A Class of Analytic Functions with Missing Coefficients

Ding-Gong Yang and Jin-Lin Liu

Abstract and Applied Analysis, 2011, vol. 2011, 1-16

Abstract:

Let 𠑇 ð ‘› ( ð ´ , ð µ , ð ›¾ , ð ›¼ ) ( − 1 ≤ ð µ < 1 , ð µ < ð ´ , 0 < ð ›¾ ≤ 1 and ð ›¼ > 0 ) denote the class of functions of the form ∑ ð ‘“ ( 𠑧 ) = 𠑧 + ∞ 𠑘 = ð ‘› + 1 ð ‘Ž 𠑘 𠑧 𠑘 ( ð ‘› ∈ ð ‘ = { 1 , 2 , 3 , … } ) , which are analytic in the open unit disk 𠑈 and satisfy the following subordination condition ð ‘“ î…ž ( 𠑧 ) + ð ›¼ 𠑧 ð ‘“ î…ž î…ž ( 𠑧 ) ≺ ( ( 1 + ð ´ ð ‘§ ) / ( 1 + ð µ ð ‘§ ) ) ð ›¾ , for ( 𠑧 ∈ 𠑈 ; ð ´ â‰¤ 1 ; 0 < ð ›¾ < 1 ) , ( 1 + ð ´ ð ‘§ ) / ( 1 + ð µ ð ‘§ ) , for ( 𠑧 ∈ 𠑈 ; ð ›¾ = 1 ) . We obtain sharp bounds on R e ð ‘“ î…ž ( 𠑧 ) , R e ð ‘“ ( 𠑧 ) / 𠑧 , | ð ‘“ ( 𠑧 ) | , and coefficient estimates for functions ð ‘“ ( 𠑧 ) belonging to the class 𠑇 ð ‘› ( ð ´ , ð µ , ð ›¾ , ð ›¼ ) . Conditions for univalency and starlikeness, convolution properties, and the radius of convexity are also considered.

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:456729

DOI: 10.1155/2011/456729

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