On the inequalities of Bernstein-Markoff for an interval
Interesting theorems on relationships between the absolute value of polynomials in a certain class, and the absolute value of their derivatives, are derived. These theorems are extensions and refinements of results by A. Markoff, W. Markoff, Duffin and Schaeffer, Schur, and others. The following is an example to the kind of theorems which are obtained: Let \(S_ n\) be the class of polynomials \(P(z)=\sum^{n}_{j=0}a_ jz^ j\), of degree at most n. Let \(P\in S\), \(P(c)=0\) where \(1\leq c\leq (2 \cos^ 2(\pi /4n)/\cos^ 2(\pi /2n))-1,\) and \(| P(x)| \leq 1\quad for\quad x=(b+\cos (\nu \Pi /n))/a,\quad 1\leq \nu \leq n,\) where \(a=[2/(c+1)]\cos^ 2 \Pi /4n;\quad b=[(c-1)/(c+1)]\cos^ 2 \Pi /4n+\sin^ 2 \Pi /4n.\) Then: \[ | p^{(k)}(x)| \leq ([2/(c+1)]\cos^ 2 \Pi /4n)^ k\frac{n^ 2(n^ 2-1^ 2)...(n\quad^ 2-(k-1)^ 2)}{1.3....(2k-1)} \] for -1\(\leq x\leq 1\) and \(k=1,...,n\). If, in addition, P(z) has real coefficients, then \[ | P^{(k)}(x+iy)| \leq ([2/(c+1)]\quad \cos^ 2 \Pi /4n)^ k\quad | T_ n^{(k)}(-1+ayi)| \] for -1\(\leq x\leq 1\), \(-\infty <y<\infty\) and \(k=1,...,n\).
scientific article; zbMATH DE number 3959852