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Abstract and Applied Analysis
Volume 2011, Article ID 456729, 16 pages
http://dx.doi.org/10.1155/2011/456729
Research Article

A Class of Analytic Functions with Missing Coefficients

1Department of Mathematics, Soochow University, Jiangsu, Suzhou 215006, China
2Department of Mathematics, Yangzhou University, Jiangsu, Yangzhou 225002, China

Received 1 March 2011; Accepted 9 May 2011

Academic Editor: Paul Eloe

Copyright © 2011 Ding-Gong Yang and Jin-Lin Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛾 , 𝛼 ) ( βˆ’ 1 ≀ 𝐡 < 1 , 𝐡 < 𝐴 , 0 < 𝛾 ≀ 1 and 𝛼 > 0 ) denote the class of functions of the form βˆ‘ 𝑓 ( 𝑧 ) = 𝑧 + ∞ π‘˜ = 𝑛 + 1 π‘Ž π‘˜ 𝑧 π‘˜ ( 𝑛 ∈ 𝑁 = { 1 , 2 , 3 , … } ) , which are analytic in the open unit disk π‘ˆ and satisfy the following subordination condition 𝑓 ξ…ž ( 𝑧 ) + 𝛼 𝑧 𝑓 ξ…ž ξ…ž ( 𝑧 ) β‰Ί ( ( 1 + 𝐴 𝑧 ) / ( 1 + 𝐡 𝑧 ) ) 𝛾 , for ( 𝑧 ∈ π‘ˆ ; 𝐴 ≀ 1 ; 0 < 𝛾 < 1 ) , ( 1 + 𝐴 𝑧 ) / ( 1 + 𝐡 𝑧 ) , for ( 𝑧 ∈ π‘ˆ ; 𝛾 = 1 ) . We obtain sharp bounds on R e 𝑓 ξ…ž ( 𝑧 ) , R e 𝑓 ( 𝑧 ) / 𝑧 , | 𝑓 ( 𝑧 ) | , and coefficient estimates for functions 𝑓 ( 𝑧 ) belonging to the class 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛾 , 𝛼 ) . Conditions for univalency and starlikeness, convolution properties, and the radius of convexity are also considered.