On a simple sufficient condition for starlikeness. (English) Zbl 0716.30003

Anisiu, Valeriu; Mocanu, Petru T.

Math., Rev. Anal. Numér. Théor. Approximation, Math. 31(54), No. 2, 97-101 (1989).

Let f be an analytic function in the unit disc \(U=\{z\in {\mathbb{C}}\), \(| z| <1\}\) with \(f(0)=0\). The basic result of the paper is the following sufficient condition for starlikeness: If f satisfies the inequality \[ | f''(z)/f'(z)| \leq M^*,\quad z\in U,\text{ where } M^*=\sqrt{1+y^ 2}=2.83..., \] y is the smallest positive root of the equation \(y \sin y+\cos y=1/e,\) then f is starlike. The bound \(M^*\) cannot be replaced by any larger number.

Reviewer: St.Walczak

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
30C80	Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination

Keywords:

starlikeness