ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-24

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A positive convex function need not be log-convex

Example

The identity function $f (x) = x$ on $(0, \infty)$ is positive and convex, but it is not log-convex.

Facts & Assumptions

Given: The identity function on the positive real axis.

[F1]

A positive function is log-convex when its logarithm is convex (Log-convex positive functions).

[F2]

The natural logarithm is strictly increasing on $(0, \infty)$ (Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm).

Verification

technique · direct

1.1givenalgebra

For $x, y > 0$ and $0 \leq λ \leq 1$ , $f ((1 - λ) x + λ y) = (1 - λ) f (x) + λ f (y)$ , so $f$ is affine and hence convex.

1.2F1F2algebra

The midpoint of $1$ and $4$ is $5 / 2 > 2$ . By [F2], $\log (5 / 2) > \log 2 = (\log 1 + \log 4) / 2$ , so $\log \circ f$ violates the midpoint convexity inequality and [F1] shows that $f$ is not log-convex.

2.1step 1.1step 1.2∎

Thus this positive function is convex but not log-convex.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

University of Toronto MAT237Y1, The Gamma Function and the Beta Function, §1.5 (standard reference, not scraped)