P-Adic meromorphic functions f′P′(f), g′P′(g) sharing a small function

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 α be a small meromorphic function with regards to f and g . If f ′ P ′ ( f ) and g ′ P ′ ( g ) share α counting multiplicity, then we show that f = g provided that the multiplicity order of zeroes of P ′ satisfy certain inequalities. If α is a Moebius function or a non-zero constant, we can obtain more general results on P .