Proof of Young's inequality

This proof leverages properties of convex functions and Hölder conjugates. I like the proof because it is small, and reasonable.

Define $𝑓 (𝑡) = 𝑡^{𝑞} / 𝑞$ , a convex function on the interval $[0, \infty)$ .

One characterization of convex functions is that they are everywhere above their tangent lines. For our convex function $𝑓 (𝑡)$ , this means that for any $𝑏, 𝑐 \geq 0$ ,

𝑓 (𝑏) \geq 𝑓 (𝑐) + 𝑓^{'} (𝑐) (𝑏 - 𝑐) .

Let $𝑝$ and $𝑞$ be Hölder conjugates (i.e. $1 / 𝑝 + 1 / 𝑞 = 1$ and $𝑝, 𝑞 > 1$ ).

Now let $𝑐 = 𝑎^{𝑝 - 1}$ .

\begin{aligned} \frac{𝑏^{𝑞}}{𝑞} & \geq \frac{𝑎^{(𝑝 - 1) 𝑞}}{𝑞} + 𝑎^{(𝑝 - 1) (𝑞 - 1)} (𝑏 - 𝑎^{𝑝 - 1}) \\ = \frac{𝑎^{𝑝}}{𝑞} + 𝑎 (𝑏 - 𝑎^{𝑝 - 1}) = \frac{𝑎^{𝑝}}{𝑞} + 𝑎 𝑏 - 𝑎^{𝑝} = (1 - 𝑞) \frac{𝑎^{𝑝}}{𝑞} + 𝑎 𝑏 \\ = - \frac{𝑎^{𝑝}}{𝑝} + 𝑎 𝑏 . \end{aligned}

The inequality follows by rearrangement. The jumps between the lines are made by recognizing Hölder conjugate identities. The jump from the first line to the second line follows after recognizing $(𝑝 - 1) 𝑞 = 𝑝$ and $(𝑝 - 1) (𝑞 - 1) = 1$ . The jump from the second line to the third line follows after seeing $(1 - 𝑞) / 𝑞 = - 1 / 𝑝$ . This fits in my head!