arXiv subjects
Publications and source records attributed to Ti-Ren Huang.
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.
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 $n\geq 1$.
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 $x\in (0,1)$.