Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-26T07:50:10.542Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Cartwright, M. L., Integral Functions, Cambridge Tracts in Mathematics and Mathematical Physics, 44 (Cambridge University Press, Cambridge, 1956).Google Scholar
[2]Chen, Z.-X. and Shon, K. H., ‘The growth of solutions of differential equations with coefficients of small growth in the disc’, J. Math. Anal. Appl. 297(1) (2004), 285–304.CrossRefGoogle Scholar
[3]Chyzhykov, I., Gundersen, G. G. and Heittokangas, J., ‘Linear differential equations and logarithmic derivative estimates’, Proc. London Math. Soc. 86(3) (2003), 735–754.CrossRefGoogle Scholar
[4]Chyzhykov, I., Heittokangas, J. and Rättyä, J., ‘On the finiteness of φ-order of solutions of linear differential equations in the unit disc’, J. Anal. Math. 209 (2009), 163–198.CrossRefGoogle Scholar
[5]Fenton, P. C. and Rossi, J., ‘ODEs and Wiman–Valiron theory in the unit disc’, J. Math. Anal. Appl. 367(1) (2010), 137–145.CrossRefGoogle Scholar
[6]Fenton, P. C. and Strumia, M. M., ‘Wiman–Valiron theory in the disc’, J. London Math. Soc. 79(2) (2009), 478–496.CrossRefGoogle Scholar
[7]Gundersen, G. G., Steinbart, E. M. and Wang, S., ‘The possible orders of solutions of linear differential equations with polynomial coefficients’, Trans. Amer. Math. Soc. 350(3) (1998), 1225–1247.CrossRefGoogle Scholar
[8]Heittokangas, J., ‘On complex differential equations in the unit disc’, Ann. Acad. Sci. Fenn. Math. Diss. 122 (2000), 1–54.Google Scholar
[9]Heittokangas, J., ‘Growth estimates for logarithmic derivatives of Blaschke products and of functions in the Nevanlinna class’, Kodai Math. J. 30 (2007), 263–279.CrossRefGoogle Scholar
[10]Heittokangas, J., Korhonen, R. and Rättyä, J., ‘Generalized logarithmic derivative estimates of Gol′dberg and Grinshtein type’, Bull. London Math. Soc. 36 (2004), 105–114.CrossRefGoogle Scholar
[11]Heittokangas, J., Korhonen, R. and Rättyä, J., ‘Linear differential equations with solutions in Dirichlet type subspace of the Hardy space’, Nagoya Math. J. 187 (2007), 91–113.CrossRefGoogle Scholar
[12]Juneja, O. P. and Kapoor, G. P., Analytic Functions—Growth Aspects, Research Notes in Mathematics, 104 (Pitman, Boston, MA, 1985).Google Scholar
[13]Korhonen, R. and Rättyä, J., ‘Linear differential equations in the unit disc with analytic solutions of finite order’, Proc. Amer. Math. Soc. 135(5) (2007), 1355–1363.CrossRefGoogle Scholar
[14]Korhonen, R. and Rättyä, J., ‘Finite order solutions of linear differential equations in the unit disc’, J. Math. Anal. Appl. 349 (2009), 43–54.CrossRefGoogle Scholar
[15]Laine, I., Nevanlinna Theory and Complex Differential Equations (Walter de Gruyter, Berlin, 1993).CrossRefGoogle Scholar
[16]Levin, B. Ja., Distribution of Zeros of Entire Functions, revised edition, Translations of Mathematical Monographs, 5 (American Mathematical Society, Providence, RI, 1980), translated by R. P. Boas et al.Google Scholar
[17]Linden, C. N., ‘The minimum modulus of functions regular and of finite order in the unit circle’, Q. J. Math. Oxford Ser. (2) 7 (1956), 196–216.CrossRefGoogle Scholar
[18]Rättyä, J., ‘Linear differential equations with solutions in Hardy spaces’, Complex Var. Elliptic Equ. 52(9) (2007), 785–795.CrossRefGoogle Scholar