is this bound on a kummer function known?

Question

Is it already known that ${}_1F_1(a;b;x) \leq \Gamma(b)(1+|x|)^{-a}$ when $a$ is an integer, $a <0,$ and $b>0?$ If it is, what is a reference?

my proof: Since the Kummer function can be written in terms of a generalized Laguerre polynomial, \begin{equation} \label{eqn:Kummer-Laguerre-equality} {}_1F_1(a;b;x) = \frac{\Gamma(1-a)\Gamma(b)}{\Gamma(b-a)} L_{-a}^{(b-1)}(x), \end{equation} when $a < 0,$ we proceed by bounding the generalized Laguerre polynomial on the right hand side.

Let $n = -a$ and $\alpha = b - 1.$ Then $$ L_{n}^{(\alpha)}(x) = \sum_{\ell=0}^n \frac{\Gamma(\alpha + n + 1)}{\Gamma(\alpha + \ell +1)(n-\ell)!\ell!} (-x)^\ell. $$ Our constraints on $a$ and $b$ ensure that each $\Gamma(\cdot)$ term in the above sum is positive. Furthermore, for $\ell=0,1,\ldots,n,$ $$ \Gamma(\alpha + \ell +1) \geq \Gamma(b) \geq \min_{x > 0} \Gamma(x) > 0.88. $$

It follows that $$ L_{n}^{(\alpha)}(x) \leq 1.14 \cdot \Gamma(\alpha + n + 1) \sum_{\ell =0}^n \frac{|x|^\ell}{(n-\ell)!\ell!} = 1.14 \cdot \Gamma(b-a) \frac{1}{(-a)!}(1 + |x|)^{-a}. $$ The last equality is a consequence of the binomial theorem.

The conclusion follows immediately when this estimate is used in the relation expressing ${}_1F_1$ in terms of the Laguerre polynomial.

Answer 1

Your alleged inequality is false. Here is a counter example.

Let $a=-0.5$, $b=1.5$ and $x=0.1$. Then, a quick calculation shows that the difference between the lhs and rhs is positive = $0.0368....$.

This gap can be made even larger as $a \to 0^-$, but not too much. This suggests that a slightly modified version of the inequality holds.