Some families of operator norm inequalities

We consider the function fα,β(t)=tγ(α,β)∏i=1nbi(tai−1)ai(tbi−1) on the interval (0,∞), where α=(a1,a2,…,an),β=(b1,b2,…,bn)∈Rn and γ(α,β)=(1−∑i=1n(ai−bi))/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 α′,β′∈Rm to hold the relation fα,β(t)⪯fα′,β′(t).