Shape Preserving Properties for -Bernstein-Stancu Operators

We investigate shape preserving for -Bernstein-Stancu polynomials introduced by Nowak in 2009. When , reduces to the well-known -Bernstein polynomials introduced by Phillips in 1997; when , reduces to Bernstein-Stancu polynomials introduced by Stancu in 1968; when , , we obtain classical Bernstein polynomials. We prove that basic basis is a normalized totally positive basis on and -Bernstein-Stancu operators are variation-diminishing, monotonicity preserving and convexity preserving on . 1. Introduction Let . For each nonnegative integer , we define the -integer as we then define -factorial as and we next define a -binomial coefficient as for integers and as zero otherwise. Also, we use the -Pochhammer symbol defined as for any For , , , and each positive integer , we will investigate the following -Bernstein-Stancu operator introduced by Nowak in 2009 [1]: where Note that empty product in (6) denotes 1. In this case, when , reduces to the well-known -Bernstein polynomials introduced by Phillips [2] in 1997: When , reduces to Bernstein-Stancu polynomials introduced by Stancu [3] in 1968: When and , we obtain the classical Bernstein polynomials defined by Now, we review and state some general properties of -Bernstein-Stancu operators. It follows directly from the definition that -Bernstein-Stancu operators possess the endpoint interpolation property, that is, and leave invariant linear function: They are also degree reducing on polynomials; that is, if is a polynomial of degree , then is a polynomials of degree ≤ . Taking , in (11), we conclude that In 2009, Nowak proved that the -Bernstein-Stancu operators can be expressed in terms of -differences [1]: where At the same time, he still showed that, for , , For a real-valued function on an interval , we define to be the number of sign changes of ; that is, where the supremum is taken over all increasing sequence in , for all . We say that is variation-diminishing if Similarly, for a matrix , we say is variation-diminishing if, for any vector for which is defined, then . Let be a sequence of positive linear operators on . We say that is monotonicity preserving if is increasing (decreasing) for an increasing (decreasing) function on . We say that is convexity preserving if is convex (concave) for a convex (concave) function on . Let , and let be the sequence of basic -Bernstein-Stancu polynomials, and denote by the sequence of all polynomials of degree at most ; then is a basis for (see [1]). Hence, there exists a nonsingular transformation matrix from to such that A matrix is said to be totally positivity