Young’s inequality

Let $𝑝, 𝑞 \in (1, \infty)$ be Hölder conjugate and $𝛼, 𝛽 \in (0, \infty)$ . Then

𝛼 𝛽 \leq 𝛼 𝑝 𝑝 + 𝛽 𝑞 𝑞

with equality iff $𝛼 𝑝 = 𝛽 𝑞$ . anal

Proof

Since the exponential function is convex,
$e x p (𝜆 𝑡 + (1 - 𝜆) 𝑢) \leq 𝜆 e x p (𝑡) + (1 - 𝜆) e x p (𝑢)$
for $𝜆 \in [0, 1]$ , with equality iff $𝜆 \in {0, 1}$ or $𝑡 = 𝑢$ . Thus setting $𝜆 = 1 𝑝$ whence $1 - 𝜆 = 1 𝑞$ , $𝑡 = l n 𝛼 𝑝$ and $𝑢 = l n 𝛽 𝑞$ , we have
$e x p (𝑝 l n 𝛼 𝑝 + 𝑞 l n 𝛽 𝑞) = 𝛼 𝛽 \leq 𝛼 𝑝 𝑝 + 𝛽 𝑞 𝑞$
with equality iff $𝑡 = 𝑢$ .

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Young’s inequality

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