Hankel determinant problem of a subclass of analytic functions

Arif Muhammad, Noor Khalida, Raza, Raza Mohsan

Description

In this article, we study the Hankel determinant problem of a subclass of analytic functions introduced recently by Arif et al. 2010 Mathematics Subject Classification: 30C45; 30C10. In this article, we study the Hankel determinant problem of a subclass of analytic functions introduced recently by Arif et al. 2010 Mathematics Subject Classification: 30C45; 30C10.

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 49
• Langue: English

Extrait

Arifet al.Journal of Inequalities and Applications2012,2012:22 http://www.journalofinequalitiesandapplications.com/content/2012/1/22

R E S E A R C HOpen Access Hankel determinant problem of a subclass of analytic functions 1* 22 Muhammad Arif, Khalida Inayat Noorand Mohsan Raza

* Correspondence: [e-mail] 1 Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan Full list of author information is available at the end of the article

Abstract In this article, we study the Hankel determinant problem of a subclass of analytic functions introduced recently by Arif et al. 2010 Mathematics Subject Classification:30C45; 30C10. Keywords:Robertson function, strongly Bazilevic functions, bounded boundary rota tions, Hankel determinant

1 Introduction Let A be the class of analytic function satisfying the conditionf(0) = 0,f’(0)  1 = 0 in ∗ the open unit disc E ={z:|z|<1} . By S , S , C , and K we means the wellknown subclasses of A which consist of univalent, starlike, convex, and closetoconvex func tions, respectively. λ π Let V (σ),k≥2, 0≤σ <1,λreal,|λ|< , denote the class of functionsf 1 (z) k 2  (0) = 1 analytic and locally univalent inℰ,f 1 (0) = 0, f 1 and satisfying 2π     (zf(z)) iλ1iθ Ree −σcosλ(1−σ) dθ≤kπcosλ,z=re . (1:1)    f(z) 1 0 This class was introduced and studied in details by Moulis [1]. Forl= 0, we obtain the class V k (σ) of analytic functions with bounded boundary rotations of ordersstu died by Padmanabhan et al. [2] and whens= 0 andl= 0, we get the class V k dis λ cussed by Paatero [3], see also [48]. Also it can eas (z)∈(σ) if ily be shown that f 1 V k and only if there exists f 2 (z)∈ V k such that −iλ ecosλ  (1−σ) (1:2) f(z) = (f(z)) . 1 2 We now consider a class of analytic functions defined by Arif et al. [9] as follows: Definition 1.1. Let f(z)∈ A inE. Then ˜ , if fork≥2, 0≤b≤ f(z)∈ B k (λ,σ,β,γ) π λ 1, 0≤g≤1,lis real with |λ|< there exists a function f 1 (z)∈ V (σ) , 0≤s< 1, 2 k such that    γ 1−γ z f(z)f(z)β π arg≤,z∈E   f(z)f(z) 2

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