Some inequalities for the beta function and its ratios

TL;DR

The paper addresses sharp inequalities for the beta function and its ratios, focusing on bounds for $\frac{B(b,y)}{B(a,y)}$ and the difference $B(b,y)-B(a,y)$ via a Generating Lemma and integral identities. It develops parameter-dependent upper and lower bounds, analyzing cases $0<y\le1$ and $y>1$ to obtain improvements over prior results, with extensive numerical validation. Additionally, the work connects these beta-function bounds to estimates for the digamma difference $\Psi(x+y)-\Psi(x)$, broadening applications in special-function analysis. An accompanying Python appendix and numerical study bolster the practical relevance of the theoretical bounds.

Abstract

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

Figures (1)

• Figure 1: Figure showing the result of the function $F(a,b,y)$ showing the parts of the plane where $A(a,b,y)>B(a,b,y)$ in white and $A(a,b,y)<B(a,b,y)$ in black. $y$ is taken from $10^{-16}$ to 1.2. The vertical axis is the $b-$axis and the horizontal axis is the $a-$axis.

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Lemma 2.1: Generating lemma