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How to prove that these functions do not intersect?

I want to prove that these two functions $f(x)$ and $g(x)$ do not intersect for $x>1$:
$$f(x)=cosh left(frac{2 sqrt{2} pi x left(x^2-1right) cosh (pi x)}{sqrt{x^4+6 x^2+left(x^2-1right)^2 cosh (2 pi x)+1}}right)$$
$$g(x)=frac{4 x^2+left(x^2-1right)^2 cosh (2 pi x)}{left(x^2+1right)^2}$$
Both functions are greater than one and strictly increasing. Subtraction and taking derivative do not work, the problem becomes more complicated.

Does anyone have an idea to prove it by assuming false assumption? Or to prove that $f(x)-g(x)$ has no real root?

Any hints or suggestions are really appreciated.

Mathematics Asked by user805770 on November 21, 2021

1 Answers

One Answer

Since you said that both functions are greater than one and strictly increasing, they vary so fast that I should instead consider $$F(x)=log(f(x)) qquad text{and} qquad G(x)=log(g(x))$$ If you plot them, you should notice that, as soon as $x > 3$, $F(x)$ and $G(x)$ looks very linear with $F(x) > G(x)$; I agree that this does not prove anything.

Now, consider a Taylor expansion of $F(x)-G(x)$ around $x=1$. This gives $$H(x)=F(x)-G(x)=(x-1)^2 left((pi ^2+1)+left(pi ^2-1right) cosh (2 pi )right)+Oleft((x-1)^3right)$$ So, starting from $0$ at $x=1$, $H(x)$ is an increasing function, which, according to Murphy's principle, will go through a maximum value. This maximum is located very close to $frac 98$ and its nature is confirmed by the second derivative test.

For sure, now the problem is : what happens for infinitely large values of $x$ ?

For the asymptotics analysis, I shall assume that, for large $t$, $cosh(t) sim frac 12 e^t$. Factoring the exponentials and ignoring the terms which are divided by $e^t$, we end with

$$f(x) sim frac 12 e^{2pi x} quad text{and} quad g(x) sim frac 12 e^{2pi x} frac{left(x^2+1right)^2}{left(x^2-1right)^2}implies H(x) sim frac{4}{x^2}+Oleft(frac{1}{x^6}right)$$ In other words, when $x$ is large $$f(x) sim g(x) , exp left({frac{4}{x^2}}right)$$

Computed exactly for $x=10$, the result is $H(10)=0.0400013$ !!

Edit

Looking at the last result, I computed as few values $$left( begin{array}{cc} x & x^2, H(x) \ 2 & 4.0859321 \ 3 & 4.0165826 \ 4 & 4.0052206 \ 5 & 4.0021354 \ 6 & 4.0010293 \ 7 & 4.0005555 \ 8 & 4.0003256 \ 9 & 4.0002032 \ 10 & 4.0001333 \ 11 & 4.0000911 \ 12 & 4.0000643 \ 13 & 4.0000467 \ 14 & 4.0000347 \ 15 & 4.0000263 \ 16 & 4.0000203 \ 17 & 4.0000160 \ 18 & 4.0000127 \ 19 & 4.0000102 \ 20 & 4.0000083 end{array} right)$$

Answered by Claude Leibovici on November 21, 2021

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