The Optimal Inequalities Of Power Means

In this paper, we mainly study some inequalities related with power means. And it includes five chapters.In the first chapter, we introduce the definitions of the inequalities, the common means and some results about mean inequalities, especially introduce many inequalities with power mean: then briefly state the application of the two-component mean in mathematical theory, especially the extensive application of power mean in various domains; finally we explain the main results we obtained in this paper.In the second chapter, firstly we prove an optimal inequality with seiffert mean in order to lead mathematic theory applied in this paper, then discuss the optimal upper power mean Mp(a,b) bound for the products of seiffert mean and arithmetic mean:Pα(a.b)A1-α(a,b).In the third chapter, using the thinking method of analysis and some tips, we mainly discuss the inequality with power mean, harmonic mean and logarithmic mean; finally we obtain optimal inequalities with power means and the convex combination of harmonic and logarithmic means, and optimal inequalities with power mean Mp(a.b) and the product of harmonic and logarithmic means: Hα(a.b)L1-α(a,b).In the forth chapter, based on monotonicity of functions and concave convex characteristics analysis.we mainly study the inequality with arithmetic mean, heron mean and power mean; finally we obtain optimal inequalities with power means and the convex combination of arithmetic and heron means, and optimal inequalities with power mean Mp(a,b) and the products of arithmetic mean and heron means: Aα(a.b)H1-αe(a.b)In the fifth chapter, based on the thinking method of analysis and by the mathematic Microsoft,we mainly study the inequality with harmonic mean, heron mean and power mean; finally we obtain optimal inequalities with power mean Mp(a,b) and the products of harmonic mean and heron means: Hα(a,b)He1-α(a,b), and optimal inequalities with power means and the convex combination of harmonic and heron means.