Bounded slope variation and generalized convexity

Author:
Frank N. Huggins

Journal:
Proc. Amer. Math. Soc. **65** (1977), 65-69

MSC:
Primary 26A45; Secondary 26A42, 26A51

DOI:
https://doi.org/10.1090/S0002-9939-1977-0466447-X

MathSciNet review:
0466447

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, the concept of bounded slope variation, that of the convexity of a function with respect to an increasing function, and the Lebesgue-Stieltjes integral are used to further generalize a theorem of F. Riesz and to give a new proof based on a weaker hypothesis that a function which has bounded slope variation with respect to an increasing function *m* over [*a, b*] can be expressed as the difference of two functions each cf which is convex with respect to *m* on [*a, b*].

**[1]**F. N. Huggins,*A generalization of a theorem of F. Riesz*, Pacific J. Math.**39**(1971), 695-701. MR**0304578 (46:3713)****[2]**-,*Bounded slope variation*, Texas J. Sci.**24**(1973), 431-437.**[3]**-,*Generalized Lipschitz conditions*, Texas J. Sci.**27**(1976), 251-256. MR**532330 (80b:26002a)****[4]**F. Riesz,*Sur certains systèmes singuliers d'équations intégrales*, Ann. École Norm. Sup. (3)**28**(1911), 33-62. MR**1509135****[5]**F. Riesz and B. Sz.-Nagy,*Functional analysis*, 2nd ed., Ungar, New York, 1953.**[6]**A. W. Roberts and D. E. Varberg,*Functions of bounded convexity*, Bull. Amer. Math. Soc.**75**(1969), 568-572. MR**0239021 (39:380)****[7]**A. M. Russell,*Functions of bounded second variation and Stieltjes-type integrals*, J. London Math. Soc. (2)**2**(1970), 193-208. MR**0274678 (43:440)****[8]**W. H. Young,*On integrals and derivates with respect to a function*, Proc. London Math. Soc. (2)**15**(1916), 35-63.**[9]**J. R. Webb,*A Hellinger integral representation for bounded linear functionals*, Pacific J. Math.**20**(1967), 327-337. MR**0208359 (34:8169)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
26A45,
26A42,
26A51

Retrieve articles in all journals with MSC: 26A45, 26A42, 26A51

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1977-0466447-X

Keywords:
Bounded slope variation,
bounded variation,
convex function,
Lebesgue-Stieltjes integral

Article copyright:
© Copyright 1977
American Mathematical Society