
Let L L denote the lower half plane and let B ( L ) B(L) denote the Banach space of analytic functions f f in L L with ‖ f ‖ L > ∞ {\left \| f \right \|_L} > \infty , where ‖ f ‖ L {\left \| f \right \|_L} is the suprenum over z ∈ L z \in L of the values | f ( z ) | ( t e x t I m z ) 2 \left | {f(z)} \right |{(text{Im} z)^2} . The universal Teichmüller space, T T , is the subset of B ( L ) B(L) consisting of the Schwarzian derivatives of conformal mappings of L L which have quasiconformal extensions to the extended plane. We denote by J J the set \[ { S f : f is conformal in L and f ( L ) is a Jordan domain } , \left \{ {{S_f}:f{\text {is conformal in }}L{\text {and }}f(L){\text {is a Jordan domain}}} \right \}, \] which is a subset of B ( L ) B(L) contained in the Schwarzian space S S . In showing S − T ¯ ≠ ∅ S - \bar T \ne \emptyset , Gehring actually proves S − J ¯ ≠ ∅ S - \bar J \ne \emptyset . We give an example which demonstrates that J − T ¯ ≠ ∅ J - \bar T \ne \emptyset .
General theory of conformal mappings, quasicircle, Jordan domain, Teichmüller space, Schwarzian derivatives of univalent mappings, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Quasiconformal mappings in the complex plane, Schwarzian derivatives of conformal mappings
General theory of conformal mappings, quasicircle, Jordan domain, Teichmüller space, Schwarzian derivatives of univalent mappings, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Quasiconformal mappings in the complex plane, Schwarzian derivatives of conformal mappings
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