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Uniform close-to-convexity radius of sections of functions in the close-to-convex family

s. V. Bharanedhar, S. Ponnusamy · 2014 📄 arXiv →

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Abstract

The authors consider the class $\F$ of normalized functions $f$ analytic in the unit disk $\ID$ and satisfying the condition $${\rm Re}\left(1+\frac{zf''(z)}{f'(z)}\right)>-\frac{1}{2},\quad z\in\D. $$ Recently, Ponnusamy et al. \cite{samy-hiroshi-swadesh} have shown that $1/6$ is the uniform sharp bound for the radius of convexity of every section of each function in the class $\F$. They conjectured that $1/3$ is the uniform univalence radius of every section of $f\in \F$. In this paper, we sol

Results & Lemmas (1)

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Theorem 1. · radius Theorem 1. Every section sn(f) of f ∈F satisfies Re (sn(f)′(z)) > 0 in the disk |z| < 1/3. In particular every section is close-to-convex in…
Theorem 1. Every section sn(f) of f ∈F satisfies Re (sn(f)′(z)) > 0 in the disk |z| < 1/3. In particular every section is close-to-convex in the disk |z| < 1/3. The radius 1/3 cannot be replaced by a greater one. We remark that this result is much stronger than the original conjecture. The following lemma is useful in the proof of Theorem 1. Lemma E. [12, Lemma 1] If f(z) = z+P∞ n=2 anzn ∈F, then the following estimates hold: (a) |an| ≤n + 1 2 for n ≥2. Equality holds for f0(z) given by (3) or it
Function classes studied:
close-to-convex

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