upper bound of contour integral

[runnnincircles]

I am asked to show that |∫ Ce z - z dz| ≤ 60
where C is the positively oriented triangle whose vertices are 0, 3i, -4.

I know that I need to find an upper bound M for e z - z
and that the upper bound for the integral will then be 12M since 12 is the length of C. I am completely lost on how to find M.

Comments

[the_s3ntinel]

Remember that because exp(z) is holomorphic, the integral around C of exp(z) is zero. So you just have to consider the integral around C of z-bar.

thank you

[runnnincircles]

okay, I see that now, but then wouldnt the upper bound for |z-bar| be 4, which would make the upper bound for the integral 48?

Re: thank you

[phlebas]

48<=60, so you'd be fine there. Is the holomorphic function result something you can assume at this point?

Re: thank you

[the_s3ntinel]

The answer must be 'no'.

To the OP: You need to use |x - y| ≤ |x| + |y|, and |exp(z)| = exp(Re(z)).