For parameters $\,c\in(0,1)\,$ and $\,\beta>0$, let $\,\ell_{ 2}(c,\beta)\,$ be the Hilbert space of real functions defined on $\,\mathbb{ N}\,$ (i.e., real sequences), for which $$ \| f \|_{ c,\beta}^2 := \sum_{ k=0}^{ \infty}\frac{ (\beta)_k}{ k!}\,c^k\,[f(k)]^2<\infty\,. $$ We study the best (i.e., the smallest possible) constant $\,\gamma_n(c,\beta)\,$ in the discrete Markov-Bernstein inequality $$ \|\Delta P\|_{ c,\beta}\leq \gamma_n(c,\beta)\,\|P\|_{ c,\beta}\,,\quad P\in\mathcal{ P}_n\,, $$ where $\,\mathcal{ P}_n\,$ is the set of real algebraic polynomials of degree at most $\,n\,$ and $\,\Delta f(x):=f(x+1)-f(x)\,$. We prove that: (i) $\displaystyle \gamma_n(c,1)\leq 1+\frac{ 1}{ \sqrt{ c}}\,$ for every $\,n\in \mathbb{ N}\,$ and $\displaystyle \lim_{ n\to\infty}\gamma_n(c,1)= 1+\frac{ 1}{ \sqrt{ c}}\,$. (ii) For every fixed $\,c\in (0,1)\,$, $\,\gamma_n(c,\beta)\,$ is a monotonically decreasing function of $\,\beta\,$ in $\,(0,\infty)\,$. (iii) For every fixed $\,c\in (0,1)\,$ and $\,\beta>0\,$, the best Markov-Bernstein constants $\,\gamma_n(c,\beta)\,$ are bounded uniformly with respect to $\,n$. A similar Markov-Bernstein unequality is proved for sequences in $\,\ell_{ 2}(c,\beta)\,$. We also establish a relation between the best Markov-Bernstein constants $\,\gamma_n(c,\beta)\,$ and the smallest eigenvalues of certain explicitly given Jacobi matrices.
Categories: math.CA