Abstract
This article explores two distinct function spaces: Hilbert spaces and mixed-Orlicz-Zygmund spaces with variable exponents. We first examine the relational properties of Hilbert spaces in a tensorial framework, utilizing self-adjoint operators to derive key results. Additionally, we extend a Maclaurin-type inequality to the tensorial setting using generalized convex mappings and establish various upper bounds. A non-trivial example involving exponential functions is also presented. Next, we introduce a new function space, the mixed-Orlicz-Zygmund space & ell;q(<middle dot>)log beta Lp(<middle dot>), which unifies Orlicz-Zygmund spaces of integrability and sequence spaces. We investigate its fundamental properties including separability, compactness, and completeness, demonstrating its significance. This space generalizes the existing structures, reducing to mixed-norm Lebesgue spaces when beta=0 and to classical Lebesgue spaces when q=infinity,beta=0. Given the limited research on such spaces, our findings contribute valuable insights to the functional analysis.