Some inequalities for the beta function and its ratios

In this paper, we prove some inequalities for the differences and ratios of the beta function.

💡 Research Summary

The paper investigates new inequalities for the beta function (B(x,y)=\int_{0}^{1}t^{x-1}(1-t)^{y-1},dt) and for products of the form (B(b,y)B(a,y)) with (0<a<b). After a brief review of classical bounds—such as Dragomir’s inequality (\frac{1}{xy-1/4}\le B(x,y)\le\frac{1}{xy}) and Ivady’s refinement—the authors introduce a central tool called the Generating Lemma (Lemma 2.1). This lemma provides an identity linking integrals of beta‑type kernels with and without a logarithmic factor: \