Small  Functions  in Disks of $\C_p$

Alain Escassut

doi:10.5644/SJM.21.01.06

Authors

Alain Escassut Universite Clermont Auvergne, UMR CNRS 6620, LMBP, F-63000 Clermont-Ferrand, France

DOI:

https://doi.org/10.5644/SJM.21.01.06

Keywords:

$p$-adic analytic functions, small functions, order, type, cotype of growth

Abstract

Small functions were defined in complex analysis and next in ultrametric analysis. Order of growth and type of growth were also defined in complex analysis and have a similar definition in ultrametric analysis. Here we compare these two notions in the same way, on a complete ultrametric algebraically closed field $\K$ of characteristic $0$ such as $\C_p$.

Small functions with respect to an entire function $f$ were studied in several articles. Inside an ''open'' disk, small functions also exist. After a general study, here we examine how two analytic functions inside an open disk can share three small functions, ignoring multiplicity and we give sufficient conditions proving that these two functions are equal.

Statistics

Abstract: 172 / PDF: 29

References

K. Boussaf, A. Boutabaa, and A. Escassut, Growth of $p$-adic entire functions and applications, Houston Journal of Mathematics, vol 40 (3), p.715-736 (2014).

K. Boussaf and A. Escassut, Growth of analytic functions in an ultrametric open disk and branched values, Bulletin of the Belgian Mathematical Society, Simon Stevin 28 p.1–16 (2021).

A. Escassut and C.C. Yang, A short note on two p-adic meromorphic functions sharing a few small ones. Rendiconti del Circolo Matematico di Palermo 70 (2), p. 623-630 (2021).

A. Escassut, p-adic Analytic Functions, World Scientific Publishing, Singapore (2021).

A. Escassut, Survey on p-adic meromorphic functions sharing ﬁve small ones on a work by Ta Thi Hoai An and Nguyen Viet Phuong, with some additional properties. Mathematics Open, Vol. 1 (2022).

A. Escassut, New properties on the growth of ultrametric entire functions and applications J. Math. Math. Sci. Vol 3, (1), 81–102 (2024).

M. Krasner, Prolongement analytique uniforme et multiforme dans les corps values complets. Les tendances geometriques en algebre et theorie des nombres, Clermont-Ferrand, p.94-141 (1964). Centre National de la Recherche Scientifique (1966), (Colloques internationaux de C.N.R.S. Paris, 143).

L.A. Rubel, Entire and meromorphic functions. Springer-Verlag, New York, (1996).

T. Thi, H. An, and P.N. Viet, Non-Archimedean Second Main Theorem sharing small functions. Taiwanese Journal of Mathematics Math. 27 (5), 913-929.(October, 2023) DOI: 10.11650/tjm/230701 Open Access

G. Valiron, Lectures on the general theory of integral functions, Chelsea Publishing Company (1949).