Some new estimates of the ‘Jensen gap’

Abstract

Abstract Let ( μ , Ω ) $( \mu,\Omega ) $ be a probability measure space. We consider the so-called ‘Jensen gap’ J ( φ , μ , f ) = ∫ Ω φ ( f ( s ) ) d μ ( s ) − φ ( ∫ Ω f ( s ) d μ ( 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.

Keywords

• Jensen’s inequality
• convex function
• γ-superconvex functions
• superquadratic functions
• Taylor expansion