Some new estimates of the ‘Jensen gap’

Let ( μ,Ω ) 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 ) ) 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.