Deficient values of solutions of linear differential equations

TL;DR

This paper investigates the existence of solutions with deficient values for linear differential equations with entire coefficients, demonstrating sharp bounds on the growth of solutions with specific deficient values.

Contribution

It constructs explicit solutions with prescribed deficient values for differential equations with entire coefficients, extending known results and providing sharp bounds on growth conditions.

Findings

Existence of solutions with Nevanlinna deficient value at 0 for order > 1/2.

Existence of solutions with Valiron deficient value at 0 for logarithmic order > 2.

Sharp bounds on growth conditions for solutions with deficient values.

Abstract

Differential equations of the form $f'' + A(z)f' + B(z)f = 0$ (*) are considered, where $A(z)$ and $B(z) \not\equiv 0$ are entire functions. The Lindel\"of function is used to show that for any $\rho \in (1/2, \infty)$, there exists an equation of the form (*) which possesses a solution $f$ of order $\rho$ with a Nevanlinna deficient value at $0$, where $f, A(z), B(z)$ satisfy a common growth condition. It is known that such an example cannot exist when $\rho \leq 1/2$. For smaller growth functions, a geometrical modification of an example of Anderson and Clunie is used to show that for any $\rho \in (2, \infty)$, there exists an equation of the form (*) which possesses a solution $f$ of logarithmic order $\rho$ with a Valiron deficient value of at $0$, where $f, A(z), B(z)$ satisfy an analogous growth condition. This result is essentially sharp. In both proofs, the separation of the zeros of the indicated solution plays a key role. Observations on the deficient values of solutions of linear differential equations are also given, which include a discussion of Wittich's theorem on Nevanlinna deficient values, a modified Wittich theorem for Valiron deficient values, consequences of Gol'dberg's theorem, and examples to illustrate possibilities that can occur.