Bounds for Bernstein basis functions and Meyer-König and Zeller basis functions (Q1269668)

The main of this paper is the following result: Let \(n,k,p\in \{0,1,2, \dots\}\) and \(C_p= (p+ 0,5)^{p+0,5} (p!)^{-1} e^{-p-0,5}\). For all \(k,x\) such that \(p\leq k\leq n-p\), \(x\in(0,1)\) \({n\choose k} x^k(1-x)^{n-k} <C_p (nx(1-x))^{-0,5}\). If \(0\leq k\leq n\) and \(x\in(0,1]\) then \({n+k-1 \choose k} x^k(1-x)^n <(2enx)^{-0,5}\). The coefficients \(C_p\), \((2e)^{-0,5} \approx 0.4288819\) and the estimate order \(n^{-0,5}\) are the best possible.