The distribution of the zeros of holomorphic functions of moderate growth in the unit disc and the representation

E. G. Kudasheva   ; B. N. Khabibullin

Sbornik. Mathematics, Tome 200 (2009) no. 9, pp. 1353-1382

Résumé

Let $\mathbb D$ be the unit disc in the complex plane $\mathbb C$ and $H$ a class of holomorphic functions in $\mathbb D$ distinguished by a restriction on their growth in a neighbourhood of the boundary of the disc which is stated in terms of weight functions of moderate growth. Some results which describe the sequences of zeros for holomorphic functions in classes $H$ of this type are obtained. The weight functions defining $H$ are not necessarily radial; however the results obtained are new even in the case of radial constraints. Conditions for meromorphic functions in $\mathbb D$ ensuring that they can be represented as a ratio of two functions in $H$ sharing no zeros are investigated. Bibliography: 28 titles.

Détail

Keywords: unit disc, holomorphic function, sequence of zeros, weighed space, representation of a meromorphic function, subharmonic functions, Green's function.

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