Albert Edrei

ON THE GROWTH OF MEROMORPHIC FUNCTIONS WITH SEVERAL DEFICIENT VALUES(

In this paper, we investigate the possibility of proving analogous theorems for meromorphic functions possessing deficient values (in the sense of R. Nevanlinna). The main interest of the results obtained lies in the fact that they provide partial answers to the three following questions. I. Under which conditions are deficiencies invariant under a change of origin? II. When are deficient values also asymptotic values? III. How does the presence of deficient values influence the gap structure of the Taylor expansion of an entire function? We leave aside questions II and III which will be treated in another paper [1]. We explain our notations in ?1 before stating our results in ?2. 1. Terminology and notations. The complex variable will be denoted by

Meromorphic functions with three radially distributed values

where 0 ^ wi < o!2 < • • • < wg < 2w (q ^ 1). We say that the roots of the equation (2) /(f) = a are distributed on the radii (1) if there exist at most a finite number of roots of the equation (2) which do not lie on the radii (1). With this definition, the main result of this paper takes the following form. Theorem 1. Letf(z) be meromorphic and such that the roots of the three equations (3) f(z) = 0, (4) f(z) = oo, (5) /«>« = 1 (1^0,/WmJ), be distributed on the radii (1). Denote by 5(a, f(l)) the deficiency of the value a, of the function fw, and assume (6) 8(0,/) +8(l,/»>) +«(«,/) >0. Then the order p, of f(z), is necessarily finite and

Zeros of Sections of Power Series

Statements of our results.- Discussion of our numerical results.- Outline of the method.- Notational conventions.- Properties of the Mittag-Leffler function of order 1 < ?<?.- Estimates for Gm(w) and Qm(w).- A differential equation.- Estimates for Jm(w) near the circumference |w|=1.- Existence and uniqueness of the Szego curve.- Crude estimates for |Um(w)| and |Qm(w)|.- Proof of Theorem 5.- Proof of Theorem 1.- Proof of Theorem 2.- The circular portion of the Szego curve (Proof of Theorem 3).- Proof of Theorem 4.- Proof of Theorem 6.- Properties of GBP-functions proof of assertion I of Theorem 7.- GBP-functions of genus zero are admissible in the sense of Hayman.- The functions Um(w), Qm(w), Gm(w) associated with GBP-functions of genus zero.- Estimates for Um(w).- Determination of lim ?m(?).- Comparison with integrals proof of assertion II of Theorem 7.- The Szego curves for GBP-functions of genus zero.- Estimates for Um(?mei?w).- Proof of assertion IV of Theorem 7.

Locally tauberian theorems for meromorphic functions of lower order less than one

(2) 8(r,f) > 0. The number of deficient values off, that is the number of distinct values of Xfor which (2) holds, will be denoted by v(f) (< + oo). Nevanlinnas fundamental results show that (3) A(f) < 2, and it is not difficult to find functions such that equality holds in (3). However, all the theorems and examples known to the author indicate that, if ,u is finite, (i) the relation A(f) = 2 is only possible for particular values of ,t; (ii) if A(f) = 2 the number of deficient values of f(z) remains finite. These remarks lead to interesting questions which may be formulated as a DEFICIENCY PROBLEM. Let f(z) be a meromorphic function of lower order ,t < + oo. I. Determine, as explicitly as possible, a function E(,ft) such that (4) /\(f)_ :!! EG be sharp for all values of ,u.