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We establish some interesting refinements of the ( p , q ) $(p,q)$ -Hölder integral inequality and the ( p , q ) $(p,q)$ -power-mean integral inequality. As applications, we show that some existing ( p , q ) $(p,q)$ -integral inequalities can be improved by the results obtained in this paper.

On the refinements of some important inequalities via $(p,q)$-calculus and their applications