On the rate of approximation in the unit disc of $H^1$-functions by logarithmic derivatives of polynomials with zeros on the boundary

2020 ◽  
Vol 84 (3) ◽  
pp. 437-448 ◽  
Author(s):  
M. A. Komarov
Keyword(s):  
Unit Disc ◽  
Rate Of Approximation ◽  
Derivatives Of ◽  
Logarithmic Derivatives

scholarly journals SHARP LOGARITHMIC DERIVATIVE ESTIMATES WITH APPLICATIONS TO ORDINARY DIFFERENTIAL EQUATIONS IN THE UNIT DISC

2010 ◽  
Vol 88 (2) ◽  
pp. 145-167 ◽  
Author(s):  
I. CHYZHYKOV ◽  
J. HEITTOKANGAS ◽  
J. RÄTTYÄ
Keyword(s):  
Differential Equations ◽  
Unit Disc ◽  
Logarithmic Derivative ◽  
Maximum Modulus ◽  
Exceptional Set ◽  
Proximate Order ◽  
Upper Density ◽  
Linear Differential ◽  
Logarithmic Derivatives ◽  
Growth Of Solutions

AbstractNew estimates are obtained for the maximum modulus of the generalized logarithmic derivatives f(k)/f(j), where f is analytic and of finite order of growth in the unit disc, and k and j are integers satisfying k>j≥0. These estimates are stated in terms of a fixed (Lindelöf) proximate order of f and are valid outside a possible exceptional set of arbitrarily small upper density. The results obtained are then used to study the growth of solutions of linear differential equations in the unit disc. Examples are given to show that all of the results are sharp.

On Logarithmic Derivatives of Functions in a Class of Starlike Mappings

1993 ◽  
Vol 36 (1) ◽  
pp. 38-44
Author(s):  
Alan D. Gluchoff
Keyword(s):  
Integral Means ◽  
Koebe Function ◽  
Starlike Mappings ◽  
Derivatives Of ◽  
Logarithmic Derivatives

AbstractThe purpose of this paper is to prove some facts about integral means of (d2/dz2)(log[f(z)/z])—or equivalently f″/f, for f in a class of starlike mappings of a "singular" nature. In particular it is noted that the Koebe function is not extremal for the Hardy means Mp(r,f″/f) for functions in this class.

scholarly journals Moments of logarithmic derivatives of L -functions

2018 ◽  
Vol 183 ◽  
pp. 40-61 ◽  
Author(s):  
Peter J. Cho ◽  
Henry H. Kim
Keyword(s):  
Derivatives Of ◽  
Logarithmic Derivatives

scholarly journals DIFFERENTIAL STRUCTURE OF ABELIAN FUNCTIONS

2008 ◽  
Vol 19 (02) ◽  
pp. 145-171 ◽  
Author(s):  
KOJI CHO ◽  
ATSUSHI NAKAYASHIKI
Keyword(s):  
Abelian Variety ◽  
Differential Operators ◽  
Free Resolution ◽  
Holomorphic Differential ◽  
Differential Structure ◽  
Linear Basis ◽  
Higher Dimensional ◽  
Abelian Functions ◽  
Derivatives Of ◽  
Logarithmic Derivatives

The space of Abelian functions of a principally polarized abelian variety (J,Θ) is studied as a module over the ring [Formula: see text] of global holomorphic differential operators on J. We construct a [Formula: see text] free resolution in case Θ is non-singular. As an application, in the case of dimensions 2 and 3, we construct a new linear basis of the space of abelian functions which are singular only on Θ in terms of logarithmic derivatives of the higher-dimensional σ-function.

OMEGA THEOREMS FOR $\frac{L'}{L}(1, \chi_D)$

2013 ◽  
Vol 09 (03) ◽  
pp. 561-581 ◽  
Author(s):  
M. MOURTADA ◽  
V. KUMAR MURTY
Keyword(s):  
Classical Result ◽  
London Math ◽  
Quadratic Field ◽  
Analogous Result ◽  
Logarithmic Derivative ◽  
Dirichlet Character ◽  
Derivative Formula ◽  
Derivatives Of ◽  
Logarithmic Derivatives

A classical result of Chowla [Improvement of a theorem of Linnik and Walfisz, Proc. London Math. Soc. (2) 50 (1949) 423–429 and The Collected Papers of Sarvadaman Chowla, Vol. 2 (Centre de Recherches Mathematiques, 1999), pp. 696–702] states that for infinitely many fundamental discriminants D we have [Formula: see text] where χD is the quadratic Dirichlet character of conductor D. In this paper, we prove an analogous result for the logarithmic derivative [Formula: see text], and investigate the growth of the logarithmic derivatives of real Dirichlet L-functions. We show that there are infinitely many fundamental discriminants D (both positive and negative) such that [Formula: see text] and infinitely many fundamental discriminants 0 < D such that [Formula: see text] In particular, we show that the Euler–Kronecker constant γK of a quadratic field K satisfies γK = Ω( log log |dK|). We get sharper results assuming the GRH. Moreover, we evaluate the moments of [Formula: see text].

INTEGRATION BY PARTS AND SMOOTHNESS OF THE LAW FOR A CLASS OF STOCHASTIC EVOLUTION EQUATIONS

2004 ◽  
Vol 07 (01) ◽  
pp. 89-129 ◽  
Author(s):  
STEFANO BONACCORSI ◽  
MARCO FUHRMAN
Keyword(s):  
Additive Noise ◽  
Evolution Equations ◽  
Transition Probabilities ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Integration By Parts ◽  
Infinite Dimensional ◽  
Integration By Parts Formula ◽  
Derivatives Of ◽  
Logarithmic Derivatives

We consider a Markov process X in a Hilbert space H, solution of a semilinear stochastic evolution equation driven by an infinite-dimensional Wiener process, occurring in the equation as an additive noise. Using techniques of the Malliavin calculus, under suitable assumptions, we prove an integration by parts formula for the transition probabilities νt, t>0 (the laws of Xt). We deduce results on differentiability (i.e. existence of logarithmic derivatives) of νt along a set of directions h∈H which can be described in terms of the coefficients of the equation. The general results are then applied to various classes of non linear stochastic partial differential equations and systems.

scholarly journals ON THE MODULUS OF THE SELBERG ZETA-FUNCTIONS IN THE CRITICAL STRIP

2015 ◽  
Vol 20 (6) ◽  
pp. 852-865
Author(s):  
Andrius Grigutis ◽  
Darius Šiaučiūnas
Keyword(s):  
Riemann Surface ◽  
Modular Group ◽  
Compact Riemann Surface ◽  
Zeta Functions ◽  
Critical Strip ◽  
The Real ◽  
Derivatives Of ◽  
Logarithmic Derivatives

We investigate the behavior of the real part of the logarithmic derivatives of the Selberg zeta-functions ZPSL(2,Z)(s) and ZC (s) in the critical strip 0 < σ < 1. The functions ZPSL(2,Z)(s) and ZC (s) are defined on the modular group and on the compact Riemann surface, respectively.

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