Deficiency for meromorphic solutions of schröder equations

Abstract

Eremenko and Sodin proved that meromorphic solution f (z) of the Schröder equation f (sz) = R (f (z)), |s| > 1, has no Valiron deficiency other than exceptional values of R(z). We consider transcendental meromorphic solutions of non-autonomous equation f (sz) =R (z, f (z)), |s| > 1. It is shown that there exists an equation of this form possessing a transcendental meromorphic solution, which has a Valiron deficiency other than a Nevanlinna deficiency. We also give some generalizations of the Eremenko and Sodin theorem for algebraic functions as targets.

Keywords:

• Schröder equation
• Meromorphic functions
• Nevanlinna theory
• Valiron deficiency
• Algebraic functions
• 2000 Mathematics Subject Classifications: Primary 30D05
• Secondary 39B32

Notes

E-mail: [email protected]

Dedicated to the memory of Professor Matts Essen.

Dedicated to the memory of Professor Matts Essen.

Additional information

Notes on contributors

Niro Yanagihara ‡

E-mail: [email protected] Dedicated to the memory of Professor Matts Essen.