Uniform Convergence of Poisson Kernel on [-π, π] minus (-a, a)

[hedipaldi]

Homework Statement

show that the integral of the poisson kernel (1-r^2)/(1-2rcos(x)+r^2) converges to 0 uniformly in x as r tend to 1 from the left ,on any closed subinterval of [-pi,pi] obtained by deleting a middle open interval (-a,a)

Homework Equations

the integral of poisson kernel from -pi to pi is 2pi

3
. The attempt at a solution:it must by a direct conclusion of the integral in 2 but somehow i fail to see it.

 

[matematikuvol]

Is $r$ always positive? Is $P(-r,x)$ defined?

 

[hedipaldi]

main guidelines for the proof in the attachment.

 

[hedipaldi]

I am hedipaldi but i did not pause this question.How come it is on my name??