Proof of an Inequality

Question

Given a sequence $(b_n)_{n=1}^p$ of positive numbers such that $b_1>b_2>\cdots>b_p>0$, define $$\beta=\bigg(p!\frac{p}{p-1}\bigg)^{\frac{1}{\min\limits_{1\leq k\leq p-1}{(b_k-b_{k+1})}}}.$$ Suppose $a_i\in\mathbb{C},i=1,2,\cdots,p$ satisfies $|a_i|>\beta|a_{i+1}|, i=1,2,\cdots p-1$ and $|a_p|>1$. Prove the following matrix $$ D= \begin{pmatrix} a_{1}^{b_1} & a_{1} ^{b_2}& \dots & a_{1} ^{b_p}& \\ a_{2} ^{b_1}& a_{2} ^{b_2}& \dots &a_{2} ^{b_p}\\ \\ a_{p} ^{b_1}& a_{p} ^{b_2}& \dots & a_{p} ^{b_p} \end{pmatrix}$$ satisfies the following inequality: $$\frac{1}{p}\prod_{i=1}^{p}|a_i|^{b_i}<|\det(D)|<2\prod_{i=1}^{p}|a_i|^{b_i}.$$