Abstract
M Vuorinen introduced the notion of \(\phi \)-uniform domains as a generalization of uniform domains. We construct an example of a quasiconvex, simply connected and bounded planar domain which fails to be \(\phi \)-uniform for any homeomorphism \(\phi :[0,\infty ]\rightarrow [0,\infty ]\). We also show that all bounded domains are \(\phi \)-John with a fixed center.
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Acknowledgements
I thank my advisor Professor Pekka Koskela for telling me about this question. I am also grateful to Professor David Herron and the referee for suggestions which made the note readable.
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Nandi, D. A note on ϕ-uniform domains. J Anal 24, 51–56 (2016). https://doi.org/10.1007/s41478-016-0004-7
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DOI: https://doi.org/10.1007/s41478-016-0004-7