Vol. 18, No. 3, 1966

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On the bounded slope condition

Philip Hartman

Vol. 18 (1966), No. 3, 495–511

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
Received: 4 June 1965
Published: 1 September 1966
Philip Hartman