he aim of this dissertation is introduce and study of some improvements of Heron mean and the refinements of Young’s inequalities for matrices. This dissertation is written based on article “Some results of Heron mean and Young’sinequalities,ChangsenandYonghuiRen(2018). Fortwopositiverealnumbera,band 0 ? ? ? 1,the equality F?(a,b) = (1??) ?ab #43;v a #43;b/2, is called Heron mean. The inequality a^? b^1?? ? ?a #43; (1??)b, is called Young’s inequality. if ? = 1 /2, then the above inequality is called arithmetic-geometric means inequality. Heron mean inequality is the interpolation between arithmetic and geometric mean. The first refinements of Young’s inequality was shown by Kittaneh and Manasrah in 2010 as follow. (a^? b^(1??))^2 #43; min{?,1??}^2 (a?b)^2 ? (?a #43; (1??)b)^2.