How to show that z/(z^2+1) is less than or equal to 1/2 when dom(z) is the line segment from 2 to 2+i? (ML formula for integration)

Question

I am having a problem solving this one :-

Use the ML formula to show that |integral over C (z/(z^2+1))dz| is less than or equal to 1/2, where C is the straight line segment from 2 to 2+i. (I am struggling to use codes to show the question in proper form here, i am sorry.)

I have calculated L which should be the distance from 2 to 2+i and it is 1 unit but I can not show that the modulus of the function f(z)= z/(z^2+1) is less than or equal to 1/2 (I am trying to find M so that M*L becomes 1/2). Can you please help me? Thank you.

My calculations are showing that |z| needs to be 1 for |z/(z^2+1)| to be less than 1/2. 

Answer 1

Hint: Take $0 \leq t \leq 1$ and show that $|\frac {2+it} {((2+it)^{2}+1}| \leq \frac 1 2$ or $4(4+t^{2})\leq ((5-t^{2})^{2}+16t^{2})$. My calculations show that this inequality is correct.