A New Subclasses of Meromorphic p-valent Functions with Positive Coefficient Defined by Fractional Calculus Operators

Transcription

1 Volume 119 No , ISSN: (on-line version) url: A New Subclasses of Meromorphic p-valent Functions with Positive Coefficient Defined by Fractional Calculus Operators Amal Mohammed Darweesh 1, Hasan Abed Hadi 2 1 Department of Mathematics, Faculty of Education for Woman, Iraq. 2 Department of Engineering Refrigeration Air- Conditioning Techniques, Islamic University, Iraq. Abstract: In our studying, we submit study new class of meromorphic p-valent functions defined in { } { } We obtain coefficient inequalities, distortion theorems, closure theorems, extreme points, radius of star like with convex functions Convolution properties. Finally we obtain application involving an Integral trans Integral operators forms for the class Keywords phrases: Meromorphic p- valent functions, analytic function, Convolution properties. Mathematics subject classification: 30C45. 1-Introduction Let be the class of meromorphic functions of the from: { } which are analytic p-valent in the punctured unit disk { } { } Let denote the subclass of containing of functions of the from A function is said to be meronorphic p-valent starlike functions of order if { } A function is said to be meromorphic p-valent convex functions of order if { } 2447

2 In our work, we discuss study a new class of meromorphic p-valent functions by making of using for the fractional differ-integral operator contained in: Definition 1: [13] { where is the generalized fractional derivative operator of order defined: { ( ) ( { } ) where is an analytic function in a simply-connected region of the cotaning the origin multiplicity of is removed by requiring to be real when provided further that is the generalized fractional integral operator of order defined by ( ) ( { } ) Where is constrained the multiplicity of is removed as above is given by the order estimate (6.1). It follows from (5.1) (7.1) that where are the familiar owa-saigo-srivastava genrlized fractional derivative integral operators (see, e.g.,[10] [14] see also [12]). Also 2448

3 where are the familiar Owa-srivastava fractional derivative integral of order, respectively (cf, Owa[9]; see also srivastava Owa[12]). Furthermore, in terms of Gamma function, we have ( { } ) ( { } ) Now using (2.1),(13.1) (14.1) in (5.1), we find that provided that It may be worth noting that, by choosing the operator reduces to the well-know Ruscheweyh derivative for meromorphic univalent function [11]. Definition 2: The function given by (2) is said to be in the class if ( ) ( ) where Some other subclass of the class were studied (for example ) by Cho et al. ([3] [4]), Liu [6], Joshi et al. [5]. 2449

4 2-Cofficient inequality: Theorem 1: Let Then if only if where The result is sharp for the function Proof: Suppose that the inequality (1.2) holds true Then we have ( ) ( ) Since the above inequality holds for all letting we have ( ) by(18). Hence. Conversely, suppose that is in the class Then ( ) ( ) Using the fact that for all, we have ( ) ( ) 2450

5 { } If we choose to be real so that ( ) is real. Upon clearing the denominator in (2.3) letting through positive values, we obtain This completes the proof of Theorem 1. Corollary 1: Let the function defined by (2.2) be in the class Then 3-Distortion theorems Theorem 2: Let the function then for we have with equality for the function { } Proof: It is easy to see from Theorem 1 that { } Then Making use of (3.3), we have 2451

6 which proves the assertion (3.1). The proof is completed. Theorem 3: Let the function then for we have ( ) ( ) with equality for the function given by (3.2). Proof : From Theorem 1 (3.3), we have The remaining part of the proof is similar to proof of Theorem 2 so, we omit the details. 4-Closure theorems Let the functions be defined, for j=1,2,,m, by ( ) Theorem 4: Let Then the function ( ) is in Proof: Since it follows from Theorem 1, that for every Hence ( ) 2452

7 ( ) From Theorem 1, it follows that Theorem 5: The class This completes the proof. is closed under convex linear combinations. Proof: Let defined by (4.1) be in the class Then it is sufficient to show that is in the class Since then, we have from Theorem 1 that so, Theorem 6: Let Then is in the class if only if can be expressed in the form where Proof :Assume that 2453

8 Then it follows that Which implies that Conversely, assume that the function defined by (1.2) be in the class Then Setting we saw that can be expressed in the form (4.6). This completes the proof of Theorem 6. Corollary 2: The extreme points of the class are the functions 5-Integral operators Theorem 7: Let the function Then the integral operator is in the class where [ 2454

9 The result is sharp for the function given by (3.2). Proof: Let, be in the class Then we have * + By Theorem 1, it is sufficient to show that Since then From (5.3) (5.4), we have Then Since [ [ is an increasing function of we obtain this completed the proof of Theorem 7. [ 2455

10 6-Radii of meromorphically p-valent strlikeness convexity Theorem 8: Let the function defined by (2.1) be in the class Then i) is meromorphically p-valent starlike of order in the disk where { } ii) is meromorphically p-valent convex of order in the disk where { } Each of these result is sharp for the function given by (2.2). Proof: We must show that Indeed we have Thus if But by Theorem 1, ansures that In view (6.5), it follows that (6.4) will be true if 2456

11 The last inequality (6.6) leads us immediately to disk where is given by (6.1). (ii)it is sufficient to show that ( ) Note that ( ) Thus ( ) if Hence, by Theorem 1, (6.9) will be true if The last inequality (6.10) readily yields the disc with defined by (6.2), the proof of Theorem 8 is completed by merely verifying that each assertion is sharp for the function given by (2.2). 7- Convolution properties For defined by (4.1), the Hadamard product of is defined by Theorem 9: Let Then where The result is sharp for functions given by Proof: Using the technique for Schild Silverman [9], m.un we need to fined the largest that such 2457

12 Since we readily see that By the Cauchy Schwarz inequality, we have Thus it is sufficient to show that or equilvalently, that Connecting with (7.7), it is sufficient to prove that It follows from (7.10) that Now defining the function by We see that is an increasing function of Therefore, we conclude that 2458

13 Which evidently completes the proof of Theorem 9. 8-Open problem The authors suggest to study the properties of the same class References [1]- M. K. Aouf, "A certain subclass of meromorphically starlike functions with positive coefficients", Rend. Mat., 9(1989), [2] - M. K. Aouf, A. O. Mostafa W. K., "Elyamany, Subclass of meromorphic functions with positive coefficients defined by Frasin Darus operator ", Int. J. Open Prorblems Complex Analysis, 8(2016), [3]- N. E. Cho, S. H. Lee S. Owa, " Aclass of meromorphic univalent functions with positive coefficient", Kobe J. Math., 4,1987, [4] - N. E. Cho, S. Owa, S. H. Lee O. Altintas, "Generalization class of certain meromorphic univalent functions with positive coefficients", Kyungpook Math. J., 29, 1989, [5] - S. B. Joshi, S. R. Kulkarni H. M. Srivastava, "Certain classes of meromorphic functions with positive missing coefficients", J. Math. Anal. Appl., 193, 1995, [6] - J. L. Liu, "Properties of some families of meromophic p-valent functions", Math. Japon., 52, 2000, [7] - J. E. Miller, "Convex meromorphic mapping related functions", Proc. Amer. Math. Soc., 25(1970), [8] - M. L. Mogra, T. Reddy O. P. Juneja, "Meromorphic univalent functions with positive coefficients", Bull. Aust. Math. Soc., 32(1985), [9]- S. Owa, "On the distortion the theorems-і", Kyungpook Math. J., 18(1978), [10] - S. Owa, M. Sago H. M. Srivastava, "Some characterization theorems for starlike convex functions involving a certain fractional integral operator", J. Math. Anal. Appl., 140(1989), [11]- S. Ruscheweyh, "New criteria for univalent functions", Proc. Amer. Math. Soc., 49(1975), [12]- H. M. Srivastava S. Owa,"Current to picsin analytic functions theory", World Scientific Publishing Company, Singapore, New Jersey, London Hong KONG, (1992). 2459

14 [13] - H. M. Srivastava S. Owa, "Current topics in analytic functions theory", World Scientific Publishing Company, Singapore, [14]- H. M. Srivastava, M. Saigo S. Owa, "A class of distortion theroems involving certain operator of fractional calculus", J. Math. Anal. Appl., 31(1988),

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