Roger-Hlder不等式的积分加强推广式及其应用

Enhancement popularization and application of Roger-Hlder's integral inequality

从离散型Roger-Hlder不等式出发,通过归纳和类比的思想方法,得到了相应的积分型不等式,又在研究积分型不等式的基础上,推广和加强了其积分型不等式,并运用算术几何平均值不等式的推广形式、Jensen不等式和构造性方法给出了十分简洁有趣的证明.最后讨论了Roger-Hlder积分不等式的加强推广式与著名的Cauchy-Schwarz积分不等式、Kantorovich积分不等式及Radon积分不等式的联系,凸显其应用的广泛性和内在规律性.

By means of induction and analogy, the paper popularized Roger-Hoelder＇s dispersed inequality, in obtaining the corresponding integral inequality. It also provides the constructive demonstration in furthering the popularization and enhancement of its integral-type inequality on the base of first period study by using the popularized form of arithmetic-geometry mean inequality, Jensen inequality and the constructive demonstration. Finally the paper discussed the relationship between the popularized and reinforced form of Roger-Hoelder＇s integral inequality, and the famous Cauchy-Schwarz integral inequality, Kantorovich integral inequality and Randon integral inequality to show the extensiveness application and inherent regularity of Roger-Hoelder＇s integral inequality