Numerical radius and p-Schatten norm inequalities for power series of operators in Hilbert spaces

Let H be a complex Hilbert space. Assume that the power series with complex coefficients f (z): =₊=₀^ a₊z^k is convergent on the open disk D (0, R), ~f₀ (z): =₊=₀^ a₊ z^k that has the same radius of convergence R and A, ~B, ~C B (H) with A R, then we have the following Schwarz type inequality C^ Af (A) Bx, y f₀ (A) A ^ B ^2x, x ^1/2 A^ ^1- C ^2y, y ^1/2 for 0, 1] and x, y H. Some natural applications for numerical radius and p-Schatten norm are also provided.