In an article I was reading, it is written:
It is well-know that for any two matrices X and Y of appropriate sizes we have
$$|tr\, X^\*Y| \leq \frac{tr \, X^\*X + tr \, Y^\*Y}{2}$$

I didn’t find it obvious and I couldn’t find a proof. 
I’ve tried to it it terms of summation but did work. How can I prove it? Or where can I find a proof?