Considering the Taylor expansion of the Bessel function of even order it seems like the following statements are true (I take $J_0$ as an example):
$1-\frac{x^2}{4}≤J_0(x)≤1$
$1 - \frac{x^2}{4} + \frac{x^4}{64} - \frac{x^6}{2304}≤J_0(x)≤1 - \frac{x^2}{4} + \frac{x^4}{64}$
Is there a general inequality saying that the Taylor series (around zero) of even order is smaller or equal $J_0$ is smaller or equal the Taylor series (around zero) of odd order? And if yes, is there a reference for that or how could one prove such a statement? Thanks for any hints.