Math 2.0

Real functions Proposition

Slopes of a convex function are increasing

Builds on Convex function

Statement

Let II be an interval and let f ⁣:I→ℝf \colon I \to \mathbb{R} be convex. Then for all x<y<zx < y < z in II, f(y)−f(x)y−x≤f(z)−f(x)z−x≤f(z)−f(y)z−y.\frac{f(y) - f(x)}{y - x} \le \frac{f(z) - f(x)}{z - x} \le \frac{f(z) - f(y)}{z - y}.

Remarks

Write y=tx+(1−t)zy = t x + (1 - t) z with t=(z−y)/(z−x)t = (z - y)/(z - x), and apply the definition of convexity. The converse holds: a function with this property is convex. The first and the last quotient give the adjacent form ConvexOn.slope_mono_adjacent of Mathlib.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Convex_function, Functions of one variable
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