264
III - Convergence: Continuous variables
f;(x) = Re[zJ'(x)),
whence If~(x)1 :::; Izl'(x) I :::; Mizi where M = 111' 111. Since fz has real values
Theorem 18 shows that
IRe(zu)1 = Ifz(b) - fz(a)1 :::; M(b - a)lzl
for any z E C. Thus lui:::; M(b - a). To sum up:
Corollary 3. Let f be a complex-valued function defined and differentiable
on an interval I. Suppose that I' is bounded on all compact subsets K c I.
Then
(16.7)
If(b) - f(a)1 :::; Ilf'IIK(b - a)
for all points a, bEl, where K = [a, b].
Still with the hypotheses of Corollary 3, let us choose an arbitrary u E C
and apply (7) to the function g(x) = f(x) - ux. We find:
Corollary 4. Under the hypotheses of Corollary 3 we have
(16.8)
If(b) - f(a) - u(b - a)1 :::; (b - a) sup 1J'(x) - ul
a::;x::;b
for all a, bEl and u E C.
For example we can choose u = I'(a), b = a + h, whence
(16.9)
If(a + h) - f(a) - J'(a)hl :::; Ihl· sup 1J'(a + th) - f'(a)l,
O::;t9
which, if I' is continuous at a, expresses more precisely the o( h) which would
appear on the right hand side than if one assumed only the existence of f' (a).
The inequality of the mean is valid for a much larger class of functions:
it is enough to assume that f is continuous and that the set of x E I where
I'(x) does not exist is countable. Since the problem is directly linked to the
construction of a "primitive" f of a regulated function g, Le. of a function such
that one has, according to taste, either f'(x) = g(x) "apart from exceptions",
or
f(x) - f(a) = l
x g(t)dt,
it will be better to delay these useless subtleties to Chap. V.