Some new estimates of the ‘Jensen gap’

Let (μ,Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$( \mu,\Omega ) $\end{document} be a probability measure space. We consider the so-called ‘Jensen gap’ J(φ,μ,f)=∫Ωφ(f(s))dμ(s)−φ(∫Ωf(s)dμ(s))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ J ( \varphi,\mu,f ) = \int_{\Omega}\varphi \bigl( f ( s ) \bigr)\,d\mu ( s ) -\varphi \biggl( \int_{\Omega }f ( s )\,d\mu ( s ) \biggr) $$\end{document} for some classes of functions φ. Several new estimates and equalities are derived and compared with other results of this type. Especially the case when φ has a Taylor expansion is treated and the corresponding discrete results are pointed out.