On the exact Markov inequality for \(k\)-monotone polynomials in uniform and \(L_1\)-norms
The authors consider the classical extremal problem of estimating norms of higher order derivatives of algebraic polynomials when their norms are given. The corresponding extremal problem for general polynomials in uniform norm was solved by A. A. Markov, while Bernstein found the exact constant in the Markov inequality for monotone polynomials. In the paper, the authors give Markov-type inequalities for higher order derivatives in the general class of \(k\)-monotone polynomials. Specifically, the authors consider the extremal problem of finding the maximal value of the uniform and \(L_r\)-norm of the \(i\)-th derivatives of normalized \(k\)-monotone polynomials \[ M^{(i)}(n,k)_r:=\sup _{p\in {P}_n^{(k)}}{\| p^{(i)}\| _r \over \| p\| _r},\quad 0\leq k\leq n,\;1\leq i\leq k,\;0<r\leq \infty, \] where \(P_n^{(k)}\) denotes the space of algebraic polynomials \(p\) of degree at most \(n\) such that \(p^{(j)}(x)\leq 0\), \(| x| \leq 1\), \(0\leq j\leq k\) and \(\| \cdot \| _r\) stands for the usual \(L_r\)-norm on \([-1,1]\). The exact value \(M^{(1)}(n,0)_{\infty }=n^2/2\) was found by A. A. Markov. Bernstein proved that \(M^{(1)}(n,1)_{\infty }\) equals \((n+1)^2/8\) or \(n(n+2)/8\) if \(n\) is odd or even, respectively. The authors consider the above problem for the general set of \(k\)-monotone polynomials \({P}_n^{(k)}\) for any \(2\leq k\leq n\), \(1\leq i<k\) and \(r=1\) or \(\infty\). For the case \(i=1\), they find the exact value of the Markov constant form \(k\)-monotone polynomials which will be given in terms of the largest zero of certain Jacobi polynomials. They also give the exact asymptotics for \(M^{(1)}(n,k)_{p}\) (\(r=1\) or \(\infty \)) with respect to \(n\) and \(k\). For \(i>1\), they give upper and lower estimates for \(M^{(i)}(n,k)_{\infty }\) in terms of zeros of Jacobi polynomials. In particular, the results imply the somewhat surprising fact that the order of Markov factors for \(k\)-monotone polynomials is \(n^2/k\).
scientific article; zbMATH DE number 5848550