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Elementary Properties of Convex Functions

Abstract

In this chapter we discuss some properties of convex functions connected with their boundedness and continuity. We start with the following Lemma 6.1.1. Let D ⊂ ℝ N be a convex and open set, and let f : D→ ℝ be a convex function. Then
$$ \frac{{f(x) - f(x - nd)}} {n} \leqslant \frac{{f(x) - f(x - md)}} {m} \leqslant \frac{{f(x + md) - f(x)}} {m} \leqslant \frac{{f(x + nd) - f(x)}} {n} $$
for every xD, d ∈ ℝ N and m, n ∈ ℕ such that 0 < m < n and x ± ndD.

Keywords

Convex Function Open Ball Real Constant Elementary Property Rational Line 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag AG 2009

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