Qiu-Ming Luo

Bounds for the ratio of two gamma functions: from Wendel’s asymptotic relation to Elezović-Giordano-Pečarić’s theorem

In the expository review and survey paper dealing with bounds for the ratio of two gamma functions, along one of the main lines of bounding the ratio of two gamma functions, the authors look back and analyze some known results, including Wendel’s asymptotic relation, Gurland’s, Kazarinoff’s, Gautschi’s, Watson’s, Chu’s, Kershaw’s, and Elezović-Giordano-Pečarić’s inequalities, Lazarević-Lupaş’s claim, and other monotonic and convex properties. On the other hand, the authors introduce some related advances on the topic of bounding the ratio of two gamma functions in recent years.MSC: 33B15, 26A48, 26A51, 26D07, 26D15, 44A10.

GENERALIZATIONS OF BERNOULLI NUMBERS AND POLYNOMIALS

The concepts of Bernoulli numbers Bn, Bernoulli polynomials Bn(x), and the generalized Bernoulli numbers Bn(a, b) are generalized to the one Bn(x; a, b, c) which is called the generalized Bernoulli polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between Bn, Bn(x), Bn(a, b) ,a ndBn(x; a, b, c) are established.

Complete monotonicity of a function involving the divided difference of digamma functions

In the paper, necessary and sufficient conditions are provided for a function involving the divided difference of two psi functions to be completely monotonic. Consequently, a class of inequalities for sums are presented, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions are derived, and two double inequalities for bounding the ratio of two gamma functions are discovered.

Generalizations of Euler numbers and polynomials

The concepts of Euler numbers and Euler polynomials are generalized and some basic properties are investigated.

Sharp Inequalities for Polygamma Functions

Abstract In the paper, the authors review some inequalities and the (logarithmically) complete monotonicity concerning the gamma and polygamma functions and, more importantly, present a sharp double inequality for bounding the polygamma function by rational functions.

A CONCISE PROOF OF OPPENHEIM'S DOUBLE INEQUALITY RELATING TO THE COSINE AND SINE FUNCTIONS

In this paper, we provide a concise proof of Oppenheims double inequality relating to the cosine and sine functions. In passing, we survey this topic.

The Function (b x − a x )∕x: Ratio’s Properties

In the present paper, after reviewing the history, background, origin, and applications of the functions \(\frac{{b}^{t}-{a}^{t}} {t}\) and \(\frac{{e}^{-\alpha t}-{e}^{-\beta t}} {1-{e}^{-t}}\), we establish sufficient and necessary conditions such that the special function \(\frac{{e}^{\alpha t}-{e}^{\beta t}} {{e}^{\lambda t}-{e}^{\mu t}}\) is monotonic, logarithmic convex, logarithmic concave, 3-log-convex, and 3-log-concave on \(\mathbb{R}\), where α, β, λ, and μ are real numbers satisfying (α, β) ≠ (λ, μ), (α, β) ≠ (μ, λ), α ≠ β, and λ ≠ μ.

Complete monotonicity of a function involving the gamma function and applications

In the article the authors present necessary and sufficient conditions for a function involving the logarithm of the gamma function to be completely monotonic and apply these results to bound the gamma function

Some generalizations of 2D Bernoulli polynomials

\Gamma (x)