On Certain Duffin and Schaeffer Type Inequalities Original Research Article

Duffin and Schaeffer type inequalities related to some ultraspherical polynomials are established. One of the results obtained reads as follows: Letfbe a real algebraic polynomial of degree at mostn, such that |f(±1)|⩽1 and|f(x)|⩽1−x2at the zeros ofTn−1(x). Then max−1⩽x⩽1|f(k)(x)|⩽T(k)n(1) for allk∈{1, …, n}. Moreover, equality holds if and only iff=±Tn.