Certain classes of starlike functions. (English) Zbl 0576.30005

Robertson, Malcolm S.

Mich. Math. J. 32, 135-140 (1985).

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

Citations:

Zbl 0057.311