Strongly regular growth of entire functions of order zero

Abstract

ABSTRACT. For entire functions of order zero we introduce a new concept of regularity of growth, which is shown to possess properties similar to those which characterize the concept of totally regular growth of entire functions of finite order in the sense of Levin-Pfliiger.KEY WORDS: entire function of order zero, regularity of growth, function of totally regular growth, function of strongly regular growth. Let n(r, a, fl) be the number of roots of the function f in the sector {z : Izl <_ r, a < argz <_ fl}. We say [1, pp. 118-119] that the set of roots of an entire function has angular density if the limitexists for all a, 1~ with the exception, perhaps, of ~ or fl belonging to some countable set. In this case, for a given a, the relation A(fl) -A(a) = A(a, fl) defines a nondecreasing function A(fl) up to an arbitrary constant. In [1, Chaps. 2 and 3] a complete description of entire functions of totally regular growth is given. In particular, if p is noninteger, then the entire function f is a function of totally regular growth if and only if the set of its roots has angular density; moreover,

Some properties of principal submodules in the module of entire functions of exponential type and polynomial growth on the real axis

“…Hence, by Theorem 1 in work [12], function 0 has a strong regular growth and it satisfies asymptotic relation (2.8).…”

“…, where constant 4 depend only on function Φ, we obtain the representation of function as the product of two entire functions 1,1 and 2,1 , and 12) constant 0 depends only on function Φ. By (3.12) and identity ln | | = ln | 1,1 | + ln | 2,1 | we obtain the estimate…”

Binomial asymptotics for the logarithmic derivative of zero-order entire functions with zeros along curves of regular rotation

“…The set E ⊂ [0, +∞) is called an E 0 -set if mes (E ∩ [0, r]) = o(r), r → +∞. By Lemmas 4 and 5 from [8] we obtain the following proposition.…”

“…was introduced by the first author of this article in the 90th of the last century for the class H 0 , which has features similar to the features of functions of c.r.g. from the class H + (ρ) ( [8]). Criteria of s.r.g.…”

“…Moving the expression (−icA k (τ, υ)) to the left-hand side of the equality and dividing both sides into the multiplier (1 + ic) we find (8).…”