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A polynomial with 2k critical values at infinity

Janusz Gwoździewicz, Maciej Sękalski (2004)

Annales Polonici Mathematici

We construct a polynomial f:ℂ² → ℂ of degree 4k+2 with no critical points in ℂ² and with 2k critical values at infinity.

A Proof of Simultaneous Linearization with a Polylog Estimate

Tomoki Kawahira (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

We give an alternative proof of simultaneous linearization recently shown by T. Ueda, which connects the Schröder equation and the Abel equation analytically. In fact, we generalize Ueda's original result so that we may apply it to the parabolic fixed points with multiple petals. As an application, we show a continuity result on linearizing coordinates in complex dynamics.

A proof of the Livingston conjecture for the fourth and the fifth coefficient of concave univalent functions

Karl-Joachim Wirths (2004)

Annales Polonici Mathematici

Let D denote the open unit disc and f:D → ℂ̅ be meromorphic and injective in D. We further assume that f has a simple pole at the point p ∈ (0,1) and an expansion f ( z ) = z + ∑ n = 2 ∞ a ₙ ( f ) z ⁿ , |z| < p. In particular, we consider f that map D onto a domain whose complement with respect to ℂ̅ is convex. Because of the shape of f(D) these functions will be called concave univalent functions with pole p and the family of these functions is denoted by Co(p). It is proved that for p ∈ (0,1) the domain of variability of the coefficient...

A pseudo-trigonometry related to Ptolemy's theorem and the hyperbolic geometry of punctured spheres

Joachim A. Hempel (2004)

Annales Polonici Mathematici

A hyperbolic geodesic joining two punctures on a Riemann surface has infinite length. To obtain a useful distance-like quantity we define a finite pseudo-length of such a geodesic in terms of the hyperbolic length of its surrounding geodesic loop. There is a well defined angle between two geodesics meeting at a puncture, and our pseudo-trigonometry connects these angles with pseudo-lengths. We state and prove a theorem resembling Ptolemy's classical theorem on cyclic quadrilaterals and three general...

A pure smoothness condition for Radó’s theorem for α -analytic functions

Abtin Daghighi, Frank Wikström (2016)

Czechoslovak Mathematical Journal

Let Ω ⊂ ℂ n be a bounded, simply connected ℂ -convex domain. Let α ∈ ℤ + n and let f be a function on Ω which is separately C 2 α j - 1 -smooth with respect to z j (by which we mean jointly C 2 α j - 1 -smooth with respect to Re z j , Im z j ). If f is α -analytic on Ω ∖ f - 1 ( 0 ) , then f is α -analytic on Ω . The result is well-known for the case α i = 1 , 1 ≤ i ≤ n , even when f a priori is only known to be continuous.

Currently displaying 261 – 280 of 702