I have proved the following bound for the Bessel function of the first kind: $$ J_0(x)=\sum_{m=0}^\infty \frac{(-1)^m\,(x/2)^{2m}}{(m!)^2} $$ which is $$ |J_0(x)|\le \frac1{\sqrt[4]{1+x^2}} $$ but I wonder if it is known in the literature. Any tips?
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3$\begingroup$ You may already know this, but I note that the bound $|J_0(x)|\le\min(1,\sqrt{2/(\pi x)})$ is known; it is weaker than your bound near the origin, for $|x|<2/\sqrt{\pi^2-4}$. See math.stackexchange.com/q/1447137 $\endgroup$– ho boon suanCommented Jun 13, 2023 at 12:47
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3$\begingroup$ To handle $x$ close to the origin, we can then use the bound $J_0(x)\le{1\over3}(1+2\cos(x\sqrt{3/4}))$ for $|x|\le\pi/2$, which follows from Theorem 2.2 in Edward Neuman, “Inequalities Involving Bessel Functions of The First Kind,” Journal of Inequalities in Pure and Applied Mathematics 5 (4) (2004), Article 94; see emis.de/journals/JIPAM/article449.html $\endgroup$– ho boon suanCommented Jun 13, 2023 at 13:26
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$\begingroup$ Thank you, I do have a proof of the statement (quite short, using Krasikov, I. (2006). Uniform Bounds for Bessel Functions), but I'm wondering if it was proven already an published somewhere in this or equivalent form... $\endgroup$– van der WolfCommented Jun 14, 2023 at 10:21
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$\begingroup$ Since @hoboonsuan points out that a sharper bound is available, for example (from his remarks) the function $$x\to \min\left(\sqrt{\frac{2}{\pi x}},\frac13 + \frac23 \cos\left(\min\left(x,\frac{\pi}{2}\right)\sqrt{\frac34}\right)\right)$$ why would you want to publish a worse bound? $\endgroup$– usernameCommented Oct 16, 2023 at 13:04
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$\begingroup$ what do you mean @username? $\endgroup$– van der WolfCommented Oct 17, 2023 at 18:53
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