I'm looking for polynomial lower estimates for beta function, and what I've found so far is this, which can be found in proposition 2.3 in this paper
Proposition 2.3 1. If $0<𝑞<1$ and $𝑝 \geq 𝑞$, then $\dfrac{2^{2(1-q)}}{p-q+1} \leq 𝐵(𝑝,𝑞)$. where $$𝐵(𝑝,𝑞)=\int_{0}^1𝑡^{𝑝−1}(1−𝑡)^{𝑞−1} 𝑑𝑡.$$ From this proposition it follows for example that $\dfrac{1}{p-q+1} \leq 𝐵(𝑝,𝑞).$ (1)
I wanted to know if this inequality is optimal, if it is the best possible polynomial estimate. In fact, the paper presents some other estimates, but of an exponential type. I find this condition $0<𝑞<1$ restrictive, are there estimates of type (1) without this restriction? I would appreciate it if you could refer me to works with such estimates.
Thanks in advance.
