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Fixed points of meromorphic functions and of their differences and shifts

Zong-Xuan Chen — 2013

Annales Polonici Mathematici

Let f(z) be a finite order transcendental meromorphic function such that λ(1/f(z)) < σ(f(z)), and let c ∈ ℂ∖0 be a constant such that f(z+c) ≢ f(z) + c. We mainly prove that m a x τ ( f ( z ) ) , τ ( Δ c f ( z ) ) = m a x τ ( f ( z ) ) , τ ( f ( z + c ) ) = m a x τ ( Δ c f ( z ) ) , τ ( f ( z + c ) ) = σ ( f ( z ) ) , where τ(g(z)) denotes the exponent of convergence of fixed points of the meromorphic function g(z), and σ(g(z)) denotes the order of growth of g(z).

On meromorphic solutions of the Riccati differential equations

Ran Ran ZhangZong Xuan Chen — 2010

Annales Polonici Mathematici

We investigate the growth and Borel exceptional values of meromorphic solutions of the Riccati differential equation w' = a(z) + b(z)w + w², where a(z) and b(z) are meromorphic functions. In particular, we correct a result of E. Hille [Ordinary Differential Equations in the Complex Domain, 1976] and get a precise estimate on the growth order of the transcendental meromorphic solution w(z); and if at least one of a(z) and b(z) is non-constant, then we show that w(z)...

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