i have exams on this course coming up in two days (i should probably write this post earlier). i was hoping to read more and try to find a solution myself but unfortunately failed.

let f be a positive function, integrable over R. prove that

and deduce that converges

prove that

this is based on the assumption that for all x in 0..pi/2 ( i have done this part)

also one mroe if p >-1 and q belongs to the natural number. prove that

(i have done this part)

deduce that

any thoughts would be appreciated! thank you!