On Meromorphic Harmonic Functions with Respect to -Symmetric Points
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Research Article On Meromorphic Harmonic Functions with Respect to k-Symmetric Points K. Al-Shaqsi and M. Darus School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Selangor D. Ehsan 43600, Malaysia Correspondence should be addressed to M. Darus, [email protected] Received 22 May 2008; Revised 20 July 2008; Accepted 23 August 2008 Recommended by Ramm Mohapatra In our previous work in this journal in 2008, we introduced the generalized derivative operator j Dm for f ∈ SH . In this paper, we introduce a class of meromorphic harmonic function with j respect to k-symmetric points defined by Dm . Coefficient bounds, distortion theorems, extreme points, convolution conditions, and convex combinations for the functions belonging to this class are obtained. Copyright q 2008 K. Al-Shaqsi and M. Darus. 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.
1. Introduction A continuous function f u iv is a complex valued harmonic function in a domain D ⊂ C if both u and v are real harmonic in D. In any simply connected domain, we write f h g where h and g are analytic in D. A necessary and sufficient condition for f to be locally univalent and orientation preserving in D is that |h | > |g | in D see 1. Hengartner and {z : |z| > 1}. Schober 2 investigated functions harmonic in the exterior of the unit disk U They showed that complex valued, harmonic, sense preserving, univalent mapping f must admit the representation fz hz gz A log |z|,
1.1
where hz and gz are defined by hz αz
∞ n1
for 0 ≤ |β| < |α|, A ∈ C and z ∈ U.
an z−n ,
gz βz
∞ bn z−n , n1
1.2
2
Journal of Inequalities and Applications For z ∈ U \ {0}, let MH denote the class of functions: fz hz gz
∞ ∞ 1 an zn bn zn , z n1 n1
1.3
which are harmonic in the punctured unit disk U \ {0}, where hz and gz are analytic in U \ {0} and U, respectively, and hz has a simple pole at the origin with residue 1 here. j In 3, the authors introduced the operator Dm for f ∈ SH which is the class of functions f h g that are harmonic univalent and sense-preserving in the unit disk U {z : |z| < 1} for which f0 h0 fz 0 − 1 0. For more details about the operator j Dm , see 4. j Now, we define Dm for f h g given by 1.3 as j
j
j
Dm fz Dm hz −1j Dm gz,
j, m ∈ N0 N ∪ {0}; z ∈ U \ {0},
1.4
where j
Dm hz j
Dm gz
∞ −1j nj Cm, nan zn , z n1 ∞ nj Cm, nbn zn ,
1.5
n1
Cm, n
nm−1 m
n m − 1! . m!n − 1!
A function f ∈ MH is said to be in the subclass MS∗H of meromorphically harmonic starlike functions in U \ {0} if it satisfies the condition zh z − zg z > 0, Re − hz gz
z ∈ U \ {0}.
1.6
Note that the class of harmonic meromorphic starlike functions has been studied by Jahangir
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