Fournier, Richard; Mocanu, Petru Differential inequalities and starlikeness. (English) Zbl 1034.30008 Complex Variables, Theory Appl. 48, No. 4, 283-292 (2003). Let \(A\) denote the class of all analytic functions \(f\) in the open unit disc \(D\) of the complex plane with \(f(0)= 0= 1- f'(0)\). Let \(S\) denote the subclass of \(A\) consisting of starlike functions. In this paper the authors are interested in finding sharp bounds for the constants \(\rho\) so that conditions such as \(f\in A\) and \(| z f''(z)- \alpha(f'(z)- 1)|<\rho\) \((z\in D)\) for some \(\alpha\) with \(0\leq \alpha< 1\) or \(f\in A\) and \(| f'(z)- \alpha(f(z)/z)+ \alpha- 1|\leq\rho\) (\(z\in D\) for some \(\alpha\in \mathbb{C}\) with \(\text{Re\,}\alpha< 2\)) imply \(f\in S\). The proofs use Hadamard convolution techniques. Reviewer: V. Karunakaran (Madurai) Cited in 6 Documents MSC: 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) PDF BibTeX XML Cite \textit{R. Fournier} and \textit{P. Mocanu}, Complex Variables, Theory Appl. 48, No. 4, 283--292 (2003; Zbl 1034.30008) Full Text: DOI OpenURL