  We consider the parabolic chemotaxis model \[ u_t=\Delta u - \chi
\nabla\cdot(\frac uv \nabla v), \qquad\qquad v_t=\Delta v - v + u\] in a
smooth, bounded, convex two-dimensional domain and show global existence and
boundedness of solutions for $\chi\in(0,\chi_0)$ for some $\chi_0>1$, thereby
proving that the value $\chi=1$ is not critical in this regard. Our main tool
is consideration of the energy functional \[ \mathcal{F}_{a,b}(u,v)=\int_\Omega
u\ln u - a \int_\Omega u\ln v + b \int_\Omega |\nabla \sqrt{v}|^2 \] for $a>0$,
$b\geq 0$, where using nonzero values of $b$ appears to be new in this context.
