Hindawi Publishing Corporation Journal of Complex Analysis Volume 2013, Article ID 397428, 7 pages http://dx.doi.org/10.1155/2013/397428
Research Article Subclass of Multivalent Harmonic Functions Defined by Wright Generalized Hypergeometric Functions M. K. Aouf, A. O. Moustafa, and E. A. Adwan Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Correspondence should be addressed to E. A. Adwan;
[email protected] Received 13 May 2013; Revised 10 September 2013; Accepted 10 September 2013 Academic Editor: J. Dziok Copyright © 2013 M. K. Aouf et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce a new class of multivalent harmonic functions defined by Wright generalized hypergeometric function. Coefficient estimates, extreme points, distortion bounds, and convex combination for functions belonging to this class are obtained.
1. Introduction A continuous complex-valued function 𝑓 = 𝑢 + 𝑖V defined in a simply connected complex domain 𝐷 is said to be harmonic in 𝐷 if both 𝑢 and V are real harmonic in 𝐷. In any simply connected domain the function 𝑓(𝑧) can be written in the form 𝑓 = ℎ + 𝑔,
(1)
where ℎ and 𝑔 are analytic in 𝐷, ℎ is called the analytic part, and 𝑔 the coanalytic part of 𝑓. A necessary and sufficient condition for 𝑓 to be locally univalent and sense-preserving in 𝐷 is that |ℎ (𝑧)| > |𝑔 (𝑧)| in 𝐷 (see [1]). Denote by 𝑆𝐻 the class of functions 𝑓 of the form (1) that are harmonic univalent and sense preserving in the unit disc 𝑈 = {𝑧 : |𝑧| < 1} for which 𝑓(0) = 𝑓𝑧 (0) − 1 = 0. Recently, Ahuja and Jahangiri [2] defined the class 𝐻𝑝 (𝑝 ∈ N = {1, 2, . . .}) consisting of all 𝑝-valent harmonic functions 𝑓 = ℎ + 𝑔 that are sense-preserving in 𝑈 and ℎ, 𝑔 are of the form 𝑝
∞
𝑛
ℎ (𝑧) = 𝑧 + ∑ 𝑎𝑛 𝑧 , 𝑛=1+𝑝
∞
𝑛
𝑔 (𝑧) = ∑ 𝑏𝑛 𝑧 , 𝑛=𝑝
𝑏𝑝 < 1. (2)
Let 𝛼1 , 𝐴 1 , . . . , 𝛼𝑞 , 𝐴 𝑞 and 𝛽1 , 𝐵1 , . . . , 𝛽𝑠 , 𝐵𝑠 (𝑞, 𝑠 ∈ N0 = N ∪ {0}) be positive real parameters such that 𝑠
𝑞
𝑗=1
𝑗=1
1 + ∑ 𝐵𝑗 − ∑ 𝐴 𝑗 ≥ 0.
(3)
The Wright generalized hypergeometric function [3] (see also [4]) 𝑞 Ψ𝑠
[(𝛼1 , 𝐴 1 ) , . . . , (𝛼𝑞 , 𝐴 𝑞 ) ; (𝛽1 , 𝐵1 ) , . . . , (𝛽𝑠 , 𝐵𝑠 ) ; 𝑧] =
𝑞 Ψ𝑠 [(𝛼𝑖 , 𝐴 𝑖 )1,𝑞 ; (𝛽𝑖 , 𝐵𝑖 )1,𝑠 ; 𝑧]
(4)
is defined by 𝑞 Ψ𝑠
[(𝛼𝑖 , 𝐴 𝑖 )1,𝑞 ; (𝛽𝑖 , 𝐵𝑖 )1,𝑠 ; 𝑧] ∞
𝑞
∏𝑖=1 Γ (𝛼𝑖 + 𝑛𝐴 𝑖 ) 𝑧𝑛 ⋅ 𝑠 𝑛! 𝑛=0 ∏𝑖=1 Γ (𝛽𝑖 + 𝑛𝐵𝑖 )
=∑
(𝑧 ∈ 𝑈) .
(5)
If 𝐴 𝑖 = 1 (𝑖 = 1, . . . , 𝑞) and 𝐵𝑖 = 1 (𝑖 = 1, . . . , 𝑠), we have the relationship Ω𝑞 Ψ𝑠 [(𝛼𝑖 , 1)1,𝑞 ; (𝛽𝑖 , 1)1,𝑠 ; 𝑧] =𝑞 𝐹𝑠 (𝛼1 , . . . , 𝛼𝑞 ; 𝛽1 , . . . , 𝛽𝑠 ; 𝑧) , (6) where 𝑞 𝐹𝑠 (𝛼1 , . . . , 𝛼𝑞 ; 𝛽1 , . . . , 𝛽𝑠 ; 𝑧) is the generalized hypergeometric function (see [4]) and Ω=
∏𝑠𝑖=1 Γ (𝛽𝑖 ) . 𝑞 ∏𝑖=1 Γ (𝛼𝑖 )
(7)
By using the generalized hypergeometric function, Dziok and Srivastava [5] introduced a linear operator. Dziok and Raina in [6] and Aouf and Dziok in [7] extended this linear operator by using Wright generalized hypergeometric function.
2
Journal of Complex Analysis
Aouf et al. [8] defined the linear operator 𝜃𝑝,𝑞,𝑠 [(𝛼𝑖 , 𝐴 𝑖 )1,𝑞 ; (𝛽𝑖 , 𝐵𝑖 )1,𝑠 ] by the Hadamard product as 𝜃𝑝,𝑞,𝑠 [(𝛼𝑖 , 𝐴 𝑖 )1,𝑞 ; (𝛽𝑖 , 𝐵𝑖 )1,𝑠 ] 𝑓 (𝑧)
(8)
= 𝑞 Φ𝑝𝑠 [(𝛼𝑖 , 𝐴 𝑖 )1,𝑞 ; (𝛽𝑖 , 𝐵𝑖 )1,𝑠 ; 𝑧] ∗ 𝜑 (𝑧) ,
where 𝐻𝑝,𝑞,𝑠 [𝛼1 ] is the modified Dziok-Srivastava operator (see [10]). For 0 ≤ 𝛾 < 1, 𝑝 ∈ N and for all 𝑧 ∈ 𝑈, let 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾) denote the family of harmonic 𝑝valent functions 𝑓(𝑧) = ℎ(𝑧) + 𝑔(𝑧), where ℎ and 𝑔 are given by (2) and satisfying the analytic criterion
where 𝑞 Φ𝑝𝑠 [(𝛼𝑖 , 𝐴 𝑖 )1,𝑞 ; (𝛽𝑖 , 𝐵𝑖 )1,𝑠 ; 𝑧] is given by 𝑝 𝑞 Φ𝑠
Re { (𝑧(𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] ℎ(𝑧))
[(𝛼𝑖 , 𝐴 𝑖 )1,𝑞 ; (𝛽𝑖 , 𝐵𝑖 )1,𝑠 ; 𝑧]
−𝑧(𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] 𝑔(𝑧)) )
(9)
= Ω 𝑧𝑞𝑝 Ψ𝑠 [(𝛼𝑖 , 𝐴 𝑖 )1,𝑞 ; (𝛽𝑖 , 𝐵𝑖 )1,𝑠 ; 𝑧] .
× (𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] ℎ (𝑧)
We observe that, for a function 𝜑(𝑧) of the form (2), we have
−1
+𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] 𝑔 (𝑧)) } ≥ 𝑝𝛾.
𝜃𝑝,𝑞,𝑠 [(𝛼𝑖 , 𝐴 𝑖 )1,𝑞 ; (𝛽𝑖 , 𝐵𝑖 )1,𝑠 ] 𝜑 (𝑧) ∞
(10)
= 𝑧𝑝 + ∑ Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑎𝑛 𝑧𝑛 , 𝑛=1+𝑝
where Ω is given by (7) and 𝜎𝑛,𝑝 (𝛼1 ) is defined by
Let 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾) be the subclass of 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾) consisting of functions 𝑓 = ℎ + 𝑔 such that ℎ and 𝑔 are of the form ∞
ℎ (𝑧) = 𝑧𝑝 − ∑ 𝑎𝑛 𝑧𝑛 ,
𝜎𝑛,𝑝 (𝛼1 ) =
(15)
𝑛=1+𝑝
Γ (𝛼1 + 𝐴 1 (𝑛 − 𝑝)) ⋅ ⋅ ⋅ Γ (𝛼𝑞 + 𝐴 𝑞 (𝑛 − 𝑝)) Γ (𝛽1 + 𝐵1 (𝑛 − 𝑝)) ⋅ ⋅ ⋅ Γ (𝛽𝑠 + 𝐵𝑠 (𝑛 − 𝑝)) (𝑛 − 𝑝)!
∞
𝑔 (𝑧) = ∑ 𝑏𝑛 𝑧𝑛 , 𝑛=𝑝
𝑏𝑝 < 1. (16)
. (11)
We note that for suitable choices of 𝑞 and 𝑠, we obtain the following subclasses: ∗ (1) 𝐻𝑝 ([1, 1, 1], 2, 1; 𝛾) = 𝑆𝐻 (𝑝, 𝛾) (see [2]);
For convenience, we write 𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] 𝑓 (𝑧)
∗ (2) 𝐻𝑝 ([𝛼1 , 1, 1], 𝑞, 𝑠; 𝛾) = 𝑇𝐻 (𝑝, 𝛼1 , 𝛾) (see [10]);
= 𝜃𝑝,𝑞,𝑠 [(𝛼1 , 𝐴 1 ) , . . . , (𝛼𝑞 , 𝐴 𝑞 ) ; (𝛽1 , 𝐵1 ) , . . . , (𝛽𝑠 , 𝐵𝑠 )]
(3) 𝐻1 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾) = 𝑊𝐻([𝛼1 ], 𝛾) (see [9]); (4) 𝐻1 ([𝛼1 , 1, 1], 𝑞, 𝑠; 𝛾) = 𝑉𝐻(𝛼1 , 𝛾) (see [11, 12]);
× 𝜑 (𝑧) . (12)
∗ (𝛾) (see [13]). (5) 𝐻1 ([1, 1, 1], 2, 1; 𝛾) = 𝑆𝐻
Now we can define the modified Wright operator as follows:
2. Coefficient Estimates
𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] 𝑓 (𝑧) = 𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] ℎ (𝑧) + 𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] 𝑔 (𝑧),
(13)
where ∞
𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] ℎ (𝑧) = 𝑧𝑝 + ∑ Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑎𝑛 𝑧𝑛 ,
Theorem 1. Let 𝑓 = ℎ + 𝑔 be such that ℎ(𝑧) and 𝑔(𝑧) are given by (2). Furthermore, let
𝑛=1+𝑝
∞
𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] 𝑔 (𝑧) = ∑ Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑏𝑛 𝑧𝑛 , 𝑛=𝑝
Unless otherwise mentioned, we will assume in the reminder of this paper that the parameters 𝛼1 , 𝐴 1 , . . . , 𝛼𝑞 , 𝐴 𝑞 and 𝛽1 , 𝐵1 , . . . , 𝛽𝑠 , 𝐵𝑠 (𝑞, 𝑠 ∈ N) are positive real numbers, 0 ≤ 𝛾 < 1, 𝑝 ∈ N, 𝑧 ∈ 𝑈, Ω is defined by (7), and 𝜎𝑛,𝑝 (𝛼1 ) is defined by (11).
𝑏𝑝 < 1. (14)
For 𝑝 = 1, 𝜃1,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ]𝑓(𝑧) = 𝑊𝑞𝑠 [𝛼1 ]𝑓(𝑧), where 𝑊𝑞𝑠 [𝛼1 ]𝑓(𝑧) is the modified Wright generalized hypergeometric functions (see [9]). We note that, for 𝐴 𝑖 = 1 (𝑖 = 1, 2, . . . , 𝑞) and 𝐵𝑖 = 1 (𝑖 = 1, 2, . . . , 𝑠), we obtain 𝜃𝑝,𝑞,𝑠 [𝛼1 , 1, 1]𝑓(𝑧) = 𝐻𝑝,𝑞,𝑠 [𝛼1 ]𝑓(𝑧),
∞
∑ (𝑛 − 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑎𝑛
𝑛=1+𝑝
∞
(17)
+ ∑ (𝑛 + 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑏𝑛 ≤ 𝑝 (1 − 𝛾) . 𝑛=𝑝
Then 𝑓(𝑧) is orientation preserving in 𝑈 and 𝑓(𝑧) ∈ 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾).
Journal of Complex Analysis
3 ∞
Proof. The inequality |ℎ (𝑧)| ≥ |𝑔 (𝑧)| is enough to show that 𝑓(𝑧) is orientation preserving. Note that
− ∑ [𝑛 − 𝑝 (1 + 𝛾)] Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑎𝑛 |𝑧|𝑛
∞ ℎ (𝑧) ≥ 𝑝|𝑧|𝑝−1 − ∑ 𝑛 𝑎𝑛 |𝑧|𝑛−1
− ∑ [𝑛 + 𝑝 (1 + 𝛾)] Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑏𝑛 |𝑧|𝑛
∞
𝑛=𝑝
𝑛=1+𝑝
∞
∞
𝑛=1+𝑝
𝑛=1+𝑝
> 𝑝 − ∑ 𝑛 𝑎𝑛 ≥ 𝑝 − ∑
(𝑛 − 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑎𝑛 𝑝 (1 − 𝛾)
∞
= 2𝑝 (1 − 𝛾) |𝑧|𝑝 − 2 ∑ (𝑛 − 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑎𝑛 |𝑧|𝑛 𝑛=1+𝑝
∞
∞
(𝑛 + 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) ∞ 𝑏𝑛 ≥ ∑ 𝑛 𝑏𝑛 ≥∑ 𝑝 (1 − 𝛾) 𝑛=𝑝
𝑛=1+𝑝
−2 ∑ (𝑛 + 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑏𝑛 |𝑧|𝑛 𝑛=𝑝
𝑛=𝑝
∞
(𝑛 − 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑛−𝑝 𝑎𝑛 |𝑧| 𝑝 (1 − 𝛾) 𝑛=1+𝑝
∞ > ∑ 𝑛 𝑏𝑛 |𝑧|𝑛−1 ≥ 𝑔 (𝑧) . 𝑛=𝑝
= 2𝑝 (1 − 𝛾) |𝑧|𝑝 (1 − ∑ (18)
∞
−∑
(𝑛 + 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑛−𝑝 𝑏𝑛 |𝑧| ) 𝑝 (1 − 𝛾)
Now we will show that 𝑓(𝑧) ∈ 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾). We only need to show that if (17) holds, then condition (15) is satisfied. Using the fact that Re{𝑤} ≥ 𝑝𝛾 if and only if |𝑝(1 − 𝛾) + 𝑤| ≥ |𝑝(1 + 𝛾) − 𝑤|, it suffices to show that
≥ 2𝑝 (1 − 𝛾) |𝑧|𝑝 {1 − ( ∑
𝐴 (𝑧) + 𝑝 (1 − 𝛾) 𝐵 (𝑧) − 𝐴 (𝑧) − 𝑝 (1 + 𝛾) 𝐵 (𝑧) ≥ 0, (19)
+∑
𝑛=𝑝
∞
𝑛=1+𝑝
(𝑛 − 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑎𝑛 𝑝 (1 − 𝛾)
∞
(𝑛 + 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑏𝑛 )} . 𝑝 (1 − 𝛾) 𝑛=𝑝
where
(21) 𝐴 (𝑧) = 𝑧(𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] ℎ(𝑧))
−𝑧(𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] 𝑔(𝑧)) ,
The last expression is nonnegative by (17). This completes the proof of Theorem 1. The harmonic 𝑝-valent function ∞
𝑝 (1 − 𝛾) 𝑋𝑛 𝑧𝑛 (𝑛 − 𝑝𝛾) Ω 𝜎 (𝛼 ) 𝑛,𝑝 1 𝑛=1+𝑝
𝑓 (𝑧) = 𝑧𝑝 + ∑
𝐵 (𝑧) = 𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] ℎ (𝑧) + 𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] 𝑔(𝑧). (20)
∞
𝑝 (1 − 𝛾) 𝑌𝑛 𝑧𝑛 , +∑ 𝑛=𝑝 (𝑛 + 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 )
Substituting for 𝐴(𝑧) and 𝐵(𝑧) in (17) yields 𝐴 (𝑧) + 𝑝 (1 − 𝛾) 𝐵 (𝑧) − 𝐴 (𝑧) − 𝑝 (1 + 𝛾) 𝐵 (𝑧) ∞ = (2𝑝 − 𝑝𝛾) 𝑧𝑝 − ∑ [𝑛 + 𝑝 (1 − 𝛾)] Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑎𝑛 𝑧𝑛 𝑛=1+𝑝 ∞ 𝑛 − ∑ [𝑛 − 𝑝 (1 − 𝛾)] Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑏𝑛 𝑧 𝑛=𝑝 ∞ − −𝑝𝛾𝑧𝑝 + ∑ [𝑛 − 𝑝 (1 + 𝛾)] Ω 𝜎𝑛,𝑝 𝑎𝑛 𝑧𝑛 𝑛=1+𝑝 ∞ − ∑ [𝑛 + 𝑝 (1 + 𝛾)] Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑏𝑛 𝑧𝑛 𝑛=𝑝
∞ where ∑∞ 𝑛=1+𝑝 |𝑋𝑛 | + ∑𝑛=𝑝 |𝑌𝑛 | = 1, shows that the coefficient bound given by (17) is sharp. It is worthy to note that the function of the form (22) belongs to the class 𝐻𝑝 ([𝛼1 , ∞ 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾) for all ∑∞ 𝑛=2 |𝑋𝑛 | + ∑𝑛=1 |𝑌𝑛 | ≤ 1 because coefficient inequality (17) holds.
Theorem 2. A function 𝑓(𝑧) is in the class 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾), if and only if 𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] ℎ (𝑧) ∗[
𝑛=1+𝑝
∞
− ∑ [𝑛 + 𝑝 (𝛾 − 1)] Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑏𝑛 |𝑧|𝑛 − 𝑝𝛾|𝑧|𝑝 𝑛=𝑝
2𝑝 (1 − 𝛾) 𝑧𝑝 + (𝜁 − 2𝑝 + 2𝑝𝛾 + 1) 𝑧𝑝+1 (1 − 𝑧)2
]
− 𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] 𝑔(𝑧)
∞
≥ (2𝑝 − 𝑝𝛾) |𝑧|𝑝 − ∑ [𝑛 + 𝑝 (1 − 𝛾)] Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑎𝑛 |𝑧|𝑛
(22)
∗[
2𝑝 (𝜁 + 𝛾) 𝑧𝑝 + (𝜁 − 2𝑝𝜁 − 2𝑝𝛾 + 1) 𝑧𝑝+1 (1 − 𝑧)2
] ≠ 0, 𝜁 = 1. (23)
4
Journal of Complex Analysis
Proof. From (15), we have the necessary and sufficient condition for 𝑓(𝑧) ∈ 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾) that is, Re {
1 𝑝 (1 − 𝛾)
Theorem 3. A function 𝑓(𝑧) = ℎ + 𝑔, where ℎ(𝑧) and 𝑔(𝑧) are given by (16), is in the class 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾), if and only if ∞
∑ (𝑛 − 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑎𝑛
𝑛=1+𝑝
× [(𝑧(𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] ℎ (𝑧))
+ ∑ (𝑛 + 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑏𝑛 ≤ 𝑝 (1 − 𝛾) . (24)
−𝑧(𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] 𝑔 (𝑧)) )
−1
𝑛=𝑝
Proof. Since 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾) ⊂ 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾), we only need to prove the “only if ” part of this theorem. To this end, for functions 𝑓(𝑧) = ℎ + 𝑔, where ℎ(𝑧) and 𝑔(𝑧) given by (16), we notice that condition
× (𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] ℎ (𝑧) +𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] 𝑔 (𝑧))
(27)
∞
− 𝑝𝛾]} ≥ 0.
Re {(𝑧(𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] ℎ(𝑧))
Hence we have the equivalent condition
− 𝑧(𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] 𝑔(𝑧)) )
1 [(𝑧(𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] ℎ(𝑧)) 𝑝 (1 − 𝛾)
(28) × (𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] ℎ (𝑧) −1
+𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] 𝑔 (𝑧)) } ≥ 𝑝𝛾
−𝑧(𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] 𝑔 (𝑧)) ) is equivalent to
× (𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] ℎ (𝑧)
∞
−1
+𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] 𝑔 (𝑧))
𝜁−1 − 𝑝𝛾] ≠ , 𝜁+1 (25)
|𝜁| = 1, 𝜁 ≠ − 1, 0 < |𝑧| < 1. Simple algebraic manipulation of (25) yields 0 ≠ [(𝑧(𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] ℎ(𝑧))
∞
− ∑ (𝑛 + 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑏𝑛 𝑧𝑛 ) 𝑛=𝑝
(29)
∞
𝑝
𝑛
− ∑ Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑏𝑛 𝑛=𝑝
× (𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] ℎ (𝑧) −1
− 𝑝𝛾]
𝑛=1+𝑝
− ∑ (𝑛 + 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑏𝑛 𝑟𝑛−1 ) 𝑛=𝑝
]
2𝑝 (𝜁 + 𝛾) 𝑧𝑝 + (𝜁 − 2𝑝𝜁 − 2𝑝𝛾 + 1) 𝑧𝑝+1
(30)
∞
𝑛−1
× (1 − ∑ Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑎𝑛 𝑟 𝑛=1+𝑝
− 𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] 𝑔(𝑧)
This completes the proof of Theorem 2.
(𝑝 (1 − 𝛾) − ∑ (𝑛 − 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑎𝑛 𝑟𝑛−1 ∞
= 𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] ℎ (𝑧)
(1 − 𝑧)2
} } ≥ 0. }
∞
(26)
2𝑝 (1 − 𝛾) 𝑧𝑝 + (𝜁 − 2𝑝 + 2𝑝𝛾 + 1) 𝑧𝑝+1
𝑧𝑛 )
The above condition must hold for all 𝑧, |𝑧| = 𝑟 < 1. Choosing the values of 𝑧 on the positive real axis where 0 ≤ 𝑟 < 1, we must have
𝜁−1 − 𝑝 (1 − 𝛾) ( ) 𝜁+1
(1 − 𝑧)2
−1
∞
− 𝑧(𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] 𝑔(𝑧)) )
∗[
𝑛=1+𝑝
𝑛=1+𝑝
∗[
Re {(𝑝 (1 − 𝛾) 𝑧𝑝 − ∑ (𝑛 − 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑎𝑛 𝑧𝑛
× (𝑧 − ∑ Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑎𝑛 𝑧
+𝜃𝑝,𝑞,𝑠 [𝛼1 , 𝐴 1 , 𝐵1 ] 𝑔 (𝑧))
∞
].
−1 𝑛−1
− ∑ Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑏𝑛 𝑟 𝑛=𝑝
)
≥ 0.
If condition (27) does not hold, then the numerator in (30) is negative for 𝑟 sufficiently close to 1. Hence there
Journal of Complex Analysis exist 𝑧0 = 𝑟0 in (0, 1) for which the quotient in (30) is negative. This contradicts the required condition for 𝑓(𝑧) ∈ 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾). This completes the proof of Theorem 3.
5 = (1 + 𝑏𝑝 ) 𝑟𝑝 + ∞
× ∑ ( 𝑛=1+𝑝
[1 + 𝑝 (1 − 𝛾)] Ω 𝜎1+𝑝,𝑝 (𝛼1 ) 𝑎𝑛 𝑝 (1 − 𝛾)
3. Distortion Theorem Theorem 4. Let the function 𝑓(𝑧) = ℎ+𝑔, where ℎ(𝑧) and 𝑔(𝑧) are given by (16), belong to the class 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾). Then for |𝑧| = 𝑟 < 1, we have 1 𝑝 𝑓 (𝑧) ≤ (1 + 𝑏𝑝 ) 𝑟 + Ω 𝜎1+𝑝,𝑝 (𝛼1 ) × (
𝑝 (1 + 𝛾) 1+𝑝 𝑝 (1 − 𝛾) 𝑏 ) 𝑟 − [1 + 𝑝 (1 − 𝛾)] [1 + 𝑝 (1 − 𝛾)] 𝑝 (32)
for |𝑏𝑝 | ≤ (1 − 𝛾)/(1 + 𝛾). The results are sharp with equality for the functions 𝑓(𝑧) defined by 𝑓 (𝑧) = (1 + 𝑏𝑝 ) 𝑧𝑝 −
1 Ω 𝜎1+𝑝,𝑝 (𝛼1 )
𝑝 (1 + 𝛾) 1+𝑝 𝑝 (1 − 𝛾) 𝑏 ) 𝑧 , − ×( [1 + 𝑝 (1 − 𝛾)] [1 + 𝑝 (1 − 𝛾)] 𝑝 𝑓 (𝑧) = (1 − 𝑏𝑝 ) 𝑧𝑝 − × (
1 Ω 𝜎1+𝑝,𝑝 (𝛼1 )
𝑝 (1 + 𝛾) 1+𝑝 𝑝 (1 − 𝛾) 𝑏 ) 𝑧 . − [1 + 𝑝 (1 − 𝛾)] [1 + 𝑝 (1 − 𝛾)] 𝑝 (33)
Proof. We only prove the first inequality. The proof for the second inequality is similar and will be omitted. Let 𝑓(𝑧) ∈ 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾). Taking the absolute value of 𝑓, we have ∞ 𝑝 𝑛 𝑓 (𝑧) ≤ (1 + 𝑏𝑝 ) 𝑟 + ∑ (𝑎𝑛 + 𝑏𝑛 ) 𝑟
∞
𝑛=1+𝑝
𝑛=1+𝑝
(𝑛 + 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 1+𝑝 𝑏𝑛 ) 𝑟 𝑝 (1 − 𝛾)
≤ (1 + 𝑏𝑝 ) 𝑟𝑝 + × (1 −
𝑝 (1 − 𝛾) [1 + 𝑝 (1 − 𝛾)] Ω 𝜎1+𝑝,𝑝 (𝛼1 )
(1 + 𝛾) 1+𝑝 𝑏 ) 𝑟 (1 − 𝛾) 𝑝
= (1 + 𝑏𝑝 ) 𝑟𝑝 + ×(
𝑝 (1 − 𝛾) [1 + 𝑝 (1 − 𝛾)] Ω 𝜎1+𝑝,𝑝 (𝛼1 )
(𝑛 − 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑎𝑛 𝑝 (1 − 𝛾) +
1 Ω 𝜎1+𝑝,𝑝 (𝛼1 )
𝑝 (1 − 𝛾) 𝑝 (1 + 𝛾) 1+𝑝 𝑏 ) 𝑟 . − [1 + 𝑝 (1 − 𝛾)] [1 + 𝑝 (1 − 𝛾)] 𝑝 (34)
This completes the proof of Theorem 4. Putting 𝐴 𝑖 = 1 (𝑖 = 1, 2, . . . , 𝑞) and 𝐵𝑖 = 1 (𝑖 = 1, 2, . . . , 𝑠) in Theorem 4, we obtain the following corollary which modifies the result obtained by Omar and Halim [10, Theorem 2.6]. Corollary 5. Let the function 𝑓(𝑧) = ℎ + 𝑔, where ℎ(𝑧) and 𝑔(𝑧) are given by (16), belong to the class 𝐻𝑝 ([𝛼1 ], 𝑞, 𝑠; 𝛾). Then, for |𝑧| = 𝑟 < 1, we have 1 𝑝 𝑓 (𝑧) ≤ (1 + 𝑏𝑝 ) 𝑟 + Ψ1+𝑝,𝑝 (𝛼1 ) ×(
𝑝 (1 − 𝛾) 𝑝 (1 + 𝛾) 1+𝑝 𝑏 ) 𝑟 , − [1 + 𝑝 (1 − 𝛾)] [1 + 𝑝 (1 − 𝛾)] 𝑝
1 𝑝 𝑓 (𝑧) ≥ (1 − 𝑏𝑝 ) 𝑟 − Ψ1+𝑝,𝑝 (𝛼1 )
𝑛=1+𝑝
∞ ≤ (1 + 𝑏𝑝 ) 𝑟𝑝 + ∑ (𝑎𝑛 + 𝑏𝑛 ) 𝑟1+𝑝
[1 + 𝑝 (1 − 𝛾)] Ω 𝜎1+𝑝,𝑝 (𝛼1 ) 1+𝑝 𝑏𝑛 ) 𝑟 𝑝 (1 − 𝛾)
≤ (1 + 𝑏𝑝 ) 𝑟𝑝 +
𝑝 (1 + 𝛾) 1+𝑝 𝑝 (1 − 𝛾) 𝑏 ) 𝑟 , − [1 + 𝑝 (1 − 𝛾)] [1 + 𝑝 (1 − 𝛾)] 𝑝 (31)
1 𝑝 𝑓 (𝑧) ≥ (1 − 𝑏𝑝 ) 𝑟 − Ω 𝜎1+𝑝,𝑝 (𝛼1 ) × (
+
× ∑ (
𝑝 (1 − 𝛾) [1 + 𝑝 (1 − 𝛾)] Ω 𝜎1+𝑝,𝑝 (𝛼1 )
×(
𝑝 (1 − 𝛾) 𝑝 (1 + 𝛾) 1+𝑝 𝑏 ) 𝑟 − [1 + 𝑝 (1 − 𝛾)] [1 + 𝑝 (1 − 𝛾)] 𝑝 (35)
6
Journal of Complex Analysis
for |𝑏𝑝 | ≤ (1 − 𝛾)/(1 + 𝛾). The results are sharp with equality for the functions 𝑓(𝑧) defined by 𝑓 (𝑧) = (1 + 𝑏𝑝 ) 𝑧𝑝 −
Then ∞
∑
1
𝑓 (𝑧) = (1 − 𝑏𝑝 ) 𝑧𝑝 −
Ψ1+𝑝,𝑝 (𝛼1 )
(
𝑝 (1 − 𝛾) 𝜇) (𝑛 − 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑛
∞
(𝑛 + 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 )
𝑛=𝑝
𝑝 (1 − 𝛾)
+∑ ∞
∞
𝑛=1+𝑝
𝑛=𝑝
(
𝑝 (1 − 𝛾) 𝜂) (𝑛 + 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑛
= ∑ 𝜇𝑛 + ∑ 𝜂𝑛 = 1 − 𝜇𝑝 ≤ 1
1 Ψ1+𝑝,𝑝 (𝛼1 )
(41)
𝑝 (1 + 𝛾) 1+𝑝 𝑝 (1 − 𝛾) 𝑏 ) 𝑧 , − [1 + 𝑝 (1 − 𝛾)] [1 + 𝑝 (1 − 𝛾)] 𝑝 (36)
× (
𝑝 (1 − 𝛾)
𝑛=1+𝑝
𝑝 (1 + 𝛾) 1+𝑝 𝑝 (1 − 𝛾) 𝑏 ) 𝑧 , − [1 + 𝑝 (1 − 𝛾)] [1 + 𝑝 (1 − 𝛾)] 𝑝
× (
(𝑛 − 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 )
and so 𝑓(𝑧) ∈ 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾). Conversely, if 𝑓(𝑧) ∈ 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾), then 𝑝 (1 − 𝛾) 𝑎𝑛 ≤ (𝑛 − 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 )
where Ψ1+𝑝,𝑝 (𝛼1 ) = Ω 𝜎1+𝑝,𝑝 (𝛼1 ) with 𝐴 𝑖 = 1 (𝑖 = 1, 2, . . . , 𝑞) and 𝐵𝑖 = 1 (𝑖 = 1, 2, . . . , 𝑠).
(𝑛 ≥ 1 + 𝑝) ,
𝑝 (1 − 𝛾) 𝑏𝑛 ≤ (𝑛 + 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 )
4. Extreme Points Theorem 6. Let 𝑓(𝑧) = ℎ + 𝑔, where ℎ(𝑧) and 𝑔(𝑧) are given by (16). Then 𝑓(𝑧) ∈ 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾), if and only if
𝑓 (𝑧) = ∑ (𝜇𝑛 ℎ𝑛 (𝑧) + 𝜂𝑛 𝑔𝑛 (𝑧)) , 𝑛=𝑝
(𝑛 ≥ 𝑝) .
Set 𝜇𝑛 =
∞
(42)
(𝑛 − 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑎𝑛 𝑝 (1 − 𝛾)
(37) 𝜂𝑛 =
(𝑛 + 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑏𝑛 𝑝 (1 − 𝛾)
(𝑛 = 1 + 𝑝, 2 + 𝑝, . . .) , (𝑛 = 𝑝, 1 + 𝑝, . . .) . (43)
where ℎ𝑝 (𝑧) = 𝑧𝑝 , ℎ𝑛 (𝑧) = 𝑧𝑝 −
𝑝 (1 − 𝛾) 𝑧𝑛 (𝑛 − 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 )
(38)
(𝑛 = 1 + 𝑝, 2 + 𝑝, . . .) , 𝑔𝑛 (𝑧) = 𝑧𝑝 +
𝑝 (1 − 𝛾) 𝑧𝑛 (𝑛 + 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 )
(𝑛 = 𝑝, 1 + 𝑝, . . .) ,
Since 0 ≤ 𝜇𝑛 ≤ 1(𝑛 = 1 + 𝑝, 2 + 𝑝, . . .) and 0 ≤ 𝜂𝑛 ≤ 1 (𝑛 = ∞ 𝑝, 1+𝑝, . . .), 𝜇𝑝 = 1−∑∞ 𝑛=1+𝑝 𝜇𝑛 +∑𝑛=𝑝 𝜂𝑛 ≥ 0, then we can see that 𝑓(𝑧) can be expressed in the form (37). This completes the proof of Theorem 6. Now we show that the class 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾) is closed under convex combinations of its members.
(39)
Theorem 7. The class 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾) is closed under convex combination.
𝜇𝑛 ≥ 0, 𝜂𝑛 ≥ 0, ∑∞ 𝑛=𝑝 (𝜇𝑛 + 𝜂𝑛 ) = 1. In particular, the extreme points of the class 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾) are {ℎ𝑛 } and {𝑔𝑛 }.
Proof. For 𝑖 = 1, 2, 3, . . ., let 𝑓𝑖 ∈ 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾), where 𝑓𝑖 is given by ∞ ∞ 𝑓𝑖 = 𝑧 − ∑ 𝑎𝑛𝑖 𝑧𝑛 + ∑ 𝑏𝑛𝑖 𝑧𝑛 . 𝑛=𝑝
Proof. Suppose that
𝑛=1+𝑝
(44)
∞
𝑓 (𝑧) = ∑ (𝜇𝑛 ℎ𝑛 (𝑧) + 𝜂𝑛 𝑔𝑛 (𝑧))
Then by using Theorem 3, we have
𝑛=𝑝
∞
∞
𝑝 (1 − 𝛾) =𝑧 − ∑ 𝜇𝑛 𝑧𝑛 (𝑛 − 𝑝𝛾) Ω 𝜎 (𝛼 ) 𝑛,𝑝 1 𝑛=1+𝑝 𝑝
∞
𝑝 (1 − 𝛾) 𝜂 𝑧𝑛 . (𝑛 + 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑛 𝑛=𝑝
+∑
(40)
∑ 𝑛=1+𝑝
(𝑛 − 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑛 𝑎𝑛𝑖 𝑧 𝑝 (1 − 𝛾) ∞
(𝑛 + 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑛 𝑏𝑛𝑖 𝑧 ≤ 1. +∑ 𝑝 (1 − 𝛾) 𝑛=𝑝
(45)
Journal of Complex Analysis
7
For ∑∞ 𝑛=1 𝑡𝑖 = 1, 0 ≤ 𝑡𝑖 ≤ 1, the convex combination of 𝑓𝑖 may be written as ∞ ∞ ∞ ∑𝑡𝑖 𝑓𝑖 (𝑧) = 𝑧𝑝 − ∑ (∑𝑡𝑖 𝑎𝑛𝑖 ) 𝑧𝑛 𝑛=1+𝑝
𝑖=1
∞
𝑖=1
(46)
∞
+ ∑ (∑𝑡𝑖 𝑏𝑛𝑖 ) 𝑧𝑛 . 𝑛=𝑝 𝑖=1
Then by (45), we have ∞
∑
(𝑛 − 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑝 (1 − 𝛾)
𝑛=1+𝑝
∞ (∑𝑡𝑖 𝑎𝑛𝑖 ) 𝑖=1
∞
(𝑛 + 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 )
𝑛=𝑝
𝑝 (1 − 𝛾)
+∑ ∞
∞
𝑖=1
𝑛=1+𝑝
= ∑𝑡𝑖 ( ∑
∞ (∑𝑡𝑖 𝑏𝑛𝑖 ) 𝑖=1
(𝑛 − 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑎𝑛𝑖 𝑝 (1 − 𝛾)
(47)
∞
(𝑛 + 𝑝𝛾) Ω 𝜎𝑛,𝑝 (𝛼1 ) 𝑏𝑛𝑖 ) 𝑝 (1 − 𝛾) 𝑛=𝑝
+∑ ∞
≤ ∑𝑡𝑖 = 1. 𝑖=1
This is the required condition and so ∑∞ 𝑖=1 𝑡𝑖 𝑓𝑖 (𝑧) ∈ 𝐻𝑝 ([𝛼1 , 𝐴 1 , 𝐵1 ], 𝑞, 𝑠; 𝛾). This completes the proof of Theorem 7.
Acknowledgment The authors would like to thank the referees of the paper for their helpful suggestions.
References [1] J. Clunie and T. Sheil-Small, “Harmonic univalent functions,” Annales Academiae Scientiarum Fennicae A, vol. 9, pp. 3–25, 1984. [2] O. P. Ahuja and J. M. Jahangiri, “Multivalent harmonic starlike functions,” Annales Universitatis Mariae Curie-Skłodowska, vol. 55, pp. 1–13, 2001. [3] E. M. Wright, “The asymptotic expansion of the generalized hypergeometric function,” Proceedings of the London Mathematical Society, vol. 46, pp. 389–408, 1946. [4] H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Ellis Horwood, Chichester, UK, 1985. [5] J. Dziok and H. M. Srivastava, “Classes of analytic functions associated with the generalized hypergeometric function,” Applied Mathematics and Computation, vol. 103, no. 1, pp. 1–13, 1999. [6] J. Dziok and R. K. Raina, “Families of analytic functions associated with the Wright generalized hypergeometric function,” Demonstratio Mathematica, vol. 37, no. 3, pp. 533–542, 2004. [7] M. K. Aouf and J. Dziok, “Certain class of analytic functions associated with the Wright generalized hypergeometric function,” Journal of Mathematics and Applications, vol. 30, pp. 23– 32, 2008.
[8] M. K. Aouf, A. Shamandy, A. O. Mostafa, and S. M. Madian, “Certain class of 𝑝-valent functions associated with the Wright generalized hypergeometric function,” Demonstratio Mathematica, vol. 43, no. 1, pp. 40–54, 2010. [9] G. Murugusundaramoorthy and R. K. Raina, “On a subclass of harmonic functions associated with Wright’s generalized hypergeometric functions,” Hacettepe Journal of Mathematics and Statistics, vol. 38, no. 2, pp. 129–136, 2009. [10] R. Omar and S. A. Halim, “Multivalent harmonic functions defined by Dziok-Srivastava operator,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 35, no. 3, pp. 601–610, 2012. [11] R. A. Al-Khal, “Goodman-Rønning-type harmonic univalent functions based on Dziok-Srivastava operator,” Applied Mathematical Sciences, vol. 5, no. 9–12, pp. 573–584, 2011. [12] H. A. Al-Kharsani and R. A. Al-Khal, “Univalent harmonic functions,” Journal of Inequalities in Pure and Applied Mathematics, vol. 8, no. 2, article 59, pp. 1–8, 2007. [13] J. M. Jahangiri, “Harmonic functions starlike in the unit disk,” Journal of Mathematical Analysis and Applications, vol. 235, no. 2, pp. 470–477, 1999.
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