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Certain classes of starlike functions. (English) Zbl 0576.30005
The author introduces a one parameter family of starlike mappings in the unit disk by proving that if f(z) is analytic in $$| z| <1$$ and if it satisfies the conditions $f(0)=f'(0)-1=0,\quad f(z)\neq 0,\quad 0<| z| <1$ and for $$| z| =r$$, $$r\in (0,1)$$ $Re\{z^ 2[(f'(z)/f(z))'+\alpha (f'(z)/f(z))^ 2]\}\geq r^ 2[(q(r)/r)'+a(q(r)/r)^ 2]$ $$q(x)\in C^ 1(0,1)$$, $$0<q(x)\leq 1=q(0)$$. Then $Re\{zf'(z)/f(z)\}\geq q(r)>0.$ The proof is based on an extended version of the author’s lemma in Trans. Am. Math. Soc. 76, 254-274 (1954; Zbl 0057.311). Some applications are also given.
Reviewer: E.Złotkiewicz

##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 30C75 Extremal problems for conformal and quasiconformal mappings, other methods
##### Keywords:
starlike mappings
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