Real meromorphic functions and a result of Hinkkanen and Rossi

Non-real zeros of derivatives of real meromorphic functions

“…The following argument is self-contained but uses some methods similar to those of [35]. If u and v are linearly independent solutions of the differential equation The equation (15) has n+2 distinct critical rays [23], namely those rays arg z = θ such that 2 arg b + (n + 2)θ = 0 (mod 2π).…”

“…Proof. The fact that all but finitely many poles of f and zeros of f ′′ are real is proved in [35], but a slightly different argument is given here for completeness. First, it is evident from ( 2) and ( 3) that there exist constants D, E and rational functions S, U, V with ( 6)…”

“…The following argument is self-contained but uses some methods similar to those of [35]. If u and v are linearly independent solutions of the differential equation Proof.…”

Non-real zeros of derivatives

“…If f is allowed multiple poles then there are further examples for which f , f ′ and f ′′ have only real zeros and poles [15]. Results from [14,18,21,23,32] show that the conjecture is true if, in addition, f ′ omits some finite value. Furthermore, Theorem 1.2 and [24] together show that there are no functions f satisfying the hypotheses of Conjecture 1.1 such that either of the following holds: f has infinite order and the zeros or poles of f have finite exponent of convergence; f has finite order and infinitely many poles but finitely many zeros.…”

“…Apply Lemma 2.3 to 1/T m = U/U (m) , with R = 2r and µ(r) = 4ε. In view of(32) this shows that, as r → ∞ in F 1 ,(m + 1)N(r, U) ≤ N(r, 1/U (m) ) + O(log r) + 88ε 1 + log 1 4ε T (2r, 1/T m ) ≤ N(r, 1/U (m) ) + O(log r) + 88ε 1 + log 1 4ε C 1 T (r, T m ) ≤ N(r, 1/U (m) ) + O(log r) + 88ε 1 + log 1 4ε C 1 N(r, T m ) ≤ N(r, 1/U (m) ) + O(log r) + 88ε 1 + log 1 4ε C 1 (m + 1)N(r, U).…”