I have questions about the integral $$F(a,b,c)=\sqrt{\frac{a}{\pi}}\int_{0}^{\infty}e^{-bx^4+cx^3-ax^2}dx$$ for $a,b,c>0$.

  1. What is the asymptotic behavior of $F(a,b,c)$ for small $a,b,c$? In particular, what is the asymptotics of $F(a,a^p,a^q)$ for positive $p,q$ when $a$ tends to $0$?

  2. When $c=0$ this integral is expressible in terms of a modified Bessel function. Is $F(a,b,c)$ expressible in terms of special functions with known asymptotics in more general situations?

  3. Are there any lower bounds for $F(a,b,c)$ in terms of elementary functions which are sharper than the bound given by Jensen's inequality? What would be good strategies to obtain them?

  • $\begingroup$ Concerning the asymptotics assuming $b=a^p$ and $c=a^q$, much will depend on how $q$ compares with $3/2$ and how $p$ compares with $2$. $\endgroup$ Apr 17, 2018 at 18:39
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    $\begingroup$ To approximate the size of the integral, one standard tool is to find where $-bx^4+cx^3-ax^2$ is maximized and compare with the closest-matching Gaussian. Also I'd expect that $\int_{-\infty}^\infty$ has been studied more thoroughly than $\int_0^\infty$ (which is not equivalent except when $c=0$). $\endgroup$ Apr 17, 2018 at 18:41
  • $\begingroup$ Noam, I've been trying for a while what you are suggesting in order to obtain an upper bound. But it seems to me that to obtain a tight lower bound something else is needed. $\endgroup$
    – S.Z.
    Apr 18, 2018 at 20:45
  • $\begingroup$ Also, $F(at^2,bt^4,ct^3)=F(a,b,c)$ for $t>0$ $\endgroup$ Apr 21, 2018 at 6:39


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