﻿ MM-凸函数及其Jensen型不等式
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 浙江大学学报(理学版)  2017, Vol. 44 Issue (4): 403-410  DOI:10.3785/j.issn.1008-9497.2017.04.004 0

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SONG Zhenyun, CHEN Shaoyuan, HU Fugao. MM-convex function & its Jensen-type inequality[J]. Journal of Zhejiang University(Science Edition), 2017, 44(4): 403-410. DOI: 10.3785/j.issn.1008-9497.2017.04.004.
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### 文章历史

MM-凸函数及其Jensen型不等式

1. 湖北职业技术学院 机电工程学院, 湖北 孝感 432000;
2. 湖北工程学院 数学与统计学院, 湖北 孝感 432000

MM-convex function & its Jensen-type inequality
SONG Zhenyun1 , CHEN Shaoyuan1 , HU Fugao2
1. School of Mechanical & Electrical Engineering, Hubei Polytechnic Institute, Xiaogan 432000, Hubei Province, China;
2. School of Mathematics & Statistics, Hubei Engineering University, Xiaogan 432000, Hubei Province, China
Abstract: Considering the general convexity of functions, the authors present the definition of MM-convex function with two variables power means within the interval. Based on the definition, this article discusses its judgment theorems and operation properties, sets up its Jensen-type inequality, and provides the equivalent form of Jensen-type inequality and the deduction. Results show that MM-convex function is an extension of all convex functions determined by the power mean value of two arbitrary points within the definition domain of comparison function and by the power mean of the value. The introduction of MM-convex function brings an effective approach to deep study and further extension of convex function.
Key words: convex function    MM-convex function    judgment theorem    operation property    Jensen-type inequality
0 引言

aiR+ti∈[0, 1](i=1, 2, …, n)，且$\sum\limits_{i = 1}^n {{t_i}} = 1$，记

 $\begin{array}{l} {\rm{M}}_n^{\left[ r \right]}\left( {{t_1},{t_2}, \cdots ,{t_n};{a_1},{a_2}, \cdots ,{a_n}} \right) = \\ \left\{ \begin{array}{l} {\left[ {\sum\limits_{i = 1}^n {{t_i}a_i^r} } \right]^{\frac{1}{r}}},\;\;\;r \ne 0,\\ \prod\limits_{i = 1}^n {a_i^{{t_i}}} ,\;\;\;\;r = 0,\\ \max \left\{ {{a_1},{a_2}, \cdots ,{a_n}} \right\},\;\;\;\;r = + \infty ,\\ \min \left\{ {{a_1},{a_2}, \cdots ,{a_n}} \right\},\;\;\;\;r = - \infty , \end{array} \right. \end{array}$ (1)

 ${\rm{M}}_n^{\left[ 1 \right]}\left( {{t_1},{t_2}, \cdots ,{t_n};{a_1},{a_2}, \cdots ,{a_n}} \right) = {\rm{A}}\left( {{a_1},{a_2}, \cdots ,{a_n}} \right),$
 ${\rm{M}}_n^{\left[ 0 \right]}\left( {{t_1},{t_2}, \cdots ,{t_n};{a_1},{a_2}, \cdots ,{a_n}} \right) = {\rm{G}}\left( {{a_1},{a_2}, \cdots ,{a_n}} \right),$
 ${\rm{M}}_n^{\left[ { - 1} \right]}\left( {{t_1},{t_2}, \cdots ,{t_n};{a_1},{a_2}, \cdots ,{a_n}} \right) = {\rm{H}}\left( {{a_1},{a_2}, \cdots ,{a_n}} \right),$
 ${\rm{M}}_n^{\left[ 2 \right]}\left( {{t_1},{t_2}, \cdots ,{t_n};{a_1},{a_2}, \cdots ,{a_n}} \right) = {\rm{SR}}\left( {{a_1},{a_2}, \cdots ,{a_n}} \right),$
 ${\rm{M}}_n^{\left[ { - 2} \right]}\left( {{t_1},{t_2}, \cdots ,{t_n};{a_1},{a_2}, \cdots ,{a_n}} \right) = {\rm{HS}}\left( {{a_1},{a_2}, \cdots ,{a_n}} \right),$
 ${\rm{M}}_n^{\left[ 3 \right]}\left( {{t_1},{t_2}, \cdots ,{t_n};{a_1},{a_2}, \cdots ,{a_n}} \right) = {\rm{CR}}\left( {{a_1},{a_2}, \cdots ,{a_n}} \right),$
 ${\rm{M}}_n^{\left[ { - 3} \right]}\left( {{t_1},{t_2}, \cdots ,{t_n};{a_1},{a_2}, \cdots ,{a_n}} \right) = {\rm{HC}}\left( {{a_1},{a_2}, \cdots ,{a_n}} \right)$

n元加权幂平均Mn[r](t1, t2, …, tn; a1, a2…, an)，由式(1) 易得如下加权幂平均恒等关系式：

(ⅰ)[Mn[r](t1, t2, …, tn; a1, a2, …, an)]r=Mn[1](t1, t2, …, tn; a1r, a2r, …, anr)；

(ⅱ)[Mn[1](t1, t2, …, tn; a1, a2, …, an)]${\frac{1}{r}}$=Mn[r](t1, t2, …, tn; a1${\frac{1}{r}}$, a2${\frac{1}{r}}$, …, an${\frac{1}{r}}$)；

(ⅲ)[Mn[r](t1, t2, …, tn; a1${\frac{1}{r}}$, a2${\frac{1}{r}}$, …, an${\frac{1}{r}}$)]r=Mn[1](t1, t2, …, tn; a1, a2, …, an)；

(ⅳ)expMn[1](t1, t2, …, tn; a1, a2, …, an)=Mn[0](t1, t2, …, tn; exp a1, exp a2, …, exp an)；

ai∈(1, +∞)，则

(ⅴ)lnMn[0](t1, t2, …, tn; a1, a2, …, an)=Mn[1](t1, t2, …, tn; ln a1, ln a2, …, ln an).

(ⅵ)M2[1](M2[1](α, 1-α; t1, t2), 1-M2[1](α, 1-α; t1, t2); x1, x2)=M2[1](α, 1-α; M2[1](t1, 1-t1; x1, x2), M2[1](t2, 1-t2; x1, x2)).

 $\begin{array}{l} f\left( {{\rm{M}}_2^{\left[ r \right]}\left( {t,1 - t;{x_1},{x_2}} \right)} \right) \le \\ \;\;\;\;\;\;\;\;\;{\rm{M}}_2^{\left[ p \right]}\left( {t,1 - t;f\left( {{x_1}} \right),f\left( {{x_2}} \right)} \right), \end{array}$ (2)

1 MM-凸函数的判定

(ⅰ)当p>0时，f(x)为I上的MM-凸(凹)函数的充要条件是(f(x${\frac{1}{r}}$))pIr上的凸(凹)函数；

(ⅱ)当p＜0时，f(x)为I上的MM-凸(凹)函数的充要条件是(f(x${\frac{1}{r}}$))pIr上的凹(凸)函数.

g(x)=(f(x${\frac{1}{r}}$))p(xIr)，则f(x${\frac{1}{r}}$)=(g(x))${\frac{1}{p}}$.

 $\begin{array}{l} f\left( {{\rm{M}}_2^{\left[ r \right]}\left( {t,1 - t;{x_1},{x_2}} \right)} \right) = f\left( {\left( {{\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;x_1^r,} \right.} \right.} \right.\\ \;\;\;\;\;\;\;\;\left. {{{\left. {\left. {x_2^r} \right)} \right)}^{\frac{1}{r}}}} \right) = {\left[ {g\left( {{\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;x_1^r,x_2^r} \right)} \right)} \right]^{\frac{1}{p}}} \le \\ \;\;\;\;\;\;\;\;{\left[ {{\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;g\left( {x_1^r} \right),g\left( {x_2^r} \right)} \right)} \right]^{\frac{1}{p}}} = \\ \;\;\;\;\;\;\;\;{\left[ {{\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;{{\left( {f\left( {{x_1}} \right)} \right)}^p},{{\left( {f\left( {{x_2}} \right)} \right)}^p}} \right)} \right]^{\frac{1}{p}}} = \\ \;\;\;\;\;\;\;\;{\rm{M}}_2^{\left[ p \right]}\left( {t,1 - t;f\left( {{x_1}} \right),f\left( {{x_2}} \right)} \right), \end{array}$

f(x)为I上的MM-凸函数.

 $\begin{array}{l} {\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;{x_1},{x_2}} \right) \in \left[ {\min \left\{ {{x_1},{x_2}} \right\},\max \left\{ {{x_1},{x_2}} \right\}} \right] \subseteq \\ {{\bf{I}}^r},{\rm{M}}_2^{\left[ r \right]}\left( {t,1 - t;x_1^{\frac{1}{r}},x_2^{\frac{1}{r}}} \right) \in \left[ {\min \left\{ {x_1^{\frac{1}{r}},x_2^{\frac{1}{r}}} \right\},\max \left\{ {x_1^{\frac{1}{r}},} \right.} \right.\\ \left. {\left. {x_2^{\frac{1}{r}}} \right\}} \right] \subseteq {\bf{I}},{\rm{且}}{\left[ {{\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;{x_1},{x_2}} \right)} \right]^{\frac{1}{r}}} \in {\bf{I}}. \end{array}$

f(x)是I上的MM-凸函数，且p>0，则由加权幂平均恒等式和MM-凸函数的定义，有

 $\begin{array}{l} g\left( {{\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;{x_1},{x_2}} \right)} \right) = \left[ {f\left( {\left( {{\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;{x_1},} \right.} \right.} \right.} \right.\\ \;\;\;\;\;\;\;\;{\left. {\left. {{{\left. {\left. {{x_2}} \right)} \right)}^{\frac{1}{r}}}} \right)} \right]^p} = {\left[ {f\left( {{\rm{M}}_2^{\left[ r \right]}\left( {t,1 - t;x_1^{\frac{1}{r}},x_2^{\frac{1}{r}}} \right)} \right)} \right]^p} \le \\ \;\;\;\;\;\;\;\;{\left[ {{\rm{M}}_2^{\left[ p \right]}\left( {t,1 - t;f\left( {x_1^{\frac{1}{r}}} \right),f\left( {x_2^{\frac{1}{r}}} \right)} \right)} \right]^p} = \\ \;\;\;\;\;\;\;\;{\left[ {{\rm{M}}_2^{\left[ p \right]}\left( {t,1 - t;{{\left( {g\left( {{x_1}} \right)} \right)}^{\frac{1}{p}}},{{\left( {g\left( {{x_2}} \right)} \right)}^{\frac{1}{p}}}} \right)} \right]^p} = \\ \;\;\;\;\;\;\;\;{\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;g\left( {{x_1}} \right),g\left( {{x_2}} \right)} \right), \end{array}$

g(x)=(f(x${\frac{1}{r}}$))pIr上的凸函数.

f(x)为I上的MM-凹函数，则以上证明中的不等号反向，因此定理1的后半部分成立.

(ⅰ)当p>0时，f(x)为I上的MM-凸(凹)函数的充要条件是exp[f((ln x)${\frac{1}{r}}$)]p(r≠0) 为exp Ir上的几何凸(凹)函数；

(ⅱ)当p＜0时，f(x)为I上的MM-凸(凹)函数的充要条件是exp[f((ln x)${\frac{1}{r}}$)]p(r≠0) 为exp Ir上的几何凹(凸)函数.

g(x)=exp[f((ln x)${\frac{1}{r}}$)]p(x∈exp Ir)，则[f((ln x)${\frac{1}{r}}$)]p=ln g(x).

 $\begin{array}{l} {\left[ {f\left( {{\rm{M}}_2^{\left[ r \right]}\left( {t,1 - t;{x_1},{x_2}} \right)} \right)} \right]^p} = \\ \;\;\;\;\;\;\;{\left[ {f\left( {{{\left( {\ln \left( {\exp {\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;x_1^r,x_2^r} \right)} \right)} \right)}^{\frac{1}{r}}}} \right)} \right]^p} = \\ \;\;\;\;\;\;\;\ln g\left( {\exp {\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;x_1^r,x_2^r} \right)} \right) = \\ \;\;\;\;\;\;\;\ln g\left( {{\rm{M}}_2^{\left[ 0 \right]}\left( {t,1 - t;\exp x_1^r,\exp x_2^r} \right)} \right) \le \\ \;\;\;\;\;\;\;\ln {\rm{M}}_2^{\left[ 0 \right]}\left( {t,1 - t;g\left( {\exp x_1^r} \right),g\left( {\exp x_2^r} \right)} \right) = \\ \;\;\;\;\;\;\;\ln {\rm{M}}_2^{\left[ 0 \right]}\left( {t,1 - t;\exp{{\left( {f\left( {{x_1}} \right)} \right)}^p},\exp{{\left( {f\left( {{x_2}} \right)} \right)}^p}} \right) = \\ \;\;\;\;\;\;\;{\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;\ln \left( {\exp{{\left( {f\left( {{x_1}} \right)} \right)}^p}} \right),\ln \left( {\exp{{\left( {f\left( {{x_2}} \right)} \right)}^p}} \right)} \right) = \\ \;\;\;\;\;\;\;{\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;{{\left( {f\left( {{x_1}} \right)} \right)}^p},{{\left( {f\left( {{x_2}} \right)} \right)}^p}} \right), \end{array}$

f(M2[r](t, 1-t; x1, x2))≤[M2[1](t, 1-t; (f(x1))p, (f(x2))p)]${\frac{1}{p}}$=M2[p](t, 1-t; f(x1), f(x2)).

f(x)为I上的MM-凸函数.

 $\begin{array}{l} g\left( {{\rm{M}}_2^{\left[ 0 \right]}\left( {t,1 - t;{x_1},{x_2}} \right)} \right) = \\ \;\;\;\;\;\;\;\exp {\left[ {f\left( {{{\left( {\ln {\rm{M}}_2^{\left[ 0 \right]}\left( {t,1 - t;{x_1},{x_2}} \right)} \right)}^{\frac{1}{r}}}} \right)} \right]^p} = \\ \;\;\;\;\;\;\;\exp {\left[ {f\left( {{{\left( {{\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;ln{x_1},ln{x_2}} \right)} \right)}^{\frac{1}{r}}}} \right)} \right]^p} = \\ \;\;\;\;\;\;\;\exp {\left[ {f\left( {{\rm{M}}_2^{\left[ r \right]}\left( {t,1 - t;{{\left( {\ln {x_1}} \right)}^{\frac{1}{r}}},{{\left( {\ln {x_2}} \right)}^{\frac{1}{r}}}} \right)} \right)} \right]^p} \le \\ \;\;\;\;\;\;\;\exp {\left[ {{\rm{M}}_2^{\left[ p \right]}\left( {t,1 - t;f\left( {{{\left( {\ln {x_1}} \right)}^{\frac{1}{r}}}} \right),f\left. {\left( {{{\left( {\ln {x_2}} \right)}^{\frac{1}{r}}}} \right)} \right)} \right)} \right]^p} = \\ \;\;\;\;\;\;\;\exp {\left[ {{\rm{M}}_2^{\left[ p \right]}\left( {t,1 - t;{{\left( {\ln g\left( {{x_1}} \right)} \right)}^{\frac{1}{p}}},{{\left( {\ln g\left( {{x_2}} \right)} \right)}^{\frac{1}{p}}}} \right)} \right]^p} = \\ \;\;\;\;\;\;\;\exp \left[ {{\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;\ln g\left( {{x_1}} \right),\ln g\left( {{x_2}} \right)} \right)} \right] = \\ \;\;\;\;\;\;\;\exp \left[ {\ln {\rm{M}}_2^{\left[ 0 \right]}\left( {t,1 - t;g\left( {{x_1}} \right),g\left( {{x_2}} \right)} \right)} \right] = \\ \;\;\;\;\;\;\;{\rm{M}}_2^{\left[ 0 \right]}\left( {t,1 - t;g\left( {{x_1}} \right),g\left( {{x_2}} \right)} \right), \end{array}$

(ⅰ)当p>0时，f(x)为I上的MM-凸(凹)函数的充要条件是(f(x))pI上的MA-凸(凹)函数；

(ⅱ)当p＜0时，f(x)为I上的MM-凸(凹)函数的充要条件是(f(x))pI上的MA-凹(凸)函数.

(ⅰ)当p>0时，f(x)为I上的MM-凸(凹)函数的充要条件是exp(f(x))pI上的MG-凸(凹)函数；

(ⅱ)当p＜0时，f(x)为I上的MM-凸(凹)函数的充要条件是exp(f(x))pI上的MG-凹(凸)函数.

(ⅰ)当p>0时，函数f(x)是I上的MM-凸(凹)函数的充分必要条件是：∀x1, x2I，函数φ(t)=[f(M2[r](t, 1－t; x1, x2))]p为[0, 1]上的凸(凹)函数；

(ⅱ)当p＜0时，函数f(x)是I上的MM-凸(凹)函数的充分必要条件是：∀x1, x2I，函数φ(t)=[f(M2[r](t, 1－t; x1, x2))]p为[0, 1]上的凹(凸)函数.

φ(t)=[f(M2[r](t, 1－t; x1, x2))]p(t∈[0, 1])，知φ(0)=(f(x2))pφ(1)=(f(x1))p.

 $\begin{array}{l} f\left( {{\rm{M}}_2^{\left[ r \right]}\left( {t,1 - t;{x_1},{x_2}} \right)} \right) = {\left[ {\varphi \left( t \right)} \right]^{\frac{1}{p}}} = \\ \;\;\;\;\;\;\;\;{\left[ {\varphi \left( {{\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;1,0} \right)} \right)} \right]^{\frac{1}{p}}} \le \\ \;\;\;\;\;\;\;\;{\left[ {{\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;\varphi \left( 1 \right),\varphi \left( 0 \right)} \right)} \right]^{\frac{1}{p}}} = \\ \;\;\;\;\;\;\;\;{\left[ {{\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;{{\left( {f\left( {{x_1}} \right)} \right)}^p},{{\left( {f\left( {{x_2}} \right)} \right)}^p}} \right)} \right]^{\frac{1}{p}}} = \\ \;\;\;\;\;\;\;\;{\rm{M}}_2^{\left[ p \right]}\left( {t,1 - t;f\left( {{x_1}} \right),f\left( {{x_2}} \right)} \right), \end{array}$

 $\begin{array}{l} \varphi \left( {{\rm{M}}_2^{\left[ 1 \right]}\left( {\alpha ,1 - \alpha ;{t_1},{t_2}} \right)} \right) = \left[ {f\left( {{\rm{M}}_2^{\left[ r \right]}\left( {{\rm{M}}_2^{\left[ 1 \right]}\left( {\alpha ,1 - \alpha ;{t_1},} \right.} \right.} \right.} \right.\\ \;\;\;\;\;\;\;\;{\left. {\left. {\left. {\left. {{t_2}} \right),1 - {\rm{M}}_2^{\left[ 1 \right]}\left( {\alpha ,1 - \alpha ;{t_1},{t_2}} \right);{x_1},{x_2}} \right)} \right)} \right]^p} = \\ \;\;\;\;\;\;\;\;\left[ {f\left( {\left( {{\rm{M}}_2^{\left[ 1 \right]}\left( {{\rm{M}}_2^{\left[ 1 \right]}\left( {\alpha ,1 - \alpha ;{t_1},{t_2}} \right),1 - {\rm{M}}_2^{\left[ 1 \right]}\left( {\alpha ,} \right.} \right.} \right.} \right.} \right.\\ \;\;\;\;\;\;\;\;{\left. {\left. {{{\left. {\left. {\left. {1 - \alpha ;{t_1},{t_2}} \right);x_1^r,x_2^r} \right)} \right)}^{\frac{1}{r}}}} \right)} \right]^p} = \left[ {f\left( {\left( {{\rm{M}}_2^{\left[ 1 \right]}\left( {\alpha ,} \right.} \right.} \right.} \right.\\ \;\;\;\;\;\;\;\;1 - \alpha ;{\rm{M}}_2^{\left[ 1 \right]}\left( {{t_1},1 - {t_1};x_1^r,x_2^r} \right),{\rm{M}}_2^{\left[ 1 \right]}\left( {{t_2},1 - {t_2};} \right.\\ \;\;\;\;\;\;\;\;{\left. {\left. {{{\left. {\left. {\left. {x_1^r,x_2^r} \right)} \right)} \right)}^{\frac{1}{r}}}} \right)} \right]^p} = \left[ {f\left( {\left[ {{\rm{M}}_2^{\left[ 1 \right]}\left( {\alpha ,1 - \alpha ;\left( {{\rm{M}}_2^{\left[ r \right]}\left( {{t_1},} \right.} \right.} \right.} \right.} \right.} \right.\\ \;\;\;\;\;\;\;\;{\left. {\left. {{{\left. {\left. {{{\left. {\left. {1 - {t_1};x_1^r,x_2^r} \right)} \right)}^r},{{\left( {{\rm{M}}_2^{\left[ r \right]}\left( {{t_2},1 - {t_2};{x_1},{x_2}} \right)} \right)}^r}} \right)} \right]}^{\frac{1}{r}}}} \right)} \right]^p} = \\ \;\;\;\;\;\;\;\;\left[ {f\left( {{\rm{M}}_2^{\left[ r \right]}\left( {\alpha ,1 - \alpha ;{\rm{M}}_2^{\left[ r \right]}\left( {{t_1},1 - {t_1};{x_1},{x_2}} \right),} \right.} \right.} \right.\\ \;\;\;\;\;\;\;\;{\left. {\left. {\left. {{\rm{M}}_2^{\left[ r \right]}\left( {{t_2},1 - {t_2};{x_1},{x_2}} \right)} \right)} \right)} \right]^p} \le \left[ {{\rm{M}}_2^{\left[ p \right]}\left( {\alpha ,1 - \alpha ;} \right.} \right.\\ \;\;\;\;\;\;\;\;f\left( {{\rm{M}}_2^{\left[ r \right]}\left( {{t_1},1 - {t_1};{x_1},{x_2}} \right)} \right),f\left( {{\rm{M}}_2^{\left[ r \right]}\left( {{t_2},1 - {t_2};} \right.} \right.\\ \;\;\;\;\;\;\;\;{\left. {\left. {\left. {\left. {{x_1},{x_2}} \right)} \right)} \right)} \right]^p} = {\rm{M}}_2^{\left[ 1 \right]}\left( {\alpha ,1 - \alpha ;\left[ {f\left( {{\rm{M}}_2^{\left[ r \right]}\left( {{t_1},1 - {t_1};} \right.} \right.} \right.} \right.\\ \;\;\;\;\;\;\;\;\left. {{{\left. {\left. {\left. {{x_1},{x_2}} \right)} \right)} \right]}^p},{{\left[ {f\left( {{\rm{M}}_2^{\left[ r \right]}\left( {{t_2},1 - {t_2};{x_1},{x_2}} \right)} \right)} \right]}^p}} \right) = \\ \;\;\;\;\;\;\;\;{\rm{M}}_2^{\left[ 1 \right]}\left( {\alpha ,1 - \alpha ;\varphi \left( {{t_1}} \right),\varphi \left( {{t_2}} \right)} \right), \end{array}$

φ(t)=[f(M2[r](t, 1－t; x1, x2))]p是[0, 1]上的凸函数.

f(x)为I上的MM-凹函数，则以上证明中不等号反向，故定理7(ⅰ)的后半部分成立.

(ⅰ)若p>0，则f(x)为I上的MM-凸(凹)函数的充要条件是：∀x1, x2, x3Ix1x2x3，当r>0时，有

 $\begin{array}{l} \left( {x_3^r - x_2^r} \right){\left( {f\left( {{x_1}} \right)} \right)^p} + \left( {x_1^r - x_3^r} \right){\left( {f\left( {{x_2}} \right)} \right)^p} + \\ \;\;\;\;\;\left( {x_2^r - x_1^r} \right){\left( {f\left( {{x_3}} \right)} \right)^p} \ge \left( \le \right)0; \end{array}$ (3)

r＜0时，有

 $\begin{array}{l} \left( {x_3^r - x_2^r} \right){\left( {f\left( {{x_1}} \right)} \right)^p} + \left( {x_1^r - x_3^r} \right){\left( {f\left( {{x_2}} \right)} \right)^p} + \\ \;\;\;\;\;\left( {x_2^r - x_1^r} \right){\left( {f\left( {{x_3}} \right)} \right)^p} \le \left( \ge \right)0. \end{array}$ (4)

(ⅱ)若p＜0，则f(x)为I上的MM-凸(凹)函数的充要条件是：∀x1, x2, x3Ix1x2x3, 当r>0时，有

 $\begin{array}{l} \left( {x_3^r - x_2^r} \right){\left( {f\left( {{x_1}} \right)} \right)^p} + \left( {x_1^r - x_3^r} \right){\left( {f\left( {{x_2}} \right)} \right)^p} + \\ \;\;\;\;\;\left( {x_2^r - x_1^r} \right){\left( {f\left( {{x_3}} \right)} \right)^p} \le \left( \ge \right)0; \end{array}$ (5)

r＜0时，有

 $\begin{array}{l} \left( {x_3^r - x_2^r} \right){\left( {f\left( {{x_1}} \right)} \right)^p} + \left( {x_1^r - x_3^r} \right){\left( {f\left( {{x_2}} \right)} \right)^p} + \\ \;\;\;\;\;\left( {x_2^r - x_1^r} \right){\left( {f\left( {{x_3}} \right)} \right)^p} \ge \left( \le \right)0. \end{array}$ (6)

f(x)为I上的MM-凸函数，则

 $\begin{array}{l} f\left( {{x_2}} \right) = f\left( {M_2^{\left[ r \right]}\left( {t,1 - t;{x_1},{x_3}} \right)} \right) \le M_2^{\left[ p \right]}\left( {t,1 - t;} \right.\\ \left. {f\left( {{x_1}} \right),f\left( {{x_3}} \right)} \right) = {\left[ {t{{\left( {f\left( {{x_1}} \right)} \right)}^p} + \left( {1 - t} \right){{\left( {f\left( {{x_3}} \right)} \right)}^p}} \right]^{\frac{1}{p}}} = \\ {\left[ {\frac{{x_3^r - x_2^r}}{{x_3^r - x_1^r}}{{\left( {f\left( {{x_1}} \right)} \right)}^p} + \frac{{x_2^r - x_1^r}}{{x_3^r - x_1^r}}{{\left( {f\left( {{x_3}} \right)} \right)}^p}} \right]^{\frac{1}{p}}}. \end{array}$

 $\begin{array}{l} \left( {x_3^r - x_2^r} \right){\left( {f\left( {{x_1}} \right)} \right)^p} + \left( {x_1^r - x_3^r} \right){\left( {f\left( {{x_2}} \right)} \right)^p} + \\ \;\;\;\;\;\;\left( {x_2^r - x_1^r} \right){\left( {f\left( {{x_3}} \right)} \right)^p} \ge 0. \end{array}$

r＜0时，类似地，有

 $\begin{array}{l} \left( {x_3^r - x_2^r} \right){\left( {f\left( {{x_1}} \right)} \right)^p} + \left( {x_1^r - x_3^r} \right){\left( {f\left( {{x_2}} \right)} \right)^p} + \\ \;\;\;\;\;\;\left( {x_2^r - x_1^r} \right){\left( {f\left( {{x_3}} \right)} \right)^p} \le 0. \end{array}$

f(x)在I上是MM-凹的，则以上证明中的不等号反向，故定理8(ⅰ)的后半部分成立.

 $\begin{array}{l} \left( {p - 1} \right){\left( {f'\left( x \right)} \right)^2}x + f\left( x \right)f''\left( x \right)x + \\ \;\;\;\;\;\;\;\left( {1 - r} \right)f\left( x \right)f'\left( x \right) \ge \left( \le \right)0\left( {p \ne 0} \right). \end{array}$ (7)

x1, x2I，不失一般性，设x1x2，令x=[tx1r+(1－t)x2r]${\frac{1}{r}}$=M2[r](t, 1－t; x1, x2)(t∈[0, 1])，则由2个正数的幂平均性质知，x∈[x1, x2]⊆I.

 $\begin{array}{l} \varphi '\left( t \right) = \frac{p}{r}{\left( {f\left( x \right)} \right)^{p - 1}}f'\left( x \right){x^{1 - r}}\left( {x_1^r - x_2^r} \right),\\ \varphi ''\left( t \right) = \frac{p}{{{r^2}}}{\left( {f\left( x \right)} \right)^{p - 1}}{x^{1 - 2r}}{\left( {x_1^r - x_2^r} \right)^2}\left[ {\left( {p - 1} \right) \times } \right.\\ \;\;\;\;\;\;\;\;\;{\left( {f'\left( x \right)} \right)^2}x + f\left( x \right)f''\left( x \right)x + \\ \;\;\;\;\;\;\;\;\;\left. {\left( {1 - r} \right)f\left( x \right)f'\left( x \right)} \right], \end{array}$

 $\begin{array}{l} \varphi ''\left( t \right) \ge \left( \le \right)0 \Leftrightarrow p\left[ {\left( {p - 1} \right){{\left( {f'\left( x \right)} \right)}^2}x + } \right.\\ \;\;\;\;\;\;\;\;\;f\left( x \right)f''\left( x \right)x + \left. {\left( {1 - r} \right)f\left( x \right)f'\left( x \right)} \right] \ge \left( \le \right)0. \end{array}$

 $\begin{array}{l} \left( {p - 1} \right){\left( {f'\left( x \right)} \right)^2}x + f\left( x \right)f''\left( x \right)x + \\ \;\;\;\;\;\;\left( {1 - r} \right)f\left( x \right)f'\left( x \right) \ge 0. \end{array}$

p＜0时，f(x)是I上的MM-凸函数⇔∀x1, x2I, 函数φ(t)=[f(M2[r](t, 1－t; x1, x2))]p是[0, 1]上的凹函数⇔φ″(t)≤0⇔p[(p-1)(f′(x))2x+f(x)f″(x)x+(1-r)f(x)f′(x)]≤0⇔(p－1)(f′(x))2x+f(x)f″(x)x+(1－r)f(x)f′(x)≥0.

2 MM-凸函数的复合运算性质

(ⅰ)若y=f(u)为I上严格递增的MM-凸函数，u=μ(x)为A上的r次幂平均凸函数(r-平均凸函数[7])，则y=f(μ(x))为A上的MM-凸函数；

(ⅱ)若y=f(u)为I上严格递减的MM-凸函数，u=μ(x)为A上的r次幂平均凹函数(r-平均凹函数)，则y=f(μ(x))为A上的MM-凸函数；

(ⅲ)若y=f(u)为I上严格递增的MM-凹函数，u=μ(x)为A上的r次幂平均凹函数(r-平均凹函数)，则y=f(μ(x))为A上的MM-凹函数；

(ⅳ)若y=f(u)为I上严格递减的MM-凹函数，u=μ(x)为A上的r次幂平均凸函数(r-平均凸函数)，则y=f(μ(x))为A上的MM-凹函数.

x1, x2AR+及∀t∈[0, 1]，显然有M2[r](t, 1－t; x1, x2)∈[min{x1, x2}, max{x1, x2}]⊆A，因为μ:ABI，所以，μ(x1), μ(x2), μ(M2[r](t, 1－t; x1, x2))∈BI, 且

 $\begin{array}{l} {\rm{M}}_2^{\left[ r \right]}\left( {t,1 - t;\mu \left( {{x_1}} \right),\mu \left( {{x_2}} \right)} \right) \in \left[ {\min \left\{ {\mu \left( {{x_1}} \right),\mu \left( {{x_2}} \right)} \right\},} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\max \left\{ {\mu \left( {{x_1}} \right),\mu \left( {{x_2}} \right)} \right\}} \right] \subseteq {\bf{B}} \subseteq {\bf{I}}, \end{array}$

u=μ(x)是A上的r次幂平均凸函数，所以

 $\begin{array}{l} \mu \left( {{\rm{M}}_2^{\left[ r \right]}\left( {t,1 - t;{x_1},{x_2}} \right)} \right) \le \\ \;\;\;\;\;\;\;{\rm{M}}_2^{\left[ r \right]}\left( {t,1 - t;\mu \left( {{x_1}} \right),\mu \left( {{x_2}} \right)} \right), \end{array}$

y=f(u)为I上严格递增的MM-凸函数，所以

 $\begin{array}{l} f\left( {\mu \left( {{\rm{M}}_2^{\left[ r \right]}\left( {t,1 - t;{x_1},{x_2}} \right)} \right)} \right) \le \\ \;\;\;\;\;\;\;\;f\left( {{\rm{M}}_2^{\left[ r \right]}\left( {t,1 - t;\mu \left( {{x_1}} \right),\mu \left( {{x_2}} \right)} \right)} \right) \le \\ \;\;\;\;\;\;\;\;{\rm{M}}_2^{\left[ p \right]}\left( {t,1 - t;f\left( {\mu \left( {{x_1}} \right)} \right),f\left( {\mu \left( {{x_2}} \right)} \right)} \right), \end{array}$

y=f(μ(x))是A上的MM-凸函数.

(ⅰ)若y=f(u)为I上严格递增的r次幂平均凸函数(r-平均凸函数)，u=μ(x)为A上的MM-凸函数，则y=f(μ(x))为A上的MM-凸函数；

(ⅱ)若y=f(u)为I上严格递减的r次幂平均凸函数(r-平均凸函数)，u=μ(x)为A上的MM-凹函数，则y=f(μ(x))为A上的MM-凸函数；

(ⅲ)若y=f(u)为I上严格递减的r次幂平均凹函数(r-平均凹函数)，u=μ(x)为A上的MM-凸函数，则y=f(μ(x))为A上的MM-凹函数；

(ⅳ)若y=f(u)为I上严格递增的r次幂平均凹函数(r-平均凹函数)，u=μ(x)为A上的MM-凹函数，则y=f(μ(x))为A上的MM-凹函数.

(ⅰ)若y=f(u)为I上严格递增的AM-凸函数，u=μ(x)为A上的MA-凸函数，则y=f(μ(x))为A上的MM-凸函数；

(ⅱ)若y=f(u)为I上严格递减的AM-凸函数，u=μ(x)为A上的MA-凹函数，则y=f(μ(x))为A上的MM-凸函数；

(ⅲ)若y=f(u)为I上严格递增的AM-凹函数，u=μ(x)为A上的MA-凹函数，则y=f(μ(x))为A上的MM-凹函数；

(ⅳ)若y=f(u)为I上严格递减的AM-凹函数，u=μ(x)为A上的MA-凸函数，则y=f(μ(x))为A上的MM-凹函数.

x1, x2A及∀t∈[0, 1], 有M2[r](t, 1－t; x1, x2)∈A，因为μ:ABI，所以，μ(x1), μ(x2)，μ(M2[r](t, 1－t; x1, x2))∈BI，且M2[r](t, 1－t; μ(x1), μ(x2))∈[min{μ(x1), μ(x2)}, max{μ(x1), μ(x2)}]⊆BI.

u=μ(x)是A上的MA-凸函数，所以

 $\begin{array}{l} \mu \left( {{\rm{M}}_2^{\left[ r \right]}\left( {t,1 - t;{x_1},{x_2}} \right)} \right) \le \\ \;\;\;\;\;\;\;\;\;{\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;\mu \left( {{x_1}} \right),\mu \left( {{x_2}} \right)} \right), \end{array}$

y=f(u)为I上严格递增的AM-凸函数，所以

 $\begin{array}{l} f\left( {\mu \left( {{\rm{M}}_2^{\left[ r \right]}\left( {t,1 - t;{x_1},{x_2}} \right)} \right)} \right) \le f\left( {{\rm{M}}_2^{\left[ 1 \right]}\left( {t,1 - t;} \right.} \right.\\ \;\;\;\;\;\;\;\left. {\left. {\mu \left( {{x_1}} \right),\mu \left( {{x_2}} \right)} \right)} \right) \le {\rm{M}}_2^{\left[ p \right]}\left( {t,1 - t;f\left( {\mu \left( {{x_1}} \right)} \right),} \right.\\ \;\;\;\;\;\;\;\left. {f\left( {\mu \left( {{x_2}} \right)} \right)} \right), \end{array}$

y=f(μ(x))是A上的MM-凸函数.

(ⅰ)若y=f(u)为I上严格递增的GM-凸函数，u=μ(x)为A上的MG-凸函数，则y=f(μ(x))为A上的MM-凸函数；

(ⅱ)若y=f(u)为I上严格递减的GM-凸函数，u=μ(x)为A上的MG-凹函数，则y=f(μ(x))为A上的MM-凸函数；

(ⅱ)若y=f(u)为I上严格递增的GM-凹函数，u=μ(x)为A上的MG-凹函数，则y=f(μ(x))为A上的MM-凹函数；

(ⅳ)若y=f(u)为I上严格递减的GM-凹函数，u=μ(x)为A上的MG-凸函数，则y=f(μ(x))为A上的MM-凹函数.

(ⅰ)若y=f(u)为I上严格递增的HM-凸函数，u=μ(x)为A上的MH-凸函数，则y=f(μ(x))为A上的MM-凸函数；

(ⅱ)若y=f(u)为I上严格递减的HM-凸函数，u=μ(x)为A上的MH-凹函数，则y=f(μ(x))为A上的MM-凸函数；

(ⅲ)若y=f(u)为I上严格递增的HM-凹函数，u=μ(x)为A上的MH-凹函数，则y=f(μ(x))为A上的MM-凹函数；

(ⅳ)若y=f(u)为I上严格递减的HM-凹函数，u=μ(x)为A上的MH-凸函数，则y=f(μ(x))为A上的MM-凹函数.

3 MM-凸函数的Jensen型不等式

 $\begin{array}{l} f\left( {{\rm{M}}_n^{\left[ r \right]}\left( {{t_1},{t_2}, \cdots ,{t_n};{x_1},{x_2}, \cdots ,{x_n}} \right)} \right) \le \left( \ge \right)\\ \;\;\;\;\;\;\;\;{\rm{M}}_n^{\left[ p \right]}\left( {{t_1},{t_2}, \cdots ,{t_n};f\left( {{x_1}} \right),f\left( {{x_2}} \right), \cdots ,f\left( {{x_n}} \right)} \right). \end{array}$

 $\begin{array}{l} g\left( {{\rm{M}}_n^{\left[ r \right]}\left( {{t_1},{t_2}, \cdots ,{t_n};{x_1},{x_2}, \cdots ,{x_n}} \right)} \right) \le \\ {\rm{M}}_n^{\left[ 1 \right]}\left( {{t_1},{t_2}, \cdots ,{t_n};g\left( {{x_1}} \right),g\left( {{x_2}} \right), \cdots ,g\left( {{x_n}} \right)} \right)\\ \Rightarrow {\left[ {f\left( {{\rm{M}}_n^{\left[ r \right]}\left( {{t_1},{t_2}, \cdots ,{t_n};{x_1},{x_2}, \cdots ,{x_n}} \right)} \right)} \right]^p} \le \\ {\rm{M}}_n^{\left[ 1 \right]}\left( {{t_1},{t_2}, \cdots ,{t_n};{{\left( {f\left( {{x_1}} \right)} \right)}^p},{{\left( {f\left( {{x_2}} \right)} \right)}^p}, \cdots ,{{\left( {f\left( {{x_n}} \right)} \right)}^p}} \right)\\ \Rightarrow f\left( {{\rm{M}}_n^{\left[ r \right]}\left( {{t_1},{t_2}, \cdots ,{t_n};{x_1},{x_2}, \cdots ,{x_n}} \right)} \right) \le \\ {\left[ {{\rm{M}}_n^{\left[ 1 \right]}\left( {{t_1},{t_2}, \cdots ,{t_n};{{\left( {f\left( {{x_1}} \right)} \right)}^p},{{\left( {f\left( {{x_2}} \right)} \right)}^p}, \cdots ,{{\left( {f\left( {{x_n}} \right)} \right)}^p}} \right)} \right]^{\frac{1}{p}}} = \\ {\rm{M}}_n^{\left[ p \right]}\left( {{t_1},{t_2}, \cdots ,{t_n};f\left( {{x_1}} \right),f\left( {{x_2}} \right), \cdots ,f\left( {{x_n}} \right)} \right). \end{array}$

p＜0时，类似可证结论成立.

f(x)是区间I上MM-凹函数的情形同理可证.

 $\begin{array}{l} f\left[ {{\rm{M}}_n^{\left[ r \right]}\left[ {\frac{{{q_1}}}{{{q_1} + {q_2} + \cdots + {q_n}}},\frac{{{q_2}}}{{{q_1} + {q_2} + \cdots + {q_n}}}, \cdots ,} \right.} \right.\\ \left. {\left. {\;\;\;\;\;\;\;\;\frac{{{q_n}}}{{{q_1} + {q_2} + \cdots + {q_n}}};{x_1},{x_2}, \cdots ,{x_n}} \right]} \right] \le \left( \ge \right)\\ \;\;\;\;\;\;\;\;{\rm{M}}_n^{\left[ p \right]}\left[ {\frac{{{q_1}}}{{{q_1} + {q_2} + \cdots + {q_n}}},\frac{{{q_2}}}{{{q_1} + {q_2} + \cdots + {q_n}}}, \cdots ,} \right.\\ \left. {\;\;\;\;\;\;\;\;\frac{{{q_n}}}{{{q_1} + {q_2} + \cdots + {q_n}}};f\left( {{x_1}} \right),f\left( {{x_2}} \right), \cdots ,f\left( {{x_n}} \right)} \right]. \end{array}$

 $\begin{array}{l} f\left( {{\rm{M}}_n^{\left[ r \right]}\left( {\frac{1}{n},\frac{1}{n}, \cdots ,\frac{1}{n};{x_1},{x_2}, \cdots ,{x_n}} \right)} \right) \le \left( \ge \right)\\ \;\;\;\;\;\;\;\;{\rm{M}}_n^{\left[ p \right]}\left( {\frac{1}{n},\frac{1}{n}, \cdots ,\frac{1}{n};f\left( {{x_1}} \right),f\left( {{x_2}} \right), \cdots ,f\left( {{x_n}} \right)} \right). \end{array}$

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