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On meromorphic solutions of the Riccati differential equations

Ran Ran Zhang, Zong 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)...

On radial behaviour and balanced Bloch functions.

Juan Jesús Donaire, Christian Pommerenke (1999)

Revista Matemática Iberoamericana

A Bloch function g is a function analytic in the unit disk such that (1 - |z|2) |g' (z)| is bounded. First we generalize the theorem of Rohde that, for every bad Bloch function, g(rζ) (r → 1) follows any prescribed curve at a bounded distance for ζ in a set of Hausdorff dimension almost one. Then we introduce balanced Bloch functions. They are characterized by the fact that |g'(z)| does not vary much on each circle {|z| = r} except for small exceptional arcs. We show e.g. that∫01 |g'(rζ)|dr <...

On ramifications divisors of functions in a punctured compact Riemann surface.

Pascual Cutillas Ripoll (1989)

Publicacions Matemàtiques

Let ν be a compact Riemann surface and ν' be the complement in ν of a nonvoid finite subset. Let M(ν') be the field of meromorphic functions in ν'. In this paper we study the ramification divisors of the functions in M(ν') which have exponential singularities of finite degree at the points of ν-ν', and one proves, for instance, that if a function in M(ν') belongs to the subfield generated by the functions of this type, and has a finite ramification divisor, it also has a finite divisor. It is also...

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