On the application of majorization principles to inequalities for polynomials and rational functions

Applications of the maximum modulus principle for holomorphic functions, Schwarz's lemma, and a theorem of Mityuk to inequalities for polynomials and rational functions are considered. Proofs or schemes of proofs of the classical inequalities of Chebyshev, Bernstein, Bernstein–Szego, Turan, as well as some modern inequalities of Li-Mohapatra–Rodriguez, Borwein–Erdelyi, Sheil-Small, Rakhmanov–Shekhtman, Lukashov will be presented. The material is available for senior university students.