Sequences of exponents in constructing strongly annular products. (English) Zbl 1280.30002
Summary: This paper gives conditions on the behavior of a sequence of holomorphic functions \(\{f{_k}(z)\}\) and a strictly increasing sequence of positive integers \(\{m{_k}\}\) that assures that the infinite product \(\prod f_k(z^{m}_k)\) is strongly annular. A constructive proof is given that shows if the sequence \(\{f{_k}(z)\}\) exhibits certain boundary behavior along with a uniform boundedness condition then a number \(p>1\) exists such that if \(\{m{_k}\}\) satisfies \(m_{k+1}/m_k\geq p\) then the above product is strongly annular.
MSC:
| 30B99 | Series expansions of functions of one complex variable |
| 30D99 | Entire and meromorphic functions of one complex variable, and related topics |
| 30E99 | Miscellaneous topics of analysis in the complex plane |
References:
| [1] | Bagemihl F., Erdös P., Seidel W.: Sur Quelques Propriétés Frontières des Fonctions Holomorphes définies par Certains Produits dans le Cercle-Unité. Ann. Sci. Ecole Norm. Sup. 70(3), 135–147 (1953) · Zbl 0053.23801 |
| [2] | Daquila R.: Strongly annular solutions of Mahler’s functional equation. Complex Var. Theory Appl. 32, 99–104 (1997) · Zbl 0865.30050 · doi:10.1080/17476939708814982 |
| [3] | Mahler K.: Arithmetische Eigenschaften einer Klasse von Funktionalgleichungen. Math. Annl. 101, 342–366 (1929) · JFM 55.0115.01 · doi:10.1007/BF01454845 |
| [4] | Binmore K.G.: Analytic functions with Hadamard gaps. Bull. Lond. Math. Soc. 1, 211–217 (1969) · Zbl 0209.10002 · doi:10.1112/blms/1.2.211 |
| [5] | Bonar D.D.: On Annular Functions. Veb Deutscher Der Wissenschaften, Berlin (1971) · Zbl 0223.30044 |
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