J. K. Langley’s research while affiliated with Technische Universität Berlin and other places

Sincc 1982, a considerable number of results have been proved concerning the frequency of zeros of solutions of second-order equations having entire coefficients. The proofs of these results were peculiar to second-order equations since they used techniques which hold only for second -order equations(e.g. the differential equalion for the product of two solutions). Surprisingly, we show in the present paper that these results also hold for higher-urder equations. The prook are far different than in the second-order case being asymptotic in nature.

Our starting point is the differential equation where A ( z ) is a transcendental entire function of finite order, and we are concerned specifically with the frequency of zeros of a non-trivial solution f ( z ) of (1.1). Of course it is well known that such a solution f ( z ) is an entire function of infinite order, and using standard notation from [7], for all , b ∈ C \{0}, at least outside a set of r of finite measure.

We consider the equation $\rm f^{\prime\prime}+{A}(z){f}=0$ with linearly independent solutions f1,2, where A(z) is a transcendental entire function of finite order. Conditions are given on A(z) which ensure that max{λ(f1),λ(f2)} = ∞, where λ(g) denotes the exponent of convergence of the zeros of g. We show as a special case of a further result that if P(z) is a non-constant, real, even polynomial with positive leading coefficient then every non-trivial solution of $\rm f^{\prime\prime}+{e}^P{f}=0$ satisfies λ(f) = ∞. Finally we consider the particular equation $\rm f^{\prime\prime}+({e}^Z-K){f}=0$ where K is a constant, which is of interest in that, depending on K, either every solution has λ(f) = ∞ or there exist two independent solutions f1, f2 each with λ(fi) ≤ 1.

The Nevanlinna characteristic of a nonconstant elliptic function (z) satisfiesT(r, )=Kr 2 (1+o(1)) asr whereK is a nonzero constant. In this paper, we completely answer the following question: For which polynomialsQ(z, u 0,...,u n ) inu 0,...,u n , having coefficientsa(z) satisfyingT(r, a)=o(r 2) asr, will the meromorphic functionh Q (z)=Q(z, (z),...,(n)(z)) either be identically zero or satisfyN(r, 1/h Q )=o(r 2) asr? In fact, we answer this question for rational functionsQ(z, u 0,...,u n ) inu 0,...,u n , and also obtain analogous results for the Weierstrass functions (z) and (z).