一个加权的Pólya-Sze积分不等式

A Weighted Integral of Pólya-Szeg Inequality

研究了Pólya-Sze不等式的加权积分推广式,并在一定程度得到了Bunziakowski-Schwarz不等式.利用归纳和类比方法,得到了Pólya-Sze不等式的加权加强推广式后,给出了一种简洁有趣的构造性证明方法.结果表明运用新的Pólya-Sze加权积分不等式,能够明显地解决Pólya-Sze不等式.通过归纳类比方法和构造性方法,确定了这两种基本方法是解决这一类解析不等式的有效手段.

The objective of the paper is to study the extension of the general expression of weighted integral of Pólya-Szeg's,and to obtain the Bunziakowski-Schwarz inequality to some extent.Methods of induction analogy are used to obtain a general expression of Pólya-Szeg's weighted strengthened inequality,with a concise,interesting and constitutive proof given.The results show that the Pólya-Szeg's inequality can be obviously solved by utilizing the new Pólya-Szeg's weighted strengthened inequality.It is concluded that the induction analogy and constructive methods are good methods for solving the analytic inequality.