arXiv subjects

Xiao-Yan Ma

Publications and source records attributed to Xiao-Yan Ma.

6 recordsLinked to original sources

arXiv2024

Turán-Type Inequalities for Gaussian Hypergeometric Functions, and Baricz's Conjecture

In 2007, Á. Baricz put forward a conjecture concerning Turán-type inequalities for Gaussian hypergeometric functions (see Conjecture \ref{ConjA} in Section \ref{Sec1}). In this paper, the authors disprove this conjecture with several methods, and present Turán-type double inequalities for Gaussian hypergeometric functions, and sharp bounds for complete and generalized elliptic integrals of the first kind.

math.CA↗

arXiv2021

On a Conjecture Concerning the Approximates of Complete Elliptic Integral of the First Kind by Inverse Hyperbolic Tangent

Let $\K$ be the complete elliptic integral of the first kind. In this paper, the authors prove that the function $r\mapsto r^{-2}\{[\log(2\K(r)/π)]/\log((\arth r)/r)-3/4\}$ is strictly increasing from $(0,1)$ onto $(1/320,1/4)$, so that $[(\arth r)/r]^{3/4+r^2/320}<2\K(r)/π<[(\arth r)/r]^{3/4+r^2/4}$ for $r\in(0,1)$, in which all the coefficients of the exponents of the two bounds are best possible, thus proving a conjecture raised by Alzer and Qiu to be true, and giving better bounds of $\K(r)$ than those they conjectured and put in an open problem. Some other analytic properties of the complete elliptic integrals, including other kind of approximates for $\K(r)$, are obtained, too.

math.CA↗

arXiv2018

Sharp Approximations for the Ramanujan Constant

In this paper, the authors present sharp approximations in terms of sine function and polynomials for the so-called Ramanujan constant (or the Ramanujan $R$-function) $R(a)$, by showing some monotonicity, concavity and convexity properties of certain combinations defined in terms of $R(a)$, $\sin(πa)$ and polynomials. Some properties of the Riemann zeta function and its related special sums are presented, too.

math.CV↗

arXiv2017

Sharp Estimates of the Generalized Euler-Mascheroni Constant

Let $a\in (0, \infty)$, $γ(a)$ be the Generalized Euler-Mascheroni Constant, and let \begin{align*} &x_n=\frac1a+\frac{1}{a+1}+\cdots+\frac{1}{a+n-1}-\ln\frac{a+n}{a},\\ &y_n=\frac1a+\frac{1}{a+1}+\cdots+\frac{1}{a+n-1}-\ln\frac{a+n-1}{a}. \end{align*} In this paper, we determine the best possible constants $α_i, β_i (i=1,2,3,4)$ such that the following inequalities \begin{align*} \frac{1}{2(n+a)-α_1}\leq &γ(a)-x_n< \frac{1}{2(n+a)-β_1},\\ \frac{1}{2(n+a)-α_2}\leq &y_n-γ(a)< \frac{1}{2(n+a)-β_2},\\ \frac{1}{2(n+a)}+\frac{α_3}{(n+a)^2}\leq &γ(a)-x_n<\frac{1}{2(n+a)}+\frac{β_3}{(n+a)^2},\\ \frac{1}{2(n+a-1)}+\frac{α_4}{(n+a-1)^2}< &y_n-γ(a)\leq\frac{1}{2(n+a-1)}+\frac{β_4}{(n+a-1)^2}. \end{align*} are valid for all integers $n\geq 1$.

math.FA↗

arXiv2016

Inequalities for Zero-Balanced Gaussian hypergeometric function

In this paper, we consider the monotonicity of certain combinations of the Gaussian hypergeometric functions $F(a-1,b;a+b;1-x^c)$ and $F(a-1-δ,b+δ;a+b;1-x^d)$ on $(0,1)$ for $δ\in(a-1,0)$, and study the problem of comparing these two functions, thus get the largest value $δ_1=δ_1(a,c,d)$ such that the inequality $F(a-1,b;a+b;1-x^c)<F(a-1-δ,b+δ;a+b;1-x^d)$ holds for all $x\in (0,1)$.

math.CA↗

arXiv2012

Sharp two parameter bounds for logarithmic and arithmetic-geometric means

For fixed $s\geq 1$ and $t_{1},t_{2}\in(0,1/2)$ we prove that the inequalities $G^{s}(t_{1}a+(1-t_{1})b,t_{1}b+(1-t_{1})a)A^{1-s}(a,b)>AG(a,b)$ and $G^{s}(t_{2}a+(1-t_{2})b,t_{2}b+(1-t_{2})a)A^{1-s}(a,b)>L(a,b)$ hold for all $a,b>0$ with $a\neq b$ if and only if $t_{1}\geq 1/2-\sqrt{2s}/(4s)$ and $t_{2}\geq 1/2-\sqrt{6s}/(6s)$. Here $G(a,b)$, $L(a,b)$, $AG(a,b)$ and $A(a,b)$ are the geometric, logarithmic, arithmetic-geometric and arithmetic means of $a$ and $b$, respectively.

math.CA↗