Majoration principles and some inequalities for polynomials and rational functions with prescribed poles

The paper considers the equality cases in the rnajoration principle for meromorphic functions established earlier by V. N. Dubinin and S. I. Kalmykov Mat. Sb., 198, No. 12, 37–46 (2007)]. As corollaries of this principle, new inequalities for the coefficients and derivatives of polynomials satisfying certain conditions on two intervals are obtained. Simple proofs of some Lukashov’s theorems on the derivatives of rational functions on several intervals are provided. Bibliography: 13 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 143–157.