Zeros of Derivatives of Real Meromorphic Functions

The reciprocal of a real entire function and non-real zeros of higher derivatives

“…The case m = 0 is already contained in Theorem 1.1, but the present proof is somewhat simpler than that in [32]. The final result is linked to the investigations of [28,30,31,32,33], which in turn followed on from earlier work [18,19,20,21,22,37] concerning the existence of non-real zeros of derivatives of real meromorphic functions in general. It seems likely that if k ≥ 2 and f is a real meromorphic function in the plane, such that f and f (k) have finitely many non-real zeros, then f has in some sense relatively few distinct non-real poles.…”

“…Since m ≥ k it then follows that T(r, L) = O(log r) as r → ∞, by standard properties of the Tsuji characteristic. Because F and 1/L have finitely many poles in the open upper half-plane H, Lemma 2.2 shows that there exist finitely many α ∈ C such that F (z) or L(z) tends to α as z tends to infinity along a path in C \ R. This implies at once that F and L satisfy the conclusions of Lemmas 2.1 and 3.2 of [33]. Since those two lemmas were the only steps in the proof of Theorem 1.4 in [33] which required the hypothesis that f ′ /f has finite lower order (see [33,Remark 3.3,p.…”

“…Because F and 1/L have finitely many poles in the open upper half-plane H, Lemma 2.2 shows that there exist finitely many α ∈ C such that F (z) or L(z) tends to α as z tends to infinity along a path in C \ R. This implies at once that F and L satisfy the conclusions of Lemmas 2.1 and 3.2 of [33]. Since those two lemmas were the only steps in the proof of Theorem 1.4 in [33] which required the hypothesis that f ′ /f has finite lower order (see [33,Remark 3.3,p. 247] for an explicit confirmation of this observation), and since f and f ′′ have finitely many non-real zeros, the remainder of the proof of Theorem 1.5 now follows exactly that of Theorem 1.4.…”

“…It seems likely that if k ≥ 2 and f is a real meromorphic function in the plane, such that f and f (k) have finitely many non-real zeros, then f has in some sense relatively few distinct non-real poles. A partial result in this spirit holds if in addition the non-real poles of f have bounded multiplicities, in which case there exists B > 0 such that if z 0 and z 1 are distinct non-real poles of f then |z 1 − z 0 | > B| Im z 0 | (see [33,Lemma 4.3]). The following theorem from [33] showed that if k = 2 and f ′ /f has finite lower order, this additional hypothesis on the multiplicities of non-real poles is not required.…”

Non-real zeros of derivatives of meromorphic functions

“…as z → ∞, and the induction is complete. ✷ Lemma 2.7 ( [36], Lemma 4.7) Let the function f be transcendental and meromorphic in the plane and let k ∈ N. Let E be an unbounded subset of [1, ∞) with the following property. For each r ∈ E there exist real θ 1 (r) < θ 2 (r) ≤ θ 1 (r) + 2π and an arc Ω r = {re iθ :…”