@article{1195.26037, author="Abramovich, S. and Ivelic, S. and Pecaric, J.E.", title="{Improvement of Jensen-Steffensen's inequality for superquadratic functions.}", language="English", journal="Banach J. Math. Anal. ", volume="4", number="1", pages="159-169", year="2010", abstract="{Jensen-Steffensen's inequality states that if $f : I \to \Bbb{R}$ is a convex function, then $$f \left ({1\over {A_n}} \sum_{i=1}^n a_i x_i \right) \leq {1\over {A_n}} \sum_{i=1}^n a_i f( x_i ) ,$$ where $I$ is an interval in $\Bbb{R}$, $( x_1,x_2,\dots,x_n)$ is any monotonic $n$-tuple in $I^n$, and $(a_1, a_2, \dots,a_n )$ is a real $n$-tuple that satisfies $$0 \leq A_j \leq A_n, \quad A_n > 0 ,\quad A_j = \sum_{i=1}^j a_i , \quad \bar{A_j} = \sum_{i=j}^n a_i , \quad j=1,2,\dots, n.$$ In this paper, the authors prove some inequalities for superquadratic functions analog to Jensen-Steffensen's inequality for convex functions. For superquadratic functions which are convex, the authors prove some improvements and extensions of Jensen-Steffensen's inequality.}", reviewer="{Prasanna Sahoo (Louisville)}", keywords="{Jensen-Steffensen's inequality; superquadratic function; convex function}", classmath="{*26D15 (Inequalities for sums, series and integrals of real functions) 46C05 (Geometry and topology of inner product spaces) }", }