Finding coefficient in contraction mapping

Question

I got a task to find for which $l$ and $m$ this mappings would be contraction mappings ($C[a,b] \rightarrow C[a,b]$)

a) $f \rightarrow (lx+m)f$

b) $f \rightarrow \int_m^x f(t)dt$

I came to point where I don't undwrstand how to proceed.

($q<1, [a,b]=Y, d(f,g)=sup_Y|f-g|$)

a)$q*sup_Y|f-g| \geq sup_Y (|f(x)-g(x)|*|lx+m|)$

b)$q*sup_Y|f-g| \geq |l|*sup_Y (|f(x)-g(x)|*|x-m|)$

Answer 1

I show you part a) and hope that you can find the solution of b).

Let us denote the mapping in a) by $T$. Hence

$$T(f)(x)=(lx+m)f(x).$$

Let $c:= \max \{|x|: x \in [a,b]\}$. Then:

$|T(f)x)-T(g)(x)|= $

$|(lx+m)(f(x)-g(x))| \le |lx+m||f(x)-g(x)|$

$\le (|l||x|+|m|)d(f,g) \le (|l|c+|m|)d(f,g) $.

Put $q=|l|c+|m|$, then we have

$$d(T(f),T(g)) \le q d(f,g).$$

Conclusion ?