Abstract
We investigate the generalization of the proximate order in the Valiron sense. The concept of proximate order is widely used in the theories of entire, meromorphic, subharmonic, and plurisubharmonic functions. We give a general interpretation of this growth estimate of the function with respect to the model function. We discuss the generalized proximate order corresponding to an arbitrary model function of growth. We also consider some properties of the generalized proximate order in the case when the model function of growth is multiplicative. In particular, we prove that the smoothness conditions on the proximate order do not matter.
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Notes
Here we use the fact \(n(r)=0\) for \(0\le t<|a_1|\).
Clearly in this inequality one can replace \(|\Pi (z)|\) by \(\max \limits _{|z|=r}|\Pi (z)|\).
Since M is entire model function that \(\displaystyle \varliminf _{t\rightarrow \infty }\frac{M(t)}{t}>0\)
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Acknowledgements
We gratefully thank the referees and Guest Associate Editor of Journal of Mathematical Sciences for careful reading of the paper and for the suggestions that have greatly improved the paper.
Funding
The research of the first author is supported by Russian Science Foundation (project No. 24-21-00006, https://rscf.ru/project/24-21-00006/).
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Malyutin, K., Kabanko, M. ON THE PROXIMATE ORDER WITH RESPECT TO THE MODEL FUNCTION. J Math Sci (2024). https://doi.org/10.1007/s10958-024-06957-w
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DOI: https://doi.org/10.1007/s10958-024-06957-w