Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 759251, 12 pages doi:10.1155/2009/759251 Research Article Certain Classes of Harmonic Multivalent Functions Based on Hadamard Product ă ă ¨ Om P Ahuja,1 H Ozlem Guney,2 and F Muge Sakar2 Department of Mathematical Sciences, Kent State University, 14111 Claridon-Troy Road, Burton, OH 44021, USA Department of Mathematics, Faculty of Science and Arts, Dicle University, 21280 Diyarbakır, Turkey ¨ Correspondence should be addressed to H Ozlem Guney, ozlemg@dicle.edu.tr ¨ Received 25 February 2009; Accepted 12 September 2009 Recommended by Yong Zhou We define and investigate two special subclasses of the class of complex-valued harmonic multivalent functions based on Hadamard product Copyright q 2009 Om P Ahuja 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 Introduction A continuous function f u iv is a complex-valued harmonic function in a complex domain C if both u and v are real harmonic in C In any simply connected domain D ⊆ C, we can write f h g, where h and g are analytic in D We call h the analytic part and g the co-analytic part of f Note that f h g reduces to h if the coanalytic part g is zero For p ≥ 1, denote by H p the set of all multivalent harmonic functions f h g defined in the open unit disc U, where h and g defined by h z zp ∞ an zn , ∞ g z n p t bn zn , bp t−1 < 1, t ∈ N : {1, 2, } 1.1 n p t−1 are analytic functions in U Let F be a fixed multivalent harmonic function given by F z H z Gz zp ∞ |An |zn n p t ∞ |Bn |zn , n p t−1 Bp t−1 < 1, t ∈ N 1.2 Journal of Inequalities and Applications A function f ∈ H p is said to be in the class HF p, t, α, k if z f ∗F Re z z f ∗F z ≥k z f ∗F z z f ∗F z pα , −p ∂/∂θ r eiθ , f z where f ∗ F is a harmonic convolution of f and F Note that z ∂/∂θ f r eiθ Using the fact Re w > k w − p pα ⇐⇒ Re 1.3 keiθ w − kpeiθ ≥ pα, 1.4 it follows that f ∈ HF p, t, α, k if and only if Re keiθ z f ∗ F z f ∗F z z − kpeiθ ≥ pα, ≤ α < 1.5 A function f in HF p, t, α, k is called k-uniformly multivalent harmonic starlike function associated with a fixed multivalent harmonic function F The set HF p, t, α, k is a comprehensive family that contains several previously studied subclasses of H p ; for example, if we let F z I z : zp 1−z zp 1−z ∞ zp zn ∞ zn , 1.6 n p n p zf z zf z ≥ pα , 1.7 then HI p, α, ≡ S∗ p, α : H f ∈ H p : Re see 1, ; HI p, 1, 0, ≡ S∗ p, ≡ S∗ p , H H 1.8 HI 1, 1, α, ≡ S∗ 1, α ≡ S∗ α , H H 1.9 HI 1, 1, 0, ≡ S∗ ≡ S∗ , H H 1.10 see ; see ; Journal of Inequalities and Applications see 5, ; HI 1, 1, α, k ≡ GH k, α : f ∈ H : Re keiθ zf z − keiθ zf z ≥α , 1.11 ≥ pα , 1.12 see ; HI p, 1, α, ≡ MH p, α : f ∈ H p : Re eiθ zf z − peiθ zf z see Finally, denote by T H p the subclass of functions f z h z zp − ∞ |an |zn , h z ∞ g z n p t g z in H p where |bn |zn 1.13 n p t−1 Let HF p, t, α, k : T H p ∩ HF p, t, α, k In this paper, we investigate coefficient conditions, extreme points, and distortion bounds for functions in the family HF p, t, α, k We observe that the results so obtained for this main family can be viewed as extensions and generalizations for various subclasses of H p and H Main Results Theorem 2.1 Let f inequality ∞ n p t n p 1−α h g be such that h and g are given by 1.1 Then f ∈ HF p, t, α, k if the k −p k α |an An | − p 1−α −1 ∞ n p t−1 n p 1−α k p k α |bn Bn | ≤ − p 1−α −1 2.1 is satisfied for some k k ≥ , p p ≥ , α ≤ α < , and t t ≥ Proof In view of 1.5 , we need to prove that Re w > 0, where keiθ w z h∗H z −z g∗G z h∗H z − p keiθ α h∗H z g∗G z g∗G z : A z B z 2.2 Using the fact that Re w > ⇔ |1 |A z w| ≥ |1 − w| , it sufficies to show that B z | − |A z − B z | ≥ 2.3 Journal of Inequalities and Applications Therefore, we obtain |A z B z | − |A z − B z | ≥ p 1−α ∞ − p − α − |z|p − n k −p k α |an An ||z|n n p t ∞ − n k n α − |an An ||z|n k −p k n p t n p t−1 ∞ − ∞ α − |bn Bn ||z|n − p k n k p k α |bn Bn ||z|n n p t−1 p 1−α − ∞ n 1 − p − α − |z|p k −p k ∞ α |an An ||z|n − n p t n k p k α |bn Bn ||z|n n p t−1 p − α − p − α − |z|p ⎧ ∞ ⎨ n k −p k α × 1− |an An | ⎩ − p 1−α −1 n p t p 1−α ∞ − n p t−1 n k p k α − p 1−α −1 p 1−α |bn Bn | ⎫ ⎬ ⎭ ≥ 2.4 By hypothesis, last expression is nonnegative Thus the proof is complete The coeficient bounds 2.1 is sharp for the function f z ∞ zp n p t ∞ where ∞ n p t |Xn | ∞ n p t−1 |Yn | n p 1−α n p t−1 − p 1−α −1 p 1−α n k −p k − p 1−α −1 k p k Xn zn α α Bn 2.5 Yn zn , Corollary 2.2 For p ≥ 1/1 − α , ≤ α < 1, if the inequality ∞ n k −p k n p t holds, then f ∈ HF p, t, α, k α |an An | ∞ n p t−1 n k p k α |bn Bn | ≤ 2.6 Journal of Inequalities and Applications Corollary 2.3 For ≤ α < and ≤ p ≤ 1/1 − α , if the inequality ∞ n k −p k ∞ α |an An | n p t n k α |bn Bn | ≤ p − α p k 2.7 n p t−1 holds, then f ∈ HF p, t, α, k Theorem 2.4 Let f ≤ α < Then h g be such that h and g are given by 1.13 Also, suppose that k ≥ and i for ≤ p ≤ 1/1 − α , f ∈ HF p, t, α, k if and only if ∞ n k −p k ∞ α |an An | n p t n k p k α |bn Bn | ≤ p − α ; 2.8 n p t−1 ii for p − α ≥ , f ∈ HF p, t, α, k if and only if ∞ n k −p k ∞ α |an An | n p t n1 k p k α |bn Bn | ≤ 2.9 n p t−1 Proof According to Corollaries 2.2 and 2.3, we must show that if the condition 2.9 does not hold, then f / HF p, t, α, k , that is, we must have ∈ ⎛ ⎜ Re⎜ ⎝ keiθ z h∗H z −z g∗G z h∗H z − p keiθ α h∗H z g∗G z g∗G z ⎞ ⎟ ⎟ ≥ ⎠ 2.10 r < 1, and using Re −eiθ ≥ Choosing the values of z r on positive real axis where ≤ z −|eiθ | −1 , the inequality 2.10 reduces to ⎧ ⎨ Re ≥ ⎩ ⎧ ⎨ ⎩ keiθ pzp − p zp − k pr p − p rp − ⎧ ⎨ p − α rp − ⎩ p rp − ∞ n p t ∞ n p t n|an An |zn − ∞ n n p t |an An |z ∞ n p t n ∞ n p t−1 ⎭ ⎫ n|bn Bn |r n − B ⎬ ⎭ ∞ n n p t−1 |bn Bn |r k −p k ∞ n n p t |an An |r ⎫ n|bn Bn |zn − A ⎬ ∞ n n p t−1 |bn Bn |z n|an An |r n − ∞ n n p t |an An |r ∞ n p t−1 ⎫ α |an An |r n − C ⎬ ∞ n n p t−1 |bn Bn |r , ⎭ 2.11 Journal of Inequalities and Applications where A denotes k eiθ α p zp − ∞ p t |an An |zn n ∞ ∞ n n n p t |an An |r n p t−1 |bn Bn |r , and C denotes Letting r → 1− , we obtain ∞ n n p t−1 |bn Bn |z ∞ k n p t−1 n p 1−α − ∞ n p t n1 k −p k ∞ n p t |an An | 1− ∞ n p t−1 α |an An | n k , B denotes k α p r p − p k α |bn Bn |r n p k α |bn Bn | ∞ n p t−1 |bn Bn | ≥ 2.12 If the condition 2.10 does not hold, then the numerator in 2.12 is negative for r sufficiently close to Hence there exists z0 r0 in 0, for which 2.12 is negative Therefore, it follows that f / HF p, t, α, k and so the proof is complete ∈ Theorem 2.5 If f ∈ HF p, t; α, k , then for |z| f z f z ≤ ≥ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ bp p − bp p − rp t−1 p p t−1 t rp t−1 t p p t−1 p 1−α k −p k t r < , |Ap t | ≤ |An | ≤ |Bn |, and Ap t / 0, α k Ap t p k k −p k α α Ap bp t t−1 Bp t−1 rp 1; p 1−α ≤1 t−1 k −p k t−1 t α k k −p k Ap t p k α α Ap ⎧ ⎪ − bp t−1 r p t−1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p 1−α ⎪ ⎪− ⎪ ⎪ ⎪ p t k − p k α Ap t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p t−1 k p k α ⎪ ⎪ ⎪ − ⎪ ⎪ ⎨ p t k − p k α Ap ⎪ ⎪ − bp t−1 r p t−1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪− ⎪ ⎪ ⎪ p t k − p k α Ap t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p t−1 k p k α ⎪ ⎪ ⎪ − ⎩ p t k − p k α Ap t t t bp t−1 Bp t−1 r p ; p − α ≥ 1, bp t−1 Bp t−1 rp 1; p 1−α ≤1 bp t−1 Bp t−1 rp 1; p − α ≥ 2.13 These bounds are sharp Journal of Inequalities and Applications Proof Suppose p − α ≤ Let f ∈ HF p, t, α, k and |Ap t | ≤ |Bn | In view of 1.13 , we get f z zp bp z t−1 p t−1 ∞ − |an |zn − |bn |zn n p t bp ≤ rp t−1 ∞ t−1 |bn | r p |an | n p t ∞ × t−1 rp p bp ≤ t−1 t t−1 rp p k −p k p 1−α n p t bp ≤ ⎛ ×⎝ ∞ bp p n ⎛ ×⎝ ∞ n n p t t t−1 p t k −p k p 1−α α Ap t t Ap α n1 |an | t k Ap α n |An an | Ap α p 1−α k −p k t |bn | r p |an | p 1−α k −p k Ap k −p k α p 1−α rp t−1 α t−1 n p t ≤ t p 1−α k −p k Ap p k α p 1−α ⎞ t |bn |⎠r p t k p k p 1−α ⎞ α |Bn bn |⎠r p 2.14 Using Theorem 2.4 i , we obtain f z bp ≤ × p 1− bp t−1 p − rp t−1 t p p α bp α Ap t−1 Bp t−1 t rp t−1 2.15 p 1−α k −p k t−1 t p k p t−1 k p 1−α rp t p 1−α k −p k t−1 k k −p k Ap α t p k α α Ap The proofs of other cases are similar and so are omitted t bp t−1 Bp t−1 rp 8 Journal of Inequalities and Applications Corollary 2.6 If f ∈ HF p, t, α, k , then w : |w z | < ⎧ p 1−α ⎪1− ⎪ ⎪ ⎪ ⎪ p t k − p k α Ap t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪− p t k −p k α − p t−1 ⎪ ⎪ ⎪ ⎪ p t k −p k ⎪ ⎪ ⎪ ⎪ ⎨ k α ⎪ ⎪ ⎪ ⎪1− ⎪ ⎪ ⎪ p t k − p k α Ap t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪− p t k −p k α − p t−1 ⎪ ⎪ ⎪ p t k −p k ⎪ ⎪ ⎩ p k Ap α t k p k α Ap α t ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Bp t−1 bp t−1 ⎪ ⎪ ⎪ ;⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ p − α ≤ 1⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Bp t−1 bp t−1 ⎪ ⎪ ⎪ ;⎪ ⎪ ⎪ ⎪ ⎪ ⎭ p 1−α ≥1 ⊂f U 2.16 Theorem 2.7 Suppose An / n p t, n Then f ∈ clco HF p, t, α, k if and only if f z ∞ p t 1, and Bn / Xn hn z Yn gn z , n z ∈ U, p t − 1, n p t, 2.17 n p t−1 where hp t−1 z hn z gn z zp ⎧ ⎪ p ⎪z − ⎪ ⎪ ⎨ n ⎧ ⎪ p ⎪z ⎪ ⎪ ⎨ ⎪ ⎪ p ⎪z ⎪ ⎩ n p t, p t 1, , p − α ≤ 1, n 1 k −p k zn ; n p t, p t 1, , p − α ≥ 1, n p 1−α zn ; k p k α |Bn | n p t − 1, p t, , p − α ≤ n ⎪ ⎪ p ⎪z − ⎪ ⎩ p 1−α zn ; k − p k α |An | k n p t − 1, p t, , p − α ≥ ⎛ Xp t−1 ≡ Xp 1−⎝ ∞ p k Xn n p t α |An | α |Bn | ∞ zn ; ⎞ Yn ⎠ , Xn ≥ 0, Yn ≥ n p t−1 2.18 In particular, the extreme points of HF p, t, α, k are {hn } and {gn } Journal of Inequalities and Applications Proof Suppose p − α ≤ For functions of the form 2.17 , we can write f z ∞ zp − n1 n p t p 1−α Xn zn k − p k α |An | ∞ n p t−1 p 1−α Yn zn 2.19 k p k α |Bn | n1 On the other hand, for ≤ Xp ≤ 1, we obtain ∞ n k − p k α |An | p 1−α n p t ∞ n k n p t−1 ⎛ ∞ ⎝ Xn n p t p 1−α Xn k − p k α |An | n p k α |Bn | p 1−α ∞ n k p 1−α Yn p k α |Bn | 2.20 ⎞ Yn ⎠ − Xp ≤ n p t−1 Thus f ∈ HF p, t, α, k , by Theorem 2.4 Conversely, suppose that f ∈ HF p, t, α, k Then, it follows Theorem 2.4 that |an | ≤ n p 1−α , k − p k α |An | |bn | ≤ n k p 1−α p k α |Bn | 2.21 Setting Xn n k − p k α |an An | , p 1−α Yn n1 k p k α |bn Bn | , p 1−α 2.22 and defining ⎛ Xp 1−⎝ ∞ Xn n p t ∞ ⎞ Yn ⎠ , n p t−1 2.23 10 Journal of Inequalities and Applications where Xp ≥ 0, we obtain f z ∞ zp − ∞ |an |zn n p t n p t−1 ∞ zp − n p t ∞ zp − p − α Xn zn k − p k α |An | n ∞ zp − hn z Xn − n p t ⎛ ⎛ ⎝1 − ⎝ ∞ n p t−1 p − α Yn z k p k α |Bn | n zp − gn z Yn ∞ ∞ ⎞⎞ ∞ Xn ∞ Xn hn z ∞ hn z Xn n p t n p t−1 n p t ∞ Yn ⎠⎠zp n 2.24 n p t−1 n p t Xp zp n |bn | z g n z Yn n p t−1 Yn g n z n p t−1 Thus f can be expressed as 2.17 The proof for the case p − α ≥ is similar and hence is omitted Theorem 2.8 The class HF p, t, α, k is closed under convex combinations Proof For j 1, 2, , let the functions fj given by ∞ zp − fj z n p t ajn zn ∞ bj n z n 2.25 n p t−1 are in HF p, t, α, k Also suppose the given fixed harmonic functions are given by Fj z zp ∞ n p t For ∞ j μj Ajn zn ∞ Bj n z n 2.26 n p t−1 1, ≤ μj ≤ the convex combinations of fj can be expressed as ∞ j μj fj z zp − ∞ ⎛ ⎝ n p t ∞ j ⎞ μj ajn ⎠zn ∞ ⎛ ⎝ n p t−1 ∞ j ⎞ μ j bj n ⎠ z n 2.27 Journal of Inequalities and Applications 11 Since ∞ n1 k −p k ∞ ajn Ajn α n p t n1 k p k bjn Bjn α n p t−1 ≤ ⎧ ⎨p − α if p − α ≥ ⎩1 if p − α ≤ 1, 2.28 2.27 yields ∞ n k −p k α n p t ∞ μj ajn Ajn j ∞ μj j ≤ ⎧ ⎨ ⎩n ∞ ∞ μj ∞ n k p k α n p t−1 k −p k p t ⎧ ⎪p − α ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ n ∞ α ∞ μ j bj n B j n j ∞ ajn Ajn n k p k α n p t−1 μj p 1−α bjn Bjn ⎫ ⎬ ⎭ if p − α ≤ j if p − α ≥ 1 j 2.29 Thus the coefficient estimate given by Theorem 2.4 holds Therefore, we obtain ∞ j μj fj z ∈ HF p, t, α, k 2.30 Acknowledgment ă This present investigation is supported with the Project no DUBAP-07-02-21 by Dicle University, The committee of the Scientific Research Projects References O P Ahuja and J M Jahangiri, “Multivalent harmonic starlike functions with missing coefficients,” Mathematical Sciences Research Journal, vol 7, no 9, pp 347–352, 2003 ă H O Guney and O P Ahuja, Inequalities involving multipliers for multivalent harmonic functions, ă Journal of Inequalities in Pure and Applied Mathematics, vol 7, no 5, article 190, pp 1–9, 2006 O P Ahuja and J M Jahangiri, “Multivalent harmonic starlike functions,” Annales Universitatis Mariae Curie-Skłodowska Sectio A, vol 55, no 1, pp 1–13, 2001 J M Jahangiri, “Harmonic functions starlike in the unit disk,” Journal of Mathematical Analysis and Applications, vol 235, no 2, pp 470–477, 1999 12 Journal of Inequalities and Applications H Silverman, “Harmonic univalent functions with negative coefficients,” Journal of Mathematical Analysis and Applications, vol 220, no 1, pp 283–289, 1998 H Silverman and E M Silvia, “Subclasses of harmonic univalent functions,” New Zealand Journal of Mathematics, vol 28, no 2, pp 275–284, 1999 O P Ahuja, R Aghalary, and S B Joshi, “Harmonic univalent functions associated with k-uniformly starlike functions,” Mathematical Sciences Research Journal, vol 9, no 1, pp 9–17, 2005 J M Jahangiri, G Murugusundaramoorthy, and K Vijaya, “On starlikeness of certain multivalent harmonic functions,” Journal of Natural Geometry, vol 24, no 1-2, pp 1–10, 2003  ... 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