On the refinements of some important inequalities with a finite set of positive numbers

In this research, a novel method for enhancing the Holder-Iscan inequality through the utilization of both integrals and sums, as well as the mean power inequality, has been introduced. This approach outperforms traditional Holder and mean power integral inequalities by employing a finite set of functions. Through the careful selection of the function phi$$ \phi $$, an entirely new category of classical inequalities emerges for both Holder and mean power inequalities.