  For a measure space $(\Omega ,\Sigma ,\mu)$ and a bijective increasing
function $\varphi :\left[ 0,\infty \right) \rightarrow \left[0,\infty \right)$
the $L^{p}$-like paranormed ($F$-normed) function space with the paranorm of
the form $\mathbf{p}_{\varphi}(x)=\varphi ^{-1}\left(\int_{\Omega}\varphi \circ
\left|x\right|d\mu \right)$ is considered. Main results give general conditions
under which this space is uniformly convex. The Clarkson theorem on the uniform
convexity of $L^{p}$-space is generalized. Under some specific assumptions
imposed on $\varphi$ we give not only a proof of the uniform convexity but also
show the formula of a modulus of convexity. We establish the uniform convexity
of all finite-dimensional paranormed spaces, generated by a strictly convex
bijection $\varphi$ of $[0, \infty)$. However, the {\it a contrario} proof of
this fact provides no information on a modulus of convexity of these spaces. In
some cases it can be done, even an exact formula of a modulus can be proved. We
show how to make it in the case when $S={\mathbb R}^2$ and $\varphi$ is given
by $\varphi(t)={\rm e}^t-1$.
