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)≢0 are entire functions. The Lindelöf function is used to show that for any ρ∈(1/2,∞), there exists an equation of the form (*) which possesses a solution f with a Nevanlinna deficient value at 0 satisfying ρ=ρ(f)≥ρ(A)≥ρ(B), where ρ(h) denotes the order of an entire function h. It is known that such an example cannot exist when ρ≤1/2. For smaller growth functions, a geometrical modification of an example of Anderson and Clunie is used to show that for any ρ∈(2,∞), there exists an equation of the form (*) which possesses a solution f with a Valiron deficient value at 0 satisfying ρ=ρlog(f)≥ρlog(A)≥ρlog(B), where ρlog(h) denotes the logarithmic order of an entire function h. 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.