  We prove two new functional inequalities of the forms\[
  \int_G \varphi (\psi - \overline{\psi}) \leq \frac{1}{a}\int_G \psi \ln
\left(\frac{\;\psi\;}{ \overline{\psi}}\right) + \frac{a}{4\beta_0} \left\{
\int_G \psi \right\}\int_G|\nabla \varphi|^2
  \] and \[
  \int_G \psi \ln \left(\frac{\;\psi\;}{ \overline{\psi}}\right) \leq
\frac{1}{\beta_0}\left\{ \int_G \psi \right\}\int_G |\nabla \ln(\psi)|^2
  \] for any finitely connected, bounded $C^2$-domain $G \subseteq
\mathbb{R}^2$, a constant $\beta_0 > 0$, any $a > 0$ and sufficiently regular
functions $\varphi$, $\psi$.
  We then illustrate their usefulness by proving long time stabilization and
eventual smoothness properties for certain generalized solutions to the
chemotaxis-Navier-Stokes system\[
  \left\{\;\;
  \begin{aligned}
  n_t + u \cdot \nabla n &\;\;=\;\; \Delta n - \nabla \cdot (nS(x,n,c) \nabla
c), \\
  c_t + u\cdot \nabla c &\;\;=\;\; \Delta c - n f(c), \\
  u_t + (u\cdot \nabla) u &\;\;=\;\; \Delta u + \nabla P + n \nabla \phi,
\;\;\;\;\;\; \nabla \cdot u = 0,
  \end{aligned}
  \right.
  \] on a smooth, bounded, convex domain $\Omega \subseteq \mathbb{R}^2$ with
no-flux boundary conditions for $n$ and $c$ as well as a Dirichlet boundary
condition for $u$. We further allow for a general chemotactic sensitivity $S$
attaining values in $\mathbb{R}^{2\times 2}$ as opposed to a scalar one.
