Weak-type inequalities for Kantorovitch polynomials and related operators

Abstract

Continuing previous investigations concerning Bernstein polynomials, the purpose of this paper is to establish the weak-type inequality (f∈L^p(0,1),n∈ℕ)${\omega }_{\varphi }(n^{-1/2},f)\le {{\rm M}}_p{n}^{-1}\sum _{k=1}^n||K_{K}f-f||p$in terms of the Kantorovitch polynomial K_kƒ and the modulus of continuity (ϕ²(x): = x(1 − x))${\omega }_{\varphi }(t,f):=\underset{0<h\le t}{sup}\left|\right|{\Delta }_{h\varphi }^{2}f|{|}_p+\underset{0<h\le t^2}{\sup }|\left|{\Delta }_h^{2}f\right|{|}_{p.}$Such estimates which immediately imply well-known inverse results are also obtained for the Kantorovitch version of the Szász-Mirakjan and Baskakov operators, respectively.