Based on "Certain Subclass of Starlike Functions" journal by Chun-Yi and Shi-Qiong Zhou in 2007 (Science Direct), I found difficulties to understand the proof in Theorem 3 where they have verified:
$$ 1+2(1-\beta) \displaystyle\sum\limits_{n=2}^\infty \frac{z^{n-1}}{n(\alpha(n-1)+1)} = 1 +\frac{2(1-\beta)}{\alpha} \int_0^1 \! t^{\frac{1}{\alpha}} \,\int_0^1 \! \frac{vz}{1-tvz} \, \mathrm{d} v \ \mathrm{d} t$$
Could someone give me the idea of how to prove it?
Thank you.