A functional inequality and its relation to convexity of vector-valued functions

Author:
Ih Ching Hsu

Journal:
Proc. Amer. Math. Soc. **58** (1976), 119-123

MSC:
Primary 46A40; Secondary 26A51

DOI:
https://doi.org/10.1090/S0002-9939-1976-0415265-6

MathSciNet review:
0415265

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Abstract: With respect to a partial ordering , the functional inequality arises naturally in the study of extending classical convex-function theory to vector-valued functions. The solution is strongly convex and has a Riemann type integral representation, even a Bochner type integral representation when the functional inequality is considered in a Banach lattice. The paper also proves the equivalence of strong and weak convexity in an ordered locally convex space whose positive cone is closed. As an application, an affirmative answer is given to an open question raised earlier by R. G. Kuller and the author.

**[1]**Garrett Birkhoff,*Lattice theory*, Third edition. American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR**0227053****[2]**Albert Wilansky,*Functional analysis*, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964. MR**0170186****[3]**B. Z. Vulikh,*Introduction to the theory of partially ordered spaces*, Translated from the Russian by Leo F. Boron, with the editorial collaboration of Adriaan C. Zaanen and Kiyoshi Iséki, Wolters-Noordhoff Scientific Publications, Ltd., Groningen, 1967. MR**0224522****[4]**Ih Ching Hsu and Robert G. Kuller,*Convexity of vector-valued functions*, Proc. Amer. Math. Soc.**46**(1974), 363–366. MR**0423076**, https://doi.org/10.1090/S0002-9939-1974-0423076-9**[5]**G. H. Hardy, J. E. Littlewood, and G. Pólya,*Inequalities*, Cambridge, at the University Press, 1952. 2d ed. MR**0046395****[6]**A. Wayne Roberts and Dale E. Varberg,*Convex functions*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Pure and Applied Mathematics, Vol. 57. MR**0442824****[7]**K. Yosida,*Functional analysis*, 3rd ed., Springer-Verlag, Berlin and New York, 1971.**[8]**Graham Jameson,*Ordered linear spaces*, Lecture Notes in Mathematics, Vol. 141, Springer-Verlag, Berlin-New York, 1970. MR**0438077**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0415265-6

Keywords:
Weak convexity,
strong convexity,
Hahn-Banach Theorem,
ordered locally convex space,
positive cone,
order-convergence,
Banach lattice,
order-bounded,
strong Lebesgue measurable,
Bochner integral,
Riesz descomposition theorem

Article copyright:
© Copyright 1976
American Mathematical Society