What is Convex function? — Mathematics

Definition 22.11 High School Mathematics · Chapter 22 — Differentiation and Convexity

A function $f$ defined on an interval $I$ is convex on $I$ if every chord of its graph lies above the graph: for all $a, b \in I$ and $t \in \intcc{0}{1}$ ,

f\bigl((1-t)a + t b\bigr) \leq (1-t) f(a) + t f(b).

It is concave if the reverse inequality holds ( $-f$ convex).

A convex function: every chord (red) lies above the graph, and the graph lies above each of its tangents (orange). The dashed segment shows the gap between f ((1-t)a+tb ) and (1-t)f(a)+tf(b). — A convex function: every chord (red) lies above the graph, and the graph lies above each of its tangents (orange). The dashed segment shows the gap between $f\bigl((1-t)a+tb\bigr)$ and $(1-t)f(a)+tf(b)$ .

Examples

Example 22.14

$f(x) = x^3$ has $f''(x) = 6x$ : $f$ is concave on $\intoc{-\infty}{0}$ , convex on $\intco{0}{+\infty}$ , with an inflection point at the origin, where the curve crosses its tangent (the $x$ -axis).

Example 22.15 (Convexity inequalities)

The exponential is convex on $\R$ (its second derivative is itself, positive). Its tangent at $0$ is $y = 1 + x$ , so

\eu^x \geq 1 + x \quad \text{for all } x \in \R .

Such tangent line inequalities are a standard product of convexity.

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Definition 14.18 University Mathematics — Year 1 · Chapter 14 — Differentiation

$f \colon I \to \R$ is convex when every chord lies above the graph:

\forall x, y \in I,\ \forall t \in \intcc{0}{1}, \quad f\bigl(tx + (1-t)y\bigr) \leq t f(x) + (1-t) f(y).

( $f$ is concave when $-f$ is convex.)

Convexity, twice: every chord (gray) lies above the graph, and the graph lies above every tangent (dashed).

Examples

Example 14.20 (Classical convexity inequalities)

$\exp$ is convex ( $\exp'' = \exp > 0$ ): its tangent at $0$ gives $\eu^x \geq 1 + x$ for all $x$ . $\ln$ is concave: its tangent at $1$ gives $\ln x \leq x - 1$ ; its chords give, for $0 < a \leq b$ , the inequality between geometric and arithmetic means: taking $t = \frac12$ in concavity,

\ln\frac{a + b}{2} \geq \frac{\ln a + \ln b}{2} = \ln\sqrt{ab}, \qquad\text{so}\qquad \sqrt{ab} \leq \frac{a+b}{2} .

The general arithmetic–geometric inequality is Exercise 14.9.

Example 14.21 (A convexity inequality from scratch)

The function $f(t) = t\ln t$ is convex on $\intoo{0}{+\infty}$ : $f''(t) = \frac1t > 0$ . Its midpoint inequality, multiplied by $2$ , reads: for all $a, b > 0$ ,

a\ln a + b\ln b \;\geq\; (a + b)\,\ln\frac{a + b}{2} ,

with equality iff $a = b$ (strict convexity). Test drive: $a = 1$ , $b = 3$ gives $3\ln 3 = 3.296$ against $4\ln 2 = 2.773$ . This innocuous inequality is the two-point case of the entropy comparison that reappears with Jensen’s inequality (Exercise 14.9) and in the information-theoretic asymptotics of the Year 3 volume. The closing insight: to manufacture an inequality, find a function whose second derivative has a sign and write down what convexity says — the differential characterization turns one sign check into infinitely many inequalities.

Example 14.24 (Young’s inequality from concavity)

Let $p, q > 1$ with $\frac1p + \frac1q = 1$ . For all $a, b > 0$ :

ab \;\leq\; \frac{a^p}{p} + \frac{b^q}{q} .

Proof by one application of the concavity of $\ln$ with weights $\frac1p, \frac1q$ (the two-point Jensen inequality, as in Exercise 14.9):

\ln\Bigl(\frac{a^p}{p} + \frac{b^q}{q}\Bigr) \;\geq\; \frac1p \ln(a^p) + \frac1q \ln(b^q) = \ln a + \ln b = \ln(ab),

and $\ln$ increasing converts the inequality of logarithms into the claim; equality iff $a^p = b^q$ (strict concavity). The case $p = q = 2$ is the arithmetic-geometric inequality $ab \leq \frac{a^2 + b^2}{2}$ in disguise. The closing insight: Young’s inequality is the algebraic seed of the Hölder and Minkowski inequalities of the Year 2 volume — one concavity statement about $\ln$ , harvested for norms.

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