arXiv subjects

Song-Liang Qiu

Publications and source records attributed to Song-Liang Qiu.

3 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↗