Some numerical radius inequalities for power series of operators in Hilbert spaces (Q2436114)

For \(T\in\mathbb{B}(H)\), the numerical range of \(T\) is defined as \[ w(T)=\sup\{|\langle Tx,x\rangle|:\|x\|=1\}. \] The author considers two power series \(f(z)=\sum_{i=1}^{\infty} a_nz^n\) and \(f_a(z)=\sum_{i=1}^{\infty} |a_n|z^n\) and finds some relations between \(w(f(T))\) and \(f_a(w(T))\). Moreover, he obtains some upper bounds for \(w(f(AB))\). Replacing \(f\) with some elementary functions, some familiar inequalities related to the numerical range are obtained.