Necessary and Sufficient Conditions for Inequalities between the Generalized Muirhead Mean and Arithmetic, Harmonic and Geometric Means

摘要

For real numbers a and b with a + b≠ 0，the generalized Muirhead mean M(a, b;x,y) of two positive real numbers x and y with parameters a and b is defined by M(a,b;x,y) = [(X~ay~b +X~by~a)/2]~1(a+b). In the article， the authors present the sufficient and necessary conditions on a and b such that the inequalities M(a,b;x,y)? A(x,y), M(a,b;x,y)?H(x,y), and M(a,b;x,y)? G(x,y) hold for all x,y > 0 with x≠y, where A(x,y),H(x,y), and G(x,y) are respectively the arithmetic, harmonic, and geometric means of x and y.