Alzer (2000) obtained the following new sharp upper bounds for Bernoulli numbers \begin{equation} |B_{2n}|\leq\frac{2(2n)!}{(2\pi)^{2n}} \cdot \frac{1}{1-2^{\beta-2n}}, n\geq1\qquad \end{equation} where $\beta=2+{\frac{\log(1-\frac{6}{\pi^2})}{\log2}}=0.6491\cdots.$ Now we compute $\mu$ as follows to obtain new data for our research $$\mu=\frac{1}{2-2^{\beta-1}}=0.833 \cdots.$$ And we define function $f$ as below $$f(x,y)=(\frac{\mu-1}{x}-\frac{1}{\pi})e^{xy}+\frac{1}{\pi-x}(e^{\pi y}-1)-\frac{\mu}{\pi-x}\sinh(\pi x)+\frac{1-\mu}{x},\\ (x,y) \in (0,\pi )\times(0,+\infty).$$ With recent investigations we observed that $f$ has a root less than 0.2. Now I have these questions:
1.Does $f$ have at most one root in its domain?
2.Is it true that $x_0 \in R$ exists such that $0< x_0<\frac{1}{2}$ and $f(x,y)>0$ holds?
3.Does $f_y(x,y)$ have any root on its domain?
Appreciate any suggestion or solution.