The growth irregularity of slowly growing entire functions

We show that entire transcendental functions f satisfying log M(r,f) = o(log 2r), r → ∞ (M(r,f): = maxf(z)| necessarily have growth irregularity, which increases as the growth diminishes. In particular, if 1 < p < 2, then the asymptotics log M(r,f) = (log pr) +0 (log2-pr), r → ∞ is impossible. It becomes possible if "o" is replaced by "O.". © Springer Science+Business Media, Inc. 2006

On power series having sections with multiply positive coefficients and a theorem of Pólya

[No abstract available

On the Titchmarsh convolution theorem

[No abstract available

Power series having tails with only real nonpositive zeroes [Les séries de puissances dont les restes ont seulement des zéros non-positifs]

If for all sufficiently large n the nth tail of the power series of an entire function f has only real nonpositive zeroes, then log M(r, f) ≤ 1/2 log 2(log r)2 + O(log r), r → ∞. This bound is close to the best possible

On an application of the hardy classes to the Riemann Zeta-function

We show that the function f(z): z/1 - zζ (1/1 - z), |z| < 1, belongs to the Hardy class Hp if and only if 0 < p < 1. © Tübi̇tak

Analytic and Asymptotic Properties of Generalized Linnik Probability Densities

[No abstract available

Non-symmetric Linnik distributions [Les distributions de Linnik non-symmétriques]

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

Non-oscillating Paley-Wiener functions

A non-oscillating Paley-Wiener function is a real entire function f of exponential type belonging to L2 (R) and such that each derivative f(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 on R 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 (as n → ∞) estimates of the number of real zeros of the n-th derivative of a function f of the class and the size of the smallest interval containing these zeros

On a conjecture of Yu. V. Linnik

A survey is given of work concerning Linnik's 1960 conjecture on the growth of entire characteristic functions (Fourier transforms) of probability measures. A proof of this conjecture is presented that is substantially shorter and more elementary than the known proofs. With the help of the same idea, new facts are established about the growth of entire characteristic functions with restrictions on the arguments of the zeros