Bounds of the Derivative of Some Classes of Rational Functions: Nuttapong Arunrat, Keaitsuda Maneeruk Nakprasit - Team, Support (Thai Journal of Mathematics, 2022)
Let r(z) be a rational function with at most n poles, a_1, a_2, …, a_n, where |a_j| > 1, 1≤ j≤ n.This paper investigates the estimate of the modulus of the derivative of a rational function r(z) on the unit circle. We establish an upper bound when all zeros of r(z) lie in |z|≥ k≥ 1 and a lower bound when all zeros of r(z) lie in |z|≤ k ≤ 1. In particular, when k=1 and r(z) has exactly n zeros, we obtain a generalization of results by A. Aziz and W. M. Shah.