CONCAVE MEROMORPHIC FUNCTIONS INVOLVING CONSTRUCTED OPERATORS. M. Al-Kaseasbeh, M. Darus

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1 Acta Universitatis Apulensis ISSN: doi: No. 52/207 pp /j.aua CONCAVE MEROMORPHIC FUNCTIONS INVOLVING CONSTRUCTED OPERATORS M. Al-Kaseasbeh, M. Darus Abstract. This paper involves constructed differential operators in concave meromorphic function and studied its properties. In particular, coefficient bounds, distortion theorem, and extreme points are obtained. 200 Mathematics Subject Classification: 33C45. Keywords: meromorphic function, concave function, constructed operator, distortion theorem, extreme points.. Introduction This paper concerns with class of functions which are analytic in the open unit disk U {z : z < } except for a simple pole at the origin. Also, this class attains certain geometrical interpretation. Explicitly, mapping U onto a domain whose complement is unbounded convex set. Back to analytic functions of the form f(z) z + a k z k, it is well-known fact, that the inequality { R + z f } (z) f > 0, (z) z U (.) characterises convex functions that map the unit disk onto convex domain. Due to the similarity, the inequality { R + z f } (z) f < 0, z U (.2) (z)

2 is used sometimes as a definition of concave analytic functions (see e.g [] and others). However, Bhowmik et al. considered another characterisation of concave analytic functions (see [6]). In 202, the condition (.2) was used again and shown to be necessary and sufficient condition of concave meromorphic mapping in the form by Chuaqui, et al. [7]. Further in [7], the coefficients inequality f(z) z + a 0 + a z + (.3) a a 2 was deduced by applied an invariant form of Schwarz s lemma involving with Schwarzian derivative. Later, Challab and Darus studied on concave meromorphic functions defined by Salagean Operator and Al-Oboudi operator respectively in [8, 9]. In [3], the authors estimated a k for k 2, 3,... for f of the form f(z) z + a k z k. (.4) Let us consider the differential operators Rα,λ n and Dn λ which introduced respectively in [0] and [4]. Then, the convoluted operator of both of them is D α,λ n f(z) Dn λ f(z) Rn α,λ ( f(z) ) z + [ + λ(k )] n a k z k ( z + [ + λ(k )] 2n C(α, k) a 2 k zk ) z + [ + λ(k )] n C(α, k) a k z k (.5) The operator D n α,λ introduced in []. In the other hand, the authors in [2] introduced new differential operator by means of linear combination of both Rα,λ n and Dn λ as follows. Dλ,α,γ n f(z) ( γ)rn αf(z) + γdλ n f(z), z U. (.6) 2

3 If f(z) is an meromorphic function of the form f(z) z + a kz k, then Dλ,α,γ n f(z) z + [ + λ(k )] n [γ + ( γ) C(α, k)] a k z k. (.7) Now let us define the classes of concave meromorphic functions involving constructed differential operator Dλ,α,γ n as follows. Definition. Let C Σ (λ, α, γ) denote the class of complex functions of the form (.4) and satisfies { } R + z (Dn λ,α,γ f(z)) (Dλ,α,γ n < 0, z U, (.8) f(z)) where λ, γ 0, and α, n N 0. For constructed differential operator D α,λ n, we define the following class of concave meromorphic functions. Definition 2. Let C Σ (λ, α) denote the class of complex functions of the form (.4) and satisfies { R + z ( D } α,λ n f(z)) ( D < 0, z U, (.9) α,λ n f(z)) where λ 0, and α, n N 0. We begin with the coefficient bounds of the classes C Σ (λ, α, γ) and C Σ (λ, α). 2. Coefficient Bounds First we obtain coefficient bounds of the normalised concave meromorphic functions of the form (.4) as follow Theorem. Let f(z) be of the form (.4) and k 2 a k. (2.) Then f(z) is concave meromorphic function. 3

4 Proof. Using the fact that Rw 0 if and only if + z f (z) f (z) + + z f (z) f (z) <, w+ w <, we need to show that and so + z f (z) f (z) + + z f (z) f (z) 2f (z) + zf (z) zf (z) z 2 ka kz k z 2 k(k )a kz k 2 + z 2 k(k )a kz k (2k + k(k )) a kz k 2 + z 2 k(k )a kz k k(k + ) a k 2 k(k ) a k. The last expression is bounded above by if k(k + ) a k < 2 k(k ) a k, which equivalent to (2.). The other side of the assertion is trivial. Therefore, f(z) is concave meromorphic function. This result was obtained by the authors in [3]. C Σ (λ, α) we provide the following theorems. For classes C Σ (λ, α, γ) and Theorem 2. Let f(z) be of the form (.4), λ, γ 0, α, n N 0 and k 2 [ + λ(k )] n [γ + ( γ) C(α, k)] a k. (2.2) Then f(z) C Σ (λ, α, γ). Proof. Using the fact that Rw 0 if and only if w+ w <, we need to show that + z (Dn λ,α,γ f) (z) (Dλ,α,γ n f) (z) + <. + z (Dn λ,α,γ f) (z) (Dλ,α,γ n f)(z) Following the steps of proof Theorem, the result is straightforward. 4

5 Theorem 3. Let f(z) be of the form (.4), λ 0, α, n N 0 and k 2 [ + λ(k )] 2n C(α, k) a k 2. (2.3) Then f(z) C Σ (λ, α). Proof. Using the fact that Rw 0 if and only if w+ w <, we need to show that + z ( D α,λ n f(z)) (z) ( D + α,λ n f(z)) (z) + z ( D α,λ n f(z)) (z) ( D <. α,λ n f(z))(z) Following the steps of proof Theorem, the result is straightforward. The following two sections are concerting on the class C Σ (λ, α, γ). 3. Distortion Theorem The forgoing theorem obtain the the bound of f(z) for the class C Σ (λ, α, γ). Theorem 4. Let f(z) be of the form (.4)and in the class C Σ (λ, α, γ). Then for z U f(z) z + 4[ + λ] n [γ + ( γ) C(α, 2)] z 2 and f(z) z + 4[ + λ] n [γ + ( γ) C(α, 2)] z 2. Proof. Using Theorem 2 we have, 4[ + λ] n [γ + ( γ) C(α, 2)] a k k 2 [ + λ(k )] n [γ + ( γ) C(α, k)] a k. That is, a k 4[ + λ] n [γ + ( γ) C(α, 2)] 5

6 f(z) z + a k z k z + a k z k z + a k z 2 z + 4[ + λ] n [γ + ( γ) C(α, 2)] z 2. The other assertion can be proved as follows f(z) z + a k z k z + a k z k This completes the proof. 3.. Extreme Points z + a k z 2 z + 4[ + λ] n [γ + ( γ) C(α, 2)] z 2. In this subsection, extreme points of the normalised concave meromorphic functions of the form (.4) are obtained. Theorem 5. Let f (z) z and f k(z) z + k 2 [+λ(k )] n [γ+( γ)c(α,k)] zk. Then f(z) concave meromorphic function of the form (.4) if and only if it can be expressed in the form f(z) δ k f k (z), where δ k 0 and k δ k. Proof. Assume that f(z) k δ k f k (z). k 6

7 Then Thus, f(z) δ k f k (z) k δ z + ( ) δ k z + k z + ( ) δ k z + k 2 [ + λ(k )] n [γ + ( γ) C(α, k)] zk δ k k 2 [ + λ(k )] n [γ + ( γ) C(α, k)] zk δ k k 2 [ + λ(k )] n [γ + ( γ) C(α, k)] zk. δ k k 2 k2 δ k δ <. Therefore, f(z) is a concave meromorphic function of the form (.4). Conversely, suppose that f(z) is concave meromorphic function of the form (.4). So a k, (k 2, 3,...). k2 We can set Then, This completes the proof. δ k : k 2 δ : δ k. f(z) z + a k z k δ f (z) + δ k f k (z) δ k f k (z). k 7

8 Corollary 6. The extreme points of concave meromorphic functions f(z) of the form (.4) are given by f (z) z and f k(z) z + z k k 2 [+λ(k )] n [γ+( γ)c(α,k)], (k, 2, 3,...). Proof. The proof follows by condition (2.2). Acknowledgement: The work here is supported by MOHE grant: FRGS//206/STG06/UKM/0/. References [] Al-Kaseasbeh, M. and Darus, M., Meromorphic functions involved constructed differential operator, Jnanabha - Vijnana Parishad of India, 47()(207), [2] Al-Kaseasbeh, M., and Darus, M., On an operator defined by the combination of both generalized operators of Salagean and Ruscheweyh, Far East Journal of Mathematical Sciences, 97(4)(205), [3] Al-Kaseasbeh, M. and Darus, M., On Concave Meromorphic Mappings, Journal of Advanced Mathematics and Applications, 6 (207), -5. (in press). [4] Al-Oboudi, F.M., On univalent functions defined by a generalized Salagean operator, International Journal of Mathematics and Mathematical Sciences, 2004(27)(2004), [5] Aouf, M. and Silverman, H., Generalizations of Hadamard products of certain meromorphic univalent functions with positive coefficients, Demonstratio Mathematica, Warsaw Technical University Institute of Mathematics, 4(2)(2008), [6] Bhowmik, B., Ponnusamy, S., and Wirths, K. J., Characterization and the pre- Schwarzian norm estimate for concave univalent functions, Monatshefte fur Mathematik, 6() (200), [7] Chuaqui, M., Duren, P., and Osgood, B., Concave conformal mappings and prevertices of Schwarz-Christoffel mappings, Proceedings of the American Mathematical Society, 40(0)(202), [8] Challab, K., and Darus, M., On certain class of meromorphic harmonic concave functions defined by Salagean operator, Journal of Quality Measurement and Analysis(JQMA), ()(205), [9] Challab, K., and Darus, M., On certain classes of meromorphic harmonic concave functions defined by Al-Oboudi operator, Journal of Quality Measurement and Analysis (JQMA), 2(-2)(206), [0] Darus, M. and K. Al-Shaqsi., Differential sandwich theorems with generalised derivative operator, International Journal of Computational and Mathematical Sciences, 2(2)(2008),

9 [] Pfaltzgraff, J. A., and Pinchuk, B., A variational method for classes of meromorphic functions, Journal d Analyse Mathematique, 24()(97), Mohammad Al-Kaseasbeh School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia UKM Bangi, Selangor, Malaysia zakariya.alkaseasbeh@gmail.com Maslina Darus School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia UKM Bangi, Selangor, Malaysia maslina@ukm.edu.my (corresponding author) 9