TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Young's inequality for conjugate real exponents

Statement

Let $p, q > 1$ satisfy $1 / p + 1 / q = 1$ . For $u, v \geq 0$ , $u v \leq \frac{u^{p}}{p} + \frac{v^{q}}{q} .$

Facts & Assumptions

Given: Conjugate real exponents $p, q > 1$ and nonnegative reals $u, v$ .

[L1]

Weighted AM-GM applies to positive entries with nonnegative weights summing to one (The weighted arithmetic-geometric mean inequality for real weights).

[L2]

Positive-base real-power laws hold, and the zero-base convention is $0^{r} = 0$ for $r > 0$ ; positive-base real powers are positive (Real powers for positive bases, with the zero-base positive-exponent convention, The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents, The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ ).

Proof

technique · direct

1.1

If $u = 0$ or $v = 0$ , the inequality is immediate from nonnegativity of the two terms on the right.

1.2

For $u, v > 0$ , apply [L1] to $u^{p}, v^{q}$ with weights $1 / p, 1 / q$ .

L1given

2.1

Its geometric side is $(u^{p})^{1 / p} (v^{q})^{1 / q} = u v$ , and its arithmetic side is $u^{p} / p + v^{q} / q$ .

step 1.2L2

3.1

Steps 1.1 and 2.1 cover all nonnegative $u, v$ .

step 1.1step 2.1∎

Depends on

Used by

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)