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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.

  • 0
    Did you try expanding the innermost integrand as a geometric series, and swapping summation and integration?2011-12-07
  • 1
    is there any bound on $|z|$? seems very similar to $\sum z^n = \frac{1}{1-z}$2011-12-07
  • 0
    @J.M.: I've tried but did not get the intended results.2011-12-07
  • 0
    @Ilya: they were inform that $|z|\leq r \rightarrow 1$2011-12-07

2 Answers 2