Electronic Journal of Differential Equations,
Vol. 2015 (2015), No. 143, pp. 1-15.
Title: Second-order complex linear differential equations with special functions
or extremal functions as coefficients
Authors: Xiubi Wu (Guizhou Univ., Guiyang, China)
Jianren Long (Guizhou Univ., Guiyang, China)
Janne Heittokangas (Univ. of Eastern Finland, Joensuu, Finland)
Ke-e Qiu (Guizhou Normal Colleage, Guiyang, China)
Abstract:
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.