数学分析中Jensen不等式由浅入深进行教学

Studing Jensen’s Inequality form Easy to Hard in Mathematical Analysis

不等式的证明一直是数学分析教学的重点和难点,运用Jensen不等式能使不等式的证明变得清晰明了.目前大多数学分析教材对Jensen不等式叙述零散且证明复杂繁琐不统一.在数学分析中由浅入深的系统学习离散型和积分型Jensen不等式,并利用凸函数的性质给出了这几种类型Jensen不等式的简单统一证明尤为重要.

The proof of inequality is always the key and difficult points in the teaching of mathematical analysis. The use of Jensen inequality can make the proof of inequality simple and clear. The current textbooks of mathematical analysis are not unified in expounding and proof of the Jensen inequalities. Through a systematic study of discrete and integral Jensen inequalities from easy to hard, by use of the property of convex function, a unified and simplified method is worked out to prove several types of Jensen inequalities.