石　磊，王智剛

(1.安陽師范學(xué)院 數(shù)學(xué)與統(tǒng)計學(xué)院，河南 安陽 455002；

兩類雙單葉非Bazilevic函數(shù)族的系數(shù)估計

石磊1，王智剛2

(1.安陽師范學(xué)院 數(shù)學(xué)與統(tǒng)計學(xué)院，河南 安陽 455002；

用A表示形如

(1)

一個函數(shù)f∈A被稱為是非Bazilevic函數(shù), 若其滿足不等式

這類函數(shù)由Obradovic[1]引入和研究, 討論的主要問題是這類函數(shù)能夠嵌入到單葉函數(shù)或其子類的必要條件, 這個問題至今尚未完全解決. Wang等[2]引入并研究了推廣的非Bazilevic函數(shù)族N(λ,μ,A,B), 將其定義為

其中:0<μ<1,λ∈C,-1≤B≤1,A≠B,A∈R.

其中

(2)

1主要結(jié)果

(3)

(4)

其中

(5)

(6)

其中

(7)

(8)

均屬于正實部函數(shù). 注意到f∈σ的Maclaurin展開式由(1)給出, 有

(9)

(10)

由(9)和(10)式易得

(11)

類似的計算可知

(12)

分別比較(5)和(6)式兩邊的系數(shù), 可以得到

(13)

(14)

(15)

(16)

結(jié)合(13)，(15)式可得

(17)

(18)

再聯(lián)立(14)，(16)和(17)式, 簡單計算可知

(19)

將(18) 式中p12+q12的值代入(19), 易見

(20)

應(yīng)用Keogh等[14]中的結(jié)果, 對任意復(fù)數(shù)ν, 有

(21)

(22)

將(17)、(18)代入(22)式, 則有

(23)

從而

(24)

另一方面, 將(17)、(19)代入(22)式, 則有

(25)

從而

(26)

類似地, 結(jié)合(17)，(20)，(22)式可知

(27)

從而

(28)

注1令λ=-1, 即得雙單葉強非Bazilevic函數(shù)族起始項的系數(shù)估計

(29)

(30)

其中：p和q分別形如(7)，(8)式.

由(29)和(30)式, 可得

應(yīng)用類似于定理1的技巧可得定理2的結(jié)果.

參考文獻：

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[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.

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[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.

(責(zé)任編輯朱夜明)

doi:10.3969/j.issn.1000-2162.2016.03.004

收稿日期：2015-03-27

基金項目:國家自然科學(xué)基金資助項目(11301008, 11426035)；河南省高等學(xué)校重點科研基金資助項目(15A110006)

作者簡介：石磊(1982-), 男, 河南信陽人, 安陽師范學(xué)院講師.

中圖分類號：O174

文獻標(biāo)志碼:A

文章編號:1000-2162(2016)03-0017-05

2.湖南第一師范學(xué)院 數(shù)學(xué)與計算科學(xué)學(xué)院，湖南 長沙 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.

關(guān)鍵詞:雙單葉函數(shù)；非Bazilevic函數(shù)；系數(shù)估計；微分從屬