An extension of an inequality of Duffin and Schaeffer (Q1772236)

For the ultraspherical polynomials \(P=P_n^{(\lambda)}\), \(\lambda \geq 0\), it is proved that \[ | P(x+iy)| \leq | P(1+iy)| , \quad x \in [-1,1], \quad y \in {\mathbb R}. \] This result is applied to obtain the following Duffin and Schaeffer type inequality. Let \(Q=P_n^{(\lambda)}\), \(\lambda \in \left[0,\frac{1}{2}\right]\) and let \(\{t_{\nu}\}_{\nu=0}^n\) be the zeros of \((1-x^2)Q'(x)\). If \(f\) is a polynomial of degree at most \(n\) with real coefficients and \[ | f(t_{\nu})| \leq | Q(t_{\nu})| , \quad \nu=0,\ldots,n, \] then, for \(k=1,\ldots,n\), \[ | f^{(k)}(x+iy)| \leq | Q^{(k)}(x+iy)| , \quad x \in [-1,1], \quad y \in {\mathbb R}. \]