Estimates of $\mathrm{\Gamma }$-functions and their $\log$-derivatives

We provide a lower bound for a weighted ratio of $\mathrm{\Gamma }-$functions and its $\log -$derivative. Then we apply this result to estimate the norm of the operator associated with the Cesáro operator in the Hardy space $H^p({\mathbb{C}}_{+})$, $1<p<\mathrm{\infty }$