1. complex number

If z is a non-zero complex number, prove that
|conjugate(z) + 1/z| >=2

2. Re: complex number

$\displaystyle |\bar{z}+1/z|\geq 2\Leftrightarrow |\bar{z}+1/z|^2\geq 4$ . Using $\displaystyle |w|^2=\bar{w}w$ we obtain

$\displaystyle |\bar{z}+1/z|^2=\ldots =|z|^2+2+1/|z|^2\geq \ldots \geq 4$ .

3. Re: complex number

Would you please derive more detail why modulus(z)^2 + 2 + 1/modulus(z)^2 >= 4?
Many thanks.

4. Re: complex number

Originally Posted by tinng
Would you please derive more detail why modulus(z)^2 + 2 + 1/modulus(z)^2 >= 4?
Many thanks.
What does the graph of $\displaystyle y = x^2 + 2 + \frac{1}{x^2}$ look like ....? What is its range ....? (Have you learned calculus and therefore know how to find coordinates of turning points?)

5. Re: complex number

Originally Posted by tinng
Would you please derive more detail why modulus(z)^2 + 2 + 1/modulus(z)^2 >= 4?
Another way, prove that $\displaystyle x^2+\dfrac{1}{x^2}\geq2\Leftrightarrow (x^2-1)^2\geq 0$ . The last inequality is trivially satisfied.