Approximations of generalized Bernstein functions

We establish sharp inequalities involving the incomplete Beta and Gamma functions. These inequalities arise in the approximation of generalized Bernstein functions by higher order Thorin-Bernstein functions. Furthermore, new properties of a related function, namely $x^{\lambda }\mathrm{\Gamma }(x)\mathrm{/}\mathrm{\Gamma }(x+\lambda )$ are derived.