利用函数凸性证明不等式

Using Function Convexity to Prove Inequality

目的不等式在高等数学中的应用非常广泛,地位举足轻重,正确使用不等式可使复杂的数学问题简单化,由于它的应用方法灵活、抽象、逻辑性较强,所以不易掌握。而在不等式的证明中,有些看似复杂的问题,利用函数的凸性可以很轻松地解决。方法从解析定义、几何解释和直观描述性定义3个方面介绍凸函数定义,再揭示凸函数的判定定理和性质,其中重点把握凸函数的Jensen不等式,在前述内容的基础上建立凸函数框架统一证明初等不等式,并推证一些著名不等式。结果通过举例的方式,巧妙地构造凸函数,利用函数凸性加以证明,确实使大部分不等式的证明更加简洁明了。结论在高等数学教学中,利用函数的单调性给出了特殊函数不等式的证明方法,使复杂问题简单化,学生在学习过程中容易接受,并增加学生学习高等数学的积极性。但不等式的证明方法繁多,难度、技巧性都很大,比如导数定义法、拉格朗日中值定理法、幂级数展开法等,把应用这些方法证明不等式和利用函数凸性证明不等式结合起来,相互补充,不断总结归纳,可以拓宽知识面,提升解题能力。

Objective Inequality is widely used in the subject of higher mathematics and plays an important role.Correct use of inequality can simplify complex mathematical problems,but its application method is flexible,abstract and logical.So it is not easy to master and control.In the proof of inequality,some seemingly complex problems can be easily solved by the convexity of functions.Methods The definition of convex function was introduced from three aspects:analytic definition,geometric interpretation and intuitive descriptive definition,and then the judgment theorem and properties of convex function were revealed.The Jensen inequality of convex function was emphasized,and the convex function frame was established to prove the elementary inequality based on content above.Some famous inequalities were deduced.Results Through examples,convex functions were ingeniously constructed and proved by using convexity of functions,which makes the proof of most inequalities more concise and clearer.Conclusion In higher mathematics teaching,the use of monotony of functions gives the proof method of special function inequalities,which simplifies complex problems and makes students easy to accept and increase students’ enthisasm in the learning process.There are many ways to prove inequalities,such as derivative definition method,Lagrange mean theorem method,power series expansion method and so on.These methods can be combined with convexity of function to prove inequalities.They complement each other.We should constantly sum up and generalize,broaden our knowledge and improve our ability to solve problems.