Searcharxiv⌕ Search

arXiv subjects

Bai-Ni Guo

Publications and source records attributed to Bai-Ni Guo.

At least 19 recordsLinked to original sources
arXiv2026

Monotonicity and absolute convexity of two functions involving Riemann zeta function

Let $ρ>0$ be a constant, let $j\ge0$ be an integer, and let $Γ(z)$ denote the Euler gamma function. With the aid of the integral representation for the Riemann zeta function $ζ(z)$, by virtue of a monotonicity rule, and by means of some properties of the function $\frac{1}{\operatorname{e}^t-1}$ and its derivatives, the authors discuss the increasing monotonicity of the function $t\mapsto\binom{t+ρ+j}ρ\frac{ζ(t+ρ)}{ζ(t)}$, where $\binom{z}{w}$ denotes the extended binomial coefficient, study the absolute convexity and logarithmic convexity of the function $t\mapstoΓ(t+j)ζ(t)$, and derive the increasing monotonicity and inequalities of some sequences involving the ratios $\bigl|\frac{B_{2n+2}} {B_{2n}}\bigr|$ of the Bernoulli numbers $B_{2n}$.

math.NT↗
arXiv2021

Maclaurin's series expansions for positive integer powers of inverse (hyperbolic) sine and related functions, specific values of partial Bell polynomials, and two applications

In the paper, the authors establish Maclaurin's series expansions and series identities for positive integer powers of the inverse sine function, for positive integer powers of the inverse hyperbolic sine function, for the composite of incomplete gamma functions with the inverse hyperbolic sine function, for positive integer powers of the inverse tangent function, and for positive integer powers of the inverse hyperbolic tangent function, in terms of the first kind Stirling numbers and binomial coefficients, apply the newly established Maclaurin's series expansion for positive integer powers of the inverse sine function to derive a closed-form formula for specific values of partial Bell polynomials and to derive a series representation of the generalized logsine function, and deduce several combinatorial identities involving the first kind Stirling numbers. Some of these results simplify and unify some known ones. All of these newly established Maclaurin's series expansions of positive integer powers of the inverse (hyperbolic) sine and tangent functions can be used to derive infinite series representations of the circular constant Pi and of positive integer powers of Pi.

math.CO↗
arXiv2020

Refinements of Young's integral inequality via fundamental inequalities and mean value theorems for derivatives

In the paper, the authors review several refinements of Young's integral inequality via several mean value theorems, such as Lagrange's and Taylor's mean value theorems of Lagrange's and Cauchy's type remainders, and via several fundamental inequalities, such as Čebyšev's integral inequality, Hermite--Hadamard's type integral inequalities, Hölder's integral inequality, and Jensen's discrete and integral inequalities, in terms of higher order derivatives and their norms, survey several applications of several refinements of Young's integral inequality, and further refine Young's integral inequality via Pólya's type integral inequalities.

math.CA↗
arXiv2020

From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions

In the paper, the authors review origins, motivations, and generalizations of a series of inequalities involving several exponential functions and sums, establish three new inequalities involving finite exponential functions and sums by finding convexity of a function related to the generating function of the Bernoulli numbers, survey the history, backgrounds, generalizations, logarithmically complete monotonicity, and applications of a series of ratios of finite gamma functions, present complete monotonicity of a linear combination of finite trigamma functions, construct a new ratio of finite gamma functions, derives monotonicity, logarithmic convexity, concavity, complete monotonicity, and the Bernstein function property of the newly constructed ratio of finite gamma functions, and suggest two linear combinations of finite trigamma functions and two ratios of finite gamma functions to be investigated.

math.CA↗
arXiv2019

A ratio of many gamma functions and its properties with applications

In the paper, the authors establish an inequality involving exponential functions and sums, introduce a ratio of many gamma functions, discuss properties, including monotonicity, logarithmic convexity, (logarithmically) complete monotonicity, and the Bernstein function property, of the newly introduced ratio, and construct two inequalities of multinomial coefficients and multivariate beta functions.

math.CA↗
arXiv2015

Six proofs for an identity of the Lah numbers

In the paper, utilizing respectively the induction, a generating function of the Lah numbers, the Chu-Vandermonde summation formula, an inversion formula, the Gauss hypergeometric series, and two generating functions of the Stirling numbers of the first kind, the authors collect and provide six proofs for an identity of the Lah numbers.

math.CO↗
arXiv2011

Complete monotonicity of a function involving a ratio of gamma functions and applications

In the paper, necessary and sufficient conditions are presented for a function involving a ratio of gamma functions to be logarithmically completely monotonic. This extends and generalizes the main result in [\emph{Inequalities and monotonicity for the ratio of gamma functions}, Taiwanese J. Math. \textbf{7} (2003), no.~2, 239\nobreakdash--247.] and others. As applications, several inequalities involving the volume of the unit ball in $\mathbb{R}^n$ are derived, which refine, generalize and extend some known inequalities.

math.CA↗
arXiv2011

An extension of an inequality for ratios of gamma functions

In this paper, we prove that for $x+y>0$ and $y+1>0$ the inequality {equation*} \frac{[Γ(x+y+1)/Γ(y+1)]^{1/x}}{[Γ(x+y+2)/Γ(y+1)]^{1/(x+1)}} <\biggl(\frac{x+y}{x+y+1}\biggr)^{1/2} {equation*} is valid if $x>1$ and reversed if $x<1$ and that the power $\frac12$ is the best possible, where $Γ(x)$ is the Euler gamma function. This extends the result in [Y. Yu, \textit{An inequality for ratios of gamma functions}, J. Math. Anal. Appl. \textbf{352} (2009), no.~2, 967\nobreakdash--970.] and resolves an open problem posed in [B.-N. Guo and F. Qi, \emph{Inequalities and monotonicity for the ratio of gamma functions}, Taiwanese J. Math. \textbf{7} (2003), no.~2, 239\nobreakdash--247.].

math.CA↗
arXiv2011

A refinement of a double inequality for the gamma function

In the paper, we present a monotonicity result of a function involving the gamma function and the logarithmic function, refine a double inequality for the gamma function, and improve some known results for bounding the gamma function.

math.CA↗
arXiv2010

Monotonicity and logarithmic convexity relating to the volume of the unit ball

Let $Ω_n$ stand for the volume of the unit ball in $\mathbb{R}^n$ for $n\in\mathbb{N}$. In the present paper, we prove that the sequence $Ω_{n}^{1/(n\ln n)}$ is logarithmically convex and that the sequence $\frac{Ω_{n}^{1/(n\ln n)}}{Ω_{n+1}^{1/[(n+1)\ln(n+1)]}}$ is strictly decreasing for $n\ge2$. In addition, some monotonic and concave properties of several functions relating to $Ω_{n}$ are extended and generalized.

math.CA↗