The Order of Starlikeness of New p-valent Meromorphic Functions

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1 Int. Journal of Math. Analysis, Vol. 6, 2012, no. 27, The Order of Starlikeness of New p-valent Meromorphic Functions Aabed Mohammed and Maslina Darus School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia Bangi Selangor D. Ehsan, Malaysia Abstract In the present paper two general integral operators of meromorphic p -valent functions in the punctured open unit disk are introduced. Two subclasses of meromorphic p -valent functions are presented. The order of starlikeness of the above operators are also determined. As an application to the above operators, two p -valent meromorphic functions are defined and studied. Mathematics Subject Classification: 30C45 Keywords: Analytic function, meromorphic function, p -valent function, starlike function, convex function, integral operators 1 Introduction Let U = {z C : z < 1}, be the open unit disc in the complex plane C, U = U\{0}, the punctured open unit disk and H(U) ={f U C : f is holomorphic in U}. For a C and n N (N = {0, 1, 2,...}), let H[a, n] ={f H(U),f = a + a n z n + a n+1 z n , z U}. Let Σ p denote the class of meromorphic functions of the form f = 1 z + a p n z n (p N = N\{0}), (1.1) n=0

2 1330 A. Mohammed and M. Darus which are analytic and p -valent in U. We say that a function f Σ p is the meromorphic p -valent starlike of order α (0 α<p) and belongs to the class f Σ p (α), if it satisfies the inequality: ( ) zf R >α. f A function f Σ p is the meromorphic p -valent convex function of order α (0 α<p), if f satisfies the following inequality ( ) R 1+ zf >α, f and we denote this class by ΣK p (α). Many important properties and characteristics of various interesting subclasses of the class Σ p of meromorphically p -valent functions were investigated extensively by (among others) Uralegaddi and Somanatha ([7] and [8]), Liu and Srivastava ([12] and [13]), Mogra ([14]and [15]), Srivastava et a1.[16], Aouf et al. ([17] and [18]), Joshi and Srivastava [19], Owa et a1. [20] and Kulkarni et al. [21]. Analogous to the integral operators defined by Breaz et a1. ([9] and [10]), Frasin [6] and Mohammed and Darus ([1], [3]) on the normalized, p -valent and meromorphic analytic functions, we now define the following two integral operators on the space meromorphic p -valent functions in the class Σ p. Definition 1.1. Let n, p N,i {1, 2, 3,..., n}, > 0. We define the integral operator F p, γ1,..., γn (f 1,f 2,..., f n ):Σ n p Σ p by F p, γ1,..., γn =I(f 1,f 2,..., f n ) = 1 z z p+1 0 (u p f 1 (u)) γ 1...(u p f n (u)) γn du. (1.2) Definition 1.2. Let n, p N,i {1, 2, 3,..., n}, > 0. We define the integral operator J p, γ1,..., γn (f 1,f 2,..., f n ):Σ n p Σ p by J p, γ1,..., γn =I(f 1,f 2,..., f n ) = 1 z ( ) u p+1 γ1 ( ) u f p+1 γn z p+1 1 p (u)... f n p (u) du. (1.3) 0

3 p-valent meromorphic functions 1331 For the sake of simplicity, from now on we shall write F p, γ1,..., γn instead of F p, γ1,..., γn ( f 1,f 2,..., f n ) and J p, γ1,..., γn instead of J p, γ1,..., γn (f 1,f 2,..., f n ). If we take p =1, we obtain the general integral operators F 1, γ1,..., γn = H and J 1, γ1,..., γn =H γ1,..., γn, introduced by the authors ([1] and [3]). For the f Σ p (p N), we introduce the following two new subclasses. Definition 1.3. Let a function f Σ p be analytic in U. Then f is in the class Ω p(β) ( 1 β<p), if, and only if, f satisfies ( ) zf f + p zf < R f + β. (1.4) Definition 1.4. Let a function f Σ p be analytic in U. Then f is in the class ΩK p (β) ( 1 β<p), if, and only if, f satisfies ( ) zf f +1+p zf < R f + β 1. (1.5) The following results will be useful in the sequel. Lemma 1.1([5]).Let n N,α,δ R,γ C with R[γ αδ] 0. If p H[p(0),n] with p(0) R and p(0) >α,then we have { } R p+ zp >α= Rp >α, z U. γ δp Theorem 1.2 [23] If f Σ p satisfies the inequality zf f zf f 1 <δ, 0 <δ<1 then f Σ p (p(1 δ)). Theorem 1.3 [23] If f Σ p satisfies the inequality ( f 1+ zf zf f zf f ( ) then f Σ p. p 1+pμ ) <μ, 0 <μ< 1 p

4 1332 A. Mohammed and M. Darus 2 Starlikeness of the operator F p, γ1,..., γn In this section we place conditions for the starlikeness of the integral operator F p, γ1,..., γn which is defined in (1.2). Theorem 2.1. For i {1,..., n}, let > 0 and f i Σ p (α i)(0 α i <p). If 0 < n (p α i ) p, then F p, γ1,..., γn is starlike by order p n (p α i ). Proof. A differentiation of F p, γ1,..., γn which is defined in (1.2), we get z p+1 F p, γ 1,..., γn +(p +1)z p F p, γ1,..., γn =(z p f 1 ) γ1 (z p f n ) γn, (2.1) and z p+1 F p, γ 1,...,γ n +2(p +1)z p F p, γ 1,...,γ n +p(p +1)z p 1 F p, γ1,...,γ n = n ( z p f ) i +pz p 1 f i [(z p f z p 1 ) γ1 (z p f n ) γn ] (2.2) f i Then from (2.1) and (2.2), we obtain z p+1 F p, γ1,...,γ n +2(p +1)z p F p, γ1,...,γ n +p(p +1)z p 1 F p, γ1,...,γ n z p+1 F p, γ 1,...,γ n +(p +1)z p F p, γ1,...,γ n = n ( fi f i + p ). (2.3) z By multiplying (2.3)with z yield, z p+1 F p, γ 1,...,γ n +2(p +1)z p F p, γ 1,...,γ n +p(p +1)z p 1 F p, γ1,...,γ n z p F p, γ1,...,γ n +(p +1)z p 1 F p, γ1,...,γ n That is equivalent to = n ( ) zfi f i + p. z p+1 F p, γ 1,...,γ n +(p +2)z p F p, γ 1,...,γ n z p F p, γ1,...,γ n +(p +1)z p 1 F p, γ1,...,γ n + p = n ( ) zfi f i + p. (2.4)

5 p-valent meromorphic functions 1333 And z ( zf p, γ 1,...,γ n +(p +2)F p, γ 1,...,γ n ) zf p, = γ 1,...,γ n +(p +1)F p, γ1,...,γ n n ( ) zfi f i + p + p. (2.5) We can write (2.5), as the following ( ) zf p,γ 1,...,γn zf p,γ 1,...,γn F p, γ1,..., γn F p, γ1,..., γn + p +2 zf p, γ1,..., γn F p, γ1,..., γn + p +1 = n ( ) zfi f i + p + p. (2.6) We define the regular function q in U by q = zf p, γ 1,..., γn F p, γ1,..., γn, (2.7) and q(0) = p. Differentiating q logarithmically, we obtain q+ zq q =1+zF p, γ1,..., γn F p, γ 1,..., γn. (2.8) From (2.6),(2.7) and (2.8) we obtain zq n q+ p +1 q = Since f i Σ p (α i), for i {1,,n}, we receive { } zq n R q+ >p p +1 q ( ) zfi f i + p + p. (2.9) (p α i ). (2.10) It is clear that q is analytic in U with q(0) = p>p n (p α i ). We also have R(γ δα) > 0,(for γ = p +1,δ = 1and α =p n (p α i ). ) Since the conditions from Lemma 1.1 are met, we obtain Rq >p n (p α i ), which is equivalent to zf p, γ1,..., γn n F p, γ1,..., γn >p (p α i ). that is F p, γ1,..., γn is starlike of order p n (p α i ).

6 1334 A. Mohammed and M. Darus Theorem 2.2. For i {1,..., n}, let > 0 and f i Ω p (β i)( 1 β i <p). If 0 < n (p β i ) p, then F p, γ1,..., γn is starlike by order p n (p β i ). Proof. Using (2.9), we have zq n ( ) q+ p +1 q = zfi f i + β i + p n (p β i ). (2.11) Since f i Ω p(β i ), for i {1,, n}, we get { } zq n zf i R q+ > γ i p +1 q f i + p n + p (p β i ). (2.12) Because n zf i f i + p > 0, we obtain that { } zq R q+ >p p +1 q n (p β i ). (2.13) The remaining part of the proof follows the pattern of those in Theorem 2.1. The proof is complete. 3 Starlikeness of the operator J p, γ1,..., γn In this section we place conditions for the starlikeness of the integral operator J p, γ1,..., γn which is defined in (1.3). Theorem 2.3. For i {1,..., n}, let > 0 and f i ΣK p (α i )(0 α i <p). If 0 < n (p α i ) p, then J p, γ1,..., γn is starlike by order p n (p α i ). Proof. A differentiation of J p, γ1,..., γn, which is defined in (1.3), we get z p+1 J p, γ 1,..., γn +(p +1)z p J p, γ1,..., γn and ( ) z p+1 γ1 ( ) z = f 1 p+1 γn p f n p, (3.1) z p+1 J p, γ1,...,γ n +2(p +1)z p J p, γ1,...,γ n +p(p +1)z p 1 J p, γ1,...,γ n =

7 p-valent meromorphic functions 1335 n ( z p+1 f i +(p +1)z p f )[( ) i z p+1 γ1 ( ) z z p+1 f i f 1 p+1 γn ] p f n p Then from (3.1) and (3.2), we obtain z p+1 J p, γ 1,...,γ n +2(p +1)z p J p, γ 1,...,γ n +p(p +1)z p 1 J p, γ1,...,γ n z p J p, γ 1,...,γ n +(p +1)z p 1 J p, γ1,...,γ n (3.2) = n That is equivalent to z ( zj p, γ 1,...,γ n +(p +2)J p, γ 1,...,γ n ) zj p, = γ 1,...,γ n +(p +1)J p, γ1,...,γ n ( ) zfi f i + p +1. (3.3) n ( ) zfi f i + p +1 + p. (3.4) We can write (3.4), as the following ( ) zj p,γ 1,...,γn zj p,γ 1,...,γn J p, γ1,..., γn + p +2 J p,γ 1,...,γn zj p, γ1,..., γn J p, γ1,..., γn + p +1 = n ( ) zfi f i + p +1 + p. (3.5) Define the regular function q in U by q = zj p, γ1,..., γn J p, γ1,..., γn, (3.6) and q(0) = p. Differentiating q logarithmically, we obtain q+ zq q From (3.5),(3.6) and (3.7) we obtain zq n q+ p +1 q = =1+zJ p, γ 1,..., γn J p, γ1,..., γn. (3.7) ( ) zfi f i + p +1 + p. (3.8) Since f i ΣK p (α i ), for i {1,,n}, we receive { } zq n R q+ >p (p α i ). (3.9) p +1 q

8 1336 A. Mohammed and M. Darus The remaining part of the proof follows the pattern of those in Theorem 2.1. Theorem 2.4. For i {1,..., n}, let > 0 and f i ΩK p (β i )( 1 β i <p). If 0 < n (p β i ) p, then J p, γ1,..., γn is starlike by order p n (p β i ). Other work that we can look at regarding these operators is on superordinationpreseving integral operator[22] and related to it (see [2],[4], [11]). 4 Application As an application to the integral operators F p, γ1,..., γn and J p, γ1,..., γn, we define the following two p -valent meromorphic functions. Definition 4.1. Let F p, γ1,..., γn be the integral operator defined as in (1.2). We define the following function, Φ =zf p, γ1,..., γn +(p +1)F p, γ1,..., γn. (4.1) Definition 4.2 Let J p, γ1,..., γn be the integral operator defined as in (1.3). We define the following function, Υ =zj p, γ1,..., γn +(p +1)J p, γ1,..., γn. (4.2) Next, we study some orders of starlikeness of Φ and Υ. Theorem 4.1. For i {1,..., n}, let R, > 0 and f i Σ p (α i)(0 α i <p). If 0 < n (p α i ) p, then Φ given by (4.1) is starlike by order p n (p α i ). Proof. A successive differentiation of F p, γ1,..., γn which is defined in (1.2), we get z p Φ =z p+1 F p,γ 1,...,γ n +(p +1)z p F p,γ1,...,γ n =(z p f 1 ) γ 1...(z p f n ) γn. (4.3) Using (4.3), we get zφ n Φ = zf i n f i + p p. (4.4)

9 p-valent meromorphic functions 1337 Since f i Σ p (α i) we receive R zφ Φ = n { R zf i f i } p n + p >p n (p α i ). But by the hypothesis, 0 p n (p α i ) < 1. Thus Φ is starlike by order p n (p α i ). Theorem 4.2. For i {1,..., n}, let R, >pand ( f i Σ p p, then Φ belong to Σ (0). ) n n +1. If Now, adopting the same technique used in Theorem 4.1 and applying Theorem 1.2 and Theorem 1.3, one can prove Theorem 4.3. For i {1,..., n}, let > 0, 0 <δ i < 1, f i Σ p, and zf i f i zf i f i 1 <δ i. If 0 < n δ i 1, then Φ is starlike by order p p n δ i. Theorem 4.4. For i {1,..., n}, let > 0, 0 <μ i < 1 p, f i Σ p and If 0 < μ i 1 p +μ i f i zf i (1+ zf i f i zf i ) <μ i. f i 1 then Φ is starlike by order p p μ i 1. p +μ i Finally, we introduce the following results for the function Υ. Theorem 4.5. For i {1,..., n}, let R, > 0 and f i ΣK p (α i )(0 α i <p). If 0 < n (p α i ) p, then Υ given by (4.2) is starlike by order p n (p α i ).

10 1338 A. Mohammed and M. Darus Theorem 4.6. For i {1,..., n}, let R, >pand ( p f i ΣK p, then Υ belong to Σ (0). ) n n +1. If Acknowledgement:The work here was supported by MOHE grant: UKM- ST-06-FRGS References [1] A. Mohammed and M. Darus, A new integral operator for meromorphic functions, Acta Universitatis Apulensis, 24 (2010,) [2] A. Mohammed and M. Darus, New properties for certain integral operators, Int. Journal of Math. Analysis, 4(42) (2010), [3] A. Mohammed and M. Darus, Starlikeness properties for a new integral operator for meromorphic functions, Journal of Applied Mathematics, 2011(Article ID ), 8 (2011). [4] A. Mohammed and M. Darus, Integral operators on new families of meromorphic functions of complex order. Journal of Inequalities and Applications 2011, 2011:121, 12 pages. [5] A. Totoi, On integral operators of meromorphic functions, General Mathematics, 18(3) (2010), [6] B.A. Frasin, New general integral operators of p -valent functions, J. Inequal. Pure Appl. Math., 10(4)(2009), Article 109, 1-9. [7] B.A.Uralegaddi and C.Somanatha, New criteria for meromorphic starlike univalent functions, Bult. Austral.Math. Soc., 43(1991), [8] B.A.Uralegaddi and C.Somanatha, Certain classes of meromorphic multivalent functions,tamkang J.Math., 23(1992), [9] D. Breaz and N. Breaz,Two integral operators, Studia Universitatis Babes- Bolyai, Mathematica, 47(3)(2002), [10] D. Breaz, S. Owa, and N. Breaz, A new integral univalent operator, Acta Univ. Apulensis Math. Inform., 16 (2008),11-16.

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12 1340 A. Mohammed and M. Darus [23] Z-G. Wang, Z-H. Liu, R-G. Xiang, Some criteria for meromorphic multivalent starlike functions, Applied Mathematics and Computation, 218(2011), pp Received: January, 2012