Iossif Ostrovskii

On a Problem of A. Eremenko

Let Pmn, 0 < Standardm < Standardn − 1, be a polynomial formed by the first m terms of the expansion of (1 + z)n according to the binomial formula. We show that, if m, n → ∞ in such a way that limm,n→∞ m/n = α ∊ (0,1), then the zeros of Pmn tend to a curve which can be explicitly described.

Anatolii asirovich gol'dberg

1. Meromorphic functions 1.1. The first result: inverse problem of value distribution theory 1.2. Structure of the set of deficient values 1.3. Invariance of the deficiencies 1.4. Non-asymptotic deficient values 1.5. The lemma on logarithmic derivative 1.6. Meromoprhic functions with separated zeros and poles 1.7. Teichmüller’s conjecture and an inequality for functions convex with respect to the logarithm 1.8. Exceptional linear combinations of entire functions 1.9. Meromorphic solutions of differential equations 2. Simply connected Riemann surfaces 2.1 Extension of the Denjoy–Carleman–Ahlfors Theorem 2.2 Riemann surfaces with finitely many “ends” 2.3 Comb-like entire functions 3. Entire functions 3.1. Connection between the Phragmén-Lindelöf indicator of an entire function and its zero distribution 3.2. Phragmén-Lindelöf indicators of Hermitian-positive entire functions ∗Supported by NSF grant DMS-9500636 †The second-named author thanks Professor Matts Essén and the Department of Mathematics of Uppsala University for their kind hospitality during his work on this paper

Hardy’s Generalization of ez and Related Analogs of Cosine and Sine

In 1904, Hardy introduced an entire function depending on two parameters being a generalization of ez. He had studied in detail its asymptotic properties and that of its zeros. We consider the two following non-asymptotic problems related to the zeros. (i) Determine values of the parameters such that all the zeros belong to the open left half-plane. For these values, the analogs of sine and cosine generated by Hardy’s function have real, simple and interlacing zeros. (ii) Determine the number of real zeros as a function of the parameters.

*Non-symmetric Linnik distributions

Abstract The aim of this Note is to study the probability density with characteristic function ϕ α,θ,ν ( t )=1/(1+ e 〈 −tθsgnt | t | α ) ν , where 0 α θ | ≤ min( πα /2, π - πα /2), and ν > 0. This density, first introduced by Linnik for θ = 0, ν = 1, received several applications later. It does not have any explicit representation. We consider here its integral and series representations and its analytical properties.

On the Lévy-Raikov-Marcinkiewicz theorem

Abstract Let μ be a finite nonnegative Borel measure. The classical Levy–Raikov–Marcinkiewicz theorem states that if its Fourier transform μ can be analytically continued to some complex half-neighborhood of the origin containing an interval (0,i R ) then μ admits analytic continuation into the strip {t: 0 I t . We extend this result to general classes of measures and distributions, assuming non-negativity only on some ray and allowing temperate growth on the whole line. To cite this article: I. Ostrovskii, A. Ulanovskii, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Distance between a Maximum Modulus Point and Zero Set of an Entire Function

Let f be an entire function of finite positive order. A maximum modulus point is a point w such that |f(w)|= max {|f(z)|:|z=|w|}. We obtain lower bounds for the distance between a maximum modulus point w and the zero set of f. These bounds are valid for all sufficiently large values of |w|.

Non-oscillating Paley-Wiener functions

A non-oscillating Paley-Wiener function is a real entire functionf of exponential type belonging toL 2(R) and such that each derivativef (n),n=0, 1, 2,…, has only a finite number of real zeros. It is established that the class of such functions is non-empty and contains functions of arbitrarily fast decay onR allowed by the convergence of the logarithmic integral. It is shown that the Fourier transform of a non-oscillating Paley-Wiener function must be infinitely differentiable outside the origin. We also give close to best possible asymptotic (asn→∞) estimates of the number of real zeros of then-th derivative of a functionf of the class and the size of the smallest interval containing these zeros.

Non-symmetric Linnik distributions

Abstract The aim of this Note is to study the probability density with characteristic function ϕ α,θ,ν ( t )=1/(1+ e 〈 −tθsgnt | t | α ) ν , where 0 α θ | ≤ min( πα /2, π - πα /2), and ν > 0. This density, first introduced by Linnik for θ = 0, ν = 1, received several applications later. It does not have any explicit representation. We consider here its integral and series representations and its analytical properties.

On a problem of H. Shapiro

Let µ be a real measure on the line such that its Poisson integral M(z) converges and satisfies |M(x+iy)|≤Ae-cyα, y → + ∞, for some constants A, c > 0 and 0 < α ≤ 1. We show that for 1/2 < α ≤ 1 the measure µ must have many sign changes on both positive and negative rays. For 0 < α ≤ 1/2 this is true for at least one of the rays, and not always true for both rays. Asymptotical bounds for the number of sign changes are given which are sharp in some sense.

On the Poisson Representation of a Function Harmonic in the Upper Half-Plane

New conditions for the validity of the Poisson representation (in usual and generalized form) for a function harmonic in the upper half-plane are obtained. These conditions differ from known ones by weaker growth restrictions inside the half-plane and stronger restrictions on the behavior on the real axis.