Intereting Posts

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Prove $BA – A^2B^2 = I_n$.
Actions of Finite Groups on Trees

i’m looking for an upper bound for the modified Bessel function of the first kind of a +ive real argument. It seems that it satisfies the inequality :

$$I_{n}(x)\leqslant \frac{x^{n}}{2^{n}n!}e^{x}$$

But i’m not able to prove this.

- $\lim_{n\to\infty} f(2^n)$ for some very slowly increasing function $f(n)$
- Since $2^n = O(2^{n-1})$, does the transitivity of $O$ imply $2^n=O(1)$?
- How fast does the function $\displaystyle f(x)=\lim_{\epsilon\to0}\int_\epsilon^{\infty} \dfrac{e^{xt}}{t^t} \, dt $ grow?
- On Shanks' quartic approximation $\pi \approx \frac{6}{\sqrt{3502}}\ln(2u)$
- Representation of $e$ as a descending series
- How best to explain the $\sqrt{2\pi n}$ term in Stirling's?
- Estimating $\sum n^{-1/2}$
- A Mathematical Coincidence, or more?
- how many ways to make change, asymptotics
- How do you prove that $n^n$ is $O(n!^2)$?

It was proved by Yudell L. Luke in 1972 that

$$

1 < \Gamma(\nu+1)\left(\frac{2}{x}\right)^\nu I_\nu(x) < \cosh x

$$

for $x > 0$ and $\nu > -1/2$. This implies your inequality since

$$

\cosh x – e^x = -\sinh x < 0

$$

for $x > 0$ and hence

$$

\cosh x < e^x

$$

for $x > 0$.

Yudell L. Luke,

Inequalities for generalized hypergeometric functions,

Journal of Approximation Theory, Volume 5, Issue 1, January 1972, pp. 41–65.(Link to the article on ScienceDirect)

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