Mean Value of Convex Real Function
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Theorem
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.
Let $f$ be convex on $\openint a b$.
Then:
- $\forall \xi \in \openint a b: \map f x - \map f \xi \ge \map {f'} \xi \paren {x - \xi}$
Proof
By the Mean Value Theorem:
- $\exists \eta \in \openint x \xi: \map {f'} \eta = \dfrac {\map f x - \map f \xi} {x - \xi}$
From Real Function is Convex iff Derivative is Increasing, the derivative of $f$ is increasing.
Thus:
- $x > \xi \implies \map {f'} \eta \ge \map {f'} \xi$
- $x < \xi \implies \map {f'} \eta \le \map {f'} \xi$
Hence:
- $\map f x - \map f \xi \ge \map {f'} \xi \paren {x - \xi}$
$\blacksquare$
Also see
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 12.21 \ (4)$