82 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 Abstract New class is introduced of meromorphic univalent functions with positive coefficient ∑ { } defined by the integral operator in the punctured unit disc { | | }, satisfying | ( ( )) ( ( )) ( ( )) ( ( )) | . Several properties were studied like coefficient estimates, convex set and weighted mean. Keywords: Meromorphic univalent function; coefficient estimates; convex set; weighted mean. 1. Introduction Let denote the class of functions of the form: ∑ { } which are analytic and meromorphic univalent in the punctured unit disc { | | } { } The Hadamard product [1]. (convolution) of function in (1) and a function : ∑ { } is defined in the class as Integral Transforms of New Subclass of Meromorphic Univalent Functions Defined by Linear Operator I Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Aqeel Ketab AL-khafaji Aqeel Ketab AL-khafaji aqeelketab@gmail.com Department of Mathematics, College of Education for Pure Sciences, University of Babylon, Babylon, Iraq. Article history: Received 15 January 2019, Accepted 18 March 2019, Publish May 2019 Doi: 10.30526/32.2.2147 mailto:tibataiee92@gmail.com mailto:tibataiee92@gmail.com 83 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 ∑ { } Let be a subclass of the class of functions of the form: ∑ { } A function in the class is said to be meromorphic starlike and meromorphic convex of order [ ] respectively if { } and { } In 2013, Juma and Zirar [3]. defined the function ̃ as follows: ̃ ∑ | | for { } is the Pochhammer symbol. Gaussian hypergeometric function ( ∑ ) was used, where ̃ and the Hadamard product for corresponding to the function ̃ the linear operator [ ] defined on by ̃ ∑ | | and ( ) and for ( ) ( ( )) ∑ | | In [4]. Darus and Frasin studied the operator ( ) Now, the condition for the function which is defined in (4) belongs to a class where according to Equation (6). Definition 1: A function of the form (1) is said to be in the class if satisfies the following condition: 84 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 | ( ( )) ( ( )) ( ( )) ( ( )) | where In this paper, A new class of meromorphic univalent functions is studied and discussed the positive coefficient defined by integral operator in the punctured unit disc . Several properties are resulted such as, coefficient estimates, convex set, extreme point and obtain some interested results. See also References [5-9]. 2. Results In this section we introduce the results of the study in the two subsections: 2.1. Coefficient Estimates In the first theorem, the necessary and sufficient condition is given to be the function in the class . Theorem 1: A function defined by (4) is in the class if and only if: ∑ | | [ ( ) ] [ ] where Proof Assume that (8) holds true. It is enough to show that: | ( ( )) ( ( )) | | ( ( )) ( ( )) | for | | from (8), that resulted: | ( ∑ | | ) ( ∑ | | )| | ( ∑ | | ) ( ∑ | | )| 85 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 |∑ | | | |( ) ∑ | | [ ] | ∑ | | ( ) ∑ | | [ ] ∑ | | [ ( ) ] [ ] Hence, Conversely, let then (7) holds true, so: we have: | ( ( )) ( ( )) ( ( )) ( ( )) | | ∑ | | ( ) ∑ | | [ ] | Since | | for all , it follows that: { ∑ | | ( ) ∑ | | [ ] } Now, we choose the value of on the real axis so that ( ) is real. Letting through real values, we obtain: ∑ | | [ ( ) ] [ ] Hence, the result follows Finally, sharpness follows if we take 86 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 [ ] | | [ ( ) ] Corollary 1: If defined by (4) is in the class , then: [ ] | | [ ( ) ] where Now, the function was defined as follows: ∑ 2.2. Convex Set Here, the class will prove as a convex set and give some result about it. Theorem 2: The class is convex set. Proof Let the functions defined by (11), be in the class then for every that showed must: [ ] Thus, we obtain: ∑[ ] and ∑ | | [ ( ) ] [ ] [ ] 87 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 ∑ | | [ ( ) ] [ ] ∑ | | [ ( ) ] [ ] Therefore, by Theorem (1), the result followed Theorem 3: Let the functions defined by (11) be in the class then ∑( ) In the class where: [ ][ ] | | [ ( ) ] [ ] Proof Since then by Theorem (1), we have: ∑ ( | | [ ( ) ] [ ] ) ∑ ( | | [ ( ) ] [ ] ) and ∑ ( | | [ ( ) ] [ ] ) ∑ ( | | [ ( ) ] [ ] ) It follows from (13) and (14), that: 88 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 ∑ ( | | [ ( ) ] [ ] ) ( ) But if and only if: ∑ | | [ ( ) ] [ ] ( ) The inequality (15) is satisfied if: | | [ ( ) ] [ ] | | [ ( ) ] [ ] Hence: [ ][ ] | | [ ( ) ] [ ] Since is an increasing function of letting in (16), we get: [ ][ ] | | [ ] [ ] This completes the proof Theorem 4: Let the functions defined by (12) be in the class then: ∑ In the class where: ∑ Proof Since , for all ( , it follows from theorem (1) that: 011 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 ∑ | | [ ( ) ] [ ] Hence: ∑ | | [ ( ) ] ∑ | | [ ( ) ] ( ∑ ) ∑ (∑ | | [ ( ) ] ) [ ] Therefore by theorem (1), we get Theorem 5: Let the functions defined by (11), be in the class Then the function: ∑ Belongs to the class where: ∑ Proof For every , it follows from theorem (1) that: ∑ | | [ ( ) ] [ ] But ∑ ∑ ( ∑ ) ∑ (∑ ) Therefore ∑ | | [ ( ) ] (∑ ) 010 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 ∑ (∑ | | [ ( ) ] ) ∑ [ ] [ ] This end of the proof Definition 2 [2]: The weighted mean of functions defined by [ ] Theorem 6. Let the functions defined by (11), be in the class then the function, then the weighted men of is also in the class Proof By Definition (2), we have [ ] [ ( ∑ ) ( ∑ )] ∑ [ ] Since in the class , then by Theorem (1), we have: ∑ | | [ ( ) ] [ ] and ∑ | | [ ( ) ] [ ] Hence ∑ | | [ ( ) ] [ ] ∑ | | [ ( ) ] 011 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 ∑ | | [ ( ) ] [ ] [ ] [ ] So 3. Conclusions From above and [10] we can use this class to generate another using the definition of meromorphic multivalent function. Also by suitable operator with meromorphic multivalent function can getting on a good class studies. References 1. Ruscheweyh, S. New criteria for univalent functions, Proc. Amer. Math. Soc.1975, 49, 109-115. 2. Duren, P.L. Springer-Verlarg, New York, Berlin, Heidleberg, Tokyo, 1983. 3. Juma, R.S.; Zirar, H. On A Class of Meromorphic Univalent Functions Defined By Hypergeomatric Function, Gen. Math. Notes. 2013, 1, 63-73. 4. Darus, M.; Frasin, B.A. On Certain Meromorphic Function with Positive Coefficient. South East Asian Bull. Math. 2004, 28, 615-623. 5. AL-khafaji, A.K.; Atshan, W.G.; Abed, S.S. On the Generalization of a Class of Harmonic Univalent Functions Defined by Differential Operator. Mathematics. 2018, 6, 12, 312. 6. AL-khafaji, A.K.; Atshan, W.G.; Abed, S.S. On a Differential Subordination of a Certain Subclass of Univalent Functions. Journal of Kufa for Mathematics and Computer. 2018, 5, 3, 11-16. 7. Abed, S.; Faraj, A. Fixed Points Results in G-Metric Spaces. Ibn AL- Haitham Journal For Pure And Applied Science. 2019, 32, 1, 139-146. 8. Abed, S.; Abdul Sada, K. Common Fixed Points in Modular Spaces. Ibn AL- Haitham Journal for Pure And Applied Science. 2018, 500-509. https://doi.org/10.30526/2017.IHSCICONF.1822. 9. Jabber, A.K.; Tawfiq, L.N.M. New Transform Fundamental Properties and Its Applications. Ibn AL- Haitham Journal for Pure And Applied Science. 2018, 31, 2, 151- 163, doi.org/10.30526/31.2.1954. 10. Tawfiq, L.N.M.; Jabber, A.K. New Transform Fundamental Properties and its Applications. Ibn Alhaitham Journal for Pure and Applied Science. 2018, 31, 1, 151- 163, doi: http://dx.doi.org/10.30526/31.2.1954 https://doi.org/10.30526/31.2.1954 http://dx.doi.org/10.30526/31.2.1954