The zeros of the first two derivatives of a meromorphic function

Abstract

Abstract. We prove a theorem which implies the following: if f is meromorphic of finite order in the plane and f and f have only finitely many zeros, then f has only finitely many poles.

Transcendental Singularities for a Meromorphic Function with Logarithmic Derivative of Finite Lower Order

“…Part (b) follows from part (a) combined with Theorem 1.1 and a well known classification theorem from [18, p.287], which shows in particular that any transcendental singularity of the inverse function over an isolated singular value is logarithmic. Theorem 1.2 was employed in [2] to prove a long-standing conjecture of Hayman [7] concerning zeros of f f ′ −1, and has found many subsequent applications, including to zeros of derivatives [12]. The reader is referred to [3,19] for further striking results on singularities of the inverse, both restricted to entire functions but independent of the order of growth.…”

“…6] implies that every neighbourhood of the singularity contains circles |z| = r with r arbitrarily large, so that a path γ as in (iii) cannot exist. [12]. ✷…”

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. 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.…”

On the zeros of the second derivative

“…The inequality (1.1), with fe = 2, c 2 = 1, would confirm the Mues conjecture [17] that for every transcendental meromorphic /. While (1.1) is known, with S(r,/) = O(logrT(r,/)), (1.2) if all but finitely many poles of/ are simple [17], no corresponding inequality in the general case, with any positive c 2 , has been proved, although such an inequality with (1.2) would imply the following conjecture from [14]. CONJECTURE 1.1.…”

“…It should be noted that Conjecture 1.1 is false for functions of infinite order, as shown in [14] by examples of form / " / / ' = eh g~1 w i t n S> n entire, for which both / ' a n d / " are zero-free. The following theorem summarises some results in the direction of Conjecture 1.1.…”