how to find upper bound of the absolute value of the line integral of z^2

Question : Find upper bound of the absolute value of the line integral of z^2 along a straight line segment C from 0 to 1 + i.

Solution :

To find upper bound of \[\left|\int_{C} z^2dz\ \ \ \right|\] Where C is along a straight line segment from 0 to 1 + i Cauchy’s inequality states that \[\int_{C}{f(z)}dz\ \ \ \le\ ML\] where L is the length of C and M is a constant such that |f(z)| ≤ M for z ∈ C Since length of line segment C from 0 to 1 + i is \[ C\ =\ L\ =\sqrt{1+1}\ =\ \sqrt2\] And \[|f(z)|\ =\ |z^2\ |\ \le\ 2\] for all z = 0 to z =1+i Substituting into Cauchy’s inequality \[\left|\int_{C} z^2dz\ \ \ \right|\ \le\ 2\sqrt2 \] Is the required value of the upper bound