Bounding a somewhat complicated integral (exponential of a polynomial)

Question

I am interested in bounding the following integral, where $a>0$ is a constant:

$$\int\limits_0^a \exp\left(\left(x^2 - \frac{2a^4+3}{4a^2}\right)^2\right) dx$$

I first conjectured that

$$\int\limits_0^a \exp\left(\left(x^2 - \frac{ba^4+c}{a^2}\right)^2 \right) dx \le a^{-1}\exp \left( \left( \frac{ba^4+c}{a^2}\right)^2 \right)$$

when $c \ge 0$ and $b > 0$. However, this is false (for example, take $c = 0$, $b = 9/20$, and plug in $a = 2$). However, this inequality does appear to hold when $b \ge \frac{1}{2}$, from the numerical tests I have done. I am not sure how to prove it, however. The expressions that I have seen for exponentials of quartics are quite complicated and do not easily admit an upper bound (see for instance Exponential of a Quartic). Does anyone have any ideas?

[Note: In fact, what I'm really interested in is the regime $a \rightarrow \infty$, but so far the upper bound I have guessed appears to hold up for all $a > 0$.]

Updated wording: I am not just looking for any bound on this integral, but I indeed would like to prove the bound that I have proposed above. This was not made clear in my original post; I apologize for that.

Answer 1

For the infinity: you can write for positive a

$$ \int_0^{a}\exp\left(\left(x^2 - \frac{2a^4+3}{4a^2}\right)^2\right)dx=\int_{0}^{\infty}\exp\left(\left(x^2 - \frac{2a^4+3}{4a^2}\right)^2\right)\chi_{[0 , a]}(x) dx $$

So you can begin by majorate the integrand only independantly of a

And apply dominated convergence theorem. You will get 0 for limit.

Just find the independant majoration (I'm thinking about..)

Answer 2

Since $\exp(z)>0 \,(\forall z \in \mathbb{R})$, you can use the following bounding method:

$$ \int_0^a f(x)dx =\int_0^a |f(x)|dx $$

Then:

$$ (a-0) \min \lbrace f(x) \rbrace \leq\int_0^a |f(x)|dx \leq (a-0) \max \lbrace f(x) \rbrace $$

Since the exponential is a monotone function you basically need to find the minimum and maximum of :

$$ g(x) = \left(x^2−\frac{2a^4+3}{4a^2}\right)^2 $$

Let $t\in[0,1]$ and $x=ta$: $$ g(x) = \left(\frac{(t^2-2)a^4-3}{4a^2}\right)^2 $$

Then the minimum value would occur for : $$ t=\sqrt{\frac{1}{2}+\frac{3}{4a^2}} $$

So the actual bounds do depend on the range for $a$, or if $a\leq\sqrt{3/8}$, if that is the case the value above allows a minimum, otherwise we pick $t=1$.

Thus the extremes occur for $x=0$ (minimum) and $x=ta$ (maximum).

$$ a \exp \left( g(ta) \right) \leq\int_0^a |f(x)|dx \leq a \exp \left( g(0) \right) $$

Do you need a tighter bound than this?

Answer 3

Another approach I can think of that could provide more insight would be computing a Taylor Remainder for the expansion around $a=0$ of:

$$ F(a) = \int_0^a f(a,x)dx = \int_0^a \exp(g(a,x))dx $$

$$ F(a) = F(0) + F'(a) + F''(\xi)/2 $$ Naturally $F(0) = 0$. So:

For that, we can use Leibniz Integral Rule: $$ \frac{dF}{da} |_{x=a} = f(a,a) \times 1 + \int_0^a \frac{\partial f(a,x)}{\partial a}dx = W(a) $$

Computing $f(a,a)$: $$ f(a,a) = \exp \left(\frac{(1-b)a^4 -c}{a^2} \right)^2 $$ Now the partial derivative: $$ \frac{\partial f(a,x)}{\partial a} = f(a,x) \times \frac{\partial g(a,x)}{\partial a} = f(a,x) \times 2 \left( \left( x^2 - \frac{ba^4+c}{a^2}\right) \left( -2ab +2\frac{c}{a^3}\right) \right) $$

Well... I'll just use Mathematica: $$ W(a) = \int_0^a \frac{\left(a^2-3\right) e^{\left(\frac{3}{4 a^4}-\frac{1}{2 a^2}+x^2\right)^2} \left(4 a^4 x^2-2 a^2+3\right)}{2 a^9} \, dx+e^{\frac{\left(4 a^6-2 a^2+3\right)^2}{16 a^8}} $$

The Remainder term will look like this: $$ \frac{e^{\left(\frac{3}{4 \xi ^4}+\xi ^2-\frac{1}{2 \xi ^2}\right)^2} \left(\xi ^6+\xi ^2-3\right) \left(4 \xi ^6-2 \xi ^2+3\right)}{\xi ^9}+\int_0^{\xi } \frac{e^{\left(\frac{3-2 \xi ^2}{4 \xi ^4}+x^2\right)^2} \left(-162 \xi ^2-24 \xi ^{14} x^2-36 \xi ^6 \left(8 x^2+1\right)+9 \xi ^4 \left(24 x^2+13\right)+4 \xi ^{12} \left(4 x^4+30 x^2+5\right)-2 \xi ^{10} \left(48 x^4+8 x^2+63\right)+2 \xi ^8 \left(72 x^4+60 x^2+83\right)+81\right)}{4 \xi ^{18}} \, dx $$

So, I call it quits on this attempt, but I do admit it is starting to look plausible, but for a very limited range on $a$, such that $a^{-1}$ is actually a big number and thus $a^{-1} K $ for big enough $K$ can be an actual bound... But thinking like this...

Yeah... Nope, I don't think this would work...