The Pre-Schwarzian Norm Estimate for Analytic Concave Functions
Let D denote the open unit disk and let S denote the class of normalized univalent functions which are analytic in D. Let Co(α) be the class of concave functions f∈S, which have the condition that the opening angle of f(D) at infinity is less than or equal to πα, α∈(1,2]. In this paper, we find a su...
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2015-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2015/814805 |
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| author | Young Jae Sim Oh Sang Kwon |
| author_facet | Young Jae Sim Oh Sang Kwon |
| author_sort | Young Jae Sim |
| collection | DOAJ |
| description | Let D denote the open unit disk and let S denote the class of normalized univalent functions which are analytic in D. Let Co(α) be the class of concave functions f∈S, which have the condition that the opening angle of f(D) at infinity is less than or equal to πα, α∈(1,2]. In this paper, we find a sufficient condition for the Gaussian hypergeometric functions to be in the class Co(α). And we define a class Co(α,A,B), (-1≤B<A≤1), which is a subclass of Co(α) and we find the set of variabilities for the functional (1-|z|2)(f″(z)/f′(z)) for f∈Co(α,A,B). This gives sharp upper and lower estimates for the pre-Schwarzian norm of functions in Co(α,A,B). We also give a characterization for functions in Co(α,A,B) in terms of Hadamard product. |
| format | Article |
| id | doaj-art-ed6c58b3453e45729243e0a25d0d628d |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-ed6c58b3453e45729243e0a25d0d628d2025-02-03T07:26:03ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252015-01-01201510.1155/2015/814805814805The Pre-Schwarzian Norm Estimate for Analytic Concave FunctionsYoung Jae Sim0Oh Sang Kwon1Department of Mathematics, Kyungsung University, Busan 608-736, Republic of KoreaDepartment of Mathematics, Kyungsung University, Busan 608-736, Republic of KoreaLet D denote the open unit disk and let S denote the class of normalized univalent functions which are analytic in D. Let Co(α) be the class of concave functions f∈S, which have the condition that the opening angle of f(D) at infinity is less than or equal to πα, α∈(1,2]. In this paper, we find a sufficient condition for the Gaussian hypergeometric functions to be in the class Co(α). And we define a class Co(α,A,B), (-1≤B<A≤1), which is a subclass of Co(α) and we find the set of variabilities for the functional (1-|z|2)(f″(z)/f′(z)) for f∈Co(α,A,B). This gives sharp upper and lower estimates for the pre-Schwarzian norm of functions in Co(α,A,B). We also give a characterization for functions in Co(α,A,B) in terms of Hadamard product.http://dx.doi.org/10.1155/2015/814805 |
| spellingShingle | Young Jae Sim Oh Sang Kwon The Pre-Schwarzian Norm Estimate for Analytic Concave Functions International Journal of Mathematics and Mathematical Sciences |
| title | The Pre-Schwarzian Norm Estimate for Analytic Concave Functions |
| title_full | The Pre-Schwarzian Norm Estimate for Analytic Concave Functions |
| title_fullStr | The Pre-Schwarzian Norm Estimate for Analytic Concave Functions |
| title_full_unstemmed | The Pre-Schwarzian Norm Estimate for Analytic Concave Functions |
| title_short | The Pre-Schwarzian Norm Estimate for Analytic Concave Functions |
| title_sort | pre schwarzian norm estimate for analytic concave functions |
| url | http://dx.doi.org/10.1155/2015/814805 |
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