Let us take $s_1, s_2, t_1, t_2 >0$ such that $ s_1 < t_1 < s_2 < t_2 $. Let us also assume that $H \in (0 , 1)$ and define
$$ a_1 = t_2 - s_1, \quad a_2 = t_2 - t_1, \quad b_1 = s_2 - s_1, \quad b_2 = s_2 -t_1 . $$
We define a function $f$ as
$$ f(x) = x^{2H}, \ x \geq 0 . $$
and we study the sign of the expression
$$ \frac{1}{2} \bigg( f(a_1) - f(a_2) - \Big( f(b_1) - f(b_2) \Big) \bigg). $$
We would like to show that for $H \in (0, 0.5)$ the sign is negative and for $H \in (0.5, 1 )$ the sign is positive.
My approach is the following. In the expression above, we compare two differences of the function $f$ on two intervals $[b_2, \ b_1 ]$ and $[a_2, a_1 ]$ of the same length, which might intersect, but the second one is shifted to the right. Since $f$ is increasing for all $H$ and for $H \in (0, 0.5)$ $f$ is concave, the change on $[a_2, \ a_1]$ must be smaller that the change on $[b_1, \ b_2]$. Hence, the inequality. The picture below gives the idea.
However, I do not know how to put it together formally. I would appreciate any advice.