Hadamard inequality and a refinement of Jensen inequality for set-valued functions. (English) Zbl 0891.26014
Let \(Y\) be a Banach space, \(I\) be an open interval and \(F:I\to\text{cl}(Y)\) be a convex set-valued function. The following analogue of the Hadamard inequality is proved:
\[
{F(a)+ F(b)\over 2}\subset {1\over b-a} \int^b_a F(x)dx\subset F\Biggl({a+ b\over 2}\Biggr)
\]
for every \(a,b\in I\), \(a<b\). Some refinements of the Jensen inequality for set-valued functions are also given.
Reviewer: K.Nikodem (Bielsko-Biała)
MSC:
| 26D15 | Inequalities for sums, series and integrals |
| 26E25 | Set-valued functions |
| 26A51 | Convexity of real functions in one variable, generalizations |
| 39B72 | Systems of functional equations and inequalities |
| 54C60 | Set-valued maps in general topology |
References:
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| [3] | R.J. Aumann: Integrals of Set-Valued Functions, Journal of Mathematical Analysis and Applications 12, (1965),1–12. · Zbl 0163.06301 · doi:10.1016/0022-247X(65)90049-1 |
| [4] | E.Hille, R.S.Phillips: Functional Analysis and Semi-Groups, American Mathematical Society Colloquium Publications, vol. XXXI, 1975. |
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