Xiubi Wu, Jianren Long, Janne Heittokangas, Ke-e Qiu
Abstract:
The classical problem of finding conditions on the entire coefficients
A(z) and B(z) guaranteeing that all nontrivial solutions of
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
,
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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