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

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.

Keywords

• Complex differential equation
• entire function
• infinite order
• Denjoy's conjecture
• Yang's inequality