Some families of operator norm inequalities

We consider the function f(alpha,beta)(t) = t(gamma(alpha,beta)) Pi(n)(i=1) bi(t(at)-1)/ai(t(bt)-1) on the interval (0, infinity), where alpha = (a(1), a(2), . . . , a(n)), beta = (b(1), b(2), . . . , b(n)) is an element of R-n and gamma(alpha, beta) = (1 - Sigma(n)(i=1)(a(i) - b(i)))/2. In [4], Hiai and Kosaki define the relation <= using positive definiteness for functions f and g with some suitable conditions and they have proved this relation implies the operator norm inequality associated with functions f and g. In this paper, we give some conditions for alpha', beta' is an element of R-m. to hold the relation f (alpha,beta)(t) <= f(alpha',beta')(t). (C) 2017 Published by Elsevier Inc.