Zeros of Differential Polynomials in Real Meromorphic Functions

Abstract

We investigate whether differential polynomials in real transcendental meromorphic functions have non-real zeros. For example, we show that if g is a real transcendental meromorphic function, c ∈ R \ {0} and n 3 is an integer, then g g n − c has infinitely many non-real zeros. If g has only finitely many poles, then this holds for n 2. Related results for rational functions g are also considered.

Green’s function and anti-holomorphic dynamics on a torus

“…The most surprising fact is that Fatou's theorem can be used sometimes to estimate the number of solutions of equations in settings where dynamics is not present. This was first noticed in [12]; the contents of this unpublished preprint is reproduced in [16,8]. The paper [6] shows that Fatou's theorem can be used to prove under some circumstances the existence of critical points of a meromorphic function.…”

“…If any of the points c and −c is in F , then it is a zero of h . By (8) the other zeros of h are g −1 (±c) = ± g −1 (c). Even though this is an infinite set on T , the critical points of h are thus contained in at most 4 grand orbits represented by c, −c, g(c) and − g(c).…”

A special case of a conjecture of Hellerstein, Shen and Williamson

“…which forces e is = −1, so that Re is lies on the negative real axis. Next, a straightforward application of the Wiman-Valiron theory [8] in (3) shows that the order of f 4 is 1/2, and so a standard generalisation of the Gauss-Lucas theorem [33] implies that all zeros of f ′ 4 are also real and non-positive. This completes the proof of the following.…”

Non-real zeros of linear differential polynomials

“…If f is defined by [10] f (z) f (z) = a + e −2az , f (z) f (z) = a 2 + e −4az , then f and f − a 2 f have no zeros at all in the plane. For an example of finite order, define a zero-free function f ∈ U 2 by setting [7] f (z) f (z) = −16z 2 + 8z + 2, f (z) − 12f (z) f (z) = 256z 3 (z − 1), so that f − 12f has only real zeros. The proof of Theorem 1.4 uses machinery developed in [25,28] for the Wiman conjecture, and refinements from [5,6,22], but departs from the earlier methods in several significant steps.…”

“…then f and f ′′ − a 2 f have no zeros at all in the plane. For an example of finite order define a zero-free function f ∈ U 2 by setting [7] f…”