• Corpus ID: 159041370

# Description of growth and oscillation of solutions of complex LDE's

@article{Chyzhykov2019DescriptionOG,
title={Description of growth and oscillation of solutions of complex LDE's},
author={Igor Chyzhykov and Janne Grohn and Janne Heittokangas and Jouni Rattya},
journal={arXiv: Classical Analysis and ODEs},
year={2019}
}
It is known that, equally well in the unit disc as in the whole complex plane, the growth of the analytic coefficients $A_0,\dotsc,A_{k-2}$ of \begin{equation*} f^{(k)} + A_{k-2} f^{(k-2)} + \dotsb + A_1 f'+ A_0 f = 0, \quad k\geq 2, \end{equation*} determines, under certain growth restrictions, not only the growth but also the oscillation of its non-trivial solutions, and vice versa. A uniform treatment of this principle is given in the disc $D(0,R)$, $0<R\leq \infty$, by using several…
4 Citations
Irregular finite order solutions of complex LDE's in unit disc
• Mathematics
• 2020
It is shown that the order and the lower order of growth are equal for all non-trivial solutions of $f^{(k)}+A f=0$ if and only if the coefficient $A$ is analytic in the unit disc and $\log^+ On the number of linearly independent admissible solutions to linear differential and linear difference equations • Mathematics Canadian Journal of Mathematics • 2020 Abstract A classical theorem of Frei states that if$A_p$is the last transcendental function in the sequence$A_0,\ldots ,A_{n-1}$of entire functions, then each solution base of the Logarithmic derivative estimates of meromorphic functions of finite order in the half-plane I. E. Chyzhykov, A. Z. Mokhon’ko. Logarithmic derivative estimates of meromorphic functions of finite order in the half-plane, Mat. Stud. 54 (2020), 172–187. We established new sharp estimates Oscillation of Solutions of LDE’s in Domains Conformally Equivalent to Unit Disc • Mathematics The Journal of Geometric Analysis • 2022 <jats:p>Oscillation of solutions of <jats:inline-formula><jats:alternatives><jats:tex-math>$$f^{(k)} + a_{k-2} f^{(k-2)} + \cdots + a_1 f' +a_0 f = 0$$</jats:tex-math><mml:math ## References SHOWING 1-10 OF 24 REFERENCES Finiteness of φ-order of solutions of linear differential equations in the unit disc • Mathematics • 2009 AbstractIf φ: [0, 1) → (0,∞) is a non-decreasing unbounded function, then the φ-order of a meromorphic function f in the unit disc is defined as$$\sigma _\phi (f) = \mathop {\lim \sup }\limits_{r Zero distribution of solutions of complex linear differential equations determines growth of coefficients • Mathematics • 2011 It is shown that the exponent of convergence λ(f) of any solution f of with entire coefficients A0(z), …, Ak−2(z), satisfies λ(f) ⩽ λ ∈ [1, ∞) if and only if the coefficients A0(z), …, Asymptotic behavior of meromorphic functions of completely regular growth Suppose X(r) is a positive continuous function on (0, ~) such that X(r) * ~ as r § ~ and is called a growth function. We will assume throughout this paper that X (2r) ~ MK (r) ( ] ) f o r some M > 0 Weighted Bergman spaces induced by rapidly incresing weights • Mathematics • 2012 This monograph is devoted to the study of the weighted Bergman space$A^p_\om$of the unit disc$\D$that is induced by a radial continuous weight$\om$satisfying {equation}\label{absteq} GROWTH ESTIMATES FOR SOLUTIONS OF LINEAR COMPLEX DIFFERENTIAL EQUATIONS • Mathematics • 2004 Two methods are used to find growth estimates (in terms of the p-characteristic) for the analytic solutions of f ( k ) + A k - 1 (z)f ( k - 1 ) + ... + A 1 (z)f' + A 0 (z)f = 0 in the disc {z E C: On the growth of a meromorphic function and its derivatives • Mathematics • 1989 The relative rates of growth of a function F meromorphic in the complex plane and its q derivative F (q) are studied via the Nevanlinna Characteristics T(r.F)and T(r.F (q)) and It is shown that lim ON INTEGRAL FUNCTIONS HAVING PRESCRIBED ASYMPTOTIC GROWTH where ^(t) is continuous, strictly increasing, and unbounded with ^(1) = 0. This involves no loss of generality since to any function which is increasing and convex in log r and not O(log r) (r —» Bergman projection induced by radial weight. • Mathematics • 2019 The question of when the Bergman projection$P_\omega$induced by a radial weight$\omega\$ on the unit disc is a bounded operator from one space into another is of primordial importance in the theory
Small weighted Bergman spaces
This paper is based on the course \lq\lq Weighted Hardy-Bergman spaces\rq\rq\, I delivered in the Summer School \lq\lq Complex and Harmonic Analysis and Related Topics\rq\rq at the Mekrij\"arvi
FAST GROWING ENTIRE SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS
We investigate the fast growing entire solutions of linear differential equations. For that we introduce a general scale to measure the growth of entire functions of infinite order including