Remez-, Nikolskii-, and Markov-type inequalities for generalized nonnegative polynomials with restricted zeros
A function of the form (1) \(f(z)=\omega\prod^ m_{j=1}(z-z_ j)^{r_ j}\) \(\omega\neq 0\), \(r_ j\in\mathbb{R}\) \((j=1,2,\dots,m)\) is said to be a generalized complex algebraic polynomial of (generalized) degree \(N=\sum^ m_{j=1}r_ j\). If \(r_ j>0\) are integers then \(f\) is an ordinary polynomial of degree \(N\). In the paper Remez-, Nikol'skij- and Markov-type inequalities are proved for polynomials (1) of degree at most \(N\) under the assumption \(\sum_{\{j:| z_ j|\leq 1\}}r_ j\leq K\), \(0\leq K\leq N\).
scientific article; zbMATH DE number 61194
scientific article