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Improvement of Jensen-Steffensen’s inequality for superquadratic functions. (English) Zbl 1195.26037
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.

26D15Inequalities for sums, series and integrals of real functions
46C05Hilbert and pre-Hilbert spaces: geometry and topology
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