Assume we have two complex variables $h_i$ and $h_d$ which satisfy the following relationship
$$ 2\ |h_i|^2\leq \ |h_d|^2$$
can we say that $$\log\left( 1+ \frac{\big||h_d| - h_i\big|^2}{2}\right) \leq\log\left(1+(1+\frac{1}{\sqrt{2}})^2 \frac{|h_d|^2}{2}\right)? $$
Many thanks
The logarithm is an increasing function, so your inequality is equivalent to $$\big||h_d|-h_i\big|^2\le |h_d|^2(1+1/\sqrt2)^2$$ But $$\big||h_d|-h_i\big|\le |h_d|+|h_i|\le |h_d|(1+1/\sqrt2)$$ So the inequality follows.