General Form of Voronovskaja’s Theorem in Terms of Weighted Modulus of Continuity

In the present paper we use weighted modulus ωφ(f;h)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega_\varphi(f;\,h)}$$\end{document} introduced by Pǎltǎnea in (Bull Transilvania Univ Brasov 4(53).67–74, 2011), and defined as ωφ(f;h)=sup|f(x)-f(y)|:x≥0,y≥0,|x-y|≤hφx+y2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega_\varphi(f;\,h)=\sup\left\{|f(x)-f(y)| : x \ge 0,y \ge 0, |x-y|\le h\varphi\left(\frac{x + y}{2} \right)\right\},$$\end{document}where h≥0φ(x)=x1+xm,x∈[0,∞),m∈N,m≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h\ge 0\varphi(x)=\frac{\sqrt{x}}{1+x^m}, x\in [0,\infty),m\in \mathbb{N},m\ge 2}$$\end{document}. We establish the general form of Voronovskaja’s theorem in terms of the modulus ωφ(f;h).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega_\varphi(f;h).}$$\end{document} In Sect. 3 applications are given for the Szász–Mirakyan, Baskakov operators and the Phillips operators.