两类双单叶非Bazilevic函数族的系数估计
2016-03-17王智刚
石 磊,王智刚
(1.安阳师范学院 数学与统计学院,河南 安阳 455002;
两类双单叶非Bazilevic函数族的系数估计
石磊1,王智刚2
(1.安阳师范学院 数学与统计学院,河南 安阳 455002;
用A表示形如
(1)
一个函数f∈A被称为是非Bazilevic函数, 若其满足不等式
这类函数由Obradovic[1]引入和研究, 讨论的主要问题是这类函数能够嵌入到单叶函数或其子类的必要条件, 这个问题至今尚未完全解决. Wang等[2]引入并研究了推广的非Bazilevic函数族N(λ,μ,A,B), 将其定义为
其中:0<μ<1,λ∈C,-1≤B≤1,A≠B,A∈R.
其中
(2)
1主要结果
(3)
(4)
其中
(5)
(6)
其中
(7)
(8)
均属于正实部函数. 注意到f∈σ的Maclaurin展开式由(1)给出, 有
(9)
(10)
由(9)和(10)式易得
(11)
类似的计算可知
(12)
分别比较(5)和(6)式两边的系数, 可以得到
(13)
(14)
(15)
(16)
结合(13),(15)式可得
(17)
(18)
再联立(14),(16)和(17)式, 简单计算可知
(19)
将(18) 式中p12+q12的值代入(19), 易见
(20)
应用Keogh等[14]中的结果, 对任意复数ν, 有
(21)
(22)
将(17)、(18)代入(22)式, 则有
(23)
从而
(24)
另一方面, 将(17)、(19)代入(22)式, 则有
(25)
从而
(26)
类似地, 结合(17),(20),(22)式可知
(27)
从而
(28)
注1令λ=-1, 即得双单叶强非Bazilevic函数族起始项的系数估计
(29)
(30)
其中:p和q分别形如(7),(8)式.
由(29)和(30)式, 可得
应用类似于定理1的技巧可得定理2的结果.
参考文献:
[1]OBRADOVIC M. A class of univalent functions[J]. Hokkaido Math J, 1998, 27(2): 329-335.
[2]WANG Z G, GAO C Y, LIAO M X. On certain generalized class of Non-Bazilevic functions[J]. Acta Math Acad Paedagog Nyhazi, 2005, 21(1): 147-154.
[3]TUNESKI N, DARUS M. Fekete-Szego functional for Non-Bazilevic functions[J]. Acta Math Acad Paedagog Nyhazi, 2002, 18(1): 63-65.
[4]LEWIN M. On a coefficient problem for bi-univalent functions[J]. Proc Amer Math Soc, 1967, 18(1): 63-68.
[5]KEDZIERAWSKI A W. Some remaks on bi-univalent functions[J]. Ann Univ Mariae Curiesklodowska Sect A, 1985, 39(1): 77-81.
[7]ALI R M, LEE S K, RAVICHANDRAN V, et al. Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions[J]. Appl Math Lett, 2012, 25(3): 344-351.
[8]FRASIN B A, AOUF M K. New subclasses of bi-univalent functions[J]. Appl Math Lett, 2011, 24(9): 1569-1573.
[9]HAYAMI T, OWA S. Coefficient bounds for bi-univalent functions[J]. Panamer Math J, 2012, 22(4): 15-26.
[10]XU Q H, SRIVASTAVA H M, LI Z. A certain subclass of analytic and close-to-convex functions[J]. Appl Math Lett, 2011, 24(3): 396-401.
[11]XU Q H, XIAO H G, SRIVASTAVA H M. A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems[J]. Appl Math Comput, 2012, 218(23): 1461-1465.
[12]SRIVASTAVA H M, BULUT S, CAGLAR M, et al. Coefficient estimates for a general subclass of analytic and bi-univalent functions[J]. Filomat, 2013, 27(5): 831-842.
[13]SRIVASTAVA H M, MISHRA A K, GOCHHHAYAT P. Certain subclasses of analytic and bi-univalent functions[J]. Appl Math Lett, 2010, 23(10): 1188-1192.
[14]KEOGH F R, MERKES E P. A coefficient inequality for certain classes of analytic functions[J]. Proc Amer Math Soc, 1969, 20(1): 8-12.
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doi:10.3969/j.issn.1000-2162.2016.03.004
收稿日期:2015-03-27
基金项目:国家自然科学基金资助项目(11301008, 11426035);河南省高等学校重点科研基金资助项目(15A110006)
作者简介:石磊(1982-), 男, 河南信阳人, 安阳师范学院讲师.
中图分类号:O174
文献标志码:A
文章编号:1000-2162(2016)03-0017-05
2.湖南第一师范学院 数学与计算科学学院,湖南 长沙 410205)
Coefficient estimates for two subclasses of bi-univalent non-Bazilevic type functions
SHI Lei1, WANG Zhigang2
(1.School of Mathematical Science and Statistics, Anyang Normal University, Anyang 455002, China;2.School of Mathematics and Computing Science, Hunan First Normal University, Changsha 410205, China)
Key words:bi-univalent; non-Bazilevic functions; coefficient estimates; differential subordination
Abstract:The class of non-Bazilevic functions was introduced and studied by obradovic and has attracted many researchers’ interest. In the present paper, we introduced and investigated two subclasses of bi-univalent functions with non-Bazilevic type. For functions belonging to these subclasses, we obtained estimates for the initial coefficients a2and a3by using the coefficient estimates for analytic functions with positive real part and differential subordination. These results generalized some earlier works.
关键词:双单叶函数;非Bazilevic函数;系数估计;微分从属