On zeros of entire functions of special form

Sufficient conditions for an entire function of special form to have no zeros in the open lower half-plane are obtained.     

On the growth of entire functions

Suppose thatα>1, 0<R<∞ and thatf is analytic in |z|≤αR with |f(0)|≥1. It is shown that for a constant dα depending only onα, . Therefore iff is entire of order λ<∞, logM(r,f)/T(r,f) has order at most λ/2. These results are shown by example to be quite precise.     

Differential operators and entire functions with simple real zeros

Let ? and f be functions in the Laguerre-Pólya class. Write ?(z)=e−αz²?₁(z) and f(z)=e−βz²f₁(z), where ?₁ and f₁ have genus 0 or 1 and α,β?0. If αβ<1/4 and ? has infinitely many zeros, then ?(D)f(z) has only simple real zeros, where D denotes differentiation.     

Asymptote of some entire exponential-type functions with zeros on spirals

The author considers the Weierstrass canonical product of the first kind II(z), all roots of which lie on a spiral with equation in polar coordinates (r, ): = log log r. With certain additional conditions on the roots, the asymptote is found for the function ln{e^AzII(z)} (A is some constant) in the complex plane cut along the spiral = log log r. The result is applied to the question of the sufficient condition for the satisfaction of an inequality for exponential-type functions, used in questions of the Dirichlet-series representation of analytic functions.Translated from Matematicheskie Zametki, Vol. 14, No. 2, pp. 173–184, August, 1973.The author wishes to express his gratitude to N. V. Govorov for his interest in this work and to I. F. Krasichkov for making certain information available to him.     

On the growth type of entire monogenic functions

In this paper we establish an explicit relation between the growth type of general entire solutions to the generalized Cauchy-Riemann system in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} and their Taylor coefficients. This formula then enables us to compute the growth type of some higher dimensional generalizations of the trigonometric and special functions that are null-solutions to this system.     

Growth of entire ridge functions with restrictions on the arguments of zeros

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 6, pp. 727–734, June, 1991.