Unicity of Meromorphic Mappings

For a given meromorphic function I(z) and an arbitrary value a, Nevanlinna's value distribution theory, which can be derived from the well known Poisson-Jensen for­ mula, deals with relationships between the growth of the function and quantitative estimations of the roots of the equation: 1 (z) - a = O. In the 1920s as an application of the celebrated Nevanlinna's value distribution theory of meromorphic functions, R. Nevanlinna [188] himself proved that for two nonconstant meromorphic func­ tions I, 9 and five distinctive values ai (i = 1,2,3,4,5) in the extended plane, 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 4, then 1 = L(g), where L denotes a suitable Mobius transformation. Then in the 19708, F. Gross and C. C. Yang started to study the similar but more general questions of two functions that share sets of values. For instance, they proved that if 1 and 9 are two nonconstant entire functions and 8 , 82 and 83 are three distinctive finite sets such 1 1 that 1- (8 ) = g-1(8 ) CM for i = 1,2,3, then 1 = g.

Contenu

Preface. 1: Nevanlinna theory. 1.1. Parabolic manifolds and Hermitian geometry. 1.2. The first main theorem. 1.3. Growths of meromorphic functions. 1.4. The lemma of logarithmic derivative. 1.5. Growth estimates of Wronskians. 1.6. The second main theorem. 1.7. Degenerate holomorphic curves. 1.8. Value distribution of differential polynomials. 1.9. The second main theorem for small functions. 1.10. Tumura-Clunie theory. 1.11. Generalizations of Nevanlinna theorem. 1.12. Generalizations of Borel theorem. 2: Uniqueness of meromorphic functions on C. 2.1. Functions that share four values. 2.2. Functions that share three values CM. 2.3. Functions that share pairs of values. 2.4. Functions that share four small functions. 2.5. Functions that share five small functions. 2.6. Uniqueness related to differential polynomials. 2.7. Polynomials that share a set. 2.8. Meromorphic functions that share the same sets. 2.9. Unique range sets. 2.10. Uniqueness polynomials. 3: Uniqueness of meromorphic functions on Cm. 3.1. Technical lemmas. 3.2. Multiple values of meromorphic functions. 3.3. Uniqueness of differential polynomials. 3.4. The four-value theorem. 3.5. The three-value theorem. 3.6. Generalizations of Rubel-Yang's theorem. 3.7. Meromorphic functions sharing one value. 3.8. Unique range sets of meromorphic functions. 3.9. Unique range sets ignoring multiplicities. 3.10. Meromorphic functions of order 4.8. Propagation theorems. 4.9. Uniqueness dealing with multiple values. 5: Algebroid functions of several variables. 5.1. Preliminaries. 5.2. Techniques of value distribution. 5.3. The second main theorem. 5.4. Algebroid reduction of meromorphic mappings. 5.5. The growth of branching divisors. 5.6. Reduction of Nevanlinna theory. 5.7. Generalizations of Malmquist theorem. 5.8. Uniqueness problems. 5.9. Multiple values of algebroid functions. References. Symbols. Index.