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

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