Version of nevanlinna five value theorem and hayman conjecture for derivatives of P-adic meromorphic functions

In this paper, we gave a version of Nevanlinna five-value theorem and Hayman Conjecture for derivatives of p-adic meromorphic functions. | Nguyễn Xuân Lai và Đtg Tạp chí KHOA HỌC & CÔNG NGHỆ 113(13): 25 - 31 VERSION OF NEVANLINNA FIVE-VALUE THEOREM AND HAYMAN CONJECTURE FOR DERIVATIVES OF P-ADIC MEROMORPHIC FUNCTIONS Nguyen Xuan Lai1,*, Tran Quang Vinh2 1 Hai Duong College, 2Dai Tu High Schools, Thai Nguyen SUMMARY In this paper, we gave a version of Nevanlinna five-value theorem and Hayman Conjecture for derivatives of p-adic meromorphic functions. Keywords: Unique problem, p-adic Meromorphic functions, derivative, Nevanlinna, Hayman Conjecture, non- Archimedean meromorphic functions, Value distribution, compensation function, characteristic function, counting function. INTRODUCTION* In 1926, Nevanlinna proved the following result (the Nevanlinna five-value theorem). Theorem A. Let f, g be two non-constant meromorphic functions such that for five distinct values a1 , a2 , a3 , a4 , a5 we have f(x) = ai ⇔ g(x) = ai , i = 1, 2, 3, 4, 5. Then f ≡ g. In 1967, Hayman also proposed the following conjecture. Hayman Conjecture. If an entire function f satisfies f n ( z) f ' ( z) ≠ 1 for a positive integer n and all z∈ ℂ , then f is a constant. It has been verified for transcendental entire functions by Hayman himself for n > 1, and by Clunie for n ≥ 1. These results and some related problems have become to be known as Hayman's Alternative, and caused increasingly attensions. In recent years the similar problems are investiged for functions in a nonArchimedean fields. In 2008, J. Ojeda[16] proved that for a transcendental meromorphic function f in an algebraically closed fields of characteristic zero, complete for a nonArchimedean absolute value K, the function f ' f n − 1 has innitely many zeros, if n ≥ 2. Ha Huy Khoai and Vu Hoai An[12] established a similar results for a differential n monomial of the form f ( f meromorphic function in ℂ P * Email: laicdhd@ (k ) . ), where f is a In this paper, by using some arguments in [12] we gave a result similar to the Nevanlinna five-value .