**
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE **

Vol. LXXIV, 2 (2005)

p. 287 - 307

Solutions of *f'' + A*(*z*)*f* = 0 with prescribed
sequences of zeros

J. Heittokangas and I. Laine

**Abstract.**
The problem of when a given sequence (resp. two sequences) of complex points can be
the zero-sequence(s) of a solution (resp. of two linearly independent solutions) of
f'' + A(z)f = 0, where A(z) is entire, has been studied by several authors during the
last two decades. However, it is not well-known that problems of this type were first
stated and studied by O. Boruvka and V. Seda almost fifty years ago. A historical
review to these studies will be given below. We then offer some remarks and
improvements on results due to S. Bank and A. Sauer found. Our
reasoning towards these improvements is based on some growth estimates for
Mittag-Leffler-type series in the complex plane. These estimates might be of
independent interest.

**Keywords**:
Prescribed zero sequences, zeros of
solutions.

**AMS Subject classification:** Primary: 34M05;
Secondary: 34M10.

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Acta Mathematica Universitatis Comenianae

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