Deficient Values of Solutions of Linear Differential Equations

Abstract

AbstractDifferential equations of the form $$f'' + A(z)f' + B(z)f = 0$$ f ′ ′ + A ( z ) f ′ + B ( z ) f = 0 (*) are considered, where A(z) and $$B(z) \not \equiv 0$$ B ( z ) ≢ 0 are entire functions. The Lindelöf function is used to show that for any $$\rho \in (1/2, \infty )$$ ρ ∈ ( 1 / 2 , ∞ ) , there exists an equation of the form (*) which possesses a solution f with a Nevanlinna deficient value at 0 satisfying $$\rho =\rho (f)\ge \rho (A)\ge \rho (B)$$ ρ = ρ ( f ) ≥ ρ ( A ) ≥ ρ ( B ) , where $$\rho (h)$$ ρ ( h ) denotes the order of an entire function h. It is known that such an example cannot exist when $$\rho \le 1/2$$ ρ ≤ 1 / 2 . For smaller growth functions, a geometrical modification of an example of Anderson and Clunie is used to show that for any $$\rho \in (2, \infty )$$ ρ ∈ ( 2 , ∞ ) , there exists an equation of the form (*) which possesses a solution f with a Valiron deficient value at 0 satisfying $$\rho =\rho _{\log }(f)\ge \rho _{\log }(A)\ge \rho _{\log }(B)$$ ρ = ρ log ( f ) ≥ ρ log ( A ) ≥ ρ log ( B ) , where $$\rho _{\log }(h)$$ ρ log ( 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.