Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity

Recent results on the growth and distribution of the values of integral and meromorphic functions of genus zero

We shall establish the following three results in more general forms. (1) The second main theorem for small functions. Let f be a meromorphic function on the complex plane C. Let a1, . . . , aq be distinct meromorphic functions on C. Assume that ai are small with respect to f ; i.e., T (r, a) 0 (Corollary 1). Here as usual in Nevanlinna theory, the terms T (r, f) and N(r, ai, f) denote for the characteristic function and the truncated counting function, respectively. (2) Application to functional equations. Let KC be the field of meromorphic functions on C. For a function ψ : R>0 → R, put KψC = {a ∈ KC; T (r, a) ≤ O(ψ(r)) ||}, which is a subfield of KC. Then the following holds: Let F (x, y) ∈ KψC [x, y] be a polynomial in two variables over K ψ C . Assume that the curve F (x, y) = 0 over KψC has genus greater than one. If ζ1, ζ2 ∈ KC satisfy the functional equation F (ζ1, ζ2) = 0, then both ζ1 and ζ2 are contained in KψC (Corollary 2). (3) Height inequality for curves over function fields. Let k be a function field of one variable over C. Let X be a smooth projective curve over k, let D ⊂ X be a reduced divisor, let L be an ample line bundle on X and let e > 0. Then we have hk,KX (D)(P ) ≤ N (1) k,S(D,P ) + dk(P ) + ehk,L(P ) +Oe(1) for all P ∈ X(k)\D (Theorem 5). Here the notations are introduced in [V1], [V3] (see also section 9). Our proof uses Ahlfors’ theory of covering surfaces and the geometry of the moduli space of q-pointed stable curves of genus 0.

/pdf/the-second-main-theorem-for-small-functions-and-related-3ul2nqtkn7.pdf

The second main theorem for small functions and related problems

A real matrix, finite or infinite, is called totally positive if and only if all its minors, of all orders = 1, 2,..., are non-negative. An infinite sequence 

$$ {a_0},{a_1},{a_2}, \ldots ,\quad \left( {{a_0} = 1} \right), $$

is said to be totally positive(2) if and only if the 2-way infinite matrix 


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is totally positive, i. e. has only non-negative minors. We are concerned with the problem of describing the nature of the generating functions 

$$f\left( z \right) = a_0 + a_1 z + a_2 z^2 + \ldots$$

(2)

of totally positive sequences. The problem will be partially solved in the present note and its solution will be completed in the immediately following note by Albert Edrei.

On the Generating Functions of Totally Positive Sequences.

https://projecteuclid.org/download/pdf_1/euclid.kmj/1138043482

E. C. Titchmarsh

Papers

Le théorème de Picard-Borel et la théorie des fonctions méromorphes. By Rolf Nevanlinna. Pp.171. 35 fr. 1929. Paris; Gauthier-Villars.