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On certain properties of some subclasses of univalent functions

Nikola Tuneski, Milutin Obradović · 2020 📄 arXiv →

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Abstract

In this paper we determine the disks $|z|<r\le1$ where for different classes of univalent functions, we have the property $${\rm Re}\left\{2\frac{zf'(z)}{f(z)}-\frac{z f''(z)}{f'(z)}\right\}>0\qquad (|z|<r).$$

Results & Lemmas (1)

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Theorem 1. Theorem 1. Let D(f; z) be defined by (3). Then Re[D(f; z)] > 0 (|z| < r∗) in each of the following cases: (i) f ∈U and r∗= r1 = 0.839... is…
Theorem 1. Let D(f; z) be defined by (3). Then Re[D(f; z)] > 0 (|z| < r∗) in each of the following cases: (i) f ∈U and r∗= r1 = 0.839 . . . is the root of the equation r3 + 2r2 −2 = 0; (ii) f ∈S⋆(1/2) and r∗= r2 = q √ 5−1 2 = 078615 . . .; (iii) f ∈G and r∗= r3 = 2 3 = 0.666 . . .; (iv) f ∈S⋆and r∗= r4 = 1 2 = 0.5; (v) f ∈S and r∗= r5 = 1
Function classes studied:
close-to-convex

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