Corollary 1

Tag 0861, permanent. Cite it as [Bourbaki, Tag 0861].

Functions of a Real Variable, chapter I, § 4. CONVEX FUNCTIONS OF A REAL VARIABLE

Let $f$ be a convex (resp. strictly convex) function on I; if $a$ and $b$ are two interior points of I such that $a < b$ one has (fig. 3)

!A graph showing a convex function with tangents at points a and b, labeled M_a and M_b, respectively.

Fig. 3

f'_d(a) \leq \frac{f(b) - f(a)}{b - a} \leq f'_g(b)

(8)

(resp.

f'_d(a) < \frac{f(b) - f(a)}{b - a} < f'_g(b)

(9)

The double inequality (8) results from (6) and (7) by a simple change of notation. On the other hand, if $f$ is strictly convex and $c$ is such that $a < c < b$ one has, from (8) and prop. 5,

f'_d(a) \leq \frac{f(c) - f(a)}{c - a} < \frac{f(b) - f(a)}{b - a} < \frac{f(b) - f(c)}{b - c} \leq f'_g(b)

from which (9).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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Cites

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