bibkey: wisniewska2025gurland
authors: Halina Wiśniewska
year: 2025
title: "Some inequalities for Gurland's ratio of the gamma functions"
doi: null
url: https://arxiv.org/html/2512.07028v1
claim: "Open Problem 1 asks whether the original modified-Gamma-ratio logarithmic remainder tends to zero for every positive real pair."
strata_touched: []
license: citation-only
triage: anchor

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Source: https://arxiv.org/html/2512.07028v1

Section 2, numbered Open Problem 1. Equations (2)–(4) define the modified Gamma ratio, the real Hurwitz series and the original finite expansion. Equations (12) and (14)–(19), together with the proof of Corollary 1, identify the logarithmic summands, finite interchange and expansion-order limit. The integer n in the remainder notation is a bound summation index.

The original whole positive-pair question

In Corollary 1 we showed that R_n^m(x,y) tends to zero under a sufficient condition. Does R_n^m(x,y) tend to zero hold for every x,y > 0?

Set a=(x+y)/2 and d=(x-y)/2. The ratio is Gamma(1+x)Gamma(1+y)/Gamma(1+a)^2, and the finite expansion is the sum, for 1 <= k < m, of d^(2k) zeta(2k,1+a)/k. The Hurwitz function here is the real series sum over j >= 0 of (j+b)^(-s), with its actual real shift b. The remainder is the difference between the logarithm of that ratio and this finite expansion. Corollary 1 explicitly takes m to infinity.

For every positive pair, c_n=d^2/(n+1+a)^2 lies in [0,1). The real Euler limit identifies the logarithm of the ratio with the sum of -log(1-c_n). The nonnegative logarithmic series can then be summed in either order. This gives the coefficient series on the entire positive quadrant, including positive integer Gamma arguments. On the diagonal all remainders are zero. This is a pointwise convergence statement.

The source’s Taylor points depend on the expansion order and must satisfy the exact Taylor remainder equality. An arbitrary point in (0,c_n) does not specify that remainder. On the diagonal that interval is empty; equation (4) still determines the zero remainder. No Taylor-point family is assumed by the formal statement.

The Euler limit, Gamma positivity, logarithmic Taylor series and p-series comparison are classical background. Merkle, Gurland’s ratio for the gamma function, DOI 10.1016/j.camwa.2004.01.016, is relevant older work. Its full-text overlap remains unverified. The versioned source and bounded bibliographic searches do not establish priority or absence of an earlier identity. Open Problems 1 and 2 are one family; Open Problem 3 remains open.

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