MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.