Abstract
In this paper, we show that if an entire function \(f(z_1,z_2)\) of two (or more) complex variables verifies \(\left| f(z_1,z_2) \right| \le K(\left| P(z_1,z_2) \right| )\), where \(P(z_1,z_2)\) is a polynomial that is not a power in \({{\mathbb {C}}}[[z_1,z_2]]\), and K is any positive-valued real function, then \(f(z_1,z_2)\) can be written as a holomorphic function of P.
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Notes
The author wants to thank the referee for providing an alternative argument that avoids the use of the continuity for h, assuming that K is locally bounded, and using Riemann extension theorem.
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Communicated by Ali Abkar.
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The author is partially supported by the Ministerio de Economía y Competitividad from Spain, under the Project “Álgebra y geometría en sistemas dinámicos y foliaciones singulares” (Ref.: MTM2016-77642-C2-1-P).
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Mozo-Fernández, J. Entire Functions Polynomially Bounded in Several Variables. Bull. Iran. Math. Soc. 46, 1117–1122 (2020). https://doi.org/10.1007/s41980-019-00316-1
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DOI: https://doi.org/10.1007/s41980-019-00316-1