I have a explicit integral of the form, $$ \int_s^t((r-s)^{-a}+(r-s)^{b-a})(t-r)^{a + b -1} d r $$,
Where $0 \leq s < t \leq T$, $b \in (\frac{1}{2},1)$ and $1-b < a <b $.
I have to Show that this integral is smaller or equal than $(t-s)^{C_1 \times b } + (t-s)^{C_2 \times b }$, where $C_1$ and $C_2$ are some real numbers. Someone sad that i have to substitute and use the beta-function, but i have no idea how. Can anybody help me?
We want to arrive at something like the beta function $B$ defined as :
$$ B(x,y)=\int_0^1 t^{x-1}(1-t)^{y-1} \operatorname d t $$ With substitution $x=\frac{r-s}{t-s}$ we get $$ x=\frac{r-s}{t-s} \iff r = x(t-s)+s \iff \operatorname d r = (t-s) \operatorname d x $$ Hence $$ \begin{split} I&=\int_s ^ t (r-s)^{-a}\left(1+ (r-s)^{-b} \right)(t-r)^{a+b-1} \operatorname d r \\ &=(t-s)^{b} \left[ \int_0^1 x^{-a} (1-x)^{a+b-1}\operatorname d x + \int_0^1 x^{-a-b} (t-s)^{-b} (1-x)^{a+b-1}\operatorname d x \right]\\ &=(t-s)^b \int_0^1 x^{-a} (1-x)^{a+b-1}\operatorname d x + \int_0^1 x^{-(a+b)}(1-x)^{a+b-1} \operatorname d x \\ &= (t-s)^{b} \cdot B(1-a,a+b) + B(1-a-b,a+b) \end{split} $$
Can you proceed from here?