On the minimum moduli of normalized polynomials with two prescribed values

WithP _n denoting the set of complex polynomials of degree at mostn (n≥1), define, for any complex numberμ, the subset $$P_n (\mu ): = \{ p_n (z) \in P_n :p_n (0) = 1 and p_n (1) = \mu \} .$$ In this paper, we determine exactly the nonnegative quantity $$S_n (\mu ): = \mathop {\sup \{ \min |p_n (z)|\} }\limits_{p_n \in P_n (\mu )|z| \leqslant 1} ,$$ as a function ofn andμ. For fixedn≥2, the three-dimensional surface, generated by the points (Reμ, Imμ,S _n(μ)) for all complex numbersμ, has the interesting shape of a volcano.