7. Suppose that I : [0,1] - [a, b] is integrable on [0,1]. Assume that ef(x)

and I/(x)iP are integrable for all 0 < p < 00 (see Exercise 11, p. 406).

(

a) Prove that efol f(x) dx -::;.11 ef(x) dx and for all 0 < r -::;. 1.

(b) If 0 < p < q, prove that

(

1 ) liT 1 11/(xW dx -::;.1 1/(x)1 dx

(111/(X)iP dX) lip -::;. (11 I/(xW dX) 11q

(c) State and prove analogues of these results for improper integrals.