s### 12.3.1: Optimality with Two Support Points

Building on earlier work by Groenen, Giaquinto, Kiers [2003], Van Ruitenburg [2005] proves that a quadratic function $g$ majorizing a differentiable function $f$ at two points must be a sharp quadratic majorizer. We summarize his argument here.

Proof: We have $$\begin{align*} g_1(x)&=f(y)+f'(y)(x-y)+\frac12 a_1(x-y)^2,\tag{1a}\\ g_2(x)&=f(y)+f'(y)(x-y)+\frac12 a_2(x-y)^2,\tag{1b} \end{align*}$$ with $a_1\not= a_2$. Subtracting $(1a)$ and $(1b)$ proves the lemma. QED

Proof: Suppose $g_1\not= g_2$. Since both $g_1$ and $g_2$ majorize $f$ at $y$, Lemma 1 applies. If $g_2$ strictly majorizes $g_1$ at $y$, then $g_1(z_2)<g_2(z_2)= f(z_2)$, and $g_1$ does not majorize $f$. If $g_1$ strictly majorizes $g_2$ at $y$, then similarly $g_2(z_1)< g_1(z_1) = f(z_1)$, and $g_2$ does not majorize $f$. Unless $g_1=g_2$, we reach a contradiction. QED

We now come to Van Ruitenburg's main result.

Proof: Suppose $g_1$ strictly majorizes $g_2$. Then $g_2(z)<g_1(z)=f(z)$ and thus $g_2$ does not majorize $f$. The result now follows from Lemma 1. QED