Results regarding the argument of certain p-valent analytic functions defined by a generalized integral operator

El-Ashwah

Description

The integral operator J p m ( λ , â„“ ) ( λ > 0 ; â„“ ≥ 0 ; p ∈ â„• ; m ∈ â„• 0 = â„• ∪ { 0 } , where â„• = {1,2,.}) for functions of the form f ( z ) = z p + ∑ k = p + 1 ∞ a k z k which are analytic and p -valent in the open unit disc U = { z ∈ â„‚: | z | < 1} was introduced by El-Ashwah and Aouf. The object of the present article is to drive interesting argument results of p -valent analytic functions defined by this integral operator. 2010 Mathematics Subject Classification : 30C45.

Sujets

• Analytic
• Integral transform
• Argument

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 10
• Langue: English

Extrait

ElAshwahJournal of Inequalities and Applications2012,2012:35 http://www.journalofinequalitiesandapplications.com/content/2012/1/35

R E S E A R C HOpen Access Results regarding the argument of certain pvalent analytic functions defined by a generalized integral operator R M ElAshwah

Correspondence: [e-mail] Department of Mathematics, Faculty of Science (Damietta Branch), Mansoura University, New Damietta 34517, Egypt

Abstract m The integral operator J(λ,)(λ >0;≥0;p ∞ ∈ N ;m∈ N 0 = N ∪ {0 , whereN=  p f(z) =z+a z {1,2,...}) for functions of the form k which are analytic andpvalent k= +1 in the open unit discU= {zÎℂ: |z| < 1} was introduced by ElAshwah and Aouf. The object of the present article is to drive interesting argument results ofpvalent analytic functions defined by this integral operator. 2010 Mathematics Subject Classification: 30C45. Keywords:analytic,pvalent, integral operator, argument

1 Introduction LetA(p) denotes the class of functions of the form: ∞  p k f(z p∈ N ={1, 2}) ) =z+a k z( ,. . . (1:1) k= +1 which are analytic andpvalent in the open unit discU= {zÎℂ: |z| < 1}. We note thatA(1) =A, the class of univalent functions. m In [1], Catas defined the linear operator (λ,)f(z as follows:  ∞  m  p++λ(k−p) m p I(λ,)f(z) =z+a k z p p+ (1:2) k=p+1 λ≥0;≥0;p∈ N ;m∈ N 0 . m Also, ElAshwah and Aouf [2] defined the integral operator (λ,)f(z as follows:   ∞ m  p+ m p J()f(z) =z+ p λ,a k z p++λ(k−p) (1:3) k=p+1 λ≥0;≥0;p∈ N ;m∈ N 0 . m The operator (λ,)f(z was studied by Srivastava et al. [3] and Aouf et al. [4]. − m From (1.2) and (1.3), we observe that p (λ,)f(z) =I(λ,)f(z)(m>0 , so the m operator (λ,)f(z is welldefined forl≥0,ℓ≥0,pÎNandmÎℤ= {..., 2, 1,0,1,2,...}.

© 2012 ElAshwah; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.