Logarithmic Derivatives

Abstract

For obvious reasons, a meromorphic function g is said to be a logarithmic derivative if \(g\frac{{f'}}{f}\) for some meromorphic function f. Not any meromorphic function can be expressed as a logarithmic derivative, and thus one might ask what special properties functions which are logarithmic derivatives posses. For example, if g = f′/ f is a logarithmic derivative, then all the poles of g must be simple poles. In terms of what to expect from a value distribution point of view, we again look at polynomials for a clue. If P is a polynomial, then P′ has degree smaller than P, and so P′/P stays small as z goes to infinity. Of course, one cannot expect exactly this behavior in general. For example, if \(f = {e^{{z^2}}} \), then f′/ f = 2z, and so |f′/f| → ∞ as |z| → ∞. However, we see here that the rate at which log |f′/f| approaches infinity is very slow by comparison to T(f, r). The main result of this chapter is what is known as the Lemma on the Logarithhmic Derivative (Theorem 3.4.1), which essentially says that if f is a meromorphic function, then the integral means of log+ | f′/f| over large circles cannot approach infinity quickly compared with the rate at which T (f, r) tends to infinity.