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Uniform convexity of paranormed generalizations of $L^p$ spaces

For a measure space $(Ω,Σ,μ)$ and a bijective increasing function $φ:\left[ 0,\infty \right) \rightarrow \left[0,\infty \right)$ the $L^{p}$-like paranormed ($F$-normed) function space with the paranorm of the form $\mathbf{p}_φ(x)=φ^{-1}\left(\int_Ωφ\circ \left|x\right|dμ\right)$ 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 $φ$ 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 $φ$ 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 $φ$ is given by $φ(t)={\rm e}^t-1$.