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幾何凸函數(shù)若干問題的探討摘 要幾何凸函數(shù)是一類非常重要的函數(shù),廣泛應(yīng)用于數(shù)學(xué)規(guī)劃、控制論等領(lǐng)域,它在判定函數(shù)的極值、研究函數(shù)的圖像以及不等式的證明諸方面都有廣泛的應(yīng)用。文章首先探究凸函數(shù)這類性質(zhì)特殊的函數(shù)的各種定義及幾何意義,探究幾何凸函數(shù)在不同學(xué)科上的應(yīng)用。討論了幾何凸函數(shù)的一些性質(zhì),并運(yùn)用其性質(zhì)證明若干不等式,介紹函數(shù)幾何凸性的幾種判別方法。文章證明幾何凸性函數(shù)的幾個生成定理,并給出一元高次多項(xiàng)式函數(shù)幾何凸性的幾個結(jié)論,最后給出幾何凸函數(shù)定義的推廣或類似,供進(jìn)一步研究之用。給出凸集的定義,借助凸集來引入凸函數(shù)的幾何直觀性定義,并借此給出幾何凸函數(shù)的解析式定義,進(jìn)行一系列的分析、類比、歸納,用實(shí)例說明用幾何凸函數(shù)解決實(shí)際問題的重要意義。由于幾何凸函數(shù)理論的廣泛性,因此對其理論的研究成果還有待進(jìn)一步的深入和推廣。如何推廣函數(shù)的凸性概念,使得在更廣泛的函數(shù)范圍內(nèi),凸函數(shù)的許多重要性質(zhì)仍然得以保留,凸規(guī)則的大多數(shù)結(jié)果能推廣到非凸規(guī)則,已構(gòu)成了數(shù)學(xué)規(guī)劃研究領(lǐng)域的當(dāng)前趨勢之一,所以研究幾何凸函數(shù)的一些定義和性質(zhì)就顯得十分必要了。文章所述內(nèi)容使我們能夠快速獲取大量有關(guān)幾何凸函數(shù)的重要內(nèi)容,從而使解決一類特別繁雜不等式證明、最優(yōu)化等問題變的別出一格。關(guān)鍵字:凸函數(shù);幾何凸函數(shù);不等式The Discussion on some Questions of Geometric Convex FunctionAbstractGeometric convex function is a very important function, widely used in mathematical programming, control theory and other fields determine the extremism of the function, the image of the research function and the inequality proof aspects have a wide range of applications. The article first explores the various definitions and geometric meaning of convex function of the nature of the special function to explore the geometric convex function on the different disciplines. Discussed some properties of the geometry of convex function, and use nature to prove certain inequalities on the function of geometric convexity several discriminate methods.Article to prove the convexity of the function of the geometry of several generation theorems, and several conclusions given a high geometric convexity of a polynomial function. Finally, promotion or similar geometric convex function defined for the purposes of further study. Given the definition of convex sets, with the convex set to the introduction of geometric definition of convex function, and to give the analytical definition of the geometric convex function, a series of analyzes, analogy, induction, illustrated by an example using Geometric Convex Function to solve practical problems of great significance.Due to the extensive geometric convex function theory, its theoretical research remains to be further indepth and promotion. How to promote the concept of convexity of the function, in a wider range of functions within many important properties of convex functions is still preserved, most of the results of the convex rules can be extended to nonconvex rules constituted the current trends of the research areas of mathematical programming one study the geometry of convex function definition and nature of it is very necessary. Articles about so that we can quickly obtain a large number of important geometric convex function, so that to solve a particular category of plex inequality proved optimization problems do not bee a grid.Keywords:convexfunction。geometricconvex function。inequality目 錄引 言 1第1章 基礎(chǔ)知識 2 凸集與凸函數(shù) 2 幾何凸函數(shù)的定義 2 幾何凸函數(shù)與凸函數(shù)的區(qū)別 3第2章 幾何凸函數(shù)的基本性質(zhì)及應(yīng)用 5 幾何凸函數(shù)的基本性質(zhì) 5 幾何凸函數(shù)的應(yīng)用 8第3章 函數(shù)幾何凸性的幾個問題 9 幾何凸函數(shù)的判別問題 9 幾何凸(凹)函數(shù)之“和積商”的幾何凸性 10 幾何凹函數(shù)之和的奇妙“凸”性 11 由凸函數(shù)生成幾何凸函數(shù) 11 由已知的較簡單的幾何凸函數(shù)生成較復(fù)雜的幾何凸函數(shù) 12 由幾何凸函數(shù)的定義域變換生成 13 一元高次多項(xiàng)式函數(shù)的幾何凸性 14第4章 幾何凸函數(shù)的積分不等式 17 介紹幾類平均 17 積分與幾何凸函數(shù) 17結(jié)論與展望 20致 謝 21參考文獻(xiàn) 22附錄 23附錄A:英文文獻(xiàn)及翻譯 23附錄B:列入的主要參考文獻(xiàn)題錄及摘要 31引 言幾何凸函數(shù)是一個與凸函數(shù)平行的概念,作為一個研究課題,它在二十幾年以前就已經(jīng)出現(xiàn)并取得了初步研究成果,但作為一個概念提出卻是十幾年的事。十幾年過去了,有關(guān)幾何凸函數(shù)的研究成果不斷涌現(xiàn)。從2003年開始,幾何凸函數(shù)逐漸成為國內(nèi)不等式研究的熱點(diǎn)之一,國內(nèi)外數(shù)學(xué)工作者在這方面已取得了不少原創(chuàng)性的研究成果,為不等式的理論發(fā)展提供了一個新的研究平臺。幾何凸函數(shù)是一類重要的函數(shù),它的概念最早見于Jensen[1905]著述中。它在純粹數(shù)學(xué)和應(yīng)用數(shù)學(xué)的眾多領(lǐng)域中具有廣泛的應(yīng)用,現(xiàn)已成為數(shù)學(xué)規(guī)劃、對策論、數(shù)理經(jīng)濟(jì)學(xué)、變分學(xué)和最優(yōu)控制等學(xué)科的理論基礎(chǔ)和有力工具。為了理論上的突破,加強(qiáng)它們在實(shí)踐中的應(yīng)用,產(chǎn)生了廣義凸函數(shù)。幾何凸函數(shù)有許多良好的性質(zhì),其中一個很重要的性質(zhì)就是:在凸集中,幾何凸函數(shù)的任何局部最小也是全局最小。它在數(shù)學(xué)的許多領(lǐng)域中都有著廣泛的應(yīng)用,現(xiàn)已成為數(shù)學(xué)規(guī)劃、對策論、數(shù)理經(jīng)濟(jì)學(xué)、變分學(xué)和最優(yōu)控制等學(xué)科的理論基礎(chǔ)和有力工具。但是凸函數(shù)的局限性也很明顯,因?yàn)樵趯?shí)際問題中,大量的函數(shù)都是非凸的。為了理論上的突破,加強(qiáng)它們在實(shí)踐中的應(yīng)用,60年代中期產(chǎn)生了凸分析,凸函數(shù)的概念也按多種途徑進(jìn)行推廣,或?qū)τ诔橄罂臻g的推廣,或?qū)τ谏厦嫣岬降牟坏仁降耐茝V,然后提出了廣義凸函數(shù)的概念。60年代后期,先是有Mangasarian把凸函數(shù)的概念推廣到擬凸函數(shù)(quasiconvex functions)和偽凸函數(shù)(pseudoconvex funct