Minimum deviation from zero for the Chebyshev mappings corresponding to an equilateral triangle (Q1360828)

The mapping \(h:(u,v) \mapsto e^{2\pi iu} +e^{2\pi iv} +e^{-2\pi i(u+v)}\) maps \(\mathbb{R}^2\) onto the Steiner deltoid \(\Delta\subset \mathbb{C}\). Following \textit{W. D. Withers} [Am. Math. Monthly 95, No. 5, 399-413 (1988; Zbl 0695.58018)], the author defines generalized Chebyshev polynomials \((A_n)_{n>0}\) in the variables \(z,\overline z\) via \(h(nu,nv) =A_n(h(u,v))\). Generalizing a classical result for the classical Chebyshev polynomials, the author shows that for all \(n\), the infimum of the norm \(\| z^n+P \|_{\infty, \Delta}\) on the space of all polynomials in \(z,\overline z\) of degree at most \(n-1\) is attained exactly for the polynomial \(P=z^n-A_n\).