Selberg and Brolin on value distribution of complex dynamics

The Brolin-Lyubich-Freire--Lopes--Ma\~n\'e equidistribution theorem for iterated preimages of a given non-exceptional value and Lyubich's periodic point version of it are foundational in the study of dynamics of rational functions of degree more than one on the complex projective line, and Drasin and the author studied a quantification of the former in a formalism of Nevanlinna theory or more specifically with the aid of Selberg's theorem. In this paper, we point out that the argument in that previous study have already yielded a better quantification of the Brolin-Lyubich-Freire--Lopes--Ma\~n\'e equidistribution theorem, and also point out that a similar argument also yields a quantification of Lyubich's theorem under an exponentwise version of the so called hypothesis H.