Convexity and inequalities of some generalized numerical radius functions

Abstract

In this paper, we prove that each of the following functions is convex on R: f(t) = wN(AtXA1?t ? A1?tXAt), g(t) = wN(AtXA1?t), and h(t) = wN(AtXAt) where A > 0, X ? Mn and N(.) is a unitarily invariant norm onMn. Consequently, we answer positively the question concerning the convexity of the function t ? w(AtXAt) proposed by in (2018). We provide some generalizations and extensions of wN(.) by using Kwong functions. More precisely, we prove the following wN(f(A)X1(A) + g(A)Xf (A)) ? wN(AX+XA) ? 2wN(X)N(A), which is a kind of generalization of Heinz inequality for the generalized numerical radius norm. Finally, some inequalities for the Schatten p-generalized numerical radius for partitioned 2 ? 2 block matrices are established, which generalize the Hilbert-Schmidt numerical radius inequalities given by Aldalabih and Kittaneh in (2019).