Please help me to prove $$ \sum\limits_{j=2}^n \frac{1}{j^\alpha (j-1)^\alpha} \int\limits_{j-1}^j \frac{dx}{x^{1-\alpha}(n-x)^\alpha} \leq \int\limits_0^1 \frac{dx}{x^{1-\alpha}(n-x)^\alpha},\quad \alpha\in(0,1/2],\quad n\geq 2. $$ This inequality appears in the study of numerical methods for problems with fractional derivatives.

I tried to let $x=j(j-1)t$ in the integrals on the left in order to remove the coefficients $\frac{1}{j^\alpha (j-1)^\alpha}$. Then we get the equivalent inequality $$ \sum\limits_{j=2}^{n} \int\limits_{1/j}^{1/(j-1)} \frac{dt}{t^{1-\alpha}\bigl(n-j(j-1)t\bigr)^\alpha} \leq \int\limits_0^1 \frac{dx}{x^{1-\alpha}(n-x)^\alpha},\quad \alpha\in(0,1/2],\quad n\geq 2. $$ But I don't see how to prove this inequality, too.