An anti-Möbius map sending $\widehat C_{m,\alpha}$ into $\widehat C_{m,\beta}$ is $z\mapsto c\bar z$ or $c/\bar z$

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An anti-Möbius map sending $\widehat C_{m,\alpha}$ into $\widehat C_{m,\beta}$ is $z\mapsto c\bar z$ or $c/\bar z$ · Prove2Me