Upperbound this difference between two log expressions I have the difference between the following log expressions and I am trying to bound the difference,
$$F=
  \log \left(1+ \left(2+\frac{1}{\sqrt{2}}\right)^2 x^2\right) -
  \log \left(1+ \left(1-\frac{1}{\sqrt{2}}\right)^2 x^2\right) $$
Can I say that 
$$F \leq \log\left(1+ \frac{ \left(2+\frac{1}{\sqrt{2}}\right)^2}{\left(1-\frac{1}{\sqrt{2}}\right)^2}\right)$$
Is this the tightest bound that exists?
Thanks in advance.

$f(a) =\log (1+ax^2) - \ln a\Rightarrow f'(a) = \dfrac{x^2}{1+ax^2}- \dfrac{1}{a}< 0\Rightarrow \text{ since } b=\left(2+\dfrac{1}{\sqrt{2}}\right)^2 > \left(1-\dfrac{1}{\sqrt{2}}\right)^2=a\Rightarrow f(a) < f(b)\text{ and} \ln\left(\dfrac{b}{a}\right) \leq \dfrac{b}{a}-1 \Rightarrow \text{the inequality holds and the tighter bound is} \dfrac{b}{a}-1$
