﻿ 强h-凸函数的若干性质

# 强h-凸函数的若干性质Some Properties of the Strongly h-Convex Function

Abstract: The strongly h-convex function is a generation of the convex function and the h-convex function, and the latter is also a common generalization of the convexity, s-convexity, the Godunova-Levin function and the P-function. In this paper, we discuss some basic properties of strongly h-convex functions, and make some presentations of them involving the notations of sup-multiplicative functions, convergence of sequence, etc.

1. 引言

2. h-凸函数和强h-凸函数的定义

2007年Varosanec [1] 引进了一类推广了的凸型函数：h-凸函数，即

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

2011年Angulo [8] 进一步推广了h-凸函数，引入了强h-凸函数的概念。

$f\left(tx+\left(1-t\right)y\right)\le h\left(t\right)f\left(x\right)+h\left(1-t\right)f\left(y\right)-ct\left(1-t\right){‖x-y‖}^{2}$ (2)

3. 强h-凸函数的基本性质

$\begin{array}{c}f\left(tx-\left(1-t\right)y\right)\le tf\left(x\right)+\left(1-t\right)f\left(y\right)-ct\left(1-t\right){‖x-y‖}^{2}\\ \le h\left(t\right)f\left(x\right)+h\left(1-t\right)f\left(y\right)-ct\left(1-t\right){‖x-y‖}^{2}\end{array}$ .

$h\left(t\right)\ge t$$t\in \left(0,1\right)$ ，且若f为I上的非负模c强凹函数，则 $\forall x,y\in I$$t\in \left(0,1\right)$ ，有

$\begin{array}{c}f\left(tx-\left(1-t\right)y\right)\ge tf\left(x\right)+\left(1-t\right)f\left(y\right)-ct\left(1-t\right){‖x-y‖}^{2}\\ \ge h\left(t\right)f\left(x\right)+h\left(1-t\right)f\left(y\right)-ct\left(1-t\right){‖x-y‖}^{2}\end{array}$ .

$\begin{array}{c}f\left(tx-\left(1-t\right)y\right)\le {h}_{2}\left(t\right)f\left(x\right)+{h}_{2}\left(1-t\right)f\left(y\right)-ct\left(1-t\right){‖x-y‖}^{2}\\ \le {h}_{1}\left(t\right)f\left(x\right)+{h}_{1}\left(1-t\right)f\left(y\right)-ct\left(1-t\right){‖x-y‖}^{2}\end{array}$ .

$f\left(tx+\left(1-t\right)y\right)\le h\left(t\right)f\left(x\right)+h\left(1-t\right)f\left(y\right)-{c}_{1}t\left(1-t\right){‖x-y‖}^{2}$

$g\left(tx+\left(1-t\right)y\right)\le h\left(t\right)g\left(x\right)+h\left(1-t\right)g\left(y\right)-{c}_{2}t\left(1-t\right){‖x-y‖}^{2}$ .

$\begin{array}{l}\left(f+g\right)\left(tx+\left(1-t\right)y\right)\\ \le h\left(t\right)\left(f\left(x\right)+g\left(x\right)\right)+h\left(1-t\right)\left(f\left(y\right)+g\left(y\right)\right)-\left({c}_{1}+{c}_{2}\right)t\left(1-t\right){‖x-y‖}^{2}\end{array}$

$\lambda f\left(tx+\left(1-t\right)y\right)\le \lambda h\left(t\right)f\left(x\right)+\lambda h\left(1-t\right)f\left(y\right)-\lambda {c}_{1}t\left(1-t\right){‖x-y‖}^{2}$

4. 上积函数，单调性与强h-凸函数的关系

$h\left(xy\right)\ge h\left(x\right)h\left(y\right)$$x,y\in J$

$f\left(\alpha x+\beta y\right)\le h\left(\alpha \right)f\left(x\right)+h\left(\beta \right)f\left(y\right)-c\alpha \beta {‖x-y‖}^{2}$ (3)

$\begin{array}{l}f\left(\alpha x+\beta y\right)=f\left(\gamma ax+\gamma by\right)\\ \le h\left(a\right)f\left(\gamma x\right)+h\left(b\right)f\left(\gamma y\right)-cab{‖\gamma x-\gamma y‖}^{2}\\ =h\left(a\right)f\left(\gamma x+\left(1-\gamma \right)\cdot 0\right)+h\left(b\right)f\left(\gamma y+\left(1-\gamma \right)\cdot 0\right)-cab{\gamma }^{2}{‖x-y‖}^{2}\end{array}$

$\begin{array}{l}\le h\left(a\right)h\left(\gamma \right)f\left(x\right)+h\left(b\right)h\left(\gamma \right)f\left(y\right)-cab{\gamma }^{2}{‖x-y‖}^{2}\\ \le h\left(a\gamma \right)f\left(x\right)+h\left(b\gamma \right)f\left(y\right)-cab{\gamma }^{2}{‖x-y‖}^{2}\\ =h\left(\alpha \right)f\left(x\right)+h\left(\beta \right)f\left(y\right)-c\alpha \beta {‖x-y‖}^{2}\end{array}$ .

$f\left(\alpha x+\beta y\right)\le h\left(\alpha \right)f\left(x\right)+h\left(\beta \right)f\left(y\right)-c\alpha \beta {‖x-y‖}^{2}$ (4)

$f\left(0\right)\le h\left(\alpha \right)f\left(0\right)+h\left(\beta \right)f\left(0\right)$ .

$f\left(0\right)\le h\left({\alpha }_{\text{0}}\right)f\left(0\right)+h\left({\alpha }_{\text{0}}\right)f\left(0\right)=2h\left({\alpha }_{\text{0}}\right)f\left(0\right)$ .

$h\left({\alpha }_{\text{0}}\right)\ge \frac{\text{1}}{\text{2}}$ .

$c|x-y|\le |f\left(x\right)-f\left(y\right)|$ ,

$\begin{array}{l}\left(f\circ g\right)\left(\alpha x+\left(1-\alpha \right)y\right)\le f\left(\alpha g\left(x\right)+\left(1-\alpha \right)g\left(y\right)\right)\\ \le h\left(\alpha \right)\left(f\circ g\right)\left(x\right)+h\left(1-\alpha \right)\left(f\circ g\right)\left(y\right)-c\alpha \left(1-\alpha \right){|g\left(x\right)-g\left(y\right)|}^{2}\\ \le h\left(\alpha \right)\left(f\circ g\right)\left(x\right)+h\left(1-\alpha \right)\left(f\circ g\right)\left(y\right)-c{\delta }^{2}\alpha \left(1-\alpha \right){|x-y|}^{2}\end{array}$ .

$\left(f\circ g\right)\left(\alpha x+\left(1-\alpha \right)y\right)\le f\left({h}_{2}\left(\alpha \right)g\left(x\right)+{h}_{2}\left(1-\alpha \right)g\left(y\right)\right)$ (5)

$\begin{array}{l}f\left({h}_{2}\left(\alpha \right)g\left(x\right)+{h}_{2}\left(1-\alpha \right)g\left(y\right)\right)\\ \le {h}_{1}\left({h}_{2}\left(\alpha \right)\right)f\left(g\left(x\right)\right)+{h}_{1}\left({h}_{2}\left(1-\alpha \right)\right)f\left(g\left(y\right)\right)-c{h}_{2}\left(\alpha \right){h}_{2}\left(1-\alpha \right){|g\left(x\right)-g\left(y\right)|}^{2}\end{array}$ (6)

$\left(f\circ g\right)\left(\alpha x+\left(1-\alpha \right)y\right)\le \left({h}_{1}\circ {h}_{2}\right)\left(\alpha \right)\left(f\circ g\right)\left(x\right)+\left({h}_{1}\circ {h}_{2}\right)\left(\text{1}-\alpha \right)\left(f\circ g\right)\left(y\right)-c{\delta }^{2}\alpha \left(1-\alpha \right){|x-y|}^{2}$ .

$f\left(\frac{1}{{W}_{n}}\underset{i=1}{\overset{n}{\sum }}{w}_{i}{x}_{i}\right)\le \underset{i=1}{\overset{n}{\sum }}h\left(\frac{{w}_{i}}{{W}_{n}}\right)f\left({x}_{i}\right)$

$\begin{array}{l}f\left(\frac{1}{{W}_{n}}\underset{i=1}{\overset{n}{\sum }}{w}_{i}{x}_{i}\right)=f\left(\frac{{w}_{n}}{{W}_{n}}{x}_{n}+\underset{i=1}{\overset{n-1}{\sum }}\frac{{w}_{i}}{{W}_{n}}{x}_{i}\right)=f\left(\frac{{w}_{n}}{{W}_{n}}{x}_{n}+\frac{{W}_{n-1}}{{W}_{n}}\underset{i=1}{\overset{n-1}{\sum }}\frac{{w}_{i}}{{W}_{n-1}}{x}_{i}\right)\\ \le h\left(\frac{{w}_{n}}{{W}_{n}}\right)f\left({x}_{n}\right)+h\left(\frac{{W}_{n-1}}{{W}_{n}}\right)f\left(\underset{i=1}{\overset{n-1}{\sum }}\frac{{w}_{i}}{{W}_{n-1}}{x}_{i}\right)-c\frac{{w}_{n}}{{W}_{n}}\frac{{W}_{n-1}}{{W}_{n}}{‖{x}_{n}-\underset{i=1}{\overset{n-1}{\sum }}\frac{{w}_{i}}{{W}_{n-1}}{x}_{i}‖}^{2}\\ \le h\left(\frac{{w}_{n}}{{W}_{n}}\right)f\left({x}_{n}\right)+\underset{i=1}{\overset{n-1}{\sum }}\left[h\left(\frac{{W}_{n-1}}{{W}_{n}}\right)h\left(\frac{{w}_{i}}{{W}_{n-1}}\right)f\left({x}_{i}\right)\right]\\ \le h\left(\frac{{w}_{n}}{{W}_{n}}\right)f\left({x}_{n}\right)+\underset{i=1}{\overset{n-1}{\sum }}h\left(\frac{{w}_{i}}{{W}_{n}}\right)f\left( x i \right)\end{array}$

5. 函数列收敛和强h-凸函数

$\begin{array}{c}{f}_{n}\left(tx+\left(1-t\right)y\right)\le {h}_{n}\left(t\right){f}_{n}\left(x\right)+{h}_{n}\left(1-t\right){f}_{n}\left(y\right)-{c}_{n}t\left(1-t\right){‖x-y‖}^{\text{2}}\\ \le {h}_{n}\left(t\right){f}_{n}\left(x\right)+{h}_{n}\left(1-t\right){f}_{n}\left(y\right)-ct\left(1-t\right){‖x-y‖}^{\text{2}}\end{array}$ .

$\underset{n\to \infty }{\mathrm{lim}}{f}_{n}\left(tx+\left(1-t\right)y\right)\le \underset{n\to \infty }{\mathrm{lim}}\left\{{h}_{n}\left(t\right){f}_{n}\left(x\right)+{h}_{n}\left(1-t\right){f}_{n}\left(y\right)\right\}-ct\left(1-t\right){‖x-y‖}^{2}$ .

$f\left(tx+\left(1-t\right)y\right)\le h\left(t\right)f\left(x\right)+h\left(1-t\right)f\left(y\right)-ct\left(1-t\right){‖x-y‖}^{2}$ .

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