Sharp Inequalities Related To One-parameter Mean And Gini Mean

Various means of two positive numbers a and b and their inequalities have wide applications in the fields of statistics, mathematics and engineering. In this paper, we consider sharp inequalities related to one-parameter mean and Gini mean. We find optimal values for p, q, s1, and s2, such that the inequalities and hold for all a, b>0with a≠b, where A(a, b)=a+b/2, G(a,b)=(?) and H(a,b)=2ab/a+b, are one-parameter mean, Gini mean, arithmetic mean, geometric mean, harmonic mean for two positive numbers a and b, respectively.