Some Inequalities Related To Power Mean

In this paper, we investigate the some equalities related to Heronian mean Hm(a, b), geometric mean G(a, b), and power mean Mp(a, b) of two positive real number a and b, and the inequality relationship of harmonic mean H(a, b), logarithmic mean L(a,b), and power mean Mp(a,b) of two positive real number a and b. Under some certain constant exponent, we obtain a suitable order p of power mean Mp(a, b). Moreover, we use Taylar expansion to examine that this exponent is the optimal one.Firstly, in Charpter 2, we study the relationship of Heronian mean Hm(a, b), geometric mean G(a, b), and power mean Mp(a, b) of two positive real number. The purpose of this charpter is to founded some inequalities of Heronian mean Hm(a,b), geometric mean G(a,b), and power mean Mp(a,b) of two positive real number. By using the method of limit comparison, together with Taylor expansion of special function, we obtain some inequalities of Heronian mean, geometric mean, and power mean under some certain constant exponent conditions. The Taylor expansion imply that the order p of these inequalities is optimal to our results if p is some special contant. At same time, we also examine that our results is in accord to the classical ones.Later, in Charpter 3, we study the relationship of harmonic mean H(a,b), logarithmic L(a,b), and power mean Mp(a,b) of two positive real number. The purpose of this charpter is to founded some inequalities of logarithmic L(a,b), harmonic mean H(a,b), and power mean Mp(a,b) of two positive real number. With the same progress, be using the method of limit comparison, together with Taylor expansion of special function, we investigate the convexity of some special function to obtain some inequalities of logarithmic, harmonic mean, and power mean under some certain constant exponent conditions. We also use the Taylor expansion to examine the order p of these inequalities is optimal to our results if p is some special contant. Moreover, we also examine that our results is in accord to the classical ones.