Energy estimate Neumann Problem with paramter

Question

Let $I=(0,1)$. Suppose that I have the following problem \begin{align}\label{eq:NBVP_FT} \left\{\begin{array}{rclclcl} -(\partial_t^2 - \lambda^2) {v} (\lambda,t) & = & {f}(\lambda,t) &&&& \text{for } t\in I, \\[6pt] \partial_t\widehat{v}(\lambda,0) & = &0, \\[6pt] \partial_t\widehat{v}(\lambda,1) & = & 0, \end{array}\right. \end{align} where $\lambda \in \mathbb{C}$ such that $\Re \lambda =\beta \notin \pi \mathbb{Z}$ and $v, f$ are complex-valued. I want to derive an estimate of the form \begin{align} (1+|\lambda|^2)^2 ||{v}(\lambda,\cdot)||^{2}_{L^2(I)} + (1+|\lambda|^2)||{\partial_{t}v}(\lambda,\cdot)||^{2}_{L^2(I)} + ||{\partial_{tt} v}(\lambda,\cdot)||^{2}_{L^2(I)} \lesssim ||{f}(\lambda,\cdot)||^{2}_{L^2(I)}. \end{align} My idea was to multiply the equation by $\bar{v}(\lambda,t)$, integrate over $I$ and do a partial integration to arrive at \begin{align} \int_I |\partial_t {v}(\lambda,t)|^2 d t + \lambda^2 \int_I |{v}(\lambda,t)|^2 d t = \int_I {f}(\lambda,t)\overline{{v}}(\lambda,\cdot) d t. \end{align} Do you have any suggestions how to proceed now? if $\lambda$ was a real number there would not be any problems, but I don't see how to go from $\lambda^2$ to $|\lambda|^2$. Moreover, I have no idea how to get the $1+|\lambda|^2$ instead of $|\lambda|^2$ as I cannot apply Poincaré's inequality here...