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

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1 El-Ashwah ournal of Inequalities and Applications 1, 1:35 RESEARCH Results regarding the arguent of certain p-valent analytic functions defined by a generalized integral operator R M El-Ashwah Open Access Correspondence: r_elashwah@yahoo.co Departent of Matheatics, Faculty of Science Daietta Branch), Mansoura University, New Daietta 34517, Egypt Abstract The integral operator p λ, l)λ >; l ; p N; N = N {}, where N = {1,,...}) for functions of the for f z) =z p + which are analytic and p-valent in the open unit disc U ={zîc: z < 1} was introduced by El-Ashwah and Aouf. The object of the present article is to drive interesting arguent results of p-valent analytic functions defined by this integral operator. 1 Matheatics Subject Classification: 3C45. Keywords: analytic, p-valent, integral operator, arguent 1 Introduction Let Ap) denotes the class of functions of the for: f z) =z p + p N = {1,,...}), 1:1) which are analytic and p-valent in the open unit disc U ={z Î C: z < 1}. We note that A1) = A, the class of univalent functions. In [1], Catas defined the linear operator p λ, l)f z) as follows: ) p + l + λk p) Ip λ, l)f z) =zp + p + l λ ; l ; p N; N ). 1:) Also, El-Ashwah and Aouf [] defined the integral operator p λ, l)f z) as follows: p λ, l)f z) =zp + ) p + l p + l + λk p) λ ; l ; p N; N ). 1:3) The operator p λ, l)f z) was studied by Srivastava et al. [3] and Aouf et al. [4]. Fro 1.) and 1.3), we observe that p λ, l)f z) =Ip λ, l)f z) > ), sothe operator p λ, l)f z) is well-defined for l, l, p Î N and Î Z = {..., -, -1,,1,,...}. 1 El-Ashwah; licensee Springer. This is an Open Access article distributed under the ters of the Creative Coons Attribution License which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited.

2 El-Ashwah ournal of Inequalities and Applications 1, 1:35 Page of 5 Fro 1.3), it is easy to verify that see, []) λzp +1 λ, l)f z)) =l + p)p λ, l)f z) [l + p1 λ)]+1 p λ, l)f z) λ >; l ; p N; N ). 1:4) We note that: i) 1 λ,)fz) =I λ f z) ) see Patel [5]); ii) p α1, 1)f z) =Iα p f z)α >) see Shas et al. [6]); iii) p 1, 1)f z) =D f z) see Patel and Sahoo [7]); iv) 1 λ,)f z) =I λ f z) see Al-Oboudi and Al-Qahtani [8]); v) 1 α1, 1)f z) =Iα f z)α >) see ung et al. [9]); vi) 1 1, 1)f z) =I f z) see Flett [1]); vii) 1 1, )f z) =L f z) see, Salagean [11]). Also we note that: ) i) p 1, )f z) = p f z) =zp + p p N; N ; z U); k ) ii) p 1, l)f z) = p l)f z) =zp + p + l k + l p N; N ; l ; z U). In this article, we drive interesting arguent results of p-valent analytic functions defined by the integral operator p λ, l)f z). Main results In order to prove our ain results, we recall the following lea. Lea 1 [1]. Let hz) beanalyticinu with h) z Î U). Further suppose that a, b Î R + =, ) and arghz)+βzh z)) < π α + π arctanβα) ) α, β>), :1) arghz)) < π α z U). :) Unless otherwise entioned we shall assue throughout the article that a, g, Î R +, l >,l, p Î N, Î Z and the powers are understood as principle values. Theore 1. Let gz) Î Ap). Suppose fz) Î Ap) satisfies the following condition γ { )}) arg p λ, l)f z) p 1+ 1 p λ, l)f z) λ, l)gz) λ p 1 p λ, l)gz) λ, l)f z) p λ, l)gz) < π α + π [ ]) γ l + p) α :3) γ arg p λ, l)f z) p λ, l)gz) < π α z U). :4)

3 El-Ashwah ournal of Inequalities and Applications 1, 1:35 Page 3 of 5 Proof. Define a function γ p λ, l)f z) hz) = p, γ :5) λ, l)gz) hz) =1+c 1 z +..., is analytic in U with h) = 1 and h ). Differentiating.5) logarithically with respect to z and ultiplying by z, we have 1 zh z) z γ hz) = p λ, l)f z)) p z p λ, l)gz)) λ, l)f z) p. :6) λ, l)gz) Using 1.4) in.6), we obtain γ { p λ, l)f z) hz)+ γ l + p) zh z) = p 1+ λ, l)gz) λ 1 p λ, l)f z) p λ, l)f z) )} λ, l)gz) p. :7) λ, l)gz) 1 p By using Lea 1, the proof of Theore 1 is copleted. Reark 1. Putting l = = p =1,l = =,andgz) =z, in Theore 1, we obtain the result obtained by Lashin [1, Theore.]. Putting g = 1 and gz) =z p in Theore 1, we obtain the following corollary: Corollary 1. Iffz) Î Ap) satisfies the following condition arg { λ p 1 λ, l)f z) z p +1 λ ) p } λ, l)f z) z p < π α + π [ ]) arctan l + p) α :8) ) arg p λ, l)f z) z p < π α z U). :9) Next, putting p = 1 in Corollary 1, we obtain the following corollary: Corollary. Iffz) Î A satisfies the following condition { arg 1 1 λ, l)f z) + 1 ) } 1 λ, l)f z) λ z λ z < π α + π [ ]) arctan l +1) α :1) ) arg 1 λ, l)f z) < π α z z U). :11) Reark. Putting l = 1 and l = = in Corollary we obtain the result obtained by Lashin [1, Exaple.]. Finally, putting g = 1 and fz) =z p in Theore 1, we obtain the following corollary: z p Corollary 3. Let p,gz) Ap) and. Suppose that λ, l)gz) { arg 1+ ) )} z p λ p λ, l)gz) p 1 λ, l)gz) z p λ p λ, l)gz) p λ, l)gz) < π α + π [ ]) :1) arctan l + p) α

4 El-Ashwah ournal of Inequalities and Applications 1, 1:35 Page 4 of 5 ) arg z p p λ, l)gz) < π α. :13) Theore. Let < 1. Suppose fz) Î Ap) satisfies the following condition ) arg p λ, l)f z) z p < π α + π [ ]) arctan γ l + p) α z U) :14) we have l + p) arg γ l + p) γ z Proof. Consider the function hz) = z t γ l + p) p +1) p λ, l)f t)dt < π α. :15) l + p) γ l + p) γ z γ l + p) p +1) z t p λ, l)f t)dt :16) hz) =1+c 1 z +..., is analytic in U with h) = 1 and h ). Differentiating.16) with respect to z, we have hz)+ γ l + p) zh z) = p λ, l)f z) z p. :17) By using Lea 1, the proof of Theore is copleted. Putting p = = g = 1 and = in Theore we obtain the following corollary: Corollary 4. Let fz) Î A satisfies the following condition f z) arg z arg l +1 z l+1) ) < π z α + π arctan [ α ] ) :18) l +1 t l 1) f t)dt < π αz U). :19) Reark 3. i) Putting l = in Corollary 4 we obtain the result obtained by Goyal and Goswai [13, Corollary 3.6]. ii) By specifying the paraeters p, l, l, and we obtain various results for different operators reinded in the introduction. Acknowledgeents The author thanks the referees for their valuable suggestions which led to iproveent of this study. Also, he would like to express his sincere gratitude to Springer Open Accounts Tea for their kind help. Copeting interests The authors declare that they have no copeting interests. Received: 8 Noveber 11 Accepted: 16 February 1 Published: 16 February 1

5 El-Ashwah ournal of Inequalities and Applications 1, 1:35 Page 5 of 5 References 1. Catas, A: On certain classes of p-valent functions defined by ultiplier transforations. Proc Book International Syposiu on Geoetric Function Theory and Applications. pp Istanbul, Turkey 7). El-Ashwah, RM, Aouf, MK: Soe properties of new integral operator. Acta Univ Apul., 4: ) 3. Srivastava, HM, Aouf, MK, El-Ashwah, RM: Soe inclusion relationships associated with a certain class of integral operators. Asian-Europ Math. 34): ). doi:1.114/s Aouf, MK, Mostafa, AO, El-Ashwah, R: Sandwich theores for p-valent functions defined by a certain integral operator. Math Coput Model ): ). doi:1.116/j.c Patel, : Inclusion relations and convolution properties of certain subclasses of analytic functions defined by a generalized Salagean operator. Bull Belg Math Soc Sion Stevin. 15, ) 6. Shas, S, Kulkarni, SR, ahangiri, M: Subordination properties of p-valent functions defined by integral operators. Internat Math Math Sci 1 3 6). Art. ID 9457, 7. Patel,, Sahoo, P: Certain subclasses of ultivalent analytic functions. Indian Pure Appl Math. 343): ) 8. Al-Oboudi, FM, Al-Qahtani, ZM: On a subclass of analytic functions defined by a new ultiplier integral operator. Far East Math Sci. 51):59 7 7) 9. ung, TB, Ki, YC, Srivastava, HM: The Hardy space of analytic functions associated with certain one-paraeter failies of integral operator. Math Anal Appl. 176, ). doi:1.16/jaa Flett, TM: The dual of an inequality of Hardy and Littlewood and soe related inequalities. Math Anal Appl. 38, ). doi:1.116/-47x7) Salagean, GS: Subclasses of univalent functions, Lecture Notes in Math. pp Springer-Verlag)113, 1983) 1. Lashin, AY: Application of Nunokawa s theore. Inequal Pure Appl Math 54):1 5 4). Art Goyal, SP, Goswai, P: Arguent estiate of certain ultivalent analytic functions defined by integral operators. Tasui Oxford Math Sci. 53):85 9 1) doi:1.1186/19-4x-1-35 Cite this article as: El-Ashwah: Results regarding the arguent of certain p-valent analytic functions defined by a generalized integral operator. ournal of Inequalities and Applications 1 1:35. Subit your anuscript to a journal and benefit fro: 7 Convenient online subission 7 Rigorous peer review 7 Iediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the field 7 Retaining the copyright to your article Subit your next anuscript at 7 springeropen.co