  In this article we study bounded operators $T$ on Banach space $X$ which
satisfy the discrete Gomilko Shi-Feng condition $$\int_{0}^{2\pi}|\langle
R(re^{it},T)^{2}x,x^*\rangle |dt \leq
\frac{C}{(r^2-1)}\norme{x}\norme{x^*},\quad r>1, x\in X, x^* \in X^*. $$ 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.
