Convexity and inequalities of some generalized numerical radius functions

Abstract

Following the recent work of Abu-Omar and Kittaneh
\cite{kittaneh-1}, we restrict our attention on presenting new
inequalities for the generalized numerical radius norm. Also
motivated by the work of Sababheh \cite{sababheh} we generalize the
convexity of the numerical radius functions and we answer positively
the question araised about the convexity of certain numerical radius
functions. Moreover, we present generalizations and extensions of
some inequalities as presented in \cite{1} by using Kwong functions.
Finally, some inequalities for the Schatten $p$-generalized
numerical radius for partitioned $ 2 \times 2$ block matrices are
established, which generalize the Hilbert-Schmidt numerical radius
inequalities given by Aldalabih and Kittaneh in \cite{a}.