The logarithmic energy of zeros and poles of a rational function

Resumo

On assuming that certain lemniscates of a rational function are connected, we establish some sharp inequality that involves the logarithmic energy of a discrete charge concentrated at the zeros and poles of this function and the absolute values of its derivatives at these points. The equality in this estimate is attained for specially arranged zeros and poles of a suitable Zolotarev fraction and for special distributions of charge.