On the growth of meromorphic functions with several deficient values.

Growth problems for subharmonic functions of finite order in space

“…There exist other related studies of this type, e.g. by Edrei and Fuchs [2], also [4], [5], [9]. This involves no restriction for the kind of asymptotic problems studied here.…”

“…For X > 1, precise results are not in general available even for entire functions. A recent result in this direction is (8) *(") > (0.9)Ç^*l (i<x<oo) (Miles and Shea [10]), and well-known examples [2] show that (8) would fail for large X if the 0.9 factor were replaced by any constant greater than unity. Inequality (8) is an easy corollary of the main result of [10], Theorem A.…”

Anatolii asirovich gol'dberg

“…On the other hand, in 1966, Anderson and Clunie proved [2] that if T (r, f ) = O(log 2 r) , r → ∞, then each deficient value must be asymptotic. It follows from the results of Edrei and Fuchs [20] that if f is an entire function of finite order and δ(a, f ) = 1, then a is an asymptotic value of f . It turns out that for functions of infinite order this is not true: in 1967, Gol'dberg [G52] (see also [G57, Chapter V]) constructed an entire function f (z) of infinite order with a non-asymptotic deficient value a with δ(a, f ) = 1.…”

Meromorphic functions with negative zeros and positive poles and a theorem of Teichmuller

“…While the functions in the clasŝ C, λ < 1, are the "extremal" functions for the solution to the "ellipse" problem for meromorphic functions of order λ < 1 (as indicated in the proof of Theorem B and by the examples showing the best possible nature of inequality (5)), this does not appear to be the case if λ > 1. Indeed, whatever the solution to the "ellipse" problem for functions of order λ, lower order μ ^ 1, estimates obtained by Edrei [1,Theorem 4a] (see also [3], Theorem 1) show that for such functions Thus, the behavior of functions in ^\ for λ near positive, even integers indicates that the class ^λ is probably not the class of "extremal" functions for the solution to the "ellipse" problem for meromorphic functions of order λ > 1. The Supporting Institutions listed above contribute to the cost of publication of this Journal, but they are not owners or publishers and have no responsibility for its content or policies.…”