On Certain Classes of Harmonic -Valent Functions Defined by an Integral Operator

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; Received 3 November 2012; Accepted 18 December 2012 Academic Editor: Frédéric 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 𝑛 𝑛 𝑓 (𝑧) = 𝐼𝑝,𝜆 ℎ (𝑧) + (−1)𝑛 𝐼𝑝,𝜆 𝑔 (𝑧), 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 𝑛 𝑓 (𝑧) 𝐼𝑝,𝜆 Re { 𝑛+1 } > 𝛼, 𝐼𝑝,𝜆 𝑓 (𝑧) (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 − H𝑝 (𝑛; 𝛼) 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). 𝑛 ∞ 𝑝 + ∑ × [( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝+1 [ 𝑛+1 −𝛼( ∞ 2. Coefficient Characterization ∞ 𝑛+1 +(−1) 𝑝 ) ∑( 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝 𝑛+1 ∞ + ((−1) ∑ [( 𝑛 ∞ 󵄨 󵄨 󵄨 󵄨 ∑ {Ψ𝑝,𝜆 (𝑛, 𝑘, 𝛼) 󵄨󵄨󵄨𝑎𝑘 󵄨󵄨󵄨 + Φ𝑝,𝜆 (𝑛, 𝑘, 𝛼) 󵄨󵄨󵄨𝑏𝑘 󵄨󵄨󵄨} ≤ 2, 𝑘=𝑝 𝑘=𝑝 (7) ] 𝑎𝑘 𝑧𝑘 𝑘=𝑝+1 Theorem 1. Let 𝑓 = ℎ + 𝑔 so that ℎ and 𝑔 are given by (2). Furthermore, let 𝑛+1 [ 𝑏𝑘 𝑧 ) 𝑛 𝑛+1 𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝) ∞ 𝑝 Ψ𝑝,𝜆 (𝑛, 𝑘, 𝛼) = (( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝+1 𝑝 −𝛼( ) 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑛+1 ) (8) 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 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) } 𝑏𝑘 𝑧 ) } } 𝑝 ) ∑( 𝑝 + 𝜆 (𝑘 − 𝑝) 𝑘=𝑝 +(−1) ∞ 𝑛 𝑛+1 𝑓 (𝑧) − 𝛼𝐼𝑝,𝜆 𝑓 (𝑧) 𝐼𝑝,𝜆 −1 𝑛+1 ∞ 𝑛+1 × (1 − 𝛼)−1 . Re { 𝑎𝑘 𝑧𝑘 = Re {((1 − 𝛼) × (1 − 𝛼)−1 , Φ𝑝,𝜆 (𝑛, 𝑘, 𝛼) = (( 𝑛+1 𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝) ] 𝑏𝑘 𝑧𝑘 ) ] where × (𝑧𝑝 + ∑ ( −1 𝑘 𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝) +𝛼( 𝑛 ] 𝑎𝑘 𝑧𝑘 ) 𝑝 ) 𝑝 + 𝜆 (𝑘 − 𝑝) × (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). 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 (...truncated)