Let $A>B>0$ I think the following inequality holds true but I can't manage to prove it
$$H(A,B):=e^{-A\frac{\log\left(\frac{A}{B}\right)}{A+B}-\frac{AB}{2}}+e^{B\frac{\log\left(\frac{A}{B}\right)}{A+B}-\frac{AB}{2}}-2<0.$$
I've tried to define $y = \frac{A}{B} >1$ and obtain $$e^{-\frac{\log\left(y\right)y}{1+y}-y\frac{B^{2}}{2}}+e^{\frac{\log\left(y\right)}{1+y}-y\frac{B^{2}}{2}}-2$$ without any success