Strong Converse Results for Linking Operators and Convex Functions

Abstract

We consider a family $B_{n,\rho }^{\left[c\right]}$ of operators which is a link between classical Baskakov operators (for $\rho =\infty$ ) and their genuine Durrmeyer type modification (for $\rho =1$ ). First, we prove that for fixed $n,c$ and a fixed convex function $f$ , $B_{n,\rho }^{\left[c\right]}f$ is decreasing with respect to $\rho$ . We give two proofs, using various probabilistic considerations. Then, we combine this property with some existing direct and strong converse results for classical operators, in order to get such results for the operators $B_{n,\rho }^{\left[c\right]}$ applied to convex functions.