Convexity of vector-valued functions
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- by Ih Ching Hsu and Robert G. Kuller PDF
- Proc. Amer. Math. Soc. 46 (1974), 363-366 Request permission
Abstract:
Let $(\mathcal {B}, \ll )$ be a Banach lattice, and $(a,b)$ be an open interval on the real line. A function $F:(a,b) \to \mathcal {B}$ is defined to be weakly convex if there exists a nonnegative nondecreasing continuous function $G:(a,b) \to \mathcal {B}$ such that $p[F(S)] + tp[G(s)] \leq p[F(s + t)]$, whenever $s$ and $s + t$ are in $(a,b)$ for each positive linear functional $p$ on $\mathcal {B}$. A representation theorem is proved as follows: If $F$ is weakly convex on $(a,b)$ and is bounded on an interval contained in $(a,b)$, then $(B)\int _{a + \epsilon }^x {G(s)dm = F(x) - F(a + \epsilon )}$, where $(B)\int _{a + \epsilon }^x {G(s)dm}$ is the Bochner integral of $G$ on $[a, \epsilon ,x]$ with $0 < \epsilon$ and $a < \epsilon < x < b$.References
- G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395 I. Hsu, Weak convexity of operator-valued functions (unpublished).
- Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053 K. Yosida, Functional analysis, Die Grundlehren der math. Wissenschaften, Band 123, Springer-Verlag, Berlin and New York, 1965. MR 31 #5054.
- Graham Jameson, Ordered linear spaces, Lecture Notes in Mathematics, Vol. 141, Springer-Verlag, Berlin-New York, 1970. MR 0438077 I. Hsu, A functional inequality and its relation to convexity of vector-valued functions (submitted).
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 46 (1974), 363-366
- MSC: Primary 46G99; Secondary 26A51
- DOI: https://doi.org/10.1090/S0002-9939-1974-0423076-9
- MathSciNet review: 0423076