Inequalities: AM-GM, Cauchy-Schwarz, and Bounding Like a Quant

Many interview questions that look like optimization — "maximize the area," "what's the best split," "show this can't exceed that" — fall to a handful of classical inequalities, no derivatives required. Recognizing which one applies is a pattern-matching skill, and this is the pattern library.

AM–GM: the workhorse

For non-negative numbers,

$$\frac{x_1 + \cdots + x_n}{n} \;\ge\; \sqrt[n]{x_1 \cdots x_n}$$

with equality iff all $x_i$ are equal. Two directions of use:

• Fixed sum → product maximized by equality. A rectangle of perimeter 20 has maximal area as a square (5×5 = 25). Splitting a budget across identical multiplicative stages: split evenly. This "symmetry wins" instinct answers a whole genre before pen touches paper.
• Fixed product → sum minimized by equality. Minimize $x + 1/x$ for $x>0$: AM-GM gives $x + 1/x \ge 2\sqrt{x \cdot 1/x} = 2$, equality at $x=1$. No calculus, no critical points, and the equality condition tells you where.

The finance echo: the AM–GM gap is volatility drag — the arithmetic mean of gross returns exceeds the geometric, with the gap growing in the spread ($\approx \sigma^2/2$, as derived in the calculus lesson). One inequality, quoted twice in this curriculum, because it prices variance both times.

Cauchy–Schwarz: the correlation bound

$$\left(\sum a_i b_i\right)^2 \le \left(\sum a_i^2\right)\left(\sum b_i^2\right)$$

— equivalently $|\langle a, b\rangle| \le \|a\|\|b\|$, equality iff the vectors are proportional. Its most important corollary for this curriculum: correlations lie in $[-1, 1]$ — Cauchy–Schwarz applied to demeaned random variables is the entire proof. It also bounds sums cleverly: e.g. $(x+y+z)^2 \le 3(x^2+y^2+z^2)$ (take $b = (1,1,1)$), the standard trick for "show the sum of squares is at least…" prompts. In portfolio terms: a signal's achievable Sharpe is bounded by its correlation with returns times the return's Sharpe-like ratio — the fundamental law of active management is Cauchy–Schwarz in a suit.

Jensen, revisited as an inequality engine

The calculus lesson introduced Jensen for options; as a bare inequality — $E[f(X)] \ge f(E[X])$ for convex $f$ — it generates one-liners: $E[X^2] \ge E[X]^2$ (variance ≥ 0), $E[1/X] \ge 1/E[X]$, $E[\max] \ge \max(E)$. Whenever an interviewer asks "which is bigger?" between an expectation-of-function and function-of-expectation, the answer is Jensen plus checking convexity — and stating the equality condition (no randomness, or linear $f$) earns the last point.

Bounding discipline

The practitioner's habits: to show $A \le B$, find a chain $A \le C \le B$ with each link simple (drop a negative term; replace each summand by the largest; integrate a monotone bound). Sanity-check every bound with an extreme case — a bound that's violated at $n=1$ or $x=0$ is wrong, and checking costs seconds. And know when the bound is tight: an inequality with unachievable equality still proves, but can't answer "what's the maximum?"

The interview version

"Positive numbers sum to 12; maximize their product if there are 3 of them." — AM-GM: equal at 4 each, product 64. "Show any portfolio's variance is non-negative given a valid covariance matrix" — that's PSD-ness (linear algebra lesson), which is Cauchy–Schwarz's big sibling. "Which is larger: the average of squares or the square of the average?" — Jensen, one word and one reason. These questions grade the reach-for-the-right-tool reflex, not algebraic stamina — build the reflex.