This question concerns Kummer's confluent Hypergeometric function $M$, also denoted as $_1F_1$. Recall that $$M(a;b;z)=\hspace{0.1cm} _1F_1(a;b;z)=\sum_{k\geq 0}\frac{(a)_n}{(b)_n}\frac{z^k}{k!}.$$ Consider $x\geq0, p\in (0,1)$, I would like to prove that there exists $x_0>0$ such that: $$f(x)=M\left(1;1+\frac{1}{p};-\frac{x^p}{p}\right)\leq x^{-p},\hspace{0.2cm} \forall x\geq x_0.$$ Plotting $f$ and $x^{-p}$ numerically, this conjecture seems to be true and sharp, nevertheless, I was unable to prove it.
$\textbf{PS:}$ This problem arises from trying to bound a generalization of the Dawson's integral $F(p,x)=e^{-x^p}\int_0^x e^{t^p}dt$ for the case $p<1$. Since $F(p,x)=xM(1;1+\frac{1}{p};-x^p)$, then $p^{1/p}F(p,p^{-1/p}x)=xf(x)$, so in these terms, the inequality is $$p^{1/p}F(p,p^{-1/p}x)\leq x^{1-p}.$$ On the other hand $$F(p,x)=e^{-x^p}\frac{x}{p}\Gamma(1/p)\gamma^*(1/p,-x^p)=\frac{x}{p}\Gamma(1/p)\sum_{k\geq 0}\frac{(-x^p)^k}{\Gamma(1+\frac{1}{p}+k)},$$ where $\gamma^*$ is the holomorphic extension of the lower incomplete Gamma function.