Uniform convexity of paranormed generalizations of Lᵖ spaces

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For a measure space (Ω,Σ,μ) and a bijective increasing function φ:[0,∞) → [0,∞) the Lᵖ-like paranormed (F-normed) function space with the paranorm of the form pφ(x)=φ⁻¹(∫Ωφ∘ |x|dμ) is considered. Main results give general conditions under which this space is uniformly convex. The Clarkson theorem on the uniform convexity of Lᵖ-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, ∞). However, the 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=R² and φ is given by φ(t)= eᵗ-1.