Inequalities for complex rational functions - Bidkham, M.; Khojastehnezhad, E. (Ukrains’kyi Matematychnyi Zhurnal, 2021)
UDC 517.5 For the rational function r(z)=p(z)/H(z) having all its zeros in |z|≤ 1, it is known that equation* |r’(z)|≥12|B’(z)||r(z)|for|z|=1, equation* where H(z)=_j=1^n(z - c_j), |c_j|>1, n is a positive integer, B(z)=H^*(z)/H(z), and H^*(z)=z^nH(1/z). In this paper, we improve the above mentioned inequality for the rational function r(z) with all zeros in |z|≤ 1 and a zero of order s at the origin. Our main results refine and generalize some known rational inequalities.