Deficient values of solutions of linear differential equations

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.