On A Class of Analytic Univalent Function Defined By Ruscheweyh Derivative. Abstract

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1 On A Class of Analytic Univalent Function Defined By Ruscheweyh Derivative A.R.S.Jum a, H. H.Ebrahim ** And SH. I. Ahmed** * * D epartment O f M athematics, A lanbar U niversity ** D epartment O f M athematics, Tikrit U niversity Received 2 December 2014 ; Accepted 3 March 2015 Abstract Making use the Ruscheweyh derivative operator, we introduce and study a certain of univalent analytic function with negative coefficient.in this paper, we obtain coefficient estimates, distortion theorem, radii of starlikness, convexity and modified Hadamard product of functions belonging to the class. Key words: Analytic function, univalent function, Hadmard product, starlike function, convex functions, Ruscheweyh derivative. AMS subject classifications: 30C45. على اصناف الدوال التحليلية أحادي التكافؤ معرفة بواسطة المشتقة الرشويه عبد الرحمن سلمان جمعة, قسم الرياضيات جامعة االنبار, الرمادي - العراق حسن حسين ابراهيم, قسم الرياضيات, جامعة تكريت, تكريت-العراق شامل ابراهيم احمد, قسم الرياضيات جامعة تكريت,تكريت-العراق الملخص من الدالة التحليلية أحادية مما يجعل استخدام المؤثر المشتقة رشويه قدمنا ودرسنا صنف معين التكافؤ مع معامل سلبي. في هذا البحث حصلنا على معامل التقدير نظرية التشويه قطر ال starlikness التحدب. الدوال والضرب الدوال التي تنتمي الى صنف الكلمات الرئيسية: الدوال التحليلية الدالة األحادية التكافؤ الضرب الدالي دالة الجمة الدالة المحدبة المشتقة رشويه. Introduction Let denotes the class of functions normalized by which are analytic and univalent in the unit disk = { : z 1}. Vol : 11 No : 4, October

2 Let be subclass of, consisting of function of the form For functions given by and given by The convolution or Hadmard product of is defined by A function is said to be univalent starlike of order if satisfies the inequality: denotes the class of all univalent starlike functions of order. Also said to be univalent convex of order if satisfies the inequality: is denotes the class of all univalent convex functions of order and Goodman [3]]. [see for example Duren [1] The classes and are studied Owa [6]. represent the Ruscheweyh derivative of order defined by [7]: Vol : 11 No : 4, October

3 We aim to study the class which consists of functions and satisfy For 1, 0 1, 0 α 1, The investigation here is motivated by M. Darus [2]. Remark: 1) when then form we get which is the class of starlike of order 2 when =1, we have the class which is the class of starlike of order studied by Owa [6] and Yamakawa [8] 3 when, we have which is the class of convex of order 4 when, =1 we have Vol : 11 No : 4, October

4 which is the class of convex of order Next we characterize the class studied by Owa [6] and Yamakawa [8]. by proving the coefficient inequality. 2. Coefficient Inequality Let the function be of the from.then if and only if where 0 α 1, 0 1,, ϵ 0, Proof: Assume inequality is true then 0 by maximum modules principle. Vol : 11 No : 4, October

5 that is using the fact that Re f ( ) f ( ) Choose on real axis and allowing 1- Thus the proof is complete. Of the form of.then be in the class with equality for Vol : 11 No : 4, October

6 3. Growth And Distortion Theorem On A Class of Analytic Univalent Function Let the function.then of the form of (2) be in the class The result is sharp for Proof: we have Similarly Combining and we get the result. The result is sharp for Vol : 11 No : 4, October

7 4. Radii Of Starlikness And Convexity The estimate is sharp for the function Proof: is starlike of order S, 0 S 1 if which simplifies to by we have Using and we get Vol : 11 No : 4, October

8 estimate is sharp for Proof: Using the arguments similar to Theorem (3.2), we get 5. Extreme Points Let then If it can be expressed in the form Vol : 11 No : 4, October

9 Proof: suppose Now since Conversely, suppose then by Setting Vol : 11 No : 4, October

10 6. Hadmard Production where Proof: We need to find largest number such that By Cauchy schwas z inequality we have Thus it enough to show that Or equivalently, that Vol : 11 No : 4, October

11 From (13) Therefor in view of and it is enough to show that Which simplifies to 7. Closuer Theorem Proof: suppose Vol : 11 No : 4, October

12 Since by 5 In view of 18 if Now Thus = 1. Proof: and hence Vol : 11 No : 4, October

13 Adding and we get We must show that Therefore, we have fined largest such that That is, that 8. Integral Operator Let function defined by number such that in the class then the function, and let c be a real defined by Also belongs the class Proof. For the representation of,it follows that Vol : 11 No : 4, October

14 Since. Hence by Theorem (2.1) Refrences. 1. P. L. Duren, Univalent Functions, "In: Grudlehren der Mathematischen Wissenschaften, Vol. 259, Springerveriag, New York G. H. Esa and M. Darus, Application of fractional calculus operators to a certain class of univalent function with negative coefficients, International Mathematical Forum, 2, (2007), 57, A. W. Goodman, Univalent Functions, Vols. I and II, Polygonal publishing House, Washington, 4. V, Kumar, S.L.Shukla, Multivalent function defined by Ruscheweyh derivatives. Indian J. Pure Appl.Math.15 (1984), V, Kumar, S.L.Shukla, Multivalent function defined by Ruscheweyh derivatives.ii, Indian J. Pure Appl.Math.15 (1984), S.Owa,''The Quasi-Hadmard products of certain Analytic Functions,' In: H.M.Sriastava and Sow, Eds.,Current Topics in Analytic Function Theory, world Scientific Publishing company,singapore,1992,pp St. Ruscheweyh, New criteria for univalent functions, Proc.Amer.Math.Soc.49 (1975), R. Yamakawa,'' Certain Subclasses of p-valent Starlike Function with Negative Coefficient,'' In: H. M.Srivastava and S. Owa, Eds,.Curren Topics in Analytic function Theory, world scientific publishing company, Singapore,1992,PP Vol : 11 No : 4, October