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Definition: A fn. f : R -> R is called convex on an interval (a,b) if
f(cx + dy) \leq cf(x) + df(y)
\forall x, y \in (a, b)
\forall c \in (0, 1), d = 1-c.
Concave is -convex.
Essentially stating that the function lies on or below a line segment
connecting f(a) and f(b) [or above in the case of concave].
Strictly convex: f(cx+dy) < cf(x) + df(y).
Jensen's Inequality
X - r.v., E|X| < infty. E|f(x)| < infty.
f(E X) \leq E(f(x)).
If f is strictly convex, \leq -> "less than" unless X is a constant r.v.
Further theorems:
(1) If f is differentiable on (a,b),
f is convex <=> f' is nondecreasing on (a,b).
f is strictly convex <=> f' is strictly increasing on (a,b)
(2) If f is twice differentiable on (a,b)
f is convex <=> f'' \geq 0 on (a,b)
f is strictly convex <=> f'' > 0 on (a,b)
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