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Let (μ,Ω)$( \mu,\Omega ) $ 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))$$ J ( \varphi,\mu,f ) = \int_{\Omega}\varphi \bigl( f ( s ) \bigr)\,d\mu ( s ) -\varphi \biggl( \int_{\Omega }f ( s )\,d\mu ( s ) \biggr) $$ 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.

Some new estimates of the ‘Jensen gap’