Bounds for an integral of the modified Bessel function of the first kind and expressions involving it

Simple upper and lower bounds are obtained for the integral ∫0xe−γttνIν(t)dt, x>0, ν>−12, 0<γ<1. Most of our this tight as x→∞. We apply one inequalities to bound some expressions involving integral. Two these appear in Stein's method variance-gamma approximation, will allow a technical advancement be made method.