Suppose that $a_1<1$, $a_1+a_2+a_3>1.$ For $x,y,z>0,$

(1) define a fucntion $$H(x,y,z)=\frac{x^{\frac{1}{2}}\int_0^{\infty}\frac{1}{t^{a_1}~ (1+t)^{a_2+1}~ (1+t+z)^{a_3}}\exp\big\{-\frac{x}{1+t}-\frac{ y}{1+t+z}\big\}dt }{\int_0^{\infty}\frac{1}{t^{a_1}~ (1+t)^{a_2}~(1+t+z)^{a_3}}\exp\big\{-\frac{x}{1+t}-\frac{ y}{1+t+z}\big\}dt}.$$ Then $H(x,y,z)$ is uniformly bounded over $x,y$, i.e. there is a constant C, such that $H(x,y)\le C.$

(2) Furthermore, define $$L(y,z)=\frac{y^{\frac{1}{2}}\int_0^{\infty}\frac{1}{t^{a_1}~ (1+t)^{a_2}~ (1+t+z)^{a_3+1}}\exp\big\{-\frac{ y}{1+t+z}\big\}dt }{\int_0^{\infty}\frac{1}{t^{a_1}~ (1+t)^{a_2}~(1+t+z)^{a_3}}\exp\big\{-\frac{y}{1+t+z}\big\}dt}.$$

Then $L(x,y)$ is also uniformly bounded over $x,y.$