In the present paper we study of some properties first - order strong differential subordination and superordination for analytic functions associated with Ruscheweyh derivative operator which are obtained by considering suitable classes of admissible functions

Alhadithi, Abdul

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ISSN 2347-1921                                                           3752 | P a g e                                                         J u n e  1 5 ,  2 0 1 5  FIRST – ORDER STRONG DIFFERENTIAL SUBORDINATION AND SUPERORDINATION PROPERTIES FOR ANALYTIC FUNCTIONS ASSOCIATED WITH LINEAR OPERATOR ABDUL RAHMAN S. JUMA1, MUSHTAQ SHAKIR A. HUSSEIN2 AND MOHAMMAD F. HANI3 1 University of Anbar, Department of Mathematics, Ramadi, Iraq, Collage of science, dr_juma@hotmail.com. 2 University of Al-Mustansiryya ,Department of Mathematics, Baghdad – Iraq, Collage of science, mushtdam8@yahoo.com. 3 University of Al-Mustansiryya, Department of Mathematics, Baghdad – Iraq, Collage of science, mohfalih17@yahoo.com. ABSTRACT In the present paper we study of some properties first - order strong differential subordination and superordination for analytic functions associated with Ruscheweyh derivative operator which are obtained by considering suitable classes of admissible functions. Indexing terms/Keywords Analytic function; univalent function; strong differential subordination; strong differential superordination; Ruscheweyh derivative; admissible functions; Hadmard product. 2010 MSC : Primary 30C80 Secondary 30C45.                       Council for Innovative Research Peer Review Research Publishing System Journal: JOURNAL OF ADVANCES IN MATHEMATICS Vol .10, No.8 www.cirjam.com , editorjam@gmail.com ISSN 2347-1921                                                           3753 | P a g e                                                         J u n e  1 5 ,  2 0 1 5  1- INTRODUCTION AND PRELIMINARIES    Let  denote the class of analytic function in the open unit disk  and  be the subclass of  consisting of the functions of the form , where  and . Also  and . Let  denote the subclass of functions of  consisting of functions of the form  which are normalized analytic univalent in . Let  we say that a function  is subordinate to  or  is said to be superordinate to , if there exists a Schwarz function  which is analytic in  with  ,  such that  In such case, we write  or .Furthermore, if the function  is univalent in , then we have the following equivalent (see [3,7]) :  and . For the function  given by (1.1) and the function  given by   the Hadmard product (or convolation) of  and  is defined by  Now, for functions  in the form (1.1) we define the Ruscheweyh derivative operator [14],  as follows   where   We note that   such type of study was carried out by several different authors, like Dinggong and Liu [4], and Lupus [5]. The notion of differential superordination was introduced in 2003 [8] by Miller and Mocanu as a dual concept differential subordination in 2000 [7]. The notion of strong differential superordination was introduced by Antonino and Romaguera in 2006 [2]  as a dual concept differential subordination in 1994 [1] which were developed by (G.I. Oros, 2007 [9]), (G.I. Oros, Oros, 2009 [10]) and (G.I. Oros, and Oros, 2009[11] ). To prove our main results, we need the following definitions and Lemmas. DIFFINTION 1.1 [9,12] Let  be analytic in  and let  analytic and univalent in . The function  is strongly subordinate to  written , or  is said to be strongly superordinate to , written  if for  as a function of  is subordinate to . We note that :  and  and if  is univalent, then :  and . DEFINITION 1.2 [9,13] Let  and let  be univalent in . If  is analytic in  for all  and satisfies the following (first – order) strong differential subordination.  ISSN 2347-1921                                                           3754 | P a g e                                                         J u n e  1 5 ,  2 0 1 5  then  is called a solution of the strong differential subordination. The univalent function  is called a dominate of the solutions of the strong differential subordination or more simply a dominant if    for all  satisfying (1.4). A dominant  that satisfy  for all dominants  of (1.4) is said to be the best dominant. DIFFINITION 1.3 [9,12] Let  and let  be analytic in . If  and  are univalent in  for all  and satisfy the (first – order) strong differential superordination   then  is called a solution of the strong differential superordination. An analytic function  called a subordinate of the solutions of the strong differential superordination, or more simply a subordinate if  for all  satisfying (1.5). A univalent subordinate  that satisfies  for all subordinants  of (1.5) is said to be the best subordinant. Note that the best subordinant is unique up to a rotation of . For  a set in  with  and  as given in Definition 1.3, suppose (1.5) is replaced by :  DIFFINITION 1.4 [7] We denoted by  the set of functions  that are analytic and injective in , where   and are such that  for  The subclass of  for which  is defined by . DIFFINITION 1.5 [13] Let  be a set in   and . The class of admissible functions  consists of those functions  that satisfy the following admissibility condition :  whenever  , where  and  We write  as  DEFINITION 1.6 [9,12] Let  be a set in  with  The class od admissible function  consists of those functions  that satisfy the  whenever  , where  and  When  we write  as  LIMMA 1.1 [13] Let , with  and let  be analytic in  with  and  If  is not subordinate to then there exist points  and  and  for which  such that : i) ; ii)  THEOREM 1.1 [12] Let  with  If  satisfies , then  THEOREM 1.2 [12] Let  with  If  satisfies  is univalent in  for ,   Implies  2- SUBORDINATION RESULTS THEOREM 2.1 Let . If  satisfies ISSN 2347-1921                                                           3755 | P a g e                                                         J u n e  1 5 ,  2 0 1 5   then  PROOF.  Let   From (1.8), we have  and differentiating (1.9), we obtain  Using the property (1.3) of the Ruscheweyh Derivative operator   Using (1.9) and (1.10) in (1.11), we obtain  Then (1.7) becomes  Assume  By Lemma 1.1 there exist points  and  and  that satisfy  Using these condition in Definition 1.5, we obtain  Since this contradicts (1.13), we must have  by Theorem 1.1 or equivalent  COROLLARY 2.1 The conclusion of Theorem 2.1 can be written in the generalized form :  then  where  is any mapping  onto  If  is a simply connected domain, then  for some conformed mapping  of  onto . In this case the class  is written as  The following result is an immediate consequence of Theorem 2.1. THEOREM 2.2 Let . If  is analytic in , and  ISSN 2347-1921                                                           3756 | P a g e                                                         J u n e  1 5 ,  2 0 1 5   then  This result can be extended to those cases in which behavior of  on the boundary of  is unknown. COROLLARY 2.2 Let  and  be univalent in  with  Let  for some  where . If  satisfy :  then  PROOF. From Theorem 2.1 yield . The result is now deduced from  THEOREM 2.3. Let  and  be univalent in  with  and set  and  Let  satisfy one of the following conditions : 1)  for some  or 2) There exist  such that  for all  If  satisfies (1.14), then  PROOF.  Case (i) . By applying Theorem 2.1 we obtain . Since , we have  i.e,  Case (ii) If we let , then  By using Corollary 2.1 with , we obtain  for  By letting  we obtain  i.e,  The next theorem yields the best dominate of the differential subordination (1.7). THEOREM 2.4 Let  be univalent in  and . Suppose that the differential equation  has a solution , whith , and one of the following conditions is satisfied : i)  and , ISSN 2347-1921                                                           3757 | P a g e                                                         J u n e  1 5 ,  2 0 1 5  ii)  is univalent in  and  for some  or iii)  is univalent in  and there exits  such that  for all  If  satisfies (1.14) and  is analytic in , then   and  is best dominant. PROOF.  By applying Theorem 2.2 and Theorem 2.3 we deduce that  is dominant of (1.14). Since  satisfies (1.15), it is also a solution of (1.14) and therefore  will be the dominant of all dominants of (1.14). Hence  will be best dominate of (1.14). 3- SUPERORDINATION AND SANDWICH – TYPE RESULTS THEOREM 3.1 Let . If  and  is univalent in , then   implies  PROOF. The same technic to proof Theorem 2.1. Next, we consider the special situation when  is analytic on  and . In this case the class  is  and the following result is an immediate consequence of Theorem 3.1. THEOREM 3.2 Let  be analytic in  and  . If  ,  and   is univalent in , Then  implies  Theorem 3.1 and Theorem 3.2 can only be used to obtain subordinations of differential superordination of the form (3.1) or (3.2). The following therefore proves the existence of the best subordinants of (3.2) for certain . THEOREM 3.3 Let  be analytic in  and . Suppose that the differential equation  has a solution . If  and  is univalent in , then  implies  and  is the best subordinant. ISSN 2347-1921                                                           3758 | P a g e                                                         J u n e  1 5 ,  2 0 1 5  PROOF. By applying Theorem 3.2, we deduce that  is a dominate of (3.2). Since  satisfies (3.3), it is also a solution of (3.2) and therefore  will be dominated by all dominates of (3.2). Hence is the best dominates of (3.2). Combining Theorem 2.2 and Theorem3.2, we obtain the following sandwich type Theorem. THEOREM 3.4 Let  and  be analytic functions in   be a univalent function in ,  with  and . If  and  is univalent in , then   implies  REFERENCES [1] Antonino J.A. and Romaguera S. 1994, Strong differential subordination to Briot – Bougquel differential equations. Journal of Differential equations, 114,  pp. 101 – 105. [2] Antonino J.A. and Romaguera S. 2006, Strong differential subordination and applications to univalency conditions. J. Korean Math. Soc. 43. No.2, pp. 311 – 322. [3] Bulbaca T. 2005, Differential subordinationsand superordinations. Recent Results, House of Scientific Book Publ. Cluj-Napoca. [4] Cho N.E. 2013, strong differential subordination properties of analytic functions involving the Komatu integral operator. (BVP), pp. 1-14. [5] Dinggong Y. and Liu J.L. 2002, On a class of analytic function involving Ruscheweyh derivative. Bull, Korean Math. Soc., 39 (1), pp. 123 – 131. [6] Lapus A.A. 2010, A note on differential subordinations using Salagean and Ruscheweyh operators. Romia J., 6 (1), 4 pages. [7] Miller  S.S. and Mocanu P.T. 2000, Differential subordinations. Theory and Application, Vol. 225, of series on Monographs and Text books in pure and Applied Mathmatics, No. 225, Marcel Dekkar, Incorporated, New York, NY, USA. [8] Miller S.S. and Mocanu P.T. 2003, Subordinates of differential superordinations. Complex Variables 48 (10), pp. 815 – 826. [9] Oros G.I 2007, First order strong differential superordinations. General Mathematics Vol. 15, Nr. 2-3, pp. 77- 87. [10] Oros G.I. and Oros  G. 2009, Strong differential subordination. Turkish Journal of Mathematics, 33 (3), pp. 249 – 257. [11] Oros and G.I. Oros  G. 2009, Second order nonlinear strong differential subordinations. Bull. Belg. Math. Soc. Simon Stevin, 16, pp. 171 – 178. [12] Oros G.I. 2010, First order differential superordinations using the Dziok – Srivatava linear operator. Math. Reports 12 (62), 1, pp. 37 – 44. [13] Oros G.I. Oros G. Kim I.H. Cho and N.E. 2011, Differential subordinations associated with the Dziok – Srivastava operator. Math. Reports 13 (63), 1,  pp. 57 – 64. [14] Ruscheweyh S., 1975, New criteria for univalent functions. Proc, Amer. Math. Soc., 49, pp. 109 – 115.         

FIRST- ORDER STRONG DIFFERENTIAL SUBORDINATION AND SUPERORDINATION PROPERTIES FOR ANALYTIC FUNCTIONS ASSOCIATED WITH LINEAR OPERATOR

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FIRST- ORDER STRONG DIFFERENTIAL SUBORDINATION AND SUPERORDINATION PROPERTIES FOR ANALYTIC FUNCTIONS ASSOCIATED WITH LINEAR OPERATOR

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