[proofplan]
We prove Young's inequality $ab \le a^p/p + b^q/q$ for conjugate exponents by exploiting the concavity of the logarithm. The conjugate relation $1/p + 1/q = 1$ makes $(1/p, 1/q)$ a convex combination, and Jensen's inequality (applied to the concave [function](/page/Function) $\log$) converts the product $ab$ into a convex combination of powers. Exponentiating gives the result.
[/proofplan]
[step:Reduce to the case $a, b > 0$]
If $a = 0$ or $b = 0$, both sides equal zero and the inequality holds with equality. For the remainder, assume $a, b > 0$.
[/step]
[step:Apply the concavity of $\log$ to obtain the inequality]
Since $\log$ is a concave function on $(0, \infty)$ and $\frac{1}{p} + \frac{1}{q} = 1$ with $\frac{1}{p}, \frac{1}{q} > 0$, Jensen's inequality gives
\begin{align*}
\log(ab) = \log a + \log b = \frac{1}{p}\log(a^p) + \frac{1}{q}\log(b^q) \le \log\!\left(\frac{a^p}{p} + \frac{b^q}{q}\right).
\end{align*}
Since $\log$ is strictly increasing, exponentiating both sides yields $ab \le \frac{a^p}{p} + \frac{b^q}{q}$.
[guided]
The argument uses Jensen's inequality in the form: for a concave function $\varphi$ and weights $\lambda_1, \lambda_2 > 0$ with $\lambda_1 + \lambda_2 = 1$,
\begin{align*}
\varphi(\lambda_1 t_1 + \lambda_2 t_2) \ge \lambda_1 \varphi(t_1) + \lambda_2 \varphi(t_2).
\end{align*}
We apply this with $\varphi = \log$, $\lambda_1 = 1/p$, $\lambda_2 = 1/q$, $t_1 = a^p$, $t_2 = b^q$. The left-hand side becomes $\log(a^p/p + b^q/q)$ (after exponentiating), while the right-hand side becomes $\frac{1}{p}\log(a^p) + \frac{1}{q}\log(b^q) = \log a + \log b = \log(ab)$.
Equivalently, one can view this geometrically: the graph of $\log$ lies above its secant line, and the point $(\frac{a^p}{p} + \frac{b^q}{q}, \log(\frac{a^p}{p} + \frac{b^q}{q}))$ lies on or above the secant through $\log(a^p)$ and $\log(b^q)$.
[/guided]
[/step]
[step:Determine the equality condition]
Equality in Jensen's inequality for a strictly concave function holds if and only if $t_1 = t_2$, i.e., $a^p = b^q$.
[/step]
