In this paper, we develop a concept of a logarithmically super-qua-dratic function. Such a class of functions is defined via superquadratic functions.These functions possess some superior properties, especially if they take values greater than or equal to one. We prove that they are convex and superadditive in the latter case.In particular, we also obtain the corresponding refinement of the Jensen inequality in a product form. Furthermore, we derive an external form of the Jenseninequality and the corresponding reverse. Finally, we give a variant of the Jensen operator inequality for logarithmically superquadratic functions.All established results are derived via the corresponding relations for superquadratic functions.