Searcharxiv⌕ Search

arXiv subjects

Bai-Ni Guo

Publications and source records attributed to Bai-Ni Guo.

At least 37 records · Page 2Linked to original sources
arXiv2010

Sharp bounds for harmonic numbers

In the paper, we first survey some results on inequalities for bounding harmonic numbers or Euler-Mascheroni constant, and then we establish a new sharp double inequality for bounding harmonic numbers as follows: For $n\in\mathbb{N}$, the double inequality -\frac{1}{12n^2+{2(7-12γ)}/{(2γ-1)}}\le H(n)-\ln n-\frac1{2n}-γ<-\frac{1}{12n^2+6/5} is valid, with equality in the left-hand side only when $n=1$, where the scalars $\frac{2(7-12γ)}{2γ-1}$ and $\frac65$ are the best possible.

math.CA↗
arXiv2010

Uniqueness of nontrivially complete monotonicity for a class of functions involving polygamma functions

For $m,n\in\mathbb{N}$, let $f_{m,n}(x)=\bigr[ψ^{(m)}(x)\bigl]^2+ψ^{(n)}(x)$ on $(0,\infty)$. In the present paper, we prove using two methods that, among all $f_{m,n}(x)$ for $m,n\in\mathbb{N}$, only $f_{1,2}(x)$ is nontrivially completely monotonic on $(0,\infty)$. Accurately, the functions $f_{1,2}(x)$ and $f_{m,2n-1}(x)$ are completely monotonic on $(0,\infty)$, but the functions $f_{m,2n}(x)$ for $(m,n)\ne(1,1)$ are not monotonic and does not keep the same sign on $(0,\infty)$.

math.CA↗
arXiv2010

A completely monotonic function involving the tri- and tetra-gamma functions

The psi function $ψ(x)$ is defined by $ψ(x)=\frac{Γ'(x)}{Γ(x)}$ and $ψ^{(i)}(x)$ for $i\in\mathbb{N}$ denote the polygamma functions, where $Γ(x)$ is the gamma function. In this paper we prove that a function involving the difference between $[ψ'(x)]^2+ψ''(x)$ and a proper fraction of $x$ is completely monotonic on $(0,\infty)$.

math.CA↗
arXiv2009

Complete monotonicity of some functions involving polygamma functions

In the present paper, we establish necessary and sufficient conditions for the functions $x^α\bigl\lvertψ^{(i)}(x+β)\bigr\lvert$ and $α\bigl\lvertψ^{(i)}(x+β)\bigr\lvert-x\bigl\lvertψ^{(i+1)}(x+β)\bigr\lvert$ respectively to be monotonic and completely monotonic on $(0,\infty)$, where $i\in\mathbb{N}$, $α>0$ and $β\ge0$ are scalars, and $ψ^{(i)}(x)$ are polygamma functions.

math.CA↗
arXiv2009

The function $(b^x-a^x)/x$: Ratio's properties

In the paper, after reviewing the history, background, origin, and applications of the functions $\frac{b^{t}-a^{t}}{t}$ and $\frac{e^{-αt}-e^{-βt}}{1-e^{-t}}$, we establish sufficient and necessary conditions such that the special function $\frac{e^{αt}-e^{βt}}{e^{λt}-e^{μt}}$ are 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 $(α,β)\ne(λ,μ)$, $(α,β)\ne(μ,λ)$, $α\neβ$ and $λ\neμ$.

math.CA↗
arXiv2009

Some logarithmically completely monotonic functions related to the gamma function

In this article, logarithmically complete monotonicity properties of some functions such as $\frac1{[Γ(x+1)]^{1/x}}$, $\frac{[{Γ(x+α+1)}]^{1/(x+α)}}{[{Γ(x+1)}]^{1/x}}$, $\frac{[Γ(x+1)]^{1/x}}{(x+1)^α}$ and $\frac{[Γ(x+1)]^{1/x}}{x^α}$ defined in $(-1,\infty)$ or $(0,\infty)$ for given real number $α\in\tr$ are obtained, some known results are recovered, extended and generalized. Moreover, some basic properties of the logarithmically completely monotonic functions are established.

math.CA↗
arXiv2009

A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function

In the article, a notion "logarithmically absolutely monotonic function" is introduced, an inclusion that a logarithmically absolutely monotonic function is also absolutely monotonic is revealed, the logarithmically complete monotonicity and the logarithmically absolute monotonicity of the function $\bigl(1+\fracαx\bigr) ^{x+β}$ are proved, where $α$ and $β$ are given real parameters, a new proof for the inclusion that a logarithmically completely monotonic function is also completely monotonic is given, and an open problem is posed.

math.CA↗
arXiv2009

Necessary and sufficient conditions for a function involving divided differences of the di- and tri-gamma functions to be completely monotonic

In the present paper, necessary and sufficient conditions are established for a function involving divided differences of the digamma and trigamma functions to be completely monotonic. Consequently, necessary and sufficient conditions are derived for a function involving the ratio of two gamma functions to be logarithmically completely monotonic, and some double inequalities are deduced for bounding divided differences of polygamma functions.

math.CA↗
arXiv2009

Sharp inequalities for polygamma functions

The main aim of this paper is to prove that the double inequality \frac{(k-1)!}{\Bigl\{x+\Bigl[\frac{(k-1)!}{|ψ^{(k)}(1)|}\Bigr]^{1/k}\Bigr\}^k} +\frac{k!}{x^{k+1}}<\bigl|ψ^{(k)}(x)\bigr|<\frac{(k-1)!}{\bigl(x+\frac12\bigr)^k}+\frac{k!}{x^{k+1}} holds for $x>0$ and $k\in\mathbb{N}$ and that the constants $\Bigl[\frac{(k-1)!}{|ψ^{(k)}(1)|}\Bigr]^{1/k}$ and $\frac12$ are the best possible. In passing, some related inequalities and (logarithmically) complete monotonicity results concerning the gamma, psi and polygamma functions are surveyed.

math.CA↗
arXiv2009

A class of completely monotonic functions involving divided differences of the psi and polygamma functions and some applications

A class of functions involving the divided differences of the psi function and the polygamma functions and originating from Kershaw's double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving ratio of two gamma functions and originating from establishment of the best upper and lower bounds in Kershaw's double inequality are derived, two sharp double inequalities involving ratios of double factorials are recovered, the probability integral or error function is estimated, a double inequality for ratio of the volumes of the unit balls in $\mathbb{R}^{n-1}$ and $\mathbb{R}^n$ respectively is deduced, and a symmetrical upper and lower bounds for the gamma function in terms of the psi function is generalized.

math.CA↗