Authorsâ€™ abstract. In a recent generalization of the Bernstein polynomials, the approximated function

$f$ is evaluated at points spaced at intervals which are in geometric progression on

$[0,1]$, instead of equally spaced points. For each positive integer

$n$, this replaces the single polynomial

${B}_{n}f$ by a one-parameter family of polynomials

${B}_{n}^{q}f$, where

$0\le q\le 1$. This paper summarizes briefly the previously known results concerning these generalized Bernstein polynomials and give new results concerning

${B}_{n}^{q}f$ when

$f$ is a monomial. The main results of the paper are obtained by using the concept of total positivity. It is shown that if

$f$ is increasing then

${B}_{n}^{q}f$ is increasing, and if

$f$ is convex then

${B}_{n}^{q}f$ is convex, generalizing well known results when

$q=1$. It is also shown that if

$f$ is convex, then for any positive integer

$n$,

${B}_{n}^{r}f\le {B}_{n}^{q}f$ for

$0<q\le r\le 1$. This supplements the well known classical result that

$f\le {B}_{n}f$ when

$f$ is convex.