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.

LOCALISATION OF LINEAR DIFFERENTIAL EQUATIONS IN THE UNIT DISC BY A CONFORMAL MAP

2015 ◽  
Vol 93 (2) ◽  
pp. 260-271
Author(s):  
JUHA-MATTI HUUSKO
Keyword(s):  
Differential Equations ◽  
Lower Bounds ◽  
Complex Plane ◽  
Exponential Growth ◽  
Unit Disc ◽  
Linear Differential Equations ◽  
Conformal Map ◽  
Conformal Maps ◽  
Linear Differential ◽  
Growth Of Solutions

We obtain lower bounds for the growth of solutions of higher order linear differential equations, with coefficients analytic in the unit disc of the complex plane, by localising the equations via conformal maps and applying known results for the unit disc. As an example, we study equations in which the coefficients have a certain explicit exponential growth at one point on the boundary of the unit disc and consider the iterated $M$-order of solutions.

DIFFERENTIAL EQUATIONS WITH COEFFICIENTS OF SLOW GROWTH IN THE UNIT DISC

2009 ◽  
Vol 07 (02) ◽  
pp. 213-224 ◽  
Author(s):  
LIPENG XIAO ◽  
ZONGXUAN CHEN
Keyword(s):  
Differential Equations ◽  
Recent Result ◽  
Unit Disc ◽  
Slow Growth ◽  
Linear Differential Equations ◽  
Fast Growing ◽  
Linearly Independent ◽  
Linear Differential ◽  
Growth Of Solutions

In this paper, the growth of solutions and the number of fast-growing linearly independent solutions of certain linear differential equations with coefficients of slow growth in the unit disc are investigated. The results we obtain are a generalization of a recent result due to Korhonen and Rättyä.

scholarly journals Some weaker sufficient conditions of $L$-index boundedness in direction for functions analytic in the unit ball

2019 ◽  
Vol 11 (1) ◽  
pp. 14-25 ◽  
Author(s):  
A.I. Bandura
Keyword(s):  
Analytic Function ◽  
Differential Equations ◽  
Unit Ball ◽  
Logarithmic Derivative ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Directional Derivatives ◽  
Local Behavior ◽  
Exceptional Set ◽  
Linear Differential

We partially reinforce some criteria of $L$-index boundedness in direction for functions analytic in the unit ball. These results describe local behavior of directional derivatives on the circle, estimates of maximum modulus, minimum modulus of analytic function, distribution of its zeros and modulus of directional logarithmic derivative of analytic function outside some exceptional set. Replacement of universal quantifier on existential quantifier gives new weaker sufficient conditions of $L$-index boundedness in direction for functions analytic in the unit ball. The results are also new for analytic functions in the unit disc. The logarithmic criterion has applications in analytic theory of differential equations. This is convenient to investigate index boundedness for entire solutions of linear differential equations. It is also apllicable to infinite products.Auxiliary class of positive continuous functions in the unit ball (so-denoted $Q_{\mathbf{b}}(\mathbb{B}^n)$) is also considered. There are proved some characterizing properties of these functions. The properties describe local behavior of these functions in the polydisc neighborhood of every point from the unit ball.

On α-Polynomial Regular Functions, with Applications to Ordinary Differential Equations

2014 ◽  
Vol 57 (2) ◽  
pp. 405-421 ◽  
Author(s):  
Peter Fenton ◽  
Janne Grohn ◽  
Janne Heittokangas ◽  
John Rossi ◽  
Jouni Rattya
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Unit Disc ◽  
Maximum Modulus ◽  
Linear Differential Equations ◽  
Real Numbers ◽  
Regular Functions ◽  
Function Classes ◽  
Finite Set ◽  
Linear Differential

AbstractThis research deals with properties of polynomial regular functions, which were introduced in a recent study concerning Wiman-Valiron theory in the unit disc. The relation of polynomial regular functions to a number of function classes is investigated. Of particular interest is the connection to the growth class Gα, which is closely associated with the theory of linear differential equations with analytic coefficients in the unit disc. If the coefficients are polynomial regular functions, then it turns out that a finite set of real numbers containing all possible maximum modulus orders of solutions can be found. This is in contrast to what is known about the case when the coefficients belong to Gα.

scholarly journals On [p, q]-order of growth of solutions of linear differential equations in the unit disc

2021 ◽  
Vol 6 (11) ◽  
pp. 12878-12893
Author(s):  
Hongyan Qin ◽  
◽  
Jianren Long ◽  
Mingjin Li
Keyword(s):  
Differential Equations ◽  
Analytic Functions ◽  
Unit Disc ◽  
Linear Differential Equations ◽  
Order Of Growth ◽  
Linear Differential ◽  
Growth Of Solutions

<abstract><p>The $ [p, q] $-order of growth of solutions of the following linear differential equations $ (**) $ is investigated,</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f^{'}+A_{0}(z)f = 0, (**) $\end{document} </tex-math></disp-formula></p> <p>where $ A_{i}(z) $ are analytic functions in the unit disc, $ i = 0, 1, ..., k-1 $. Some estimations of $ [p, q] $-order of growth of solutions of the equation $ (\ast*) $ are obtained when $ A_{j}(z) $ dominate the others coefficients near a point on the boundary of the unit disc, which is generalization of previous results from S. Hamouda.</p></abstract>

scholarly journals Possible Intervals forT- andM-Orders of Solutions of Linear Differential Equations in the Unit Disc

2011 ◽  
Vol 2011 ◽  
pp. 1-25
Author(s):  
Martin Chuaqui ◽  
Janne Gröhn ◽  
Janne Heittokangas ◽  
Jouni Rättyä
Keyword(s):  
Unit Disc ◽  
Bergman Spaces ◽  
Maximum Modulus ◽  
Weighted Bergman Spaces ◽  
Nevanlinna Characteristic ◽  
Weighted Hardy Spaces ◽  
Polynomial Coefficients ◽  
Finite Set ◽  
Linear Differential ◽  
Growth Of Solutions

In the case of the complex plane, it is known that there exists a finite set of rational numbers containing all possible growth orders of solutions off(k)+ak-1(z)f(k-1)+⋯+a1(z)f′+a0(z)f=0with polynomial coefficients. In the present paper, it is shown by an example that a unit disc counterpart of such finite set does not contain all possibleT- andM-orders of solutions, with respect to Nevanlinna characteristic and maximum modulus, if the coefficients are analytic functions belonging either to weighted Bergman spaces or to weighted Hardy spaces. In contrast to a finite set, possible intervals forT- andM-orders are introduced to give detailed information about the growth of solutions. Finally, these findings yield sharp lower bounds for the sums ofT- andM-orders of functions in the solution bases.

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