An application of hypergeometric functions to a problem in function theory

Daniel S. Moak

International Journal of Mathematics and Mathematical Sciences, 1984, vol. 7, 1-4

Abstract:

In some recent work in univalent function theory, Aharonov, Friedland, and Brannan studied the series ( 1 + x t ) α ( 1 − t ) β = ∑ n = 0 ∞ A n ( α , β ) ( x ) t n . Brannan posed the problem of determining S = { ( α , β ) : | A n ( α , β ) ( e i θ ) | < | A n ( α , β ) ( 1 ) | , 0 < θ < 2 π , α > 0 , β > 0 , n = 1 , 2 , 3 , … } . Brannan showed that if β ≥ α ≥ 0 , and α + β ≥ 2 , then ( α , β ) ∈ S . He also proved that ( α , 1 ) ∈ S for α ≥ 1 . Brannan showed that for 0 < α < 1 and β = 1 , there exists a θ such that | A 2 k ( α , 1 ) e ( i θ ) | > | A 2 k ( α , 1 ) ( 1 ) | for k any integer. In this paper, we show that ( α , β ) ∈ S for α ≥ 1 and β ≥ 1 .

Date: 1984
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:450915

DOI: 10.1155/S0161171284000545

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