Quantitative Korovkin theorems for monotone sublinear and strongly translatable operators in $L_{p}([0, 1])$, $1\le p\le \infty $

By extending the classical quantitative approximation results for positive linear operators in $L_{p}([0, 1])$, $1\le p \le \infty $, of Berens and DeVore in 1978 and of Swetits and Wood in 1983 to the more general case of monotone sublinear and strongly translatable operators, we obtain quantitative estimates in terms of the second order and third order moduli of smoothness, in Korovkin type theorems. Applications to concrete examples are included and an open question concerning interpolation theory for sublinear, monotone and strongly translatable operators is raised.