Rational decompositions of complex meromorphic functions

Abstract

Let h be a complex meromorphic function. The problem of decomposing h in two different ways, P (f)  and Q(g) with f, g two other meromorphic functions and P, Q polynomials, was studied by C.-C. Yang, P. Li and H.K. Ha. Here we consider the problem when we replace the polynomials P, Q by rational functions F, G. Let deg(F ) be the maximum degree of numerator and denominator of F. Assume some zeros c ₁, … ,c _k of

satisfy a pack of five conditions particularly involving G(d,) ≠ F(c _j ) and D(d) ≠ 0 for every zero d of

, with G = C/D,  (j = 1,…,k). First, we show that if f, g are entire functions such that F(f) = G(g), then k deg (G) ≤  deg(F). Now, let u be the number of distinct zeros of the denominator of G and assume that meromorphic functions f, g satisfy F(f) = G(g), then k deg (G) ≤  deg (F) + kγ (D). When zeros c ₁, …, c _k of

satisfy a stronger condition, then we show that k deg (G) ≤  deg (F) + k min (γ (C), γ (D)).

E-mail: [email protected]

Keywords:

• Nevanlinna theory
• Meromorphic functions
• Functional equations
• Mathematics Subject Classifications: 30D35
• 30D30
• 39B32

Acknowledgments

The authors are very grateful to C.-C. Yang for informing them of the problem of the decomposition and the first results obtained in this domain by himself and his colleagues P. Li and H.K. Ha.

Notes

E-mail: [email protected]