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Feng Qi

Publications and source records attributed to Feng Qi.

At least 91 records · Page 5Linked to original sources
arXiv2014

Bounds for the combination of Toader mean and the arithmetic mean in terms of the contraharmonic mean

In the paper, the authors find the greatest value $λ$ and the least value $μ$ such that the double inequality \begin{multline*} C(λa+(1-λ)b,λb+(1-λ)a)<αA(a,b)+(1-α)T(a,b)\\ < C(μa+(1-μ)b,μb+(1-μ)a) \end{multline*} holds for all $α\in(0,1)$ and $a,b>0$ with $a\ne b$, where $$ C(a,b)=\frac{a^{2}+b^{2}}{a+b},\quad A(a,b)=\frac{a+b}2, $$ and $$ T(a,b)=\frac{2}π\int_{0}^{π/{2}}\sqrt{a^2\cos^2θ+b^2\sin^2θ}\,dθ$$ denote respectively the contraharmonic, arithmetic, and Toader means of two positive numbers $a$ and $b$.

math.CA↗
arXiv2013

Some best approximation formulas and inequalities for Wallis ratio

In the paper, the authors establish some best approximation formulas and inequalities for Wallis ratio. These formulas and inequalities improve an approximation formula and a double inequality for Wallis ratio recently presented in ``S. Guo, J.-G. Xu, and F. Qi, \textit{Some exact constants for the approximation of the quantity in the Wallis' formula}, J. Inequal. Appl. 2013, \textbf{2013}:67, 7 pages; Available online at \url{http://dx.doi.org/10.1186/1029-242X-2013-67}''.

math.CA↗
arXiv2013

A unified proof of inequalities and some new inequalities involving Neuman-Sándor mean

In the paper, by finding linear relations of differences between some means, the authors supply a unified proof of some double inequalities for bounding Neuman-Sándor means in terms of the arithmetic, harmonic, and contra-harmonic means and discover some new sharp inequalities involving Neuman-Sándor, contra-harmonic, root-square, and other means of two positive real numbers.

math.CA↗
arXiv2013

Extended fermionic $p$-adic integrals on $\mathbb{Z}_p$

In the paper, using the extended fermionic $p$-adic integral on $\mathbb{Z}_p$, the authors find some applications of the umbral calculus. From these applications, the authors derive some identities on the weighted Euler numbers and polynomials. In other words, the authors investigate systematically the class of Sheffer sequences in connection with the generating function of the weighted Euler polynomials.

math.NT↗
arXiv2013

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

In the paper the author provides necessary and sufficient conditions on $a$ for the function $\frac{1}{2}\ln(2π)-x+\bigl(x-\frac{1}{2}\bigr)\ln x-\lnΓ(x)+\frac1{12}{ψ'(x+a)}$ and its negative to be completely monotonic on $(0,\infty)$, where $a\ge0$ is a real number, $Γ(x)$ is the classical gamma function, and $ψ(x)=\frac{Γ'(x)}{Γ(x)}$ is the di-gamma function. As applications, some known results and new inequalities are derived.

math.CA↗
arXiv2013

The best bounds for Toader mean in terms of the centroidal and arithmetic means

In the paper, the authors discover the best constants $α_{1}$, $α_{2}$, $β_{1}$, and $β_{2}$ for the double inequalities $$ α_{1}\bar{C}(a,b)+(1-α_{1}) A(a,b)< T(a,b) <β_{1} \bar{C}(a,b)+(1-β_{1})A(a,b) $$ and $$ \frac{α_{2}}{A(a,b)}+\frac{1-α_{2}}{\bar{C}(a,b)}<\frac1{T(a,b)} <\frac{β_{2}}{A(a,b)}+\frac{1-β_{2}}{\bar{C}(a,b)} $$ to be valid for all $a,b>0$ with $a\ne b$, where $$ \bar{C}(a,b)=\frac{2(a^{2}+ab+b^{2})}{3(a+b)},\quad A(a,b)=\frac{a+b}2, $$ and $$ T(a,b)=\frac{2}π\int_{0}^{π/{2}}\sqrt{a^2{\cos^2θ}+b^2{\sin^2θ}}\,\tdθ$$ are respectively the centroidal, arithmetic, and Toader means of two positive numbers $a$ and $b$. As an application of the above inequalities, the authors also find some new bounds for the complete elliptic integral of the second kind.

math.CA↗