All Questions

Let $$J_{f;y}(u):=3 u^3 \int_u^1\frac{dt}{t^4} \,e^{-i y t}f(t)- e^{-i u y}f(u),$$ where $y\in(0,\infty)$, $u\in(0,1)$, and $$f(t):=t+\pi (1-t) t \cot (\pi t).$$ Here are the graphs of $f$ (black), ...

Let $$h(u):=u^3 \left|\int_u^\infty \frac{e^{-i t}}{t^3} \, dt\right|$$ for $u>0$. Is the function $h$ concave on $(0,\infty)$? (For context, see Proposition 4.4.4 and formula (4.4.21) in this ...

Given $a>b>0$, is there any upper bound of the following ratio of hypergeometric function? $$\frac{_2F_1(a,1-b;a+1;x)}{_2F_1(a,1-b;a+1;y)}$$ for $1>x>y>0$ ideally in the form like some ...

I'm considering a ratio of incomplete Selberg integral: $$f_n(a,b)=\frac{\int_{\Delta_a}\prod_{i=1}^nx_i^{\alpha-\frac{n+1}{2}}\prod_{i=1}^n(1-x_i)^{-1/2}\prod_{i<j}|x_i-x_j|}{\int_{\Delta_b}\prod_{...

Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...