From 32f4af5f369fa9f0b2988ecad7797f4bec3661c3 Mon Sep 17 00:00:00 2001
From: Holden Rohrer <hr@hrhr.dev>
Date: Tue, 21 Sep 2021 17:12:46 -0400
Subject: notes and homework

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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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cgit