I have the following function
$$ -\int_A^{\infty} \left(mRr - mR^2/2\right){r^{p-1}} \exp\left( -\frac{\lambda r^2}{2}\right) dr $$ where $m, R, A, \lambda > 0$. I am trying to find an upper bound on it. There might be two ways to do it, first is by assuming it difference of moment function of chi-squared distribution, another one is by using the below inequality for incomplete gamma function- Let $B > 1$, and $q \geq \frac{p+1}{2}$. Then, for all $x \geq \frac{B}{B-1}(q-1)$. \begin{align*} \int_{x}^{+\infty}y^{q-1}\exp(-y)dy \leq Bx^{q-1}\exp(-x). \end{align*} The above function will help with the positive part but I don't know any good lower bound on the incomplete gamma function. I am not sure how to work around the differences, should I consider them separately?