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

By the help of power series f(z) = Sigma(infinity)(n=0) a(n)z(n), we can naturally construct another power series that has as coefficients the absolute values of the coefficients of namely, f(a)(z) := Sigma(infinity)(n=0)vertical bar a(n)vertical bar z(n). Utilizing these functions, we show among others that w[f(T)] <= f(a)[w(T)] and w[f(T)] <= 1/2[f(a)(parallel to T parallel to) + f(a)(parallel to T-2 parallel to(1/2))], where w(T) denotes the numerical radius of the bounded linear operator T on a complex Hilbert space, while parallel to T parallel to is its norm.