Monotonicity and Inequalities in Special Functions

Special functions such as the gamma function and its derivatives (collectively known as polygamma functions) occupy a central position in mathematical analysis, with applications ranging from probability theory to physics and geometry. Monotonicity properties and sharp inequalities for these functions not only yield insight into their intrinsic structure but also underpin practical tasks such as numerical approximation, error estimation and the analysis of complex models. Researchers have developed systematic methods for proving that certain combinations of special functions or their ratios are increasing, decreasing or convex over specified domains. In particular, the emerging focus on complete monotonicity—an infinite sequence of alternating derivative signs—has provided a unifying framework for establishing families of inequalities. This has led to refined bounds on classical quantities, novel characterisations of asymptotic behaviour and the discovery of connections between seemingly disparate branches of analysis.

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Monotonicity and Inequalities in Special Functions publication trend

The graph below shows the total number of articles in monotonicity and inequalities in special functions across all publications each year (not limited to Nature Index journals).

Technical terms

Monotonicity: A property of a function that is either non‐increasing or non‐decreasing throughout its domain.

Complete monotonicity: A function is completely monotonic if all derivatives exist and alternate in sign.

Gamma function: A continuous extension of the factorial function to real or complex arguments.

Polygamma functions: The derivatives of the logarithm of the gamma function, including the digamma, trigamma and higher‐order functions.

Bernstein function: A function whose derivative is completely monotonic, often used in probability and potential theory.

Logarithmically convex: A function whose logarithm is convex, implying the function itself satisfies certain geometric mean inequalities.

References

1. Monotonicity and inequalities for the gamma function. Journal of Inequalities and Applications (2017).
2. Several Functions Originating from Fisher–Rao Geometry of Dirichlet Distributions and Involving Polygamma Functions. Mathematics (2023).
3. Decreasing properties of two ratios defined by three and four polygamma functions. Comptes Rendus Mathématique (2022).
4. Monotonicity properties for a ratio of finite many gamma functions. Advances in Continuous and Discrete Models (2020).
5. Complete monotonicity involving some ratios of gamma functions. Journal of Inequalities and Applications (2017).
6. Necessary and sufficient conditions for a difference constituted by four derivatives of a function involving trigamma function to be completely monotonic. Mathematical Inequalities & Applications (2021).