On the fixpoints of composite meromorphic functions and generalizations (Q1922213)

The main purpose of this paper is to give lower bounds for the counting function of the zeros of meromorphic functions of the form \(f(g) - \gamma\). Here \(\gamma \) is nonconstant and satisfies \(T(r, \gamma) = o(T(r,g))\). We mention only two results. If \(f\) is a rational function of degree \(m\geq 1\) and \(g\) is a transcendental meromorphic function, then \[ \liminf_{r \to \infty,r \notin E} {N \biggl(r,1/ \bigl(f(g) - \gamma \bigr) \biggr) \over T(r,g)} \geq m- 2+ \delta (\infty,f) + \delta (0,f) \] for some set \(E\) of finite measure (Corollary 3.3). If \(f\) and \(g\) are transcendental entire functions and \(\gamma\) is a (nonconstant) polynomial, then \[ \lim_{r \to \infty,r \in I} {N\biggl(r+ \eta,1/ \bigl(f(g) - \gamma \bigr) \biggr) \over T(r,g)} = \infty \] for some set \(I\) of logarithmic density one. Here \(\eta = Kr \nu(r,g)^{-\delta}\) with certain constants \(K\) and \(\delta\), and \(\nu(r,g)\) is the central index (Theorem 4.2). The paper also contains growth estimates of canonical products which are used in the proofs, but which are also of independent interest.