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- Publisher
- Springer Journals
- Copyright
- Copyright © 2017 by Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia
- Subject
- Mathematics; Mathematics, general; Applications of Mathematics
- ISSN
- 0126-6705
- eISSN
- 2180-4206
- DOI
- 10.1007/s40840-017-0546-0
- Publisher site
- See Article on Publisher Site
Abstract
Let $${{\mathcal {A}}}$$ A denote the family of all functions that are analytic in the unit disk $${\mathbb {D}} := \{ z:\, |z| < 1 \}$$ D : = { z : | z | < 1 } and satisfy $$f(0)=0= f'(0)-1$$ f ( 0 ) = 0 = f ′ ( 0 ) - 1 . Let $${\mathcal {U}} $$ U denote the subset of functions $$f\in {{\mathcal {A}}}$$ f ∈ A which satisfy $$\begin{aligned} \left| \left( \frac{z}{f(z)} \right) ^{2}f'(z)-1\right| < 1,\quad z\in {\mathbb {D}}, \end{aligned}$$ z f ( z ) 2 f ′ ( z ) - 1 < 1 , z ∈ D , and let $${\mathcal {P}}(2)$$ P ( 2 ) be the subclass of all functions $$f\in {\mathcal {A}}$$ f ∈ A such that $$f(z)\ne 0$$ f ( z ) ≠ 0 for $$0<|z|<1$$ 0 < | z | < 1 and $$\begin{aligned} \left| \left( \frac{z}{f(z)}\right) ''\right| \le 2,\quad z\in {\mathbb {D}}. \end{aligned}$$ z f ( z ) ′ ′ ≤ 2 , z ∈ D . In this paper, a conjecture on the class $${\mathcal {U}} $$ U and $${\mathcal {P}}(2)$$ P ( 2 ) has been resolved. Furthermore, two sufficient conditions for functions to be univalent are presented.
Journal
Bulletin of the Malaysian Mathematical Sciences Society – Springer Journals
Published: Sep 20, 2017