bibkey: mahmoudmortici2026wallisratio
authors: M. Mahmoud and C. Mortici
year: 2026
title: "Structural Properties, Complete Monotonicity, Bounds, and Analytic Approximations with Bounded Remainders for a Wallis-Type Gamma Ratio"
doi: 10.1007/s00025-026-02697-2
url: https://link.springer.com/article/10.1007/s00025-026-02697-2
claim: "Section 4 asks whether mu and chi are strictly increasing and theta is strictly decreasing for every positive real argument."
strata_touched:
  - D5/S3/Analytic/WallisRemainder/Defs
  - D5/S3/Analytic/WallisRemainder/Result
license: CC-BY-4.0
triage: anchor

Wallis gamma remainder monotonicity

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DOI: 10.1007/s00025-026-02697-2

Source: https://link.springer.com/article/10.1007/s00025-026-02697-2

Version of record: Results in Mathematics 81 (2026), article 141, published 2026-06-30. Section 4, Open Question; Section 5 restates it. The article is open access under Creative Commons Attribution 4.0.

Mathematical correspondence

For $x>0$, let $B(x)=\sqrt{x}\Gamma (x+1/2)/\Gamma (x+1)$ and define

$$\mu (x)=-x^2-{\left\{\frac{3}{x}[\ln B(x){]}^{-1}+24\right\}}^{-1},$$

$$\theta (x)=\frac{23040x^5}{31}\left\{\frac{3x}{24x^2+1}+\ln B(x)\right\},$$

$$\chi (x)=\frac{358400x^7}{257}\left\{\frac{120x^2+31}{96x(10x^2+3)}+\ln B(x)\right\}.$$

Defs.wallisB, Defs.mu, Defs.theta and Defs.chi in D5/S3/Analytic/WallisRemainder/Defs are these exact real functions. Defs.claim includes the two nonvanishing assertions implicit in the reciprocals and all three strict monotonicity comparisons. D5/S3/Analytic/WallisRemainder/Result.result proves this claim. The source proves continuity and boundedness of these remainders in Theorems 2.1–2.3 and leaves their strict monotonicity as the question above.

The bounded literature scope and its limitations are stated in Problems/mahmoud-mortici-2026-wallis-remainder-monotonicity.md.