I have a map $f:\mathbb{R}^n\mapsto\mathbb{R}^m$ where $n<m$. Let $D_f\in\mathbb{R}^{m\times n}$ denote the Jacobian matrix. I want to bound the smallest singular value of $J_f$ from above, for example $\sigma_{\text{min}}\left(J_f\right)\geq C$. Therefore I have
\begin{align} \sigma_{\text{min}}\left(J_f\right)^2=&\min_{\mathbf{v}=1}\lVert J_f\cdot\mathbf{v}\rVert_2^2.\\ \end{align}
I wanted to try bound using inequality manipulations as the Jacobian is non-square so I didn't want to use the spectral matrix norm of $J_f^\dagger$. What inequality could I use from here?
Thanks in advance