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Boundedness of $L$ $L$ -index in joint variables for composition of analytic functions in the unit ball

Department of Advanced Mathematics, Ivano-Frankivsk National Technical University of Oil and Gas, 15 Karpatska Street, 76019 Ivano-Frankivsk, Ukraine

E-mail Address: andriykopanytsia@gmail.com

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https://doi.org/10.1142/S1793557121500546Cited by:1 (Source: Crossref)

Abstract

In this paper, the following composite analytic functions $F (z) = f (Φ (z))$ $F (z) = f (Φ (z))$ and $H (z) = G (Φ (z), \dots, Φ (z))$ $H (z) = G (Φ (z), \dots, Φ (z))$ are considered, where $f : ℂ \to ℂ,$ $Φ : 𝔹^{n} \to ℂ,$ $G : ℂ^{m} \to ℂ .$ We established conditions which provide equivalence of boundedness of the $l$ -index of the function $f$ and boundedness of the $L$ -index in joint variables of the function $F,$ where $l : ℂ \to ℝ_{+}$ is a continuous function, $L (z) = (l (Φ (z)) | \frac{\partial Φ (z)}{\partial z_{1}} |, \dots,$ $l (Φ (z)) | \frac{\partial Φ (z)}{\partial z_{n}} |) .$ For the function $H$ with additional restrictions, the function $L$ is constructed such that $H$ has bounded $L$ -index in joint variables in the case when the function $G$ has bounded $L$ -index in the direction $1 = (1, \dots, 1)$ , where $L : ℂ^{n} \to ℝ_{+}^{n}$ is a positive continuous function. Our proofs are based on the application of analog of Hayman’s theorem for these classes of functions.

Communicated by A. Escassut

Keywords:

AMSC: 32A10, 32A17, 32A37

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