An application of hypergeometric functions to a problem in function theory
In some recent work in univalent function theory, Aharonov, Friedland, and Brannan studied the series (1+xt)α(1−t)β=∑n=0∞An(α,β)(x)tn. Brannan posed the problem of determining S={(α,β):|An(α,β)(eiθ)|<|An(α,β)(1)|, 0<θ<2π, α>0, β>0, n=1,2,3,…}. Brannan showed that if β≥α≥0, and...
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| Language: | English |
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Wiley
1984-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171284000545 |
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| author | Daniel S. Moak |
| author_facet | Daniel S. Moak |
| author_sort | Daniel S. Moak |
| collection | DOAJ |
| description | In some recent work in univalent function theory, Aharonov, Friedland, and Brannan studied the series (1+xt)α(1−t)β=∑n=0∞An(α,β)(x)tn. Brannan posed the problem of determining S={(α,β):|An(α,β)(eiθ)|<|An(α,β)(1)|, 0<θ<2π, 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 |A2k(α,1)e(iθ)|>|A2k(α,1)(1)| for k any integer. In this paper, we show that (α,β)∈S for α≥1 and β≥1. |
| format | Article |
| id | doaj-art-040a3e2eaa594e1ab60ffb049b4927db |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1984-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-040a3e2eaa594e1ab60ffb049b4927db2025-08-20T03:55:32ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251984-01-017350350610.1155/S0161171284000545An application of hypergeometric functions to a problem in function theoryDaniel S. Moak0Department of Mathematics, Texas Tech University, Lubbock 79409, Texas, USAIn some recent work in univalent function theory, Aharonov, Friedland, and Brannan studied the series (1+xt)α(1−t)β=∑n=0∞An(α,β)(x)tn. Brannan posed the problem of determining S={(α,β):|An(α,β)(eiθ)|<|An(α,β)(1)|, 0<θ<2π, 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 |A2k(α,1)e(iθ)|>|A2k(α,1)(1)| for k any integer. In this paper, we show that (α,β)∈S for α≥1 and β≥1.http://dx.doi.org/10.1155/S0161171284000545hypergeometric functionsJacobi polynomialsmaximum propertyand positive maximum property. |
| spellingShingle | Daniel S. Moak An application of hypergeometric functions to a problem in function theory International Journal of Mathematics and Mathematical Sciences hypergeometric functions Jacobi polynomials maximum property and positive maximum property. |
| title | An application of hypergeometric functions to a problem in function theory |
| title_full | An application of hypergeometric functions to a problem in function theory |
| title_fullStr | An application of hypergeometric functions to a problem in function theory |
| title_full_unstemmed | An application of hypergeometric functions to a problem in function theory |
| title_short | An application of hypergeometric functions to a problem in function theory |
| title_sort | application of hypergeometric functions to a problem in function theory |
| topic | hypergeometric functions Jacobi polynomials maximum property and positive maximum property. |
| url | http://dx.doi.org/10.1155/S0161171284000545 |
| work_keys_str_mv | AT danielsmoak anapplicationofhypergeometricfunctionstoaprobleminfunctiontheory AT danielsmoak applicationofhypergeometricfunctionstoaprobleminfunctiontheory |