SHARP WEIGHTED HOLDER MEAN BOUNDS FOR SEIFFERT'S MEANS

Let P(a, b) and T(a, b) be the first and second Seiffert's means for two positive numbers a and b , in this paper, for any fixed p is an element of R , we present the optimal parameters alpha(p), beta(p), lambda(p), mu(p) is an element of [0, 1] such that the inequalities H-p(a, b; alpha(p)) <= P(a, b) H-p(a, b; beta(p)), H-p(a, b; lambda(p)) <= T(a, b) H-p(a, b; mu(p)) hold true for all a, b > 0, where H-p(a, b; omega) is the weighted p-order Holder (power) mean with the weight omega is an element of [0, 1]. As applications, various sharp inequalities for P(a, b) and T(a, b), including the sharp power mean bounds, will be established.