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Generalizations of the Jensen inequality. (English) Zbl 0639.26009

The authors offer proofs to the following theorems. Let f be a real valued convex function on a convex subset U of a real linear space. Define on \(I\cup (-I)\), where I is an interval of positive numbers, the function g by \[ g(x)=\sum^{n}_{i=1}\frac{1}{q_ i}f(q_ ixA_ i+(r-x)\sum^{n}_{k=1}A_ k), \] where \(\sum^{n}_{k=1}1/q_ k=1\), \(q_ i>0\), r real and the arguments of f on the right hand side are in U. Then g is convex and increasing on I, decreasing on (-I). Of course, for all convex functions on real intervals, the slopes of the chords are increasing. A function is Wright-convex if the slopes of those chords increase, whose horizontal projections are of equal length. Let f be defined on [a,2c-a], Wright-convex on [a,c] and \(f(x)=-f(2c-x)\quad (x\in [a,c]).\) Then f is midpoint-convex (or Jensen-convex: \(f(\frac{x+y}{2})\leq \frac{f(x)+f(y)}{2})\) on [a,c]. (The paper’s statement is somewhat more general.) Several consequences are listed, many of them covering previous results.
Reviewer: J.Aczél

MSC:

26A51 Convexity of real functions in one variable, generalizations
26D15 Inequalities for sums, series and integrals
26A48 Monotonic functions, generalizations
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