构造法思想与琴生不等式融合在高中不等式证明中的应用研究

A Study on the Application of Construction Methods and the Qin-Sheng Inequality in the Proof of High School Inequalities

摘要: 琴生不等式是高中数学中证明不等式的重要工具之一，其根据凸函数的核心性质，将复杂的不等式问题转化为较为简单的函数值与平均值的关系问题。而构造法是衔接待证不等式与琴生不等式的关键所在，根据高中不等式的结构特征构造合适的凸函数、权重系数，可快速满足琴生不等式的应用条件。本文以高中数学知识为背景，梳理凸函数、琴生不等式的基础内容，结合典型的高中不等式例题，分析构造法与琴生不等式融合的具体应用方法，总结解题规律，为高中数学中不等式的证明提供更高效思路。

Abstract: The Quines-Schoenberg inequality is one of the key tools for proving inequalities in high school mathematics. Based on the core properties of convex functions, it transforms complex inequality problems into simpler problems involving the relationship between function values and their averages. The construction method is the key to linking the proof of inequalities with the Quines-Schoenberg inequality; by constructing appropriate convex functions and weighting coefficients based on the structural characteristics of high school inequalities, one can quickly satisfy the application conditions of the Quines-Schoenberg inequality. Against the backdrop of high school mathematics, this paper reviews the fundamentals of convex functions and the Kinesis inequality. By examining typical high school inequality problems, it analyzes specific application methods that integrate the construction method with the Kinesis inequality, summarizes problem-solving patterns, and provides a more efficient approach to proving inequalities in high school mathematics.