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28 CHAPTER 1. 40 ([32]). Take r E jR+ BASIC FACTS IN P-ADIC ANALYSIS and let = L anz n E Ar(K:), 00 f(z) s = sup lanlr n- l > O. n;:::l n=O Then the following statements are equivalent: 1) lall> lanlr n- l whenever n > 1; 2) If(x) - f(y)1 = Ix - Yllall whenever x,y E K:[O;r]; 3) fis injective in K:[O;r] and j'(z) f=. 0 whenever z E K:[O;r]. Proof. (1) =? (2). Since lanlr n -+ 0 as n -+ 00, then the condition (1) implies Note that f(x) - f(y) = (x - y) 00 ( n-l al + ~ an j;xjyn-l-j ) . and hence If(x) - f(y)1 = Ix - Yllall· (2) =?

33. Assume that j is a non-constant entire function. Then tor any b E ", we have N (1", j ~ b) = N (1", 7) + 0(1). H. H. KhOlii [44] was the fist to begin a systematic development of one variable Nevanlinna theory over non-Archimedean fields. He gave some preliminary definitions for the Nevanlinna functions and took the first steps toward recognizing that the information coming from the Newton polygon could be used to prove a non-Archimedean analog of the Jensen formula. These valence functions, which appear in the works of Khoai [44], CorralesRodrigaiiez [28], Khoai-Quang[50], and Boutabaa [10], are exact analogs ofthose in classical Nevanlinna theory.

Write f = h/ fo, where fo, h Fo = fo, Fi = h - a;Jo (i = 1,2, ... k(z)1 ::; Amax{lFo(z)l, lFi(Z)I} (k = 0, 1). , W 2i =W2, Next we fix z E K - i=1,2, ... ,q. K[O; Pol such that W 2 (z),h(z),Fi (z) =1O, i=O,l, ... ,q. 15, we can similarly obtain 1F0(z)··· F (z)1 (q-1)loglf(z)1 ::;log IW2 (z)! 4. 15, wc obtain and hence logD 2j (z) ::::: -2logr. 15. Similarly, for a non-constant meromorphic functlon f in of order k by Nk,Ram(r, f) -_ (k + 1)N(r, f) - N ((k)) r, f K, define the mmification term +N (1) r, f(k) .