Electron. J. Diff. Equ., Vol. 2015 (2015), No. 143, pp. 1-15.

Second-order complex linear differential equations with special functions or extremal functions as coefficients

Xiubi Wu, Jianren Long, Janne Heittokangas, Ke-e Qiu

The classical problem of finding conditions on the entire coefficients A(z) and B(z) guaranteeing that all nontrivial solutions of $f''+A(z)f'+B(z)f=0$ are of infinite order is discussed. Two distinct approaches are used. In the first approach the coefficient A(z) itself is a solution of a differential equation $w''+P(z)w=0$, where P(z) is a polynomial. This assumption yields stability on the behavior of A(z) via Hille's classical method on asymptotic integration. In this case A(z) is a special function of which the Airy integral is one example. The second approach involves extremal functions. It is assumed that either A(z) is extremal for Yang's inequality or B(z) is extremal for Denjoy's conjecture. A combination of these two approaches is also discussed.

Submitted November 17, 2014. Published May 22, 2015.
Math Subject Classifications: 34M10, 30D35.
Key Words: Complex differential equation; entire function; infinite order; Denjoy's conjecture; Yang's inequality.

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Xiubi Wu
School of Science, Guizhou University
Guiyang 550025, China
email: basicmath@163.com
Jianren Long
School of Mathematics and Computer Science
Guizhou Normal University
Guiyang 550001, China
email: longjianren2004@163.com, jianren.long@uef.fi
Janne Heittokangas
Department of Physics and Mathematics
University of Eastern Finland
P.O. Box 111, 80101 Joensuu, Finland
email: janne.heittokangas@uef.fi
  Ke-e Qiu
School of Mathematics and Computer Science
Guizhou Normal Colleage
Guiyang 550018, China
email: qke456@sina.com

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