An Interpolating Family of Means

This paper is concerned with a new family of binary symmetric means mp of two positive numbers a and b: 1 mp(a, b) = cp ∫ ∞ 0 dx [(xp + ap)(xp + bp)]1/p , 0 < p < ∞, where the constant cp, depending on p, is chosen to have mp(a, a) = a. Two distinctive members in the family are the well-known logarithmic mean (p = 1) and arithmeticgeometric mean (p = 2). Different expressions for mp are obtained to establish its other properties, including m2(a, b) ≤ m∞(a, b) and the relation between mp and the power difference mean. Through investigating the induced operator norm of the integral operator with m−1 p as its kernel, a generalization of the Hilbert inequality is obtained. Finally positive definiteness of certain matrices as implications of inequalities between two means is also investigated.