Hindawi Publishing Corporation International Journal of Analysis Volume 2013, Article ID 352823, 7 pages http://dx.doi.org/10.1155/2013/352823
Research Article On Certain Classes of Harmonic 𝑝-Valent Functions Defined by an Integral Operator T. M. Seoudy Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt Correspondence should be addressed to T. M. Seoudy;
[email protected] Received 3 November 2012; Accepted 18 December 2012 Academic Editor: Fr´ed´eric Robert Copyright © 2013 T. M. Seoudy. 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 obtain coefficient characterization, extreme points, and distortion bounds of certain classes of harmonic 𝑝-valent functions defined by an integral operator.
1. Introduction
where
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, we can write 𝑓 = ℎ + 𝑔,
(1)
where ℎ and 𝑔 are analytic in 𝐷. We call ℎ 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, Jahangiri and Ahuja [2] defined the class H𝑝 (𝑝 ∈ N = {1, 2, 3, . . .}), consisting of all 𝑝-valent harmonic functions 𝑓 = ℎ + 𝑔 that are sense preserving in 𝑈 and ℎ, and 𝑔 are of the form ∞
ℎ (𝑧) = 𝑧𝑝 + ∑ 𝑎𝑘 𝑧𝑘 , 𝑘=𝑝+1
∞
𝑔 (𝑧) = ∑ 𝑏𝑘 𝑧𝑘 , 𝑘=𝑝
𝑏𝑝 < 1. (2)
For 𝑓 = ℎ + 𝑔 given by (2), we define the modified 𝑝𝑛 of 𝑓 (see [3] and also valent Salagean integral operator 𝐼𝑝,𝜆 [4] when 𝑝 = 1) as follows: 𝑛 𝑓 (𝑧) 𝐼𝑝,𝜆
=
𝑛 𝐼𝑝,𝜆 ℎ (𝑧)
+
𝑛 𝑔 (𝑧), (−1)𝑛 𝐼𝑝,𝜆
(3)
∞
𝑛 ℎ (𝑧) = 𝑧𝑝 + ∑ ( 𝐼𝑝,𝜆 𝑘=𝑝+1
𝑛
𝑝 ) 𝑎𝑘 𝑧𝑘 𝑝 + 𝜆 (𝑘 − 𝑝)
(𝑝 ∈ N; 𝜆 > 0; 𝑛 ∈ N0 = N ∪ {0}) , 𝑛
∞
𝑝 𝑛 𝐼𝑝,𝜆 𝑔 (𝑧) = ∑ ( ) 𝑏𝑘 𝑧𝑘 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝
(4)
(𝑝 ∈ N; 𝜆 > 0; 𝑛 ∈ N0 ). For 𝑝 ∈ N, 𝜆 > 0, 𝑛 ∈ N0 , 0 ≤ 𝛼 < 1, and 𝑧 ∈ 𝑈, we let H𝑝,𝜆 (𝑛; 𝛼) denote the family of harmonic functions 𝑓 of the form (2) such that Re {
𝑛 𝑓 (𝑧) 𝐼𝑝,𝜆 𝑛+1 𝐼𝑝,𝜆 𝑓 (𝑧)
} > 𝛼,
(5)
𝑛 𝑓 is defined by (3). where 𝐼𝑝,𝜆 We let the subclass H−𝑝,𝜆 (𝑛; 𝛼) consists of harmonic functions 𝑓𝑛 = ℎ + 𝑔𝑛 in H𝑝,𝜆 (𝑛; 𝛼) so that ℎ and 𝑔𝑛 are of the form ∞
ℎ (𝑧) = 𝑧𝑝 − ∑ 𝑎𝑘 𝑧𝑘 , 𝑘=𝑝+1
∞
𝑔𝑛 (𝑧) = (−1)𝑛 ∑ 𝑏𝑘 𝑧𝑘 , 𝑘=𝑝
𝑎𝑘 , 𝑏𝑘 ≥ 0.
(6)
2
International Journal of Analysis
We note that H−𝑝,1 (𝑛; 𝛼) = H−𝑝 (𝑛; 𝛼), where the class was defined and studied by Cotirla [5]. In this paper, we obtain coefficient characterization of the classes H𝑝,𝜆 (𝑛; 𝛼) and H−𝑝,𝜆 (𝑛; 𝛼). We also obtain extreme points and distortion bounds for functions in the class H−𝑝,𝜆 (𝑛; 𝛼). H−𝑝 (𝑛; 𝛼)
𝑛
∞ 𝑝 + ∑ × [( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝+1 [
𝑛+1
−𝛼( ∞
2. Coefficient Characterization
× (𝑧𝑝 + ∑ (
Unless otherwise mentioned, we assume throughout this paper that 𝑝 ∈ N, 𝑛 ∈ N0 , 0 ≤ 𝛼 < 1, 𝑎𝑝 = 1, and 𝜆 > 0. We begin with a sufficient condition for functions in H𝑝,𝜆 (𝑛; 𝛼). Theorem 1. Let 𝑓 = ℎ + 𝑔 so that ℎ and 𝑔 are given by (2). Furthermore, let
𝑘=𝑝+1
+(−1)
∞
𝑘=𝑝
(7)
𝑛+1
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
]
𝑎𝑘 𝑧𝑘
∞
𝑝 ) ∑( 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝
𝑛+1
𝑛+1
∞
× (𝑧𝑝 + ∑ ( 𝑘=𝑝+1
𝑝 −𝛼( ) 𝑝 + 𝜆 (𝑘 − 𝑝)
𝑛+1
)
(8)
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝) +𝛼(
]
𝑛
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
𝑛
𝑛+1
)
(9) −𝛼(
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
𝑘=𝑝+1
𝑛+1
+ (−1)
It follows that
Re {
𝑛 𝑛+1 𝑓 (𝑧) − 𝛼𝐼𝑝,𝜆 𝑓 (𝑧) 𝐼𝑝,𝜆 𝑛+1 𝐼𝑝,𝜆 𝑓 (𝑧)
= Re {((1 − 𝛼) 𝑧𝑝
(𝑧 ∈ 𝑈) .
(10)
𝑛+1
] 𝑎𝑘 𝑧𝑘−𝑝 ) ]
𝑛+1
× (1 + ∑ (
Proof. According to (2) and (3), we only need to show that
}≥0
} 𝑏𝑘 𝑧 ) } } 𝑘
∞ 𝑝 ) + ∑ × [( 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝+1 [
Then, 𝑓 is sense preserving in 𝑈 and 𝑓 ∈ H𝑝,𝜆 (𝑛; 𝛼).
𝑛+1 𝐼𝑝,𝜆 𝑓 (𝑧)
−1
𝑛+1
𝑝 ) ∑( 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝
+(−1)
∞
𝑛 𝑛+1 𝑓 (𝑧) − 𝛼𝐼𝑝,𝜆 𝑓 (𝑧) 𝐼𝑝,𝜆
𝑎𝑘 𝑧𝑘
∞
𝑛+1
× (1 − 𝛼)−1 .
Re {
𝑛+1
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
] 𝑏𝑘 𝑧𝑘 )
= Re {((1 − 𝛼)
× (1 − 𝛼)−1 , Φ𝑝,𝜆 (𝑛, 𝑘, 𝛼) = ((
𝑘
𝑏𝑘 𝑧 )
where 𝑝 Ψ𝑝,𝜆 (𝑛, 𝑘, 𝛼) = (( ) 𝑝 + 𝜆 (𝑘 − 𝑝)
−1
𝑛+1
𝑛
∞
+𝛼(
𝑛
] 𝑎𝑘 𝑧𝑘 )
𝑝 + ((−1) ∑ [( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝 [ 𝑛
∑ {Ψ𝑝,𝜆 (𝑛, 𝑘, 𝛼) 𝑎𝑘 + Φ𝑝,𝜆 (𝑛, 𝑘, 𝛼) 𝑏𝑘 } ≤ 2,
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
𝑎𝑘 𝑧𝑘−𝑝 −1
𝑛+1
∞
𝑝 ) ∑( 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝
𝑘 −𝑝
𝑏𝑘 𝑧 𝑧 )
𝑛
∞ 𝑝 + ((−1)𝑛 ∑ [( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝 [
𝑛+1
+ 𝛼( } × 𝑏𝑘 𝑧𝑘 𝑧−𝑝 )
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
] ]
International Journal of Analysis
3 𝑛+1
𝑛+1
∞
𝑝 −( ) 𝑝 + 𝜆 (𝑘 − 𝑝)
𝑝 × (1 + ∑ ( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝+1 𝑘−𝑝
× 𝑎𝑘 𝑧
+ (−1)
𝑛+1
∞
𝑝 ) ∑( 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝
𝑛+1
∞
× (2 (1 − 𝛼) + ∑ 𝑐𝑘 𝑎𝑘 𝑟𝑘−𝑝 𝑒𝑖(𝑘−𝑝)𝜃 𝑘=𝑝+1
−1
} × 𝑏𝑘 𝑧 𝑧 ) } }
−1
∞
𝑛
𝑘−𝑝 −𝑖(𝑘+𝑝)𝜃
+(−1) ∑ 𝑑𝑘 𝑏𝑘 𝑟
𝑘 −𝑝
= Re {
∞
+ ((−1)𝑛 ∑ [( 𝑘=𝑝
(11)
𝑛+1
𝑘=𝑝+1
] 𝑎𝑘 𝑟𝑘−𝑝 𝑒𝑖(𝑘−𝑝)𝜃 ]
𝑝 + (−1)𝑛 ∑ [( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝 [
𝑛+1
𝑝 +𝛼( ) 𝑝 + 𝜆 (𝑘 − 𝑝)
𝑘=𝑝
𝑘−𝑝 −𝑖(𝑘+𝑝)𝜃
+(−1) ∑ 𝑑𝑘 𝑏𝑘 𝑟
𝑒
∞ 𝑛 𝑝 ≤ ( ∑ [( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝+1 𝑛+1
]
−(
]
× 𝑏𝑘 𝑟𝑘−𝑝 𝑒−𝑖(𝑘+𝑝)𝜃 ,
+ (−1)𝑛+1 ∑ (
∞
𝑛
𝑘=𝑝
𝑛
∞
∞
]
∞
𝑛+1
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
× (2 (1 − 𝛼) + ∑ 𝑐𝑘 𝑎𝑘 𝑟𝑘−𝑝 𝑒𝑖(𝑘−𝑝)𝜃
𝑝 −𝛼( ) 𝑝 + 𝜆 (𝑘 − 𝑝)
𝑘=𝑝+1
𝑛
× 𝑏𝑘 𝑟𝑘−𝑝 𝑒−𝑖(𝑘+𝑝)𝜃 )
𝑛
𝑝 𝐴 (𝑟𝑒𝑖𝜃 ) = ∑ [ ( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝+1 [
𝐵 (𝑟𝑒𝑖𝜃 ) = ∑ (
)
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝) +𝛼(
For 𝑧 = 𝑟𝑒𝑖𝜃 , we have
∞
𝑒
𝑘=𝑝
1 − 𝛼 + 𝐴 (𝑧) }. 1 + 𝐵 (𝑧)
∞
] 𝑎𝑘 𝑟𝑘−𝑝 𝑒𝑖(𝑘−𝑝)𝜃 )
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
−1
)
] 𝑎𝑘 𝑟𝑘−𝑝 )
∞
𝑛+1
× (2 (1 − 𝛼) − ∑ 𝑐𝑘 𝑎𝑘 𝑟𝑘−𝑝
𝑎𝑘 𝑟𝑘−𝑝 𝑒𝑖(𝑘−𝑝)𝜃 𝑛+1
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
𝑘=𝑝+1
∞
𝑏𝑘 𝑟𝑘−𝑝 𝑒−𝑖(𝑘+𝑝)𝜃 . (12)
Setting that
− ∑ 𝑑𝑘 𝑏𝑘 𝑟𝑘−𝑝 )
−1
𝑘=𝑝
∞
𝑝 + ( ∑ [( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝
𝑛
𝑛+1
1 − 𝛼 + 𝐴 (𝑧) 1 + 𝑤 (𝑧) = (1 − 𝛼) , 1 + 𝐵 (𝑧) 1 − 𝑤 (𝑧)
(13)
+(
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝) ∞
the proof will be complete if we can show that |𝑤(𝑧)| < 1. Using the condition (7), we can write 𝐴 (𝑧) − (1 − 𝛼) 𝐵 (𝑧) |𝑤 (𝑧)| = 𝐴 (𝑧) + (1 − 𝛼) 𝐵 (𝑧) + 2 (1 − 𝛼) 𝑛 ∞ 𝑝 = ( ∑ [( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝+1
× (2 (1 − 𝛼) − ∑ 𝑐𝑘 𝑎𝑘 𝑟𝑘−𝑝 𝑘=𝑝+1
∞
− ∑ 𝑑𝑘 𝑏𝑘 𝑟𝑘−𝑝 )
−1
𝑘=𝑝
∞ 𝑛 𝑝 = ( ∑ [( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝+1
] 𝑏𝑘 𝑟𝑘−𝑝 )
4
International Journal of Analysis 𝑛+1
𝑝 −( ) 𝑝 + 𝜆 (𝑘 − 𝑝)
∞
1 𝑦𝑘 𝑧𝑘 , Φ 𝑘, 𝛼) (𝑛, 𝑘=𝑝 𝑝,𝜆
] 𝑎𝑘 𝑟𝑘−𝑝 )
+∑
(16)
−1
∞
× (4 (1 − 𝛼) − ∑ {𝑐𝑘 𝑎𝑘 + 𝑑𝑘 𝑏𝑘 } 𝑟𝑘−𝑝 )
∞ where ∑∞ 𝑘=𝑝+1 |𝑥𝑘 | + ∑𝑘=𝑝 |𝑦𝑘 | = 1 show that the coefficient bound given by (7) is sharp. The functions of the form (8) are in the class H𝑝,𝜆 (𝑛; 𝛼) because
𝑘=𝑝
∞
𝑛
+ ( ∑ [( 𝑘=𝑝
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
+(
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
𝑛+1
∞
∑ {Ψ𝑝,𝜆 (𝑛, 𝑘, 𝛼) 𝑎𝑘 + Φ𝑝,𝜆 (𝑛, 𝑘, 𝛼) 𝑏𝑘 }
] 𝑏𝑘 𝑟𝑘−𝑝 )
𝑘=𝑝
−1
×(4 (1 − 𝛼) − ∑ {𝑐𝑘 𝑎𝑘 + 𝑑𝑘 𝑏𝑘 } 𝑟𝑘−𝑝 ) 𝑘=𝑝 ∞ 𝑛 𝑝 < ( ∑ [( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝+1 ∞
−(
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
𝑛+1
𝑘=𝑝
In the following theorem, it is shown that the condition (7) is also necessary for functions 𝑓𝑛 = ℎ + 𝑔𝑛 , where ℎ and 𝑔𝑛 are of the form (6).
−1
Theorem 2. Let 𝑓𝑛 = ℎ + 𝑔𝑛 , where ℎ and 𝑔𝑛 are given by (6). Then, 𝑓𝑛 ∈ H−𝑝,𝜆 (𝑛; 𝛼) if and only if
𝑘=𝑝
𝑛
𝑝 + ( ∑ [( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝
∞
∑ {Ψ𝑝,𝜆 (𝑛, 𝑘, 𝛼) 𝑎𝑘 + Φ𝑝,𝜆 (𝑛, 𝑘, 𝛼) 𝑏𝑘 } ≤ 2,
(18)
𝑘=𝑝 𝑛+1
𝑝 +( ) 𝑝 + 𝜆 (𝑘 − 𝑝)
] 𝑏𝑘 )
∞
where Ψ𝑝,𝜆 (𝑛, 𝑘, 𝛼) and Φ𝑝,𝜆 (𝑛, 𝑘, 𝛼) are given by (8) and (9), respectively.
−1
×(4 (1 − 𝛼) − ∑ {𝑐𝑘 𝑎𝑘 + 𝑑𝑘 𝑏𝑘 }) 𝑘=𝑝
≤ 1, (14)
Proof. Since H−𝑝,𝜆 (𝑛; 𝛼) ⊂ H𝑝,𝜆 (𝑛; 𝛼), we only need to prove the “only if ” part of the theorem. To this end, for functions 𝑓𝑛 = ℎ + 𝑔𝑛 , where ℎ and 𝑔𝑛 are given by (6), we notice that 𝑛 𝑛+1 the condition Re{𝐼𝑝,𝜆 𝑓(𝑧)/𝐼𝑝,𝜆 𝑓(𝑧)} > 𝛼 is equivalent to Re {((1 − 𝛼) 𝑧𝑝
where 𝑛
𝑐𝑘 = (
𝑘=𝑝+1
(17)
This completes the proof of Theorem 1.
× (4 (1 − 𝛼) − ∑ {𝑐𝑘 𝑎𝑘 + 𝑑𝑘 𝑏𝑘 }) ∞
∞
= 1 + ∑ 𝑥𝑘 + ∑ 𝑦𝑘 = 2.
] 𝑎𝑘 )
∞
∞
𝑛+1
𝑝 𝑝 ) + (1 − 2𝛼) ( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑛
𝑝 𝑝 ) − (1 − 2𝛼) ( ) 𝑑𝑘 = ( 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑝 + 𝜆 (𝑘 − 𝑝)
,
∞ 𝑝 − ∑ × [( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝+1 [
𝑛
𝑛+1
𝑝 −𝛼( ) 𝑝 + 𝜆 (𝑘 − 𝑝)
𝑛+1
. (15) ∞
The harmonic functions are as follows: ∞
1 𝑓 (𝑧) = 𝑧 + ∑ 𝑥𝑘 𝑧𝑘 𝑘, 𝛼) Ψ (𝑛, 𝑘=𝑝+1 𝑝,𝜆 𝑝
× (𝑧𝑝 − ∑ ( 𝑘=𝑝+1
+ (−1)
2𝑛
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝) ∞
𝑛+1
] 𝑎𝑘 𝑧𝑘 ) ]
𝑎𝑘 𝑧𝑘 𝑛+1
𝑝 ) ∑( 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝
−1 𝑘
𝑏𝑘 𝑧 )
International Journal of Analysis
5
∞
𝑧0 = 𝑟0 in (0, 1) for which the quotient in (20) is negative. This contradicts the required condition for 𝑓𝑛 ∈ H−𝑝,𝜆 (𝑛; 𝛼), and so the proof of Theorem 2 is completed.
+ ((−1)2𝑛−1 ∑ 𝑘=𝑝 𝑛
𝑝 × [( ) 𝑝 + 𝜆 (𝑘 − 𝑝) [
3. Extreme Points and Distortion Theorem
𝑝 +𝛼( ) 𝑝 + 𝜆 (𝑘 − 𝑝) ∞
𝑝 × (𝑧 − ∑ ( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝+1 𝑝
∞
+ (−1)2𝑛 ∑ ( 𝑘=𝑝
𝑛+1
Our next theorem is on the extreme points of convex hulls of the class H−𝑝,𝜆 (𝑛; 𝛼) denoted by 𝑐𝑙𝑐𝑜H−𝑝,𝜆 (𝑛; 𝛼).
] 𝑏𝑘 𝑧𝑘 )
Theorem 3. Let 𝑓𝑛 = ℎ + 𝑔𝑛 , where ℎ and 𝑔𝑛 are given by (6). Then, 𝑓𝑛 ∈ H−𝑝,𝜆 (𝑛; 𝛼) if and only if
]
𝑛+1
𝑎𝑘 𝑧𝑘
∞
𝑓𝑛 (𝑧) = ∑ [𝑥𝑘 ℎ𝑘 (𝑧) + 𝑦𝑘 𝑔𝑛𝑘 (𝑧)] ,
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
𝑛+1
𝑘=𝑝
−1
} 𝑏𝑘 𝑧𝑘 ) } }
where ℎ1 (𝑧) = 𝑧𝑝 ,
≥ 0.
ℎ𝑘 (𝑧) = 𝑧𝑝 −
(19) The previous required condition (19) must hold for all values of 𝑧 in 𝑈. Upon choosing the values of 𝑧 on the positive real axis where 0 ≤ 𝑧 = 𝑟 < 1, we must have ∞
𝑝 ((1 − 𝛼) − ∑ [ ( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝+1 [ −𝛼(
𝑘=𝑝+1
+ ∑( 𝑘=𝑝 ∞
+ (− ∑ [( 𝑘=𝑝
[
𝑛+1
] 𝑎𝑘 𝑟𝑘−𝑝 )
𝑎𝑘 𝑟𝑘−𝑝
∞
𝑘=𝑝+1
∞
∞
𝑘=𝑝+1
𝑘=𝑝
𝑥𝑝 = 1 − ∑ 𝑥𝑘 − ∑ 𝑦𝑘 .
∞
−1
𝑓𝑛 (𝑧) = ∑ (𝑥𝑘 ℎ𝑘 (𝑧) + 𝑦𝑘 𝑔𝑛𝑘 (𝑧))
𝑏𝑘 𝑟𝑘−𝑝 )
𝑘=𝑝 ∞
∞
1 𝑥𝑘 𝑧𝑘 𝑘, 𝛼) Ψ (𝑛, 𝑝,𝜆 𝑘=𝑝+1
= ∑ (𝑥𝑘 + 𝑦𝑘 ) 𝑧𝑝 − ∑
(20)
𝑘=𝑝
(23)
∞
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
𝑛+1
𝑛+1
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
𝑛+1
𝑝 + ∑( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝
(22)
Proof. Suppose that
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
× (1 − ∑ (
1 𝑧𝑘 Φ𝑝,𝜆 (𝑛, 𝑘, 𝛼)
In particular, the extreme points of the class H−𝑝,𝜆 (𝑛; 𝛼) are {ℎ𝑘 } and {𝑔𝑛𝑘 }.
]
𝑛
+𝛼(
∞
𝑥𝑘 , 𝑦𝑘 ≥ 0,
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
𝑔𝑛𝑘 (𝑧) = 𝑧𝑝 + (−1)𝑛−1
(𝑘 = 𝑝, 𝑝 + 1, 𝑝 + 2, . . .) ,
𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝)
𝑛+1
∞
1 𝑧𝑘 Ψ𝑝,𝜆 (𝑛, 𝑘, 𝛼)
(𝑘 = 𝑝 + 1, 𝑝 + 2, 𝑝 + 3, . . .) ,
𝑛
𝑛+1
∞
× (1 − ∑ (
(21)
1 𝑦𝑘 𝑧𝑘 . Φ 𝑘, 𝛼) (𝑛, 𝑘=1 𝑝,𝜆
+ (−1)𝑛−1 ∑ ] 𝑏𝑘 𝑟𝑘−𝑝 ) ]
Then, ∞
𝑎𝑘 𝑟𝑘−𝑝 −1 𝑘−𝑝
𝑏𝑘 𝑟
)
≥ 0. If the condition (18) does not hold, then the numerator in (20) is negative for 𝑟 sufficiently close to 1. Hence there exists
∑ Ψ𝑝,𝜆 (𝑛, 𝑘, 𝛼) (
𝑘=𝑝+1
1 𝑥 ) Ψ𝑝,𝜆 (𝑛, 𝑘, 𝛼) 𝑘
∞
+ ∑ Φ𝑝,𝜆 (𝑛, 𝑘, 𝛼) ( 𝑘=1
∞
∞
𝑘=𝑝+1
𝑘=𝑝
1 𝑦) Φ𝑝,𝜆 (𝑛, 𝑘, 𝛼) 𝑘
= ∑ 𝑥𝑘 + ∑ 𝑦𝑘 = 1 − 𝑥𝑝 ≤ 1, and so 𝑓𝑛 ∈ 𝑐𝑙𝑐𝑜H−𝑝,𝜆 (𝑛; 𝛼).
(24)
6
International Journal of Analysis ∞
Conversely, if 𝑓𝑛 ∈ 𝑐𝑙𝑐𝑜H−𝑝,𝜆 (𝑛; 𝛼), then 1 , 𝑎𝑘 ≤ Ψ𝑝,𝜆 (𝑛, 𝑘, 𝛼)
× ∑ {Ψ𝑝,𝜆 (𝑛, 𝑘, 𝛼) 𝑎𝑘 + Φ𝑝,𝜆 (𝑛, 𝑘, 𝛼) 𝑏𝑘 }
1 𝑏𝑘 ≤ . Φ𝑝,𝜆 (𝑛, 𝑘, 𝛼)
𝑘=𝑝+1
(25)
≤ (1 + 𝑏𝑝 ) 𝑟𝑝 + {Γ𝑝,𝜆 (𝑛, 𝑘, 𝛼) − Δ 𝑝,𝜆 (𝑛, 𝑘, 𝛼) 𝑏𝑝 } 𝑟𝑝+1 .
Set that 𝑥𝑘 = Ψ𝑝,𝜆 (𝑛, 𝑘, 𝛼) 𝑎𝑘
(𝑘 = 𝑝 + 1, 𝑝 + 2, 𝑝 + 3, . . .) ,
𝑦𝑘 = Φ𝑝,𝜆 (𝑛, 𝑘, 𝛼) 𝑏𝑘
(𝑘 = 𝑝, 𝑝 + 1, 𝑝 + 2, . . .) . (26)
Then note that by Theorem 2, 0 ≤ 𝑥𝑘 ≤ 1, (𝑘 = 𝑝 + 1, 𝑝 + 2, 𝑝 + 3, . . .), and 0 ≤ 𝑦𝑘 ≤ 1, (𝑘 = 𝑝, 𝑝 + 1, 𝑝 + 2, . . .). ∞ We define that 𝑥𝑝 = 1 − ∑∞ 𝑘=𝑝+1 𝑥𝑘 − ∑𝑘=𝑝 𝑦𝑘 and note that by Theorem 2, 𝑥𝑝 ≥ 0. Consequently, we obtain 𝑓𝑛 (𝑧) = ∑∞ 𝑘=𝑝 {𝑥𝑘 ℎ𝑘 (𝑧) + 𝑦𝑘 𝑔𝑘 (𝑧)} as required.
(29) The bounds given in Theorem 4 for functions 𝑓𝑛 = ℎ + 𝑔𝑛 , whereℎ and 𝑔𝑛 of form (6), also hold for functions of form (2) if the coefficient condition (7) is satisfied. The upper bound given for 𝑓 ∈ H−𝑝,𝜆 (𝑛; 𝛼) is sharp, and the equality occurs for the functions 𝑓 (𝑧) = 𝑧𝑝 + 𝑏𝑝 𝑧𝑝 −(
1−𝛼 𝑛
The following theorem gives the distortion bounds for functions in the class H−𝑝,𝜆 (𝑛; 𝛼) which yields a covering result for this class. Theorem 4. Let 𝑓𝑛 (𝑧) ∈ H−𝑝,𝜆 (𝑛; 𝛼). Then, for |𝑧| = 𝑟 < 1, we have 𝑝
𝑝+1
(1 − 𝑏𝑝 ) 𝑟 − {Γ𝑝,𝜆 (𝑛, 𝛼) − Δ 𝑝,𝜆 (𝑛, 𝛼)} 𝑟
≤ 𝑓𝑛 (𝑧)
≤ (1 + 𝑏𝑝 ) 𝑟𝑝 + {Γ𝑝,𝜆 (𝑛, 𝛼) − Δ 𝑝,𝜆 (𝑛, 𝛼) 𝑏𝑝 } 𝑟𝑝+1 ,
−
1−𝛼 𝑛
𝑛+1
(𝑝/(𝑝 + 𝜆)) + 𝛼(𝑝/ (𝑝 + 𝜆))
𝑏𝑝 ) 𝑧𝑝+1 , (30)
𝑓 (𝑧) = 𝑧𝑝 + 𝑏𝑝 𝑧𝑝 −(
(27)
𝑛+1
(𝑝/(𝑝 + 𝜆)) − 𝛼(𝑝/ (𝑝 + 𝜆))
1−𝛼 𝑛
𝑛+1
(𝑝/(𝑝 + 𝜆)) − 𝛼(𝑝/ (𝑝 + 𝜆)) −
where
1−𝛼 𝑛
𝑛+1
(𝑝/(𝑝 + 𝜆)) + 𝛼(𝑝/ (𝑝 + 𝜆))
𝑏𝑝 ) 𝑧𝑝+1 (31)
1−𝛼 , (𝑝/(𝑝 + 𝜆)) − 𝛼(𝑝/(𝑝 + 𝜆))𝑛+1
Γ𝑝,𝜆 (𝑛, 𝛼) =
showing that the bounds given in Theorem 4 are sharp.
𝑛
Δ 𝑝,𝜆 (𝑛, 𝛼) =
1+𝛼 𝑛+1
(𝑝/(𝑝 + 𝜆))𝑛 − 𝛼(𝑝/(𝑝 + 𝜆))
(28)
.
Remark 5. (i) Putting 𝜆 = 1 in the previous results, we obtain the results of Cotirla [5]. (ii) Putting 𝜆 = 1 in the previous results, we obtain the results of Cotirla [6], when 𝛽 = 0.
The result is sharp. Proof. We only prove the right-hand inequality. The proof for the left-hand inequality is similar and will be omitted. Let 𝑓𝑛 (𝑧) ∈ H−𝑝,𝜆 (𝑛; 𝛼). Taking the absolute value of 𝑓𝑛 , we have ∞
𝑝 𝑘 𝑓𝑛 (𝑧) ≤ (1 + 𝑏𝑝 ) 𝑟 + ∑ (𝑎𝑘 + 𝑏𝑘 ) 𝑟 𝑘=𝑝+1 ∞
≤ (1 + 𝑏𝑝 ) 𝑟𝑝 + ∑ (𝑎𝑘 + 𝑏𝑘 ) 𝑟𝑝+1 𝑘=𝑝+1
= (1 + 𝑏𝑝 ) 𝑟𝑝 + Γ𝑝,𝜆 (𝑛, 𝛼) ∞
1 (𝑎𝑘 + 𝑏𝑘 ) 𝑟𝑝+1 𝛼) Γ (𝑛, 𝑘=𝑝+1 𝑝,𝜆
× ∑
≤ (1 + 𝑏𝑝 ) 𝑟𝑝 + Γ𝑝,𝜆 (𝑛, 𝛼) 𝑟𝑝+1
Acknowledgment The author is grateful to the referees for their valuable suggestions.
References [1] J. Clunie and T. Sheil–Small, “Harmonic univalent functions,” Annales Academiæ Scientiarum Fennicæ Mathematica A, vol. 9, p. 3–25, 1984. [2] J. M. Jahangiri and O. P. Ahuja, “Multivalent harmonic starlike functions,” Annales Mariae Curie-SkThlodowska A, vol. 56, pp. 1–13, 2001. [3] M. K. Aouf, A. O. Mostafa, and R. El-Ashwah, “Sandwich theorems for p-valent functions defined by a certain integral operator,” Mathematical and Computer Modelling, vol. 53, no. 9-10, pp. 1647–1653, 2011. [4] F. M. Al-Oboudi, “On univalent functions defined by a generalized S˘al˘agean operator,” International Journal of Mathematics
International Journal of Analysis and Mathematical Sciences, vol. 2004, no. 27, pp. 1429–1436, 2004. [5] L.-I. Cotirla, “Harmonic multivalent functions defined by integral operator,” Studia Universitatis Babes¸-Bolyai, vol. 54, no. 1, pp. 65–74, 2009. [6] L.-I. Cotirla, “A new class of harmonic multivalent functions defined by an integral operator,” Acta Universitatis Apulensis, vol. 21, pp. 55–63, 2010.
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