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Entire Bivariate Functions of Unbounded Index in Each Direction

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We study a class of entire functions ƒ(z1; z2) with the following property: ∀ b = (b1, b2) ∈ ℂ2 \ {0} and ∀ \( {z}_1^0 \),\( {z}_2^0 \) ∈ ℂ, the function ƒ(\( {z}_1^0 \)+tb1,\( {z}_2^0 \)+tb2) regarded as a function of one variable t ∈ ℂ has a bounded index, while the function ƒ(z1; z2) has an unbounded index in each direction b: In particular, we prove that, for any even entire function ƒ(t), which has an infinite sequence of complex zeros, the corresponding function \( f\left(\sqrt{z_1{z}_2}\right) \) has an unbounded index in each direction b: This improves our similar result obtained in [A. I. Bandura, Mat. Stud., 44, No. 1, 107–112 (2015)] for even entire functions ƒ(t) with complex zeros ck such that \( {c}_k^2 \) ∈ ℝ.

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Authors and Affiliations

  1. Ivano-Frankivs’k National Technical University of Oil and Gas, Karpatska Str., 15, Ivano-Frankivs’k, 76019, Ukraine

    A. Bandura

  2. I. Franko National University Lviv, Universytets’ka Str.1, Lviv, 79000, Ukraine

    O. Skaskiv

Authors
  1. A. Bandura
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  2. O. Skaskiv
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Corresponding authors

Correspondence to A. Bandura or O. Skaskiv.

Additional information

Published in Neliniini Kolyvannya, Vol. 21, No. 4, pp. 435–443, October–December, 2018.

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Bandura, A., Skaskiv, O. Entire Bivariate Functions of Unbounded Index in Each Direction. J Math Sci 246, 293–302 (2020). https://doi.org/10.1007/s10958-020-04739-8

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  • DOI: https://doi.org/10.1007/s10958-020-04739-8

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