Majorization Properties for Subclass of Analytic p-valent Functions Defined by the Generalized Hypergeometric Function

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1 Tamsui Oxford Journal of Information and Mathematical Sciences 284) 2012) Aletheia University Majorization Properties for Subclass of Analytic p-valent Functions Defined by the Generalized Hypergeometric Function R. M. El-Ashwah Department of Mathematics, Faculty of Science Damietta Branch) Mansoura University, New Damietta 34517, Egypt Received April 25, 2011, Accepted April 24, Abstract The object of the present paper is to investigate the majorization properties of certain subclass of analytic and p-valent functions defined by the generalized hypergeometric function. Keywords and Phrases: Analytic, p-valent, Majorization. 1. Introduction Let f and g be analytic in the open unit disc U = {z C : z < 1}. We say that f is majorized by g in U see [11]) and write if there exists a function ϕ, analytic in U such that fz) << gz) z U), 1.1) ϕz) < 1 and fz) = ϕz)gz) z U). 1.2) 2000 Mathematics Subject Classification. Primary 30C45. r elashwah@yahoo.com

2 396 R. M. El-Ashwah It may be noted that 1.1) is closely related to the concept of quasi-subordination between analytic functions. For fz) and gz) are analytic in U, we say that fz) is subordinate to gz) written symbolically as follows: f g or fz) gz), if there exists a Schwarz function wz), which by definition) is analytic in U with w0) = 0 and wz) < 1 z U), such that fz) = gwz)) z U). Further, if the function gz) is univalent in U, then we have the following equivalent see [12, p. 4]) fz) gz) f0) = g0) and fu) gu). Let Ap) denote the class of functions of the form: fz) = z p + a k+p z k+p p N = {1, 2,...}), 1.3) k=1 which are analytic and p-valent in U. For complex parameters α 1,..., α q and β 1,..., β s β j / Z 0 = {0, 1, 2,...}; j = 1, 2,..., s), we now define the generalized hypergeometric function qf s α 1,..., α q ; β 1,..., β s ; z) by see, for example, [5] and [19, p. 20]) qf s α 1,..., α q ; β 1,..., β s ; z) = k=0 α 1 ) k...α q ) k β 1 ) k...β s ) k. zk k! 1.4) q s + 1; q, s N 0 = N {0}; z U), where θ) ν is the Pochhammer symbol defined, in terms of the Gamma function Γ, by { Γθ + ν) 1 ν = 0; θ C θ) ν = = = C\{0}), Γθ) θθ + 1)...θ + ν 1) ν N; θ C). 1.5) Corresponding to the function h p α 1,..., α q ; β 1,..., β s ; z), defined by h p α 1,..., α q ; β 1,..., β s ; z) = z p qf s α 1,..., α q ; β 1,..., β s ; z), 1.6)

3 Majorization Properties for Subclass of Analytic p-valent Functions 397 we consider a linear operator H p α 1,..., α q ; β 1,..., β s ; z) : Ap) Ap), which is defined by the following Hadamard product or convolution): H p α 1,..., α q ; β 1,..., β s ; z)fz) = h p α 1,..., α q ; β 1,..., β s ; z) fz). 1.7) We observe that, for a function fz) of the form 1.3), we have H p α 1,..., α q ; β 1,..., β s )fz) = z p + If, for convenience, we write α 1 ) k...α q ) k. a k+p z k+p. 1.8) β 1 ) k...β s ) k k! k=1 H p,q,s α 1 ) = H p α 1,..., α q ; β 1,..., β s ), 1.9) then one can easily verify from the definition 1.8) that see [5]) zh p,q,s α 1 )fz)) = α 1 H p,q,s α 1 + 1)fz) α 1 p)h p,q,s α 1 )fz). 1.10) It should be remarked that the linear operator H p,q,s α 1 ) is a generalization of many other linear operators considered earlier. In particular, for fz) Ap) we have the following observations: i) H p,2,1 a, 1; c)fz) = L p a; c)fz) a R; c R\Z 0 ), this linear operator studied by Saitoh [18] which yields the operator La, c)fz) introduced by Carlson and Shaffer [3] for p = 1; ii) H p,2,1 n + p, 1; 1)fz) = D n+p 1 fz) n N; n > p), this linear operator studied by Goel and Sohi [6]. In the case when p = 1, D n fz) is the Ruscheweyh derivative [17] of fz) A1); iii) H p,2,1 c, λ + p; a)fz) = Ip λ a, c)fz)a, c N\Z 0 ; λ > p), the Cho Kwon Srivastava operator [4]; iv) H p,2,1 1, p + 1; n + p)fz) = I n,p fz)n Z; n > p), the extended Noor integral operator considered by Liu and Noor [10]; v) H p,2,1 p + 1, 1; p + 1 λ)fz) = Ω λ,p) z fz) < λ < p + 1), the extended fractional differintegral operator considered by Patel and Mishra [15]. Now, by making use of the operator H p,q,s α 1 ), we define a new subclass of functions f Ap) as follows.

4 398 R. M. El-Ashwah Definition 1. Let 1 B < A 1, p N, j N 0 = N {0}, γ C, γa B) + α 1 B < α 1 and f Ap). Then f Sp,q,s j γ; α 1; A, B), the class of p-valent functions of complex order γ in U, if and only if { γ z H p,q,s α 1 )fz)) j+1) H p,q,s α 1 )fz)) j) p + j )} 1 + Az 1 + Bz, 1.11) Clearly, we have the following relationships: i) Sp,q,s j γ; α 1; 1, 1) = Sp,q,s j γ; α 1 ) ; ii) S j p,1,0 γ; 1; 1, 1) = Sp j γ) ; ii) S1,1,0 0 γ; 1; 1, 1) = S γ) γ C ) see [13]); iii) S1,1,0 0 1 α; 1, 1, 1) = S α) 0 α < 1) see [16]). Also, we note that: i) For j = 0, q = s + 1, α 1 = β 1 = p, α i = 1i = 2, 3,..., s + 1) and β i = 1i = 2, 3,...s), Sp,q,s j γ; α 1 ) reduces to the class S p γ) γ C ) of p valently starlike functions of order γ γ C ) in U, where S p γ) = { fz) Ap) : Re γ zf z) fz) p )) } > 0, p N, γ C ; ii) For j = 0, q = s+1, α 1 = p+1, β 1 = p, α i = 1i = 2, 3,..., s+1) and β i = 1i = 2, 3,...s), Sp,q,s j γ; α 1 ) we get the class K p γ) γ C ) of p valently convex functions of order γ γ C ) in U, where, K p γ) = { fz) Ap) : Re γ 1 + zf z) f z) p )) } > 0, p N, γ C. We shall need the following lemma. Lemma 1 [1]. Let γ C and f Kpγ). j Then f Sp j 1 γ), that is, 2 K j pγ) S j p 1 2 γ) γ C ). 1.12) A majorization problem for the class Sγ)γ C ) has been investigated by Altintas et al. [1]. Also, majorization problem for the class S = S 0) has been investigated by MacGregor [11]. Recently Goyal and Goswami [8] and Goyal et al. [9] generalized these results for classes of multivalent function defined by fractional derivatives operator and Saitoh operator, respectively. In this paper we investigate majorization problem for the class Sp,q,s j γ; α 1; A, B) and other related subclasses.

5 Majorization Properties for Subclass of Analytic p-valent Functions Main Results Unless otherwise mentioned we shall assume throughout the paper that 1 B < A 1, γ, α 1 C, j N 0 and p N. Theorem 1. Let the function f Ap) and suppose that g S j p,q,s γ; α 1; A, B). If H p,q,s α 1 )fz)) j) is majorized by H p,q,s α 1 )gz)) j) in U, then H p,q,s α 1 + 1)fz)) j) Hp,q,s α 1 + 1)gz)) j) z < r0 ), 2.1) where r 0 = r 0 γ, α 1, A, B) is the smallest positive root of the equation γa B) + α 1 B r 3 2 B + α 1 )r γa B) + α 1 B )r + α 1 = ) Proof. Since g Sp,q,s j γ; α 1; A, B), we find from 1.11) that γ z H p,q,s α 1 )gz)) j+1) H p,q,s α 1 )gz)) j) p + j ) = 1 + Awz) 1 + Bwz), 2.3) where w is analytic in U with w0) = 0 and wz) < 1 z U). From 2.3), we have z H p,q,s α 1 )gz)) j+1) H p,q,s α 1 )gz)) j) = Also from 1.10), we have p j) + γa B) + p j)b)wz). 2.4) 1 + Bwz) z H p,q,s α 1 )gz)) j+1) = α 1 H p,q,s α 1 + 1)gz)) j) α 1 +j p) H p,q,s α 1 )gz)) j). 2.5) From 2.4) and 2.5), we have H p,q,s α 1 )gz)) j) α B z ) α 1 γa B) + α 1 B z H p,q,s α 1 + 1)gz)) j). 2.6) Next, since H p,q,s α 1 )fz)) j) is majorized by H p,q,s α 1 )gz)) j) in U, from 1.2), we have H p,q,s α 1 )fz)) j) = ϕz) H p,q,s α 1 )gz)) j). 2.7)

6 400 R. M. El-Ashwah Differentiating 2.7) with respect to z and multiplying by z, we have z H p,q,s α 1 )fz)) j+1) = zϕ z) H p,q,s α 1 )gz)) j) +zϕz) H p,q,s α 1 )gz)) j+1), 2.8) using 2.5) in 2.8), we have H p,q,s α 1 + 1)fz)) j) = zϕ z) α 1 H p,q,s α 1 )gz)) j) +ϕz) H p,q,s α 1 + 1)gz)) j). Thus, by noting that ϕz) satisfies the inequality see [14]), 2.9) ϕ z) 1 ϕz) 2 1 z 2 z U), 2.10) and making use of 2.6) and 2.10) in 2.9), we have H p,q,s α 1 + 1)fz)) j) ) ϕz) + 1 ϕz) B z ) z 1 z 2. Hp,q,sα 1 + 1)gz)) j), α 1 γa B) + α 1 B z 2.11) which upon setting leads us to the inequality H p,q,s α 1 + 1)fz)) j) where z = r and ϕz) = ρ 0 ρ 1), Ψρ) H 1 r 2 p,q,s α 1 + 1)gz)) j), ) α 1 γa B) + α 1 B r) Ψρ) = r 1 + B r) ρ r 2 ) α 1 γa B) + α 1 B r)ρ +r 1 + B r), 2.12)

7 Majorization Properties for Subclass of Analytic p-valent Functions 401 takes its maximum value at ρ = 1, with r 0 = r 0 γ, α 1, A, B), where r 0 γ, α 1, A, B) is the smallest positive root of 2.2), then the function Φρ) defined by Φρ) = σ 1 + B σ) ρ σ 2 ) [ α 1 γa B) + α 1 B σ] ρ +σ 1 + B σ) 2.13) is an increasing function on the interval 0 ρ 1, so that Φρ) Φ1) = 1 σ 2 ) α 1 γa B) + α 1 B σ) 0 ρ 1; 0 σ r 0 p, γ, α 1, A, B)). 2.14) Hence upon setting ρ = 1 in 2.13), we conclude that 2.1) holds true for z r 0 = r 0 γ, α 1, A, B), where r 0 γ, α 1, A, B) is the smallest positive root of 2.2). This completes the proof of Theorem 1. Putting A = 1 and B = 1 in Theorem 1, we obtain the following corollary. Corollary 1. Let the function f Ap) and suppose that g S j p,q,s γ; α 1 ). If H p,q,s α 1 )fz)) j) is majorized by H p,q,s α 1 )gz)) j) in U, then H p,q,s α 1 + 1)fz)) j) Hp,q,s α 1 + 1)gz)) j) z < r0 ), where r 0 = r 0 γ; α 1 ) is given by r 0 = r 0 γ; α 1 ) = k k 2 4 2γ α 1 α 1, 2 2γ α 1 where k = 2 + α 1 + 2γ α 1, γ, α 1 C ). Putting q = s + 1, α 1 = β 1 = p, α i = 1i = 2, 3,..., s + 1) and β i = 1i = 2, 3,...s) in Corollary 1, we obtain the following corollary. Corollary 2 [1, Theorem 1]. Let the function f Ap) and suppose that g S j pγ). If f j) z) is majorized by g j) z) in U, then f j+1) z) g j+1) z) z < r 0 ),

8 402 R. M. El-Ashwah where r 0 = r 0 γ, p, j) is given by r 0 = r 0 γ, p, j) = k k 2 4p 2γ p + j, 2 2γ p + j where k = 2 + p j + 2γ p + j, p N, j N 0, γ C ). Putting j = 0 in Corollary 2, we obtain the following corollary. Corollary 3. Let the function f Ap) and suppose that g S p γ). If fz) is majorized by gz) in U, then where r 0 = r 0 γ, p) is given by f z) g z) z < r 0 ), r 0 = r 0 γ, p) = k k 2 4p 2γ p, 2 2γ p where k = 2 + p + 2γ p, p N, γ C ). Putting j = 0, q = s + 1, α 1 = p + 1, β 1 = p, α i = 1i = 2, 3,..., s + 1) and β i = 1i = 2, 3,...s), in Corollary 1, with the aid of Lemma 1 with j = 0), we obtain the following corollary. Corollary 4. Let the function f Ap) and suppose that g K p γ). If fz) is majorized by gz) in U, then where r 0 = r 0 γ, p) is given by f z) g z) z < r 0 ), r 0 = r 0 γ, p) = k k 2 4p γ p, 2 γ p where k = 2 + p + γ p, p N, γ C ). Putting A = 1, B = 1, p = 1, j = 0 and q = 2, s = 1, α 1 = α 2 = β 1 = 1, in Theorem 1, we obtain the following corollary.

9 Majorization Properties for Subclass of Analytic p-valent Functions 403 Corollary 5 [2, Theorem 1]. Let the function f A and suppose that g Sγ). If fz) is majorized by gz) in U, then where r 0 = r 0 γ) is given by where k = 3 + 2γ 1, γ C ). f z) g z) z < r 0 ), r 0 = r 0 γ) = k k 2 4 2γ 1, 2 2γ 1 Letting γ 1 in Corollary 5, we obtain the following corollary. Corollary 6 [11]. Let the function f A and suppose that g S. If fz) is majorized by gz) in U, then f z) g z) z < r 0 ), where r 0 is given by r 0 = 2 3. Remarks. i) Putting q = 2, s = 1, α 1 = p + 1, α 2 = 1 and β 1 = p + 1 λ, in Theorem 1 we obtain the result obtained by Goswami and Wang [7, Theorem 1]; ii) Putting A = 1, B = 1, q = 2, s = 1, α 1 = p + 1, α 2 = 1 and β 1 = p + 1 λ, in Corollary 1 we obtain the result obtained by Goyal and Goswami [8, Theorem 1]; iii) Putting q = 2, s = 1, α 1 R, α 2 = 1 and β 1 R\Z 0, in Theorem 1 we obtain the result obtained by Goyal et al. [9, Theorem 1]; iv) Also by specializing the parameters p,α i i = 1, 2,..., q) and β j j = 1, 2,..., s), we obtain various results corresponding to various operators defined in the introduction. Acknowledgments. The author thanks the referees for their valuable suggestions which led to improvement of this paper.

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