arXiv subjects
Publications and source records attributed to Feng Qi.
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.
In this article, a necessary and sufficient condition and a necessary condition are established for a function involving the gamma function to be logarithmically completely monotonic on $(0,\infty)$. As applications of the necessary and sufficient condition, some inequalities for bounding the psi and polygamma functions and the ratio of two gamma functions are derived.
In this expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some inequalities, several complete monotonicity of functions involving ratios of two gamma or $q$-gamma functions, and necessary and sufficient conditions for functions involving ratios of two gamma or $q$-gamma functions to be logarithmically completely monotonic.
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 $λ\neμ$.
In this expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some inequalities, the complete monotonicity of several functions involving ratios of two gamma or $q$-gamma functions, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions, some new bounds for the ratio of two gamma functions and divided differences of polygamma functions, and related monotonicity results.
In this paper, by a concise and elementary approach, we sharpen and generalize Shafer's inequality for the arc sine function, and some known results are extended and generalized.
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.
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.
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.
In the paper, some lower bounds for polygamma functions are refined.
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.
In this note, we present monotonicity results of a function involving to the inverse hyperbolic sine. From these, we derive some inequalities for bounding the inverse hyperbolic sine.
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.
In the paper, the monotonicity and logarithmic convexity of Gini means and related functions are investigated.
In the present note, an alternative proof is supplied for Theorem~1 in [N. Elezović, C. Giordano and J. Pečarić, \textit{The best bounds in Gautschi's inequality}, Math. Inequal. Appl. \textbf{3} (2000), 239\nobreakdash--252.].
In this paper, some monotonicity and concavity results of several functions involving the psi and polygamma functions are proved, and then some known inequalities are extended and generalized.
In the note, the functions $\abs{ψ^{(i)}(e^x)}$ for $i\in\mathbb{N}$ are proved to be sub-additive on $(\lnθ_i,\infty)$ and super-additive on $(-\infty,\lnθ_i)$, where $θ_i\in(0,1)$ is the unique root of equation $2\abs{ψ^{(i)}(θ)}=\abs{ψ^{(i)}(θ^2)}$.
In this paper, we sharpen and generalize Shafer's inequality for the arc tangent function. From this, some known results are refined.