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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 259205, 11 pages doi:10.1155/2008/259205 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, maslina@ukm.my 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 D j m for f ∈S H . In this paper, we introduce a class of meromorphic harmonic function with respect to k-symmetric points defined by D j m .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 Schober 2 investigated functions harmonic in the exterior of the unit disk  U  {z : |z| > 1}. They showed that complex valued, harmonic, sense preserving, univalent mapping f must admit the representation fzhz gzA log |z|, 1.1 where hz and gz are defined by hzαz  ∞  n1 a n z −n ,gzβz  ∞  n1 b n z −n , 1.2 for 0 ≤|β| < |α|,A∈ C and z ∈  U. 2 Journal of Inequalities and Applications For z ∈ U \{0}, let M H denote the class of functions: fzhz gz 1 z  ∞  n1 a n z n  ∞  n1 b n z n , 1.3 which are harmonic in the punctured unit disk U \{0}, where hz and gz are analytic in U \{0} and U, respectively, and hz has a simple pole at the origin with residue 1 here. In 3, the authors introduced the operator D j m for f ∈S H 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 f0h0f z 0 − 1  0. For more details about the operator D j m ,see4. Now, we define D j m for f  h  g given by 1.3 as D j m fzD j m hz−1 j D j m gz, j, m ∈ N 0  N ∪{0}; z ∈ U \{0}, 1.4 where D j m hz −1 j z  ∞  n1 n j Cm, na n z n , D j m gz ∞  n1 n j Cm, nb n z n , Cm, n  n  m − 1 m   n  m − 1! m!n − 1! . 1.5 A function f ∈M H is said to be in the subclass MS ∗ H of meromorphically harmonic starlike functions in U \{0} if it satisfies the condition Re  − zh  z − zg  z hzgz  > 0, z ∈ U \{0}. 1.6 Note that the class of harmonic meromorphic starlike functions has been studied by Jahangiri and Silverman 5, and Jahangiri 6. Now, we have the following definition. Definition 1.1. For j, m ∈ N 0 , 0 ≤ α<1andk ≥ 1, let MHS k s j, m, α denote the class of meromorphic harmonic functions f of the form 1.3 such that Re  − D j1 m fz D j m f k z  > 0, z ∈ U \{0}, 1.7 where D j m f k zD j m h k −1 j D j m g k j, m ∈ N 0 ,k ≥ 1, 1.8 h k z −1 j z  ∞  n1 a n Φ n z n ,g k z ∞  n1 Φ n z n , 1.9 Φ n  1 k k−1  ν0 ε n−1ν ,  k ≥ 1; ε  exp  2πi k  . 1.10 K. Al-Shaqsi and M. Darus 3 For more details about harmonic functions with respect to k-symmetric points, see papers 7, 8 given by the authors. Also, note that MHS 2 s j, 0,α ⊂ MHS ∗ S n, α was introduced by Bostancı and ¨ Ozt ¨ urk 9. Finally, let MHS k s j, m, α denote the subclass of MHS k s j, m, α consist of harmonic functions f j  h j  g j such that h j and g j are of the form h j z −1 j z  ∞  n1 |a n |z n ,g j z−1 j ∞  n1 |b n |z n . 1.11 Also, let f k j  h k j  g k j where h k j and g k j are of the form h k j z −1 j z  ∞  n1 Φ n |a n |z n ,g k j z−1 j ∞  n1 Φ n |b n |z n , 1.12 where Φ n is given by 1.10. In this paper, we will give a sufficient condition for functions f  h  g, where h and g given by 1.3 to be in the class MHS k s j, m, α. Indeed, it is shown that this coefficient condition is also necessary for functions to be in the class MHS k s j, m, α. Also, we obtain distortion bounds and characterize the extreme points for functions in MHS k s j, m, α. Convolution and closure theorems are also obtained. 2. Coefficient bounds First, we prove a sufficient coefficient bound. Theorem 2.1. Let f  h  g be of the form 1.3 and f k  h k  g k where h k and g k are given by 1.9.If ∞  n1 n − 1k  1  α|a n−1k1 | n − 1k  1 − α|b n−1k1 |Ω j m n, k  ∞  n2 n /  lk1 n j1 Cm, n|a n |  |b n | ≤ 1 − α, 2.1 where j, m ∈ N 0 , 0 ≤ α<1,k≥ 1 and Ω j m n, kn − 1k  1 j Cm, nk  1,thenf is harmonic univalent, sense preserving in U \{0} and f ∈ MHS k s j, m, α. Proof. For 0 < |z 1 |≤|z 2 | < 1, we have   f  z 1  − f  z 2    ≥   h  z 1  − h  z 2    −   g  z 1  − g  z 2    ≥   z 1 − z 2     z 1     z 2   −   z 1 − z 2   ∞  n1    a n      b n      z n−1 1  ··· z n−1 2   >   z 1 − z 2     z 1     z 2    1 −   z 2   2 ∞  n1 n    a n      b n     4 Journal of Inequalities and Applications >   z 1 − z 2     z 1     z 2    1 −   z 2   2  ∞  n1 n    a n      b n     ∞  n1  n − 1k  1    a n−1k1      b n−1k1      >   z 1 − z 2     z 1     z 2    1 − ∞  n1  n − 1k  1  α   a n−1k1   − n − 1k  1 − α   b n−1k1    Ω j m n, k − ∞  n2 n /  lk1 n j1 Cm, n    a n      b n    . 2.2 This last expression is nonnegative by 2.1,andsof is univalent in U \{0}. To show that f is sense preserving in U \{0}, we need to show that |h  z|≥|g  z| in U \{0}. We have |h  z|≥ 1 |z| 2 − ∞  n1 n|a n ||z| n−1  1 r 2 − ∞  n1 n|a n |r n−1 > 1 − ∞  n1 n|a n | ≥ 1 − ∞  n1 n − 1k  1  α|a n−1k1 |Ω j m n, k − ∞  n2 n /  lk1 n j1 Cm, n|a n | ≥ ∞  n1 n − 1k  1 − α|b n−1k1 |Ω j m n, k ∞  n2 n /  lk1 n j1 Cm, n|b n | ≥ ∞  n1 2n|b 2n |  ∞  n1 2n − 1|b 2n−1 | > ∞  n1 n|b n |r n−1  ∞  n1 n|b n ||z| n−1 ≥|g  z|. 2.3 Now, we will show that f ∈ MHS k s j, m, α. According to 1.4 and 1.7,for0≤ α<1, we have Re  − D j1 m fz D j m f k z   Re ⎧ ⎨ ⎩ − D j1 m hz − −1 j D j1 m gz D j m h k z−1 j D j m g k z ⎫ ⎬ ⎭ ≥ α. 2.4 Using the fact that Re{w}≥α if and only if |1 − α  w|≥|1  α − w|,itsuffices to show that     1 − α − D j1 m fz D j m f k z     ≥     1  α  D j1 m fz D j m f k z     , 2.5 which is equivalent to   D j1 m fz − 1 − αD j m f k z   −   D j1 m fz1  αD j m f k z   ≥ 0. 2.6 K. Al-Shaqsi and M. Darus 5 Substituting D j m fz, D j1 m fz, and D j m f k z in 2.6 yields     D j1 m hz − −1 j D j1 m gz − 1 − α  D j m h k z−1 j D j m g k z      −     D j1 m hz − −1 j D j1 m gz1  α  D j m h k z−1 j D j m g k z            −1 j z − ∞  n1 n j1 Cm, na n z n −1 j ∞  n1 n j1 Cm, nb n z n 1 − α  −1 j z  ∞  n1 n j Cm, nΦ n a n z n −1 j ∞  n1 n j Cm, nΦ n b n z n       −      −1 j z − ∞  n1 n j1 Cm, na n z n −1 j ∞  n1 n j1 Cm, nb n z n − 1  α  −1 j z  ∞  n1 n j Cm, nΦ n a n z n −1 j ∞  n1 n j Cm, nΦ n b n z n             2 − α−1 j z − ∞  n1 n j Cm, nn−1−αΦ n a n z n −1 j ∞  n1 n j Cm, nn1 −αΦ n b n z n      −      α−1 j z − ∞  n1 n j Cm, nn 1  αΦ n a n z n −1 j ∞  n1 n j Cm, nn − 1  αΦ n b n z n      ≥ 2 − α |z| − ∞  n1 n j Cm, nn − 1 − αΦ n |a n ||z n |− ∞  n1 n j Cm, nn 1 − αΦ n |b n ||z n | − α |z| − ∞  n1 n j Cm, nn 1  αΦ n |a n ||z n |− ∞  n1 n j Cm, nn − 1  αΦ n |b n ||z n |  21 − α |z|  1 − ∞  n1 n j Cm, nn  αΦ n  1 − α |a n |   z n1   − ∞  n1 n j Cm, nn − αΦ n  1 − α |b n |   z n1    ≥ 21 − α  1 − ∞  n1 n j Cm, nn  αΦ n  1 − α |a n |− ∞  n1 n j Cm, nn − αΦ n  1 − α |b n |  . 2.7 From the definition of Φ n , we know that Φ n  ⎧ ⎨ ⎩ 1,n lk  1, 0,n /  lk  1, n ≥ 2,k,l≥ 1. 2.8 6 Journal of Inequalities and Applications Substituting 2.8 in 2.7, then 2.7 is equivalent to   D j1 m fz − 1 − αD j m f k z   −   D j1 m fz1  αD j m f k z   ≥ 21 − α  1 − ∞  n1 nk  1 j Cm, nk  1nk  1  α 1 − α |a nk1 | − ∞  n1 nk  1 j Cm, nk  1nk  1 − α 1 − α |b nk1 |− ∞  n2 n /  lk1 n j Cm, n 1 − α |a n | − ∞  n2 n /  lk1 n j Cm, n 1 − α |b n |− 1  α 1 − α |a 1 |−|b 1 |   21 − α  1 − ∞  n1  n − 1k  1  α 1 − α |a n−1k1 |− n − 1k  1 − α 1 − α |b n−1k1 |  Ω j m n, k − ∞  n2 n /  lk1 n j1 Cm, n 1 − α |a n |  |b n |  ≥ 0, by 2.6. 2.9 Thus, this completes the proof of the t heorem. We next show that condition 2.1 is also necessary for functions in MHS k s j, m, α. Theorem 2.2. Let f j  h j  g j ,whereh j and g j are given by 1.11, and f k j  h k j  g k j where h k j and g k j are given by 1.12. Then, f j ∈ MHS k s j, m, α, if and only if the inequality 2.1 holds for the coefficient of f j  h j  g j and f k j  h k j  g k j . Proof. In view of Theorem 2.1, we need only to show that f j / ∈ MHS k s j, m, α if condition 2.1 does not hold. We note that for f j ∈ MHS k s j, m, α, then by 1.7 the condition 2.4 must be satisfied for all values of z in U \{0}. Substituting for h j ,g j ,h k j , and g k j given by 1.11 and 1.12, respectively, in 2.4 and choosing 0 <z r<1, we are required to have Re{Ψz/Υz}≥0, where Ψz−D j1 m h j z−1 n D j1 m g j z − αD j m h k j z − α−1 j D j m g k j z  1 − α z − ∞  n1 n j Cm, nn  αΦ n |a n |z n  ∞  n1 n j Cm, nn − αΦ n |b n |z n , ΥzD j m h k j z−1 j D j m g k j z  1 z  ∞  n1 n j Cm, nΦ n |a n |z n  ∞  n1 n j Cm, nΦ n |b n |z n . 2.10 Then, the required condition Re{Ψz/Υz}≥0 is equivalent to 1 − α/z −  ∞ n1 n j Cm, nn  αΦ n |a n |r n   ∞ n1 n j Cm, nn − αΦ n |b n |r n 1/z   ∞ n1 n j Cm, nΦ n |a n |r n   ∞ n1 n j Cm, nΦ n |b n |r n . 2.11 K. Al-Shaqsi and M. Darus 7 By using 2.8, and if condition 2.1 does not hold, then the numerator of 2.11 is negative for r sufficiently close to 1. Thus, there exists a z 0  r 0 in 0, 1 for which the quotient in 2.11  is negative. This contradicts the required condition for f j ∈ MHS k s j, m, α and so the proof is complete. 3. Distortion bounds and extreme points In this section, we will obtain distortion bounds for functions f j ∈ MHS k s j, m, α and also provide extreme points for the class MHS k s j, m, α. Theorem 3.1. If f j  h j  g j ∈ MHS k s j, m, α and 0 < |z|  r<1,then 1 r − 1 − α 2 j m  12 − α r ≤|f j z|≤ 1 r  1 − α 2 j m  12 − α r. 3.1 Proof. We will prove the left side of the inequality. The argument for the right side of the inequality is similar to the left side, and thus the details will be omitted. Let f j  h j  g j ∈ MHS k s j, m, α. Taking the absolute value of f,weobtain |f j |       −1 j z  ∞  n1 a n z n −1 n ∞  n1 b n z n      ≥ 1 r − ∞  n1 |a n |  |b n |r n ≥ 1 r − ∞  n1 |a n |  |b n |r ≥ 1 r − 1 − α 2 j m  12 − αΦ 2  ∞  n1 2 j m  12 − αΦ 2  1 − α |a n |  |b n |r ≥ 1 r − 1 − α 2 j m  12 − α ∞  n1  n j Cm, nn  αΦ n  1 − α |a n |  n j Cm, nn − αΦ n  1 − α |b n |  r ≥ 1 r − 1 − α 2 j m  12 − α r, by 2.7. 3.2 The bounds given in Theorem 3.1 hold for functions f j  h  g j of the form 1.11. And it is also discovered that the bounds hold for functions of the form 1.3, if the coefficient condition 2.1 is satisfied. The following covering result follows from the left-hand side of the inequality in Theorem 3.1. Corollary 3.2. If f j ∈ MHS k s j, m, α,then f j U \{0} ⊂  w : |w| < 2 j m  12 − α − 1 − α 2 j m  12 − α  . 3.3 8 Journal of Inequalities and Applications Next, we determine the extreme points of closed convex hulls of MHS k s j, m, α denoted by clco MHS k s j, m, α. Theorem 3.3. Let f j  h j  g j where h j and g j are given by 1.11. Then, f j ∈ MHS k s j, m, α if and only if f j,n z ∞  n0 x n h j n zy n g j n z, 3.4 where h j,0  g j,0 z−1 j /z, h j,n z−1 j /z 1 − α/n j Cm, nn  αΦ n z n n  1, 2, 3, ,g j,n z−1 j /z−1 j 1− α/n j Cm, nn− αΦ n z k n  1, 2, 3, ,  ∞ n0 x n  y n 1,x n ≥ 0,y n ≥ 0. In particular, the extreme points of MHS k s j, m, α are {h j,n } and {g j,n }. Proof. For functions f j  h j  g j , where h j and g j are given by 1.11, we have f j,n z ∞  n0 x n h j,n zy n g j,n z  ∞  n0 x n  y n  −1 j z  ∞  n1 1 − α n j Cm, nn  αΦ n  x n z n −1 j ∞  n1 1 − α n j Cm, nn − αΦ n  y n z k . 3.5 Now, the first part of the proof is complete, and Theorem 2.2 gives ∞  n1 1 − α n j Cm, nn  αΦ n  n j Cm, nn  αΦ n  1 − α x n  ∞  n1 1 − α n j Cm, nn − αΦ n  n j Cm, nn − αΦ n  1 − α y n  ∞  n0 x n  y n − x 0  y 0 1 − x 0  y 0  ≤ 1. 3.6 Conversely, suppose that f j ∈ clcoMHS k s j, m, α. For n  1, 2, 3, ,set x n  n j Cm, nn  αΦ n  1 − α |a n | 0 ≤ x n ≤ 1, y n  n j Cm, nn − αΦ n  1 − α |b n | 0 ≤ y n ≤ 1, 3.7 K. Al-Shaqsi and M. Darus 9 x 0  1 −  ∞ n1 x n  y n . Therefore, f can be written as f j,n z −1 j z  ∞  n1 |a n |z n −1 j ∞  n1 |b n |z n  −1 j z  ∞  n1 1 − αx n n j Cm, nn  αΦ n  z n −1 j ∞  n1 1 − αy n n j Cm, nn − αΦ n  z n  −1 j z  ∞  n1  h j,n z − −1 j z  x n  ∞  n1  g j,n z − −1 j z  y n  ∞  n1 h j,n zx n  ∞  n1 g j,n zy n  −1 j z  1 − ∞  n1 x n − ∞  n1 y n   ∞  n0 h j,n zx n  g j,n zy n , as required. 3.8 4. Convolution and convex combination In this section, we show that the class MHS k s j, m, α is invariant under convolution and convex combination of its member. For harmonic functions f j z−1 j /z   ∞ n1 |a n |z n −1 j  ∞ n1 |b n |z n and F j z −1 j /z   ∞ n1 |A n |z n −1 j  ∞ n1 |B n |z n , the convolution of f j and F j is given by f j ∗F j zf j z∗F j z −1 j z  ∞  n1 |a n ||A n |z n −1 j ∞  n1 |b n ||B n |z n . 4.1 Theorem 4.1. For 0 ≤ β ≤ α<1,letf j ∈ MHS k s j, m, α and F j ∈ MHS k s j, m, β. Then, f j ∗F j ∈ MHS k s j, m, α ⊂ MHS k s j, m, β. Proof. We wish to show that the coefficients of f j ∗F j satisfy the required condition given in Theorem 2.2. For F j ∈ MHS k s j, m, β, we note that |A n |≤1and|B n |≤1. Now, for the convolution function f j ∗F j ,weobtain ∞  n1 n j Cm, nn  βΦ n  1 − β |a n ||A n |  ∞  n1 n j Cm, nn − βΦ n  1 − β |b n ||B n | ≤ ∞  n1 n j Cm, nn  βΦ n  1 − β |a n |  ∞  n1 n j Cm, nn − βΦ n  1 − β |b n | ≤ ∞  n1 n j Cm, nn − αΦ n  1 − α |a n |  ∞  n1 n j Cm, nn − αΦ n  1 − α |b n |≤1, 4.2 since 0 ≤ β ≤ α<1andf j ∈ MHS k s j, m, α. Therefore f j ∗F j ∈ MHS k s j, m, α ⊂ MHS k s j, m, β. 10 Journal of Inequalities and Applications We now examine the convex combination of MHS k s j, m, α. Let the functions f j,t be defined, for t  1, 2, ,ρ,by f j,t z −1 j z  ∞  n1 |a n,t |z n −1 j ∞  n1 |b n,t |z n . 4.3 Theorem 4.2. Let the functions f j,t defined by 4.3 be in the class MHS k s j, m, α for every t  1, 2, ,ρ. Then, the functions ξ t z defined by ξ t z ρ  t1 c t f j n z, 0 ≤ c t ≤ 1, 4.4 are also in the class MHS k s j, m, α, where  ρ t1 c t  1. Proof. According to the definition of ξ t , we can write ξ t z −1 j z  ∞  n1  ρ  t1 c t a n,t  z n −1 j ∞  n1  ρ  t1 c t b n,t  z n . 4.5 Further, since f j,t z are in MHS k s j, m, α for every t  1, 2, ,ρ. Then by 2.7, we have ∞  n1   n  αΦ n   ρ  t1 c t |a n,t |  n − αΦ n   ρ  t1 c t |b n,t |   n j Cm, n   ρ  t1 c t  ∞  n1 n  αΦ n |a n,t | n − αΦ n |b n,t |n j Cm, n  ≤ ρ  t1 c t 1 − α ≤ 1 − α. 4.6 Hence, the theorem follows. Corollary 4.3. The class MHS k s j, m, α is close under convex linear combination. Proof. Let the functions f j,t zt  1, 2 defined by 4.3 be in the class MHS k s j, m, α. Then, the function ψz defined by ψzμf j,1 z1 − μf j,2 z, 0 ≤ μ ≤ 1, 4.7 is in the class MHS k s j, m, α. Also, by taking ρ  2,ξ 1  μ, and ξ 2 1 − μ in Theorem 4.2, we have the corollary. Acknowledgment The work here was fully supported by Fundamental Research Grant SAGA: STGL-012-2006, Academy of Sciences, Malaysia. [...]... Society, vol 37, no 2, pp 291–301, 2000 7 K Al-Shaqsi and M Darus, On subclass of harmonic starlike functions with respect to K-symmetric points,” International Mathematical Forum, vol 2, no 57–60, pp 2799–2805, 2007 8 K Al-Shaqsi and M Darus, On harmonic univalent functions with respect to k-symmetric points,” International Journal of Contemporary Mathematical Sciences, vol 3, no 1–4, pp 111–118, 2008... 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Convolution and convex combination In this section, we show that the class MHS k s j, m, α is invariant under convolution and convex combination of its member. For harmonic functions f j z−1 j /z. 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