an: Zbl 1195.26037
au: Abramovich, S.; Ivelic, S.; Pecaric, J.E.
ti: Improvement of Jensen-Steffensen's inequality for superquadratic functions.
la: EN
so: Banach J. Math. Anal. 4, No. 1, 159-169, electronic only (2010).
py: 2010
dt: J
cc: *26D15 46C05
ut: Jensen-Steffensen's inequality; superquadratic function; convex function
ab: 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.
rv: Prasanna Sahoo (Louisville)