arXiv subjects
Publications and source records attributed to Feng Qi.
In the paper the authors alternatively prove that the function $x^α\big[\ln\frac{px}{x+p+1}-ψ_p(x)\big]$ is completely monotonic on $(0,\infty)$ if and only if $α\le 1$, where $p\in\mathbb{N}$ and $ψ_p(x)$ is the $p$-analogue of the classical psi function $ψ(x)$. This generalizes a known result.
In this paper, some new integral inequalities on time scales are presented by using elementarily analytic methods in calculus of time scales.
In the article we 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 $Γ(x)$, the $n$-th harmonic number $\sum_{k=1}^n\frac1k$, and the factorial $n!$.
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.
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.].
In the paper, we establish an inequality involving the gamma and digamma functions and use it to prove the negativity and monotonicity of a function involving the gamma and digamma functions.
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.
In the note, two alternative proofs are provided for a monotonicity result that the function $ψ(x)+\ln\bigl(e^{1/x}-1\bigr)$ is strictly increasing on $(0,\infty)$, where $ψ(x)$ is the psi function.
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.
In the paper, we extend Binet's first formula for the logarithm of the gamma function and investigate some properties, including inequalities, star-shaped and sub-additive properties and the complete monotonicity, of the extended remainder of Binet's first formula for the logarithm of the gamma function and related functions.
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.
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 $(0,\infty)$.
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 $(0,\infty)$.
In present paper, we prove the monotonicity of two functions involving the gamma function $Γ(x)$ and relating to the $n$-dimensional volume of the unit ball $\mathbb{B}^n$ in $\mathbb{R}^n$.
In this paper, we sharpen and generalize Shafer-Fink's double inequality for the arc sine function.
In the present paper, we first prove the logarithmic convexity of the elementary function $\frac{b^x-a^x}x$, where $x\ne0$ and $b>a>0$. Basing on this, we then provide a simple proof for Schur-convex properties of the extended mean values, and, finally, discover some convexity related to the extended mean values.
In this paper, we sharpen and generalize Carlson's double inequality for the arc cosine function.
In the article, the complete elliptic integrals of the first and second kinds are bounded by using the power series expansions of some functions, the celebrated Wallis' inequality, and an integral inequality due to R. P. Agarwal, P. Cerone, S. S. Dragomir and F. Qi.