Second Order Linear Differential Polynomials and Real Meromorphic Functions

Linear differential polynomials in zero-free meromorphic functions

“…If Q is a constant d in (1.7) then integration shows that f is a constant multiple of y 1 (v/u − 1) d . Conclusion (1.5) may be compared with that of Theorem 1.2, and links closely to (1.2) of Theorem 1.1 and [21,Theorem 1.3(II)]. Examples II and III in Section 2 demonstrate that in (1.7) the multiplicities of poles of f may be unbounded, in sharp contrast to the situation in Theorem 1.2, where any poles of f must all have the same multiplicity m. Example III also shows that T ′ need not be rational at infinity in (1.7).…”

“…A refinement of this theorem for meromorphic functions in the plane may be found in [21,Theorem 1.3]. For k ≥ 3 and f, F zero-free in the whole plane, the case of constant coefficients was solved in full by Steinmetz in [23], while polynomial coefficients were treated in [4] for entire f , and for meromorphic f by Brüggemann in [2].…”

“…21) that B(z) = (B(z) + dC(z))z η 1 −η 2 has an asymptotic series on Σ in descending integer powers of z 1/2 ; thus(17.22) holds unless this series for B(z) vanishes identically, in which case B(z) + dC(z) tends to zero transcendentally fast on Σ, and so does B(z) 2 − C(z) 2 , by the second equation of(17.19), which forces b 3 = 0 in(17.19), contrary to assumption. ✷Lemma 17.5 For d = ±1 there exists a polynomial Q d ≡ 0 such that 2dCA P ′ − B ′ + dC ′ B + dC −k = Q d (P − log(B + dC)).…”

Non-real zeros of linear differential polynomials

“…To deduce Corollary 1.1 from Theorem 1.3, set L = f ′ /f , so that L ′ +L 2 +ω = (f ′′ +ωf )/f ; because f has infinite order, [3, Lemma 5.1] applied to f or 1/f shows that φ is transcendental in (1). Since f (z) = sec z gives f ′ (z)/f (z) = tan z and f ′′ + f = 2f 3 = 0, the assumption that f has infinite order cannot be deleted in Corollary 1.1 (see also [14,Theorem 1.5]).…”

Non-real zeros of derivatives

“…is not new [22], but the present proof is considerably simpler than that of [22] and the result substantially more general. For results on non-real zeros of f ′′ + ωf when ω ≥ 0 and f has finite order, the reader is referred to [22,24,27] and [26,Theorem 1.5].…”