Derivative bounded functional calculus of power bounded operators on Banach spaces

In this article we study bounded operators $T$ on Banach space $X$ which satisfy the discrete Gomilko Shi-Feng condition $${\int }_0^{2\pi }\mathrm{\mid }⟨R(r{e}^{it},T{)}^{2}x,x^{\ast }⟩\mathrm{\mid }dt\le \frac{C}{(r^2-1)}\text{norme}x\text{norme}{x}^{\ast },r>1,x\in X,x^{\ast }\in X^{\ast }\mathrm{.}$$ We show that it is equivalent to a certain derivative bounded functional calculus and also to a bounded functional calculus relative to Besov space. Also on Hilbert space discrete Gomilko Shi-Feng condition is equivalent to power-boundedness. Finally we discuss the last equivalence on general Banach space involving the concept of $\gamma$-boundedness.