Optimal Inequalities For One-parameter Mean

Analytic inequalities are widely used in mathematics, physics and engineering technology and other related fields. As an important part of the basic inequalities, the mean inequality plays a very important role in academic research, for example, it can be used in solving maximum value problems,the proof of convergence of series and design optimization problems. In the present paper, we answer the question:for0<α<1fixed, what are the greatest value p1(α), p2(α),p3(α),p4(α) and the least values q1(α),q2(α),q3(α),q4(α), such that the inequalities hold for all a,b>0with a≠b? where for p∈R, the one-parameter mean Jp(a,b),arithmetic mean A(a,b) and geometric mean G(a,b) of two positive real numbers a and b are defined by and G(a,b)=(?)ab, respectively.