Ohlin-Type Theorem for Convex Set-Valued Maps
Abstract: A counterpart of the Ohlin theorem for convex set-valued maps is proved. An application of this result to obtain some inclusions related to convex set-valued maps in an alternative unified way is presented. In particular counterparts of the Jensen integral and discrete inequalities, the converse Jensen inequality and the Hermite–Hadamard inequalities are obtained.
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Cited by 2 publications
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We study the connections between the equation $$\begin{aligned} \sum _{i=1}^na_if(\alpha _ix+(1-\alpha _i)y)=0 \end{aligned}$$ ∑ i = 1 n a i f ( α i x + ( 1 - α i ) y ) = 0 and the corresponding inequality. $$\begin{aligned} \sum _{i=1}^na_if(\alpha _ix+(1-\alpha _i)y)\ge 0. \end{aligned}$$ ∑ i = 1 n a i f ( α i x + ( 1 - α i ) y ) ≥ 0 . At present, it is clear that linear functional equations should be solved with the use of the results due to L. Székelyhidi. We show that the simplest and most efficient way of dealing with the (continuous) solutions of linear inequalities of the above form is connected with the use of stochastic orderings tools. It will be shown that, in order to solve the inequality, we need to know the continuous solutions of the corresponding equation. In the last part of the paper, we obtain some simple but unexpected connections in the other direction.
We study the connections between the equation $$\begin{aligned} \sum _{i=1}^na_if(\alpha _ix+(1-\alpha _i)y)=0 \end{aligned}$$ ∑ i = 1 n a i f ( α i x + ( 1 - α i ) y ) = 0 and the corresponding inequality. $$\begin{aligned} \sum _{i=1}^na_if(\alpha _ix+(1-\alpha _i)y)\ge 0. \end{aligned}$$ ∑ i = 1 n a i f ( α i x + ( 1 - α i ) y ) ≥ 0 . At present, it is clear that linear functional equations should be solved with the use of the results due to L. Székelyhidi. We show that the simplest and most efficient way of dealing with the (continuous) solutions of linear inequalities of the above form is connected with the use of stochastic orderings tools. It will be shown that, in order to solve the inequality, we need to know the continuous solutions of the corresponding equation. In the last part of the paper, we obtain some simple but unexpected connections in the other direction.
Let n be an odd positive integer. It was proved by Brass and Schmeisser that for every quadrature $$\mathcal {Q}=\alpha _1f(x_1)+\dots +\alpha _mf(x_m)$$ Q = α 1 f ( x 1 ) + ⋯ + α m f ( x m ) (with positive weights) of order at least $$n+1$$ n + 1 and for every $$n-$$ n - convex function f, the value of Q on f lies between the values of Gauss-Legendre and Gauss-Lobatto quadratures of order $$n+1$$ n + 1 calculated for the same function f. We generalize this result in two directions, replacing Q by an integral with respect to a given measure and allowing the number n to any positive integer (for even n Gauss-Radau quadratures replace Gauss-Legendre and Gauss-Lobatto ones).
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