A non-quasicircle with almost smooth mapping functions (Q1083574)

Let \(\Gamma\) be a closed Jordan curve with interior \(\Omega\) and exterior \(\Omega^*\). Let D and \(D^*\) be the interior and exterior of the unit circle, and suppose that f and \(f^*\) are respectively conformal mappings of D and \(D^*\) onto \(\Omega\) and \(\Omega^*\), extended continuously to \(\partial D\). We say that a function is Lip(\(\alpha)\) on its domain if it is uniformly Hölder continuous with exponent \(\alpha\). If \(\Gamma\) is a quasicircle then f, \(f^*\), \(f^{*-1}\) are uniformly Hölder continuous on their domains, with exponents depending on the geometry of \(\Gamma\). In the opposite direction, if f and \(f^*\) are Lip(1) on \(\partial D\), then \(\Gamma\) is of bounded arclength-chordlength ratio, and is thus a quasicircle. The question at hand is, how smooth can all of the mapping functions be without \(\Gamma\) being a quasicircle ? We construct a non-quasicircle such that \(| f^*{}'|\), \(1/| f^*{}'|\) and 1/\(| f'|\) are all uniformly bounded on \(\partial D\), while \(| f'(e^{i\theta})|\) is exponentially integrable. Consequently, \(f^*\) is Lip(1) while f, \(f^*\), \(f^{-1}\) are Lip(\(\alpha)\) for all \(\alpha <1\) on the closures of their domains.