On the zeros of the second derivative

Abstract

Suppose that f is meromorphic of finite order in the plane, and that f″ has only finitely many zeros. We prove a strong estimate for the frequency of distinct poles of f. In particular, if the poles of f have bounded multiplicities, then f has only finitely many poles.

The Second Derivative of a Meromorphic Function

“…On the other hand, Conjecture 1.2 is false for functions of infinite order, as shown in [21] by examples of the form f /f = e h g −1 with g, h entire, for which both f and f are zero-free. Theorem 1.3 (see [22,23]). Theorem 1.3 (see [22,23]).…”

“…This is more complicated than in [22] because of the different way that J was chosen. Then J divides its complement in We consider the components of the sets F −1 (B m ).…”

“…The following theorem, in which the notation is that of [9], summarizes some results in the direction of Conjecture 1.2. [22,23]). Suppose that f is meromorphic of finite order ρ in the plane and that f has finitely many zeros.…”

“…Fix conformal mappings We consider the components of the sets F −1 (B m ). This is more complicated than in [22] because of the different way that J was chosen. It is convenient to take a quasiconformal homeomorphism ψ 1 of the extended plane onto itself such that ψ 1 (∞) = ∞ and ψ 1 (B 1 ) = ∆.…”

“…see[17,22]). Suppose that h(z) =∞ j=1 a j z j maps the disc B(0, s) conformally onto a simply connected domain D of finite area A.…”