Nevanlinna’s Theory of Value Distribution pp 89-106 | Cite as

# 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* = 2*z*, 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.

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