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Differential inequalities and starlikeness. (English) Zbl 1034.30008

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.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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