琴生(Jensen)不等式的应用

Application of Jensen inequality

通过对凸函数的描述 ,凸函数与不等式的关系 ,得到了琴生 (Jensen)不等式 .利用凸函数或微积分中二阶导数符号可以直接给出一连串不等式 .如由 (0 ,π)内 (sinx )″<0得到在 ΔABC中有 sin A +sin B +sin C≤3 32 ;0 ,π2 内 (tgx )″>0得到在锐角 ΔABC中有 tg A +tg B +tg C≥ 3 3 .从而说明凸函数或函数在某区间上二阶导数符号不变时应用琴生不等式可得到一系列不等式 ,为数学竞赛和初等数学构造一些不等式问题提供了理论依据 .

This paper discusses the acquisition of Jensen Inequation by expatiating on convex functions and the relationship between convex functions and inequations. That is, a succession of inequations can be obtained directly by using convex functions or quadratic derivative symbols in calculous. Thereby, it can be concluded that, in the case of the stability of quadratic derivative symbol, a series of subsequent inequality will be obtained on the basis of Jensen Inequation, thus providing some theoretical basis for some inequation problems in mathematic contests and elementary mathematics constructions as well.