Starlikeness and convexity of a class of analytic functions
Let be the class of analytic functions in the unit disk that are normalized with f(0)=f′(0)−1=0 and let −1≤B<A≤1. In this paper we study the class Gλ,α={f∈:|(1−α+αzf″(z)/f′(z))/zf′(z)/f(z)−(1−α)|<λ,z∈},0≤α≤1, and give sharp sufficient conditions that embed it into the classes S∗[A,B]={f∈:zf′...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2006-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/IJMMS/2006/38089 |
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| Summary: | Let be the class of analytic functions in the unit disk that
are normalized with f(0)=f′(0)−1=0 and let −1≤B<A≤1. In this paper we study the class Gλ,α={f∈:|(1−α+αzf″(z)/f′(z))/zf′(z)/f(z)−(1−α)|<λ,z∈},0≤α≤1, and give sharp sufficient conditions that embed it into the classes
S∗[A,B]={f∈:zf′(z)/f(z)≺(1+Az)/(1+Bz)}
and K(δ)={f∈:1+zf″(z)/f′(z)≺(1−δ)(1+z)/(1−z)+δ}, where “≺” denotes the usual subordination. Also, sharp upper
bound of |a2| and of the Fekete-Szegö functional |a3−μa22| is given for the class Gλ,α. |
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| ISSN: | 0161-1712 1687-0425 |