DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 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.

Log-convex positive functions

Definition

A positive function $f : I \to (0, \infty)$ is log-convex when $\log \circ f$ is convex on $I$ .

Equivalently, for $x, y \in I$ and $0 \leq λ \leq 1$ with $(1 - λ) x + λ y \in I$ ,

$f ((1 - λ) x + λ y) \leq f (x)^{1 - λ} f (y)^{λ} .$

It is strictly log-convex when this inequality is strict for $x \neq y$ and $0 < λ < 1$ . Positivity ensures that every logarithm and every real power in these formulas is defined.

Log-convex positive functions

Definition

Depends on

Dependency tree · 2 levels

Sources