By Pei-Chu Hu

For a given meromorphic functionality I(z) and an arbitrary worth a, Nevanlinna's price distribution conception, which are derived from the well-known Poisson-Jensen for mula, offers with relationships among the expansion of the functionality and quantitative estimations of the roots of the equation: 1 (z) - a = O. within the Nineteen Twenties as an software of the distinguished Nevanlinna's worth distribution idea of meromorphic capabilities, R. Nevanlinna [188] himself proved that for 2 nonconstant meromorphic func tions I, nine and 5 particular values ai (i = 1,2,3,4,5) within the prolonged aircraft, if 1 1- (ai) = g-l(ai) 1M (ignoring multiplicities) for i = 1,2,3,4,5, then 1 = g. Fur 1 thermore, if 1- (ai) = g-l(ai) CM (counting multiplicities) for i = 1,2,3 and four, then 1 = L(g), the place L denotes an appropriate Mobius transformation. Then within the 19708, F. Gross and C. C. Yang began to learn the same yet extra basic questions of 2 features that percentage units of values. for example, they proved that if 1 and nine are nonconstant complete services and eight , eighty two and eighty three are 3 unique finite units such 1 1 that 1- (8 ) = g-1(8 ) CM for i = 1,2,3, then 1 = g.

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**Example text**

Let f : em -- P(V) be a linearly non-degenerate meromorphic mapping. Let I be the index of f. Then Q (q-2u+n)Tf(f) $ 'LNf(r,aj)-ONRam(r,f) j=O +lO log { (;) 2m-l holds for any ro ;~~ } + 0(1) < r < p < R, where 0 2':: 1 is the Nochka constant. Proof. 48, and w. l. o. , assume lIajll = 1 for j = 0, ... , q. (j) II ' CHAPTER 1. NEVANUNNA THEORY 56 which yields where c is a positive constant. 48, we obtain q Lw(aj)mf(r,aj) j=O < (n+l)Tf(r)-NRam(r,f) +llog { (;/m-l ;~R:} + 0(1). 1) and the properties of the Nochka weights, we have q Lmf(r,aj) j=O < (2u-n+l)Tf(r)-ONRam(r,f) and, hence, the theorem follows from this and the first main theorem.

CHAPTER 1. 5 Growth estimates of Wronskians Following Vitter [265] and Fujimoto [62], first of all, we introduce some basic properties of generalized Wronskians. 39 (cf. [265], [62]). Let 10, ft, ... , In be linearly independent meromorphic functions in em. Write 1= (fo, ft, ... , In). Then there are multi-indices Vi E Z+ (i = 1, ... , n) such that 0 < IVil :::; i and I, 8 V1 I, ... , 8 v " I are linearly independent over em. Proof. For any positive integer k, write Fo = {f}, Fk = {8 I I v E Z+, Ivi = k}.

Proof. By the concavity of logarithmic function (cf. 14) Put

0),