Abstract
A theorem of A. and C. Rényi on periodic entire functions states that an entire function f(z) must be periodic if P(f(z)) is periodic, where P(z) is a nonconstant polynomial. By extending this theorem, we can answer some open questions related to the conjecture of C. C. Yang concerning periodicity of entire functions. Moreover, we give more general forms for this conjecture and we prove, in particular, that f(z) is periodic if either P(f (z))f(k)(z) or P(f(z))/f(k)(z) is periodic, provided that f(z) has a finite Picard exceptional value. We also investigate the periodicity of f(z) when f(z)n + a1f′(z) + ⋯ + akf(k)(z) is periodic. In all our results, the possibilities for the period of f(z) are determined precisely.
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The authors would like to thank the referees for their comments and suggestions.
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Z. Latreuch is supported by the Directorate General for Scientific Research and Technological Development (DGRSDT), Algeria.
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Latreuch, Z., Zemirni, M.A. On a theorem of A. and C. Rényi and a conjecture of C. C. Yang concerning periodicity of entire functions. Anal Math 48, 111–125 (2022). https://doi.org/10.1007/s10476-022-0118-x
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DOI: https://doi.org/10.1007/s10476-022-0118-x
Key words and phrases
- differential polynomial
- entire function
- Nevanlinna theory
- order of growth
- periodic function
- Yang’s conjecture