Performance bounds for low-pass filters
As far as I know there are no known tight upper and lower bounds, but there's a quite accurate empirical formula for estimating the required filter order $M$ given maximum passband and stopband deviations $\delta_1$ and $\delta_2$, and the width of the transition band $\Delta\omega$:
$$M=\frac{-10\log_{10}(\delta_1\delta_2)-13}{2.324\;\Delta\omega}\tag{1}$$
Formula $(1)$ can be found as Eq. $(7.117)$ in Oppenheim and Schafer's textbook Discrete-Time Signal Processing (3rd ed, p.570). According to the authors, this formula is due to Kaiser, who developed it based on experimental results published by Herrmann, Rabiner, and Chan.
Also Matlab's firpmord.m uses a similar but more complex formula to estimate the required filter order.