[Paper Review] Bounds for the ratio of two gamma functions--From Wendel's and related inequalities to logarithmically completely monotonic functions

Feng Qi|arXiv (Cornell University)|Apr 7, 2009

Mathematical Inequalities and Applications40 references4 citations

TL;DR

This survey paper systematically reviews bounds for ratios of gamma and q-gamma functions, establishing necessary and sufficient conditions for such ratios to be logarithmically completely monotonic. It unifies classical inequalities—Wendel's, Gautschi's, Kershaw's—within a framework of complete monotonicity, proving that specific product ratios of gamma functions are logarithmically completely monotonic under explicit parameter conditions.

ABSTRACT

In this expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some inequalities, several complete monotonicity of functions involving ratios of two gamma or $q$-gamma functions, and necessary and sufficient conditions for functions involving ratios of two gamma or $q$-gamma functions to be logarithmically completely monotonic.

Motivation & Objective

• To consolidate and analyze decades of research on bounding ratios of gamma functions, particularly focusing on inequalities and monotonicity properties.
• To clarify the conditions under which ratios of gamma or q-gamma functions are logarithmically completely monotonic, a key property in analysis and special functions.
• To unify and extend results from classical inequalities (e.g., Wendel’s, Gautschi’s) within the broader framework of complete monotonicity and logarithmic complete monotonicity.
• To present new sufficient conditions for logarithmic complete monotonicity of ratios involving products of gamma or q-gamma functions with multiple parameters.
• To provide a comprehensive reference for researchers in analysis, special functions, and applied mathematics seeking rigorous bounds and monotonicity criteria.

Proposed method

• Surveying and synthesizing foundational inequalities: Wendel’s double inequality, Kazarinoff’s refinement, Watson’s monotonicity, Gautschi’s bounds, and Kershaw’s first double inequality.
• Analyzing the complete monotonicity of functions involving ratios of gamma and q-gamma functions using integral representations and derivative sign conditions.
• Applying the concept of logarithmically completely monotonic functions—defined by non-negative alternating derivatives of the logarithm—to derive necessary and sufficient conditions.
• Using the integral representation of completely monotonic functions (via Theorem 1.1) to characterize the structure of such functions.
• Extending results to q-gamma functions by leveraging q-analogues of classical identities and measures, including the discrete measure representation in (1.6).
• Proving that specific symmetric product ratios of gamma functions (e.g., (5.9) and (5.11)) are logarithmically completely monotonic under explicit parameter constraints.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for a ratio of two gamma functions to be logarithmically completely monotonic?
• RQ2 How do classical inequalities like Wendel’s and Gautschi’s relate to the broader framework of complete monotonicity?
• RQ3 What conditions ensure that a product of ratios of gamma or q-gamma functions remains logarithmically completely monotonic?
• RQ4 How do the q-gamma function analogues of classical results preserve or generalize monotonicity properties?
• RQ5 Can the monotonicity of the function $ q_{\alpha,\beta}(t) $ be used to derive sufficient conditions for logarithmic complete monotonicity of more complex ratios?

Key findings

• The function $ F(x) = \frac{\prod_{\sigma \in E_n} \Gamma(x + a_{\sigma(2)} + 2a_{\sigma(3)} + \cdots + (n-1)a_{\sigma(n)})}{\prod_{\sigma \in O_n} \Gamma(x + a_{\sigma(2)} + 2a_{\sigma(3)} + \cdots + (n-1)a_{\sigma(n)})} $ is logarithmically completely monotonic on $ (0, \infty) $ after an appropriate shift.
• The function $ F_n(x) = \frac{\Gamma(x) \prod_{k=1}^{[n/2]} \left[ \prod_{m \in P_{n,2k}} \Gamma\left(x + \sum_{j=1}^{2k} a_{m_j} \right) \right]}{\prod_{k=1}^{[(n+1)/2]} \left[ \prod_{m \in P_{n,2k-1}} \Gamma\left(x + \sum_{j=1}^{2k-1} a_{m_j} \right) \right]} $ is logarithmically completely monotonic on $ (0, \infty) $ for any $ a_k > 0 $.
• For $ 0 < q < 1 $, the q-analogue $ F_{n,q}(x) $ defined similarly to $ F_n(x) $ is also logarithmically completely monotonic on $ (0, \infty) $, with analogous results for $ F_q(x) $.
• Theorem 5.1 establishes that if $ (b_i - a_i)(1 - a_i - b_i) \geq 0 $, $ (b_i - a_i)(|a_i - b_i| - a_i - b_i) \geq 0 $, and $ \sum b_i \geq \sum a_i $, then $ h_{\boldsymbol{a},\boldsymbol{b};n}(x) $ is logarithmically completely monotonic on $ (-\rho_n, \infty) $.
• The q-analogue (Theorem 5.2) confirms that $ h_{q;\boldsymbol{a},\boldsymbol{b};n}(x) = \prod_{i=1}^n \frac{\Gamma_q(x + a_i)}{\Gamma_q(x + b_i)} $ is logarithmically completely monotonic on $ (-\rho_n, \infty) $ under the same conditions.
• The paper confirms that logarithmically completely monotonic functions are a strict subclass of completely monotonic functions, and that they correspond to infinitely divisible completely monotonic functions as noted by Berg and Horn.