Deficient Values of Solutions of Linear Differential Equations

• Published in: Computational methods and function theory Vol. 21; no. 1; pp. 145 - 177
• Main Authors: Gundersen, Gary G., Heittokangas, Janne, Wen, Zhi-Tao
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2021; Springer Nature B.V
• Subjects: Analysis; Computational Mathematics and Numerical Analysis; Differential equations; Entire functions; Functions of a Complex Variable; Mathematical analysis; Mathematics; Mathematics and Statistics; Theorems; Valiron deficient value; Logarithmic order; Nevanlinna deficient value; Linear differential equation; Finite order; Primary 34M05; Secondary 34M10; 30D35
• ISSN: 1617-9447; 2195-3724
• DOI: 10.1007/s40315-020-00320-1

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 = ρ log ( f ) ≥ ρ log ( A ) ≥ ρ log ( B ) , where 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.