Shape Preserving Properties for q-Bernstein-Stancu Operators

We investigate shape preserving for q-Bernstein-Stancu polynomials B-n(q,alpha)(f; x) introduced by Nowak in 2009. When alpha= 0, B-n(q,alpha)(f; x) reduces to the well- known q-Bernstein polynomials introduced by Phillips in 1997; when q = 1, B-n(q,alpha)(f;x) reduces to Bernstein-Stancu polynomials introduced by Stancu in 1968; when q = 1, alpha = 0, we obtain classical Bernstein polynomials. We prove that basic B-n(q,alpha)(f; x) basis is a normalized totally positive basis on [0, 1] and q-Bernstein-Stancu operators are variationdiminishing, monotonicity preserving and convexity preserving on [0, 1].