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Solution of a problem of M. Katz concerning the optimization of a functional. (English) Zbl 0545.49001
Let F(m) be the class of measurable functions f: (0,1)\(\to(0,1)\), such that \(\int^{1}_{0}f dx=m\) and \(\lambda f(x)=meas\{y;f(y)\geq x\}\). The paper consists in an exact solution of the optimization problem \[ \sup_{f\in F(m)}\{\alpha \int^{1}_{0}f(x)\lambda_ f(x)dx+\int^{1}_{0}(f^ 2(x)+\lambda^ 2_ f(x))dx\}. \]
Reviewer: Yu.A.Brudnyj
MSC:
49J05 Existence theories for free problems in one independent variable
26E25 Set-valued functions
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
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References:
[1] KATZ M.: Optimization of functionals containing functions and their inverses. J. of Math. Analysis and Applications, 59, 1977, 163-168. · Zbl 0353.26012
[2] KATZ M.: Rearrangements of (0-1) matrices. Isr. J. Math., 9, No. 1, 1971, 53-72. · Zbl 0215.33405
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