New One-parameter Integral Formulas and Inequalities of the Logarithmic Type

This article deals with two fundamental topics in mathematical analysis: the formulation of integral expressions and the derivation of integral inequalities. In particular, it introduces new one-parameter integral formulas and inequalities of the logarithmic type, where the integrands involve the logarithmic function in one way or another. Among the results are weighted Hölder-type integral inequalities and two different forms of Hardy-Hilbert-type integral inequalities. These results are illustrated by various examples and accompanied by rigorous proofs.