Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 985-1003

COMPLETELY REGULAR GROWTH SOLUTIONS OF SECOND ORDER COMPLEX LINEAR DIFFERENTIAL EQUATIONS

Janne Heittokangas, Ilpo Laine, Kazuya Tohge and Zhi-Tao Wen

University of Eastern Finland, Department of Physics and Mathematics
P.O. Box 111, 80101 Joensuu, Finland; janne.heittokangas 'at' uef.fi

University of Eastern Finland, Department of Physics and Mathematics
P.O. Box 111, 80101 Joensuu, Finland; ilpo.laine 'at' uef.fi

Kanazawa University, College of Science and Engineering
Kakuma-machi, Kanazawa 920-1192, Japan; tohge 'at' se.kanazawa-u.ac.jp

Taiyuan University of Technology, Department of Mathematics
Yingze West Street, No. 79, Taiyuan 030024, China; zhitaowen 'at' gmail.com

Abstract. If A(z) and B(z) are transcendental entire functions, then all solutions of the differential equation f'' + A(z)f' + B(z)f = 0 are entire and typically of infinite order. Simple examples show that finite order solutions are also possible. Assuming that A(z) and B(z) are of completely regular growth, Gol'dberg-Ostrovskii-Petrenko asked whether all solutions of finite order are of completely regular growth also. This problem remains unsolved, but several aspects of the problem are addressed. Exponential polynomials form an important subclass of functions of completely regular growth, and they are known to always satisfy certain linear differential equations. Hence cases where the coefficients and/or the solutions are exponential polynomials are also discussed.

2010 Mathematics Subject Classification: Primary 34M05; Secondary 30D35.

Key words: Completely regular growth, complex differential equation, exponential polynomial, Phragmén-Lindelöf indicator function.

Reference to this article: J. Heittokangas, I. Laine, K. Tohge and Z.-T. Wen: Completely regular growth solutions of second order complex linear differential equations. Ann. Acad. Sci. Fenn. Math. 40 (2015), 985-1003.

Full document as PDF file

doi:10.5186/aasfm.2015.4057