New Refinements of the Generalized Versions of Hölder’s Inequality

In this paper, we present new refinements of the generalized versions of Hölder’s inequality. Such refinements are rare. Our results are novel and clearly illustrate the essence of some recent specific refinements. As for applications, we present new inequalities for integral power means, give a new refinement of the generalized Opial–Olech inequality, refine the well-known inequality between Rényi’s entropies with different parameters by utilizing the concept of “useful” information, and finally, we obtain a refinement of the Cauchy–Bunyakovsky–Schwarz inequality.