{I}}^{0}$. If f is right (resp. left) convex, let ${k}_{r}\in \mathbb{R}$ (resp. ${k}_{l}$) be the slope of the right (resp. left) tangent line at c.
Then the following three conditions are equivalent:

(1)
The inequality
$$f\left(\sum _{i=1}^{n}{p}_{i}{x}_{i}\right)\le \sum _{i=1}^{n}{p}_{i}f({x}_{i})$$holds for all convex combinations from ℐ satisfying ${\sum}_{i=1}^{n}{p}_{i}{x}_{i}=c$.

(2)
The inequality
$$f(px+qy)\le pf(x)+qf(y)$$holds for all binomial convex combinations from ℐ satisfying $px+qy=c$.

(3)
The tangent line inequality
$$f(x)\ge k(xc)+f(c)$$with $k={k}_{r}$ or $k={k}_{l}$ holds for all points $x\in \mathcal{I}$.
If f is concave on ${\mathcal{I}}_{x\ge c}$ or ${\mathcal{I}}_{x\le c}$ for some point $c\in {\mathcal{I}}^{0}$, then the reverse inequalities are valid in (1), (2), and (3).
4 Application to quasiarithmetic means
In the applications of convexity to quasiarithmetic means, we use strictly monotone continuous functions $\phi ,\psi :\mathcal{I}\to \mathbb{R}$ such that ψ is φconvex, that is, the function $f=\psi \circ {\phi}^{1}$ is convex (according to the terminology in [[2], Definition 1.19]). A similar notation is used for concavity.
A concept of quasiarithmetic mean refers to convex combinations and strictly monotone continuous functions. The φquasiarithmetic mean of the convex combination ${\sum}_{i=1}^{n}{p}_{i}{x}_{i}$ from ℐ is the point
which belongs to ℐ, because the convex combination ${\sum}_{i=1}^{n}{p}_{i}\phi ({x}_{i})$ belongs to $\phi (\mathcal{I})$. If φ is the identity function on ℐ, then its quasiarithmetic mean is just the convex combination ${\sum}_{i=1}^{n}{p}_{i}{x}_{i}$.
Right convexity and concavity for quasiarithmetic means can be obtained using Lemma 3.1.
Corollary 4.1 Let $\phi ,\psi :\mathcal{I}\to \mathbb{R}$ be strictly monotone continuous functions, and $\mathcal{J}=\phi (\mathcal{I})$ be the φimage of the interval ℐ.
If ψ is either φconvex on ${\mathcal{J}}_{z\ge d}$ for some point $d=\phi (c)\in {\mathcal{J}}^{0}$ and increasing on ℐ or φconcave on ${\mathcal{J}}_{z\ge d}$ and decreasing on ℐ, then the inequality
holds for all convex combinations from $\mathcal{J}$ satisfying ${\sum}_{i=1}^{n}{p}_{i}\phi ({x}_{i})\ge d$ if, and only if, the inequality
holds for all binomial convex combinations from $\mathcal{J}$ satisfying $p\phi (x)+q\phi (y)=d$.
If ψ is either φconvex on ${\mathcal{J}}_{z\ge \phi (c)}$ for some point $d=\phi (c)\in {\mathcal{J}}^{0}$ and decreasing on ℐ or φconcave on ${\mathcal{J}}_{z\ge d}$ and increasing on ℐ, then the reverse inequalities are valid in equations (4.2) and (4.3).
Proof We prove the case that the function ψ is φconvex on the interval ${\mathcal{J}}_{z\ge d}$ and increasing on the interval ℐ. Put $f=\psi \circ {\phi}^{1}$.
In the first step, applying Lemma 3.1 to the function $f:\mathcal{J}\to \mathbb{R}$, convex on the interval ${\mathcal{J}}_{z\ge d}$, we get the equivalence: the inequality
holds for all convex combinations from $\mathcal{J}$ satisfying ${\sum}_{i=1}^{n}{p}_{i}\phi ({x}_{i})\ge d$ if, and only if, the inequality
holds for all binomial convex combinations from $\mathcal{J}$ satisfying $p\phi (x)+q\phi (y)=d$.
In the second step, assigning the increasing function ${\psi}^{1}$ to the above inequalities, the equivalence follows. The inequality
holds for all convex combinations from $\mathcal{J}$ satisfying ${\sum}_{i=1}^{n}{p}_{i}\phi ({x}_{i})\ge d$ if, and only if, the inequality
holds for all binomial convex combinations from $\mathcal{J}$ satisfying $p\phi (x)+q\phi (y)=d$. □
Combining ‘right’ and ‘left’ similarly as in Theorem 3.3, we get the following characterization of half convexity and concavity for quasiarithmetic means.
Corollary 4.2 Let $\phi ,\psi :\mathcal{I}\to \mathbb{R}$ be strictly monotone continuous functions, and let $\mathcal{J}=\phi (\mathcal{I})$ be the φimage of the interval ℐ.
If ψ is either φconvex on ${\mathcal{J}}_{z\ge d}$ or ${\mathcal{J}}_{z\le d}$ for some point $d=\phi (c)\in {\mathcal{J}}^{0}$ and increasing on ℐ or φconcave on ${\mathcal{J}}_{z\ge d}$ or ${\mathcal{J}}_{z\le d}$ and decreasing on ℐ, then the inequality
holds for all convex combinations from $\mathcal{J}$ satisfying ${\sum}_{i=1}^{n}{p}_{i}\phi ({x}_{i})=d$ if, and only if, the inequality
holds for all binomial convex combinations from $\mathcal{J}$ satisfying $p\phi (x)+q\phi (y)=d$.
If ψ is either φconvex on ${\mathcal{J}}_{z\ge \phi (c)}$ or ${\mathcal{J}}_{z\le d}$ for some point $d=\phi (c)\in {\mathcal{J}}^{0}$ and decreasing on ℐ or φconcave on ${\mathcal{J}}_{z\ge d}$ or ${\mathcal{J}}_{z\le d}$ and increasing on ℐ, then the reverse inequalities are valid in equations (4.4) and (4.5).
For a basic study of means and their inequalities the excellent book in [9] is recommended. Very general forms of discrete and integral quasiarithmetic means and their refinements were studied in [6]. A simple transformation for deriving new means was introduced in [10].
In further research of half convexity it would be useful to include the functions of several variables. Their geometric characterization using support surfaces would be especially interesting.
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Pavić, Z. Half convex functions. J Inequal Appl 2014, 13 (2014). https://doi.org/10.1186/1029242X201413
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Keywords
 convex combination center
 half convex function
 Jensen’s inequality
 quasiarithmetic mean