Asymptotic behavior of the logarithmic derivative of an entire function of zero order

1979 ◽  
Vol 30 (3) ◽  
pp. 221-226
Author(s):  
A. A. Gol'dberg ◽  
N. E. Korenkov
Keyword(s):  
Asymptotic Behavior ◽  
Entire Function ◽  
Logarithmic Derivative ◽  
Zero Order

scholarly journals Convergence in $L^p[0,2\pi]$-metric of logarithmic derivative and angular $\upsilon$-density for zeros of entire function of slowly growth

2015 ◽  
Vol 7 (2) ◽  
pp. 209-214
Author(s):  
M.R. Mostova ◽  
M.V. Zabolotskyj
Keyword(s):  
Entire Function ◽  
Logarithmic Derivative ◽  
Zero Order

The subclass of a zero order entire function $f$ is pointed out for which the existence of angular $\upsilon$-density for zeros of entire function of zero order is equivalent to convergence in $L^p[0,2\pi]$-metric of its  logarithmic derivative.

ASYMPTOTIC BEHAVIOR OF THE LOGARITHMIC DERIVATIVE OF ENTIRE FUNCTION OF IMPROVED REGULAR GROWTH IN THE METRIC OF $L^q[0,2\pi]$

2021 ◽  
Vol 9 (1) ◽  
pp. 49-55
Author(s):  
R. Khats’
Keyword(s):  
Asymptotic Behavior ◽  
Entire Function ◽  
Convex Function ◽  
Logarithmic Derivative ◽  
Finite System ◽  
Regular Growth ◽  
Finite Sum

Let $f$ be an entire function with $f(0)=1$, $(\lambda_n)_{n\in\mathbb N}$ be the sequence of its zeros, $n(t)=\sum_{|\lambda_n|\le t}1$, $N(r)=\int_0^r t^{-1}n(t)\, dt$, $r>0$, $h(\varphi)$ be the indicator of $f$, and $F(z)=zf'(z)/f(z)$, $z=re^{i\varphi}$. An entire function $f$ is called a function of improved regular growth if for some $\rho\in (0,+\infty)$ and $\rho_1\in (0,\rho)$, and a $2\pi$-periodic $\rho$-trigonometrically convex function $h(\varphi)\not\equiv -\infty$ there exists a set $U\subset\mathbb C$ contained in the union of disks with finite sum of radii and such that \begin{equation*} \log |{f(z)}|=|z|^\rho h(\varphi)+o(|z|^{\rho_1}),\quad U\not\ni z=re^{i\varphi}\to\infty. \end{equation*} In this paper, we prove that an entire function $f$ of order $\rho\in (0,+\infty)$ with zeros on a finite system of rays $\{z: \arg z=\psi_{j}\}$, $j\in\{1,\ldots,m\}$, $0\le\psi_1<\psi_2<\ldots<\psi_m<2\pi$, is a function of improved regular growth if and only if for some $\rho_3\in (0,\rho)$ \begin{equation*} N(r)=c_0r^\rho+o(r^{\rho_3}),\quad r\to +\infty,\quad c_0\in [0,+\infty), \end{equation*} and for some $\rho_2\in (0,\rho)$ and any $q\in [1,+\infty)$, one has \begin{equation*} \left\{\frac{1}{2\pi}\int_0^{2\pi}\left|\frac{\Im F(re^{i\varphi})}{r^\rho}+h'(\varphi)\right|^q\, d\varphi\right\}^{1/q}=o(r^{\rho_2-\rho}),\quad r\to +\infty. \end{equation*}

scholarly journals Mocking the u-plane integral

2021 ◽  
Vol 8 (3) ◽  
Author(s):  
Georgios Korpas ◽  
Jan Manschot ◽  
Gregory W. Moore ◽  
Iurii Nidaiev
Keyword(s):  
Asymptotic Behavior ◽  
Gauge Theory ◽  
Entire Function ◽  
Gauge Group ◽  
Strong Coupling ◽  
Modular Forms ◽  
Correlation Functions ◽  
Coulomb Branch ◽  
Mock Modular Forms ◽  
Donaldson Invariants

AbstractThe u-plane integral is the contribution of the Coulomb branch to correlation functions of $${\mathcal {N}}=2$$ N = 2 gauge theory on a compact four-manifold. We consider the u-plane integral for correlators of point and surface observables of topologically twisted theories with gauge group $$\mathrm{SU}(2)$$ SU ( 2 ) , for an arbitrary four-manifold with $$(b_1,b_2^+)=(0,1)$$ ( b 1 , b 2 + ) = ( 0 , 1 ) . The u-plane contribution equals the full correlator in the absence of Seiberg–Witten contributions at strong coupling, and coincides with the mathematically defined Donaldson invariants in such cases. We demonstrate that the u-plane correlators are efficiently determined using mock modular forms for point observables, and Appell–Lerch sums for surface observables. We use these results to discuss the asymptotic behavior of correlators as function of the number of observables. Our findings suggest that the vev of exponentiated point and surface observables is an entire function of the fugacities.

On a conjecture of Fuchs

2001 ◽  
Vol 131 (5) ◽  
pp. 1209-1216 ◽  
Author(s):  
Joseph Miles ◽  
John Rossi
Keyword(s):  
Entire Function ◽  
Logarithmic Derivative ◽  
Image Position ◽  
Nevanlinna Deficiency

If f is an entire function of order ρ, 0 < ρ < 2−11, it is shown that the Nevanlinna deficiency d(0, f′/f) of the logarithmic derivative of f satisfies For small positive ρ, this result strengthens an earlier estimate of Eremenko et al. concerning a conjecture of Fuchs.

scholarly journals Painlevé V, Painlevé XXXIV and the degenerate Laguerre unitary ensemble

2019 ◽  
Vol 09 (02) ◽  
pp. 2050016 ◽  
Author(s):  
Chao Min ◽  
Yang Chen
Keyword(s):  
Asymptotic Behavior ◽  
Compatibility Condition ◽  
Logarithmic Derivative ◽  
Hankel Determinant ◽  
Ladder Operators ◽  
Coupled Riccati Equations ◽  
Coulomb Fluid ◽  
Laguerre Unitary Ensemble ◽  
Soft Edge ◽  
Unitary Ensemble

In this paper, we study the Hankel determinant associated with the degenerate Laguerre unitary ensemble (dLUE). This problem originates from the largest or smallest eigenvalue distribution of the dLUE. We derive the ladder operators and its compatibility condition with respect to a general perturbed weight function. By applying the ladder operators to our problem, we obtain two auxiliary quantities [Formula: see text] and [Formula: see text] and show that they satisfy the coupled Riccati equations, from which we find that [Formula: see text] satisfies the Painlevé V equation. Furthermore, we prove that [Formula: see text], a quantity related to the logarithmic derivative of the Hankel determinant, satisfies both the continuous and discrete Jimbo–Miwa–Okamoto [Formula: see text]-form of the Painlevé V. In the end, by using Dyson’s Coulomb fluid approach, we consider the large [Formula: see text] asymptotic behavior of our problem at the soft edge, which gives rise to the Painlevé XXXIV equation.

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