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):=∑ₖ₌₀∞aₖzᵏ is convergent on the open disk D(0,R),~fₐ(z):=∑ₖ₌₀∞ aₖ zᵏ 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 $ CAf(A)Bx,y ≤ fₐ( A ) A αB ²x,x 1/2 A 1-αC ²y,y 1/2 $ for $α ∈ 0,1]$ and $x,y∈ H.$ Some natural applications for numerical radius and p-Schatten norm are also provided.