I believe the following bound to be correct:

${}_1F_1(-a;1+a;-z) a z^{-a} \gamma(a,z) \leq 1$

for real-valued a > 0 and z $\geq 0$. $\gamma(a,z)$ is the lower incomplete gamma function.

Apart from special cases for $a$ (e.g. $a = 1$), I have not been able to find a general proof.

Proving that the derivative of the lhs is always negative is just as hard, if not harder. I have also tried to find a tight upper bound (in terms of a function that behaves similarly) of the lhs, such that both coincide at $z = 0$ (where the lhs equals 1), but to no avail.

Has anyone got any ideas, or references that I could look into? I have found a number of papers with bounds for general hypergeometric functions, but none have helped (so far).

Note: the inequality can be rewritten in terms of hypergeometric functions only as:

${}_1F_1(-a;1+a;-z) _1F_1(a,1+a,-z) \leq 1$

This may help. The bound might actually hold for any (positive) value of the second argument, but I have not tested this thoroughly. Thanks.