In some biological modeling work I am doing, I end up with an expression involving Kummer's confluent hypergeometric function ${}_1F_1(a;b;z)$. The expression is as follows:
$f(x,n) = \frac{1}{(n+2)} {}_1F_1\left(1;\frac{n}{2}+2;-x \right) - \left[ \frac{\Gamma\left(\frac{n}{2} +\frac{1}{2} \right)}{\sqrt{2} \Gamma\left(\frac{n}{2}+1\right)}{}_1F_1\left(\frac{1}{2};\frac{n}{2}+1;-x\right) \right]^2$
We have that $n$ is an integer with $n >1$, and $x>0$. Because of the intuition behind the system I'm studying, as well as simulations over a wide range of $n$ and $x$, I strongly suspect that $f(x,n)$ is negative for all allowed values of $n$ and $x$. However, I can't figure out a way to prove this.
I tried thinking of the relation between the hypergeometric functions of each term, but because the second term has a squared hypergeometric function, I'm not sure that will actually be useful.