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

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 regard to f and g. If f P (f) and g P (g) share α ignoring multiplicity orders, then we show that f = g provided that the multiplicity order of zeros of P satisfies certain inequalities. If α is a Moebius function or a non-zero constant, we can obtain more general results on P and if f is an analytic funtion in K or in the disk, we also obtain more precise results. All results follow previous ones obtained for similar meromorphic functions sharing a small function, counting multiplicity. That comes after similar results obtained by Buy Thi Kieu Oanh and Ngo Thi Thu Thuy for complex functions and results obtained by the present authors with Jacqueline Ojeda, counting multiplicity.
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Submitted on : Tuesday, November 13, 2018 - 11:21:55 AM
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Kamal Boussaf, Alain Escassut. p-adic meromorphic functions f P (f ), g P (g) sharing a small function, ignoring multiplicity. Contemporary Mathematics, Amer. Math. Soc., 2018. ⟨hal-01920413⟩



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