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Let f be a function transcendental and meromorphic in the plane, and define g(z) by g(z) = ∆f (z) = f (z + 1) − f (z). A number of results are proved concerning the existence of zeros of g(z) or g(z)/f (z), in terms of the growth and the poles of f . The results may be viewed as discrete analogues of existing theorems on the zeros of f ′ and f ′ /f . MSC 2000: 30D35.

We prove that if f is a real entire function of infinite order, then f f has infinitely many non-real zeros. In conjunction with the result of Sheil-Small for functions of finite order this implies that if f is a real entire function such that f f has only real zeros, then f is in the Laguerre-Pólya class, the closure of the set of real polynomials with real zeros. This result completes a long line of development originating from a conjecture of Wiman of 1911.

We prove the following, which confirms a conjecture of W. K. Hayman from 1959. If f is meromorphic in the plane such that f and f″ have only finitely many zeros, then f(Z) = R(z) exp (P(Z)), where R is rational and P is a polynomial. The theorem is related to earlier results of Frank, Mues and others.

We show that if the maximum modulus of a quasiregular mapping f grows sufficiently rapidly then there exists a non-empty escaping set I(f) consisting of points whose forward orbits under iteration tend to infinity. This set I(f) has an unbounded component but, in contrast to the case of entire functions on the complex plane, the closure of I(f) may have a bounded component.Comment: 10 page

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