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

TL;DR

This paper investigates conditions under which two p-adic meromorphic functions sharing a small function must be identical, extending uniqueness results in p-adic analysis with specific polynomial and multiplicity conditions.

Contribution

It establishes new uniqueness theorems for p-adic meromorphic functions sharing a small function, involving derivatives and polynomial conditions, generalizing previous results.

Findings

f=g when sharing a small function under certain multiplicity conditions

More general results obtained when the shared function is a Möbius function or constant

Conditions on zero multiplicities of P' are crucial for the uniqueness conclusion

Abstract

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.