Abstract
Let \(-1 \le B<A \le 1\) and \(F= f/f'\) for normalized locally univalent functions f. For analytic function p on the open unit disc \(\mathbb {D}\) with \(p(0)=1\) and satisfying the subordination \(p(z) \prec (1+A z)/(1+B z)\), we determine bounds for \(|p(z)- z p'(z)-1|\) and \(|z p'(z)|/{{\mathrm{Re\,}}}p(z)\). As an application, we investigate the radius problem for F to satisfy \(|F'(z)(z/F(z))^2-1| <1\), where f is a Janowski starlike function and the radius of univalence of F when f is a starlike function of order \(\alpha \). Also, we have discussed the radius of starlikeness of F when f is a Janowski starlike function with fixed second coefficient. Apart from the radius problems, we give the sharp coefficient bounds of F when f is a Janowski starlike function and a sufficient condition for starlikeness of f when F is a Janowski starlike function. Our results generalize some of the earlier known results.
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The first author is supported by a Grant from NBHM, Mumbai. The authors are thankful to the referee for the useful suggestions.
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Communicated by See Keong Lee.
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Verma, S., Ravichandran, V. Radius Problems for Ratios of Janowski Starlike Functions with Their Derivatives. Bull. Malays. Math. Sci. Soc. 40, 819–840 (2017). https://doi.org/10.1007/s40840-016-0363-x
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DOI: https://doi.org/10.1007/s40840-016-0363-x