急寻数学论文摘要翻译!

急寻数学论文摘要翻译!
摘要如下：
凹凸性是函数的一种独特性质,它有几何意义,也有代数意义,凹凸性与凸（凹）函数在不等式、泛函分析、最优理论、数学规划、运筹学、控制论及数理经济学等数学领域的理论研究与应用当中都有着广泛的应用,这方面的研究已经和凸集的研究一起形成了一个专门的方向,即凸分析.它提出的背景非常浅近,仅仅从函数图像的研究中就可获得凹凸性提出的必要性,但是此处所体现的数形结合的数学思想却是非常重要的.
本文首先介绍了函数凹凸性的一些基本概念,其次给出了判断函数凹凸性的一些条件,在这里也探讨了低阶导数的几何意义,最后介绍了函数凹凸性的简单应用,在引出了拐点定义的条件下展示了凹凸性在函数作图中的作用,在引出了詹森不等式之后展示了凹凸性在不等式证明当中的妙用.
关键词 函数；凹凸性；导数；拐点；詹森不等式

The concavity is a charateristic feature of a function.It has geometric and algebraic meanings.The concavity and convex(concave) functions has massive applications in inequality,functional analysis,optimization theorem,mathematical planning,operational research,cybernetics and many other mathematical,physical and economical aspects for theoretical research and application.With convex set,this aspect of study has formed a particular research direction,convex analysis.The background of its appearance is relatively simple,which is that merely from function graph studies,the importance of concavity can be drawn,but it is significant to be noted that the mathematical thought of the combination of figure and chart shown here is very important.
This article firstly describes several fundamental concepts of function concavity,secondly provides a few conditions for determining the concavity of a function,where the geometric meaning of low-level derivatives is also discussed,and finally introduces a few simple applications of function concavity.With the definition of inflexion,it displays the function of concavity in a function plot.With Jensen's Inequality,it showcases the effective usage of concavity in inequality deduction.
Key words:function,concavity,derivative,inflexion,Jensen's Inequality