On the Log-convexity of two-parameter homogeneous functions

Suppose f (x, y) is a positive homogeneous function defined on U((C) double under barR(+) x R+) then call f ((a(P),a(P))/f (a(q), b(q))) (1/p-q) two-parameter homogeneous function and denote by Hf (a, b;p, q) If f (x, y) is third differentiable, then the log-convexity with respect to parameters p and q of H-f (p, q) depend on the sign of J = (x - y) (xI)(x), where I = (Inf)(xy). As applications a group of chains of inequalities for homogeneous means are established, which generalize, strengthen and unify Tong-po Ling 's and Stolarsky's inequalities, and a reversed chain of inequalities for exponential mean (identic mean) is derived, which contains a reversed Stolarsky's inequality. Several estimations of lower and upper bounds of extended mean are presented.