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

Part of: Differential equations in the complex domain

Published online by Cambridge University Press:  07 April 2010

I. CHYZHYKOV ,
J. HEITTOKANGAS  and
J. RÄTTYÄ
I. CHYZHYKOV
Affiliation:
Department of Mechanics and Mathematics, Lviv National University, Universytetska 1, Lviv 79000, Ukraine (email: ichyzh@lviv.farlep.net)
J. HEITTOKANGAS*
Affiliation:
Mathematics, University of Joensuu, PO Box 111, 80101 Joensuu, Finland (email: janne.heittokangas@joensuu.fi)
J. RÄTTYÄ
Affiliation:
Mathematics, University of Joensuu, PO Box 111, 80101 Joensuu, Finland (email: jouni.rattya@joensuu.fi)
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Abstract

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New 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.

Keywords

logarithmic derivativeproximate orderorder of growthlinear differential equationgrowth of solutions

MSC classification

Secondary: 34M10: Oscillation, growth of solutions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 88 , Issue 2 , April 2010 , pp. 145 - 167
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

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