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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.

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1) Cf. inequalities in the paper: D. Karp, S.M. Sitnik, Log-convexity and log-concavity of hypergeometric-like functions, Journal of Mathematical Analysis and Applications, Volume 364, Issue 2, P. 384-394.

There is an inequality cited in it due to Barnard, Gordy and Richards of so called Turan type: $$ _{1}F_{1}(a+b,c,x)_{1}F_{1}(a-b,c,x)\le \left(_{1}F_{1}(a,c,x)\right)^2, $$ which formally gives the inequality in need for $a=0$, as $_{1}F_{1}(0,c,x)=1$.

Unfortunately the parameters in it are out of needed range, but may be it will be all the same somehow useful???

2) Also it seems that the function $$ f(x)={}_1F_1(-a;1+a;-x) _1F_1(a,1+a,-x) $$ is decreasing for $x\ge 0, a\ge 0 $ and so $f(x)\le f(0)=1$ but it is another inequality with four ${}_1F_1$ functions to prove.

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