TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-24

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The real Gamma function is strictly log-convex

Statement

The real Gamma function is strictly log-convex on $(0, \infty)$ .

Facts & Assumptions

Given: Distinct $x, y > 0$ and a weight $0 < λ < 1$ .

[F1]

For $0 < λ < 1$ , exponential convexity is strict unless its two arguments are equal (The two-point convexity inequality for the exponential function).

[F2]

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

Proof

technique · direct

1.1

Fix $c > 0$ and write $c^{s} Γ (s) = \int_{0}^{\infty} t^{- 1} e^{- t} \exp (s \log (c t)) d t$ . By [F1], the integrand at $(1 - λ) x + λ y$ is at most the corresponding convex combination, with strict inequality except at the single point $t = 1 / c$ ; integration makes $s \mapsto c^{s} Γ (s)$ strictly convex.

givenF1

1.2

Choose $c = (Γ (x) / Γ (y))^{1 / (y - x)} > 0$ . Then $c^{x} Γ (x) = c^{y} Γ (y)$ .

constructalgebra

2.1

Apply step 1.1 at $x, y$ with the $c$ from step 1.2. After cancelling $c^{(1 - λ) x + λ y}$ , one gets $Γ ((1 - λ) x + λ y) < Γ (x)^{1 - λ} Γ (y)^{λ}$ , which is strict log-convexity by [F2].

step 1.1step 1.2F2algebra∎

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Sources

University of Toronto MAT237Y1, The Gamma Function and the Beta Function, §1.5 (standard reference, not scraped)
K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 7 §4 (standard reference, not scraped)