Inequalities for a Rational Function with Prescribed Poles

By considering the rational function $ r(z)=\frac{P(z)}{W(z)} $ (where $ P(z) $ and $ W(z) $ are polynomials of degree $ n $ )with prescribed poles, we establish some estimates for $ \operatorname{Re}\left(\frac{zr^{\prime}(z)}{r(z)}\right) $ on the unit disk. The estimates strength a few familiar inequalitiesof Mir, Hans, Aziz, and Zargar as well as extend some recent results of Dubinin.