Is $\limsup_{a\rightarrow0^+}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{1}{1-|-a+(1-a)e^{it}|}\operatorname{d}t<\infty$?

Question

Is $\limsup_{a\rightarrow0^+}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{1}{1-|-a+(1-a)e^{it}|}\operatorname{d}t<\infty$?

I've found this integral doing estimatation on the integral over circles tangent to the unit disk of the Poisson's kernel on the disk, but I can't show if this estimate is useful (if the statement is true) or useless (if the statement is false).

Answer 1

If $-\pi/2\le t\le\pi/2$ and $0<a<1$, then \begin{align} \bigl|-a+(1-a)\,e^{it}\bigr|^2&=(-a+(1-a)\cos t)^2+(1-a)^2\sin^2t\\ &=(1-a)^2+a^2-2\,a(1-a)\cos t\\ &\ge(1-a)^2+a^2-2\,a(1-a)\\ &=(1-2\,a)^2. \end{align} Thus $$ \frac{1}{1-\bigl|-a+(1-a)\,e^{it}\bigr|}\ge\frac{1}{2\,a}. $$