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Linear Differential Equations and Logarithmic Derivative Estimates

Published online by Cambridge University Press:  09 June 2003

Igor Chyzhykov ,
Gary G. Gundersen  and
Janne Heittokangas
Igor Chyzhykov
Affiliation:
Department of Mechanics and Mathematics, Lviv National University, Universytetska 1, Lviv 79000, Ukraine. E-mail: matstud@uli2.franko.lviv.ua
Gary G. Gundersen
Affiliation:
Department of Mathematics, University of New Orleans, New Orleans, LA 70148, USA. E-mail: ggunders@math.uno.edu
Janne Heittokangas
Affiliation:
Department of Mathematics, University of Joensuu, P.O. Box 111, FIN–80101 Joensuu, Finland. E-mail: heittoka@cc.joensuu.fi Department of Mathematics, University of Illinois at Urbana-Champaign, Altgeld Hall, 1409 W. Green Street, Urbana, IL 61801, USA. E-mail: janne@math.uiuc.edu
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Abstract

We prove two sharp inequalities for the growth of solutions of certain linear differential equations in the unit disk. For the proofs of these inequalities, we use the method of successive approximations and sharp estimates for the logarithmic derivatives of finite order meromorphic functions in the unit disk. These techniques can also be used to give an alternate proof of a well-known result in the plane. The sharp logarithmic derivative estimates are a corollary of general estimates, and all these estimates have independent interest.

Keywords

34M1030D35
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 86 , Issue 3 , May 2003 , pp. 735 - 754
Copyright
2003 London Mathematical Society

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Footnotes

The first and third authors were partially supported by INTAS, project 99-00089.