How do I upperbound this expression?

Question

With a given condition such as

$$|x|^2 > |y|^2$$

Is there any way I can upper bound the following expression

$$\log\left(1+\big||y|-x\big|^2\right) \leq \,\,\, ? $$

Thank you

Answer 1

You always have $\ln (1+x)\le x$, so $$\ln\left(1+\big||y|-x\big|^2\right) \le \big||y|-x\big|^2=(x-|y|)^2.$$

Answer 2

You can find the upper limit for the absolute value of y, this happens to be the absolute value of x.

Sub this into your expression and it is easy to find the upper bound.