Vol. 18, No. 3, 1966

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 325: 1
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Vol. 321: 1  2
Vol. 320: 1  2
Vol. 319: 1  2
Vol. 318: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Contacts
 
Submission Guidelines
Submission Form
Policies for Authors
 
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
 
Other MSP Journals
On the bounded slope condition

Philip Hartman

Vol. 18 (1966), No. 3, 495–511
Abstract

Let Ω be a bounded open set in Rn and let φ(x), x Ω, satisfy a “bounded slope condition”. The latter reduces to the classical “3-point condition” if n = 2 and occurs in papers on partial differential equations. The properties of φ(x) are studied. It is shown, for example, that if Ω C1 or C1, 0 < λ 1, then φ(x) C1 or C1. Hence, if Ω C1,1 is uniformly convex, then φ(x), x Ω, satisfies a bounded slope condition if and only if φ(x) C1,1. The proofs use generalized convex functions of Beckenbach and, if n > 2, the equivalence of the bounded slope condition and an (n+1)-point condition”.

Mathematical Subject Classification
Primary: 26.52
Milestones
Received: 4 June 1965
Published: 1 September 1966
Authors
Philip Hartman