## 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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