Approximation of Values of the Gauss Hypergeometric Function by Rational Fractions

We consider a new approach to estimating the irrationality measure of numbers that are values of the Gauss hypergeometric function. Some of the previous results are improved, in particular, those concerning irrationalities of the form $\sqrt{k}\ln((\sqrt{k}+1)/(\sqrt{k}-1))$ with $k\in\mathbb N$.