Annales Academić Scientiarum Fennicć
Mathematica
Volumen 36, 2011, 341-351

A UNIT DISC ANALOGUE OF THE BANK-LAINE CONJECTURE DOES NOT HOLD

Janne Heittokangas and Kazuya Tohge

University of Eastern Finland, Department of Physics and Mathematics
P.O. Box 111, 80101 Joensuu, Finland; janne.heittokangas 'at' uef.fi

Kanazawa University, College of Science and Engineering
Kakuma-machi, Kanazawa 920-1192, Japan; tohge 'at' t.kanazawa-u.ac.jp

Abstract. The 1982 conjecture due to Bank and Laine claims the following: If A(z) is a transcendental entire function of order of growth \rho(A) \in [0,\infty) \ N, then max{\lambda(f₁),\lambda(f₂)} = \infty, where f₁, f₂ are linearly independent solutions of f'' + A(z)f = 0 and \lambda(g) stands for the exponent of convergence of the zeros of g. This conjecture has been verified in the case \rho(A) \leq 1/2, while counterexamples have been found in the cases \rho(A) \in N \cup {\infty}. The aim of this paper is to illustrate that no growth condition on A(z) alone yields a unit disc analogue of the Bank-Laine conjecture. The main discussion yields solutions to two open problems recently stated by Cao and Yi.

2000 Mathematics Subject Classification: Primary 34M10; Secondary 30D35.

Key words: Asymptotic growth, Bank-Laine conjecture, exponent of convergence, linear differential equation, Nevanlinna theory, oscillation, unit disc, zero-free solution base.

Reference to this article: J. Heittokangas and K. Tohge: A unit disc analogue of the Bank-Laine conjecture does not hold. Ann. Acad. Sci. Fenn. Math. 36 (2011), 341-351.

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doi:10.5186/aasfm.2011.3622