Rational decompositions of complex meromorphic functions

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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: eberhard.mayerhofer@univie.ac.at