p-adic meromorphic functions f'P'(f), g'P'(g) sharing a small function

Details p-adic meromorphic functions f'P'(f), g'P'(g) sharing a small function p-adic meromorphic functions f'P'(f), g'P'(g) sharing a small function

2011-05-30

arxiv.org/abs/1105.6006v1

Mathematics

Number Theory

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Scientific paper

Let K be a complete algebraically closed p-adic field of characteristic zero. Let f, g be two transcendental meromorphic functions in the whole field K or meromorphic functions in an open disk that are not quotients of bounded analytic functions. Let P be a polynomial of uniqueness for meromorphic functions in K or in an open disk and let $\alpha$ be a small meromorphic function with regards to f and g. If f'P'(f) and g'P'(g) share $\alpha$ counting multiplicity, then we show that f=g provided that the multiplicity order of zeroes of P' satisfy certain inequalities. If $\alpha$ is a Moebius function or a non-zero constant, we can obtain more general results on P.

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