arXiv subjects
Publications and source records attributed to Feng Qi.
In the paper, the authors concisely survey and review some functions involving the gamma function and its various ratios, simply state their logarithmically complete monotonicity and related results, and find necessary and sufficient conditions for a new function involving the ratio of two gamma functions and originating from the coding gain to be logarithmically completely monotonic.
In the paper, the author presents diagonal recurrence relations for the Stirling numbers of the first kind. As by-products, the author also recovers three explicit formulas for special values of the Bell polynomials of the second kind.
In the paper, the authors establish three kinds of double inequalities for the trigamma function in terms of the exponential function to powers of the digamma function. These newly established inequalities extend some known results. The method in the paper utilizes some facts from the asymptotic theory and is a natural way to solve problems for approximating some quantities for large values of the variable.
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.
In the paper, the author derives several "diagonal" recurrence relations, constructs some inequalities, finds monotonicity, and poses a conjecture related to Stirling numbers of the second kind.
By using Cauchy integral formula in the theory of complex functions, the authors establish some integral representations for the principal branches of several complex functions involving the logarithmic function, find some properties, such as being operator monotone function, being complete Bernstein function, and being Stieltjes function, for these functions, and verify a conjecture on complete monotonicity of a function involving the logarithmic function.
In the paper, the authors review some explicit formulas and establish a new explicit formula for Bernoulli and Genocchi numbers in terms of Stirling numbers of the second kind.
In the paper, the authors introduce a new concept "extended $s$-convex functions", establish some new integral inequalities of Hermite-Hadamard type for this kind of functions, and apply these inequalities to derive some inequalities of special means.
In the paper, the author establishes inequalities, monotonicity, convexity, and unimodality for functions concerning the modified Bessel functions of the first kind and compute the completely monotonic degrees of differences between the exponential and trigamma functions.
Let $ψ(x)$ be the di-gamma function, the logarithmic derivative of the classical Euler's gamma function $Γ(x)$. In the paper, the author shows that the completely monotonic degree of the function $[ψ'(x)]^2+ψ''(x)$ is $4$, surveys the history and motivation of the topic, supplies a proof for the claim that a function $f(x)$ is strongly completely monotonic if and only if the function $xf(x)$ is completely monotonic, conjectures the completely monotonic degree of a function involving $[ψ'(x)]^2+ψ''(x)$, presents the logarithmic concavity and monotonicity of an elementary function, and poses an open problem on convolution of logarithmically concave functions.
In the paper, the authors provide four alternative proofs of an explicit formula for computing Bernoulli numbers in terms of Stirling numbers of the second kind.
In the paper, the author elementarily unifies and generalizes eight identities involving the functions $\frac{\pm1}{e^{\pm t}-1}$ and their derivatives. By one of these identities, the author establishes two explicit formulae for computing Euler polynomials and two-parameter Euler polynomials, which are a newly introduced notion, in terms of Stirling numbers of the second kind.
In the paper, the author establishes an integral representation for Cauchy numbers of the second kind, finds the complete monotonicity, minimality, and logarithmic convexity of Cauchy numbers of the second kind, and presents some inequalities for determinants and products of Cauchy numbers of the second kind.
Let $ψ_q(x)$, $ψ_q'(x)$, and $ψ_q''(x)$ for $q>0$ stand respectively for the $q$-digamma, $q$-trigamma, and $q$-tetragamma functions. In the paper, the author proves along two different approaches that the functions $[ψ'_q(x)]^2+ψ''_q(x)$ for $q>1$ and $[ψ_{q}'(x)-\ln q]^2 +ψ''_{q}(x)$ for $0<q<1$ are completely monotonic on $(0,\infty)$. Applying these results, the author derives monotonic properties of four functions involving the $q$-digamma function $ψ_q(x)$ and two double inequalities for bounding the $q$-digamma function $ψ_q(x)$.
In the paper, the authors first inductively establish explicit formulas for derivatives of the arc sine function, then derive from these explicit formulas explicit expressions for a family of Bell polynomials related to the square function, and finally apply these explicit expressions to find explicit formulas for derivatives of some elementary functions.
In the paper, the authors find some integral representations and discover some properties of Lah numbers.
In the paper, the authors establish integral representations of some functions related to the remainder of Burnside's formula for the gamma function and find the (logarithmically) complete monotonicity of these and related functions. These results extend and generalize some known conclusions.
In the paper, the authors find the best numbers $α$ and $β$ such that $$ \overline{C}\bigl(αa+(1-α)b,αb+(1-α)a\bigr) 0$ with $a\ne b$, where $\overline{C}(a,b)={2\bigl(a^2+ab+b^2\bigr)}{3(a+b)}$ and $T(a,b)=\frac{2}π\int_{0}^{π/{2}}\sqrt{a^2{\cos^2θ}+b^2{\sin^2θ}}\,dθ$ denote respectively the centroidal mean and Toader mean of two positive numbers $a$ and $b$.