On solutions of linear differential equations with entire coefficients
The first result of the paper concerns the effect of a perturbation of the entire coefficients~\(A_j\) of certain homogeneous linear differential equations \[ y^{(k)}+\sum_{j=0}^{k-2} A_{j}y^{(j)}=0 \quad \text{with } k\geq 2 \tag{1} \] on the oscillation of the solutions~\(y\). In fact, Theorem~1.1 considers small perturbations of the coefficients \(A_j\) (for \(1\leq j\leq k-2\)) of equation~(1) with a solution \(f\) whose zeros have finite exponent of convergence, say \(\lambda(f)\), into some entire coefficients~\(B_j\) (again for \(1\leq j\leq k-2\)) of the equation \[ y^{(k)}+\sum_{j=0}^{k-2}B_{j}y^{(j)}=0 \tag{2} \] and concludes that equation~(2) cannot have any solution whose zeros have finite exponent of convergence, if, for example, either of the following two hypotheses hold: (a) The function \(A_0\) is transcendental of order~\(0\), while \(B_0-A_0\) and the \(A_j\) and \(B_j\) (for \(1\leq j\leq k-2\)) are polynomials, and \(f\) has finitely many zeros. (b) The function \(A_0\) has order, say \(\rho(f)\), less than~\(1/2\), while \(B_0-A_0\) and the \(A_j\) and \(B_j\) (again for \(1\leq j\leq k-2\)) have order less than \(\rho(A_0)\), and \(\lambda(f)<\rho(A_0)\). Let us try to present the statement of Theorem~1.1 without any specific expressions. In this theorem, the authors treat the case when \(A_0,\dots,A_{k-2}\) are entire functions of finite order and the transcendental function \(A_0\) dominates the other functions \(A_j\) in a sense and equation (1) has a certain solution~\(f\) with \(\lambda(f)<\infty\) and, in addition, the distribution of its zeros is negligible in comparison with the growth of the dominant coefficient \(A_0\). Then, they show that any small perturbed equation (2) of (1) with entire coefficients \(B_j\) (for \(0\leq j\leq k-2\)) cannot have a solution \(g\) with \(\lambda(g)<\infty\), provided that \(A_0\) dominates again all the \(B_j\) (for \(1\leq j\leq k-2\)) and the difference \(B_0-A_0\) does not vanish identically but its derivative is dominated by \(A_0\), too. In particular this theorem for hypothesis (b) above improves a result of the first author [Result. Math. 47, No. 3--4, 165--175 (2005; Zbl 1101.34071)] when \(k=2\). Subsequent results involve the separation of the zeros of a Bank-Laine function of finite order. An entire function \(E\) is called a Bank-Laine function if \(E(z)=0\) implies \(E'(z)=\pm 1\) in relation to the celebrated conjecture by Bank and Laine concerning the second order equation (1), that is for \(k=2\). \textit{L. C. Shen} [Proc. Am. Math. Soc. 95, 544--546 (1985; Zbl 0596.30048)] has shown that, in general, a Bank-Laine function~\(E\) may have an arbitrary zero sequence \(\{z_n\}\), \(|z_n|\to\infty\), without repetition. Contrary, if it is assumed that \(E\) has finite order, then it is possible to prove that certain zero sequences cannot arise for \(E\). According to \textit{S. M. ElZaidi} [Complex Variables, Theory Appl. 38, 201--220 (1999; Zbl 1036.34099)], there is then a separation between each pair of the zeros \(\{z_n\}\) of a Bank-Laine function, that is to say there exist \(M>0\) such that \[ |z_m-z_n|\geq \exp(-|z_n|^M) \quad \text{ for \(n\neq m\) and \(n\) large}. \] The authors of the paper under review conjecture that the separation theorem due to ElZaidi holds with positive constant~\(M\) depending only on the exponent of convergence of the zero-sequence \(\{z_n\}\). As a first result concerning this conjecture, they prove in Theorem~1.3 that it is certainly true if the zeros are all real numbers. Note that, results on the zero distribution of Bank-Laine functions with real zeros appeared in [\textit{J. K. Langley}, Proc. Am. Math. Soc. 129, No.~7, 1969--1978 (2001; Zbl 0969.30015) and \textit{D. Drasin} and \textit{J. K. Langley}, in: Transcendental dynamics and complex analysis. In honour of Noel Baker. Cambridge: Cambridge University Press. London Mathematical Society Lecture Note Series 348, 165--178 (2008; Zbl 1158.30019)]. The second result answers the question whether a similar construction as Langley's method in the above paper, for Bank-Laine functions of exceptional type, could be employed to produce a Bank-Laine function of finite order whose zeros occur in sparsely distributed small discs, with at least two zeros in each disc. In fact, Theorem~1.4 proves a lower bound for the radius of any such discs, depending on the exponent of convergence of the zeros, and lends some support to the conjecture. It should be noted that the hypothesis that \(E\) has finite order is not redundant in both Theorems~1.3 and~1.4, as the result of Shen already mentioned it. Among others, the proof of Theorem~1.1 requires growth estimates for logarithmic derivatives of finite-order meromorphic functions and of solutions to~(1) with finite exponent of convergence of zeros by means of the coefficients~\(A_j\). The proofs of Theorems~1.3 and~1.4 are based, respectively, on the second author's observation in his paper cited above on the number of zeros that some Bank-Laine function can have in a sector around the positive real axis and on an application of a well-known univalence criterion of Nehari, plus standard estimates for logarithmic derivatives, again.
scientific article; zbMATH DE number 5706215
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