Minimizing and Evaluating Weighted Means of Special Mappings
A constrained optimization problem is solved, as an application of minimum principle for a sum of strictly concave continuous functions, subject to a linear constraint, firstly for finite sums of elementary such functions. The motivation of solving such problems is minimizing and evaluating the (unknown) mean of a random variable, in terms of the (known) mean of another related random variable. The corresponding result for infinite sums of the such type of functions follows as a consequence, passing to the limit. Note that in our statements and proofs the condition on the positive numbers is not essential for the interesting part of the results. So, our work refers not only to means of random variables, but to more general weighted means. A related example is given. A corresponding result for special concave mappings taking values into an order-complete Banach lattice of self-adjoint operators is also proved. Namely, one finds a lower bound for a sum of special concave mappings with ranges in the above mention order-complete Banach lattice, under a suitable linear constraint.
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