\begin{multline*}
\exp(-2\beta \theta \ell_2 - \beta^2 \sigma_r^2/2) D_1 = \max_u \biggl| \frac{d}{du} \biggl\{ \frac{2\ell_2}{\sigma_r} \phi( 2\ell_2( u - \mu_{\ell}) /\sigma_r)
\biggr\} \biggr|
= \max_u \biggl|  \frac{4\ell_2^2}{\sigma_r^2} \phi'( 2\ell_2( u - \mu_{\ell}) /\sigma_r)
\biggr\} \biggr| 
= \\ 
\max_u \biggl|  \frac{4\ell_2^2}{\sigma_r^2} 
\{ (2\ell_2/\sigma_r) ( u - \mu_{\ell}) \phi( 2\ell_2( u - \mu_{\ell}) /\sigma_r) \}
 \biggr|
\leq c_2 \frac{4\ell_2^2}{\sigma_r^2} 
=c_2  \frac{\ell_2^2}{2 \ell_2 \theta (1 - \theta) ( 1 - y_{n,2})} \\
\leq c_3  \theta^{-1}(1- y_{n,2})^{-1} \ell^2
\leq c_4 (n/k) (1 - y_{n,2})^{-1} \ell^2
\end{multline*} 
