Complex Analysis: Theory and Applications · Problem 3.1.13

Analytic and Harmonic Functions — Howell & Matthews

Mathematics — Complex Analysis · mean-value-theorem-complex

Problem

Consider the differentiable function

f(z) = z^3

and the two points

z_1 = 1

and

z_2 = i

. Show that there does not exist a point

c

on the line

y = 1-x

between

1

and

i

such that

\dfrac{f(z_2)-f(z_1)}{z_2-z_1} = f'(c)

. This result shows that the mean value theorem for derivatives does not extend to complex functions.