Lawrence Zalcman’s conjecture states that if f(z)=z+∑n=2∞anzn\documentclass[12pt]{minimal}
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\begin{document}$${f(z)=z+\sum\nolimits_{n=2}^{\infty}a_{n}z^{n}}$$\end{document} is analytic and univalent in the unit disk |z|<1\documentclass[12pt]{minimal}
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\begin{document}$${|z|<1}$$\end{document}, then |an2-a2n-1|≤(n-1)2,\documentclass[12pt]{minimal}
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\begin{document}$${|a_n^2-a_{2n-1}|\leq (n-1)^2,}$$\end{document} for each n≥2\documentclass[12pt]{minimal}
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\begin{document}$${n\geq 2}$$\end{document}, with equality only for the Koebe function k(z)=z/(1-z)2\documentclass[12pt]{minimal}
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\begin{document}$${k(z)=z/(1-z)^2}$$\end{document} and its rotations. This conjecture remains open although it has been verified for a few geometric subclasses of the class of univalent analytic functions. In this paper, we consider this problem for the family of normalized functions f analytic and univalent in the unit disk |z| < 1 satisfying the condition
Re1+zf′′(z)f′(z)>-12for|z|<1.\documentclass[12pt]{minimal}
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\begin{document}$${\rm Re }\left( 1+\frac{zf''(z)}{f'(z)}\right) > -\frac{1}{2}\,\,\,\,\,{\rm for}\,\,\,\,\,|z|<1.$$\end{document}Functions satisfying this condition are known to be convex in some direction (and hence close-to-convex and univalent) in |z| < 1. A few other related basic results and remarks about the Hayman index of functions in this family are also presented.
