A. Then p P( ) if and only if there exists w Ω such that p(z)= (z U). (1.4)

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1 American International Journal of Research in Science, Technology, Engineering & Mathematics Available online at ISSN (Print): , ISSN (Online): , ISSN (CD-ROM): AIJRSTEM is a refereed, indexed, peer-reviewed, multidisciplinary and open access journal published by International Association of Scientific Innovation and Research (IASIR), USA (An Association Unifying the Sciences, Engineering, and Applied Research) Differential Subdination Results F Meromphic Multivalent Functions Applying A Linear Operat Associated with Generalized Hypergeometric Functions, Abstract: Our purpose in this paper is to define a linear operat, and taking the advantage of the Hadamard product ( convolution) to associate it with the generalized hypergeometric function, obtaining the operat and then using this operat to get some subdination results f these meromphic multivalent functions belonging to the subclasses [ and [ ]. Keywds: Analytic functions, Multivalent functions, Meromphic functions, Hadamard product ( convolution), Generalized hypergeometric function, Differential subdination, Linear operat. AMS Subject Classifications: 30C45 1. INTRODUCTION Let A = A(U) denote the class of all analytic functions in the open unit disc U ={z C: z < 1}. Consider Ω={w A: w(0) = 0 and w(z) <1} ( z U}, (1.1) the class of Schwarz functions. F 0 l, let P( ) = {p A: p(0)=1 and Re(p(z)) > } (z U). (1.2) Note that P = P(0) is the well-known Carath ody class of functions. The classes of Schwarz and Carath ody functions play an extremely imptant role in the they of analytic functions and have been studied by many auths. It is easy to see that p P( ) if and only if. (1.3) The following lemmas are needed f proving our results Lemma 1.1 [10] Let p A. Then p P( ) if and only if there exists w Ω such that p(z)= (z U). (1.4) Lemma 1.2 [10] (Herglotz fmula) A function p A belongs to the class P if and only if here exists a probability measure (x) on U, such that p(z)= (z U). (1.5) Let f and g be functions in A. The function f is said to be subdinate to g, g is said to be superdinate to f if there exists a function w Ω, such that f(z) = g(w(z)). In such a case, we write f g f(z) g(z) (z U). If the function g is univalent in U, then we have ( see [9] ) f(z) g(z (z U) f(0) = g(0) and f(u) g(u). AIJRSTEM ; 2013, AIJRSTEM All Rights Reserved Page 127

2 Let denote the class of all meromphic functions f of the fm f(z) = + ( p N = {1,2,3,... }), (1.6) which are analytic and p-valent in the punctured unit disc = U\{0}. Denote by the subclass of consisting of functions of the fm f(z) = +, ( z (1.7) A function f is meromphically multivalent starlike of der (0 < 1) see [1] if -Re }> (z U). The class of all such functions is denoted ( If f is given by (1.6) and g is given by g(z) = + ( z, then, the Hadamard product ( convolution) of f and g is defined by (f*g)(z)= + = (g*f)(z), (p N, z ). Now, f functions f(z) in the fm (1.6) we define the linear operat : as follows f(z) = f(z) and }f(z) = f(z) (0 < p; p N, n U{0}; z ), f(z)= f(z)= (1+ f(z)+ pzf (z), = +, therefe = + = + =. By simple calculations and using (1.9), we can easily verify that pz[ = - ( Following the same manner as in H. Orhan et al. [10], we use the linear operat the class as follows (1.8) (1.9) (1.10) to define a subclass of Definition 1.3. A function f is said to be in the subclass if it satisfies the condition <, (1.11) f some (0 <1), (0 1) and (z. F parameters C, (j=1,...,q) and C\{0,-1,-2,...} (j=1,...,s), the generalized hypergeometric function qfs( ;z) is defined by the following infinite series (see [2,3,6]): qfs( ;z)= (q s+1; q,s =N {0}; z U), = = is the Pochhammer symbol ( the shifted factial) defined (in terms of the Gamma function) by Cresponding to a function ;z) defined by ;z)= qfs ;z), Liu-Srivastava [8] defined the operat ; by the following Hadamard product ( convolution) ;z)*f(z). We also note that the definition of the operat was motivated essentially by Dziok and Srivastava [2]. Some interesting developments involving the Dziok-Srivastava operat were considered by (f example) Dziok and Srivastava [4], also see [6,7,11,13]. F functions belonging to and satisfying the subclass, we define the linear operat as the Hadamard product ( convolution) of the operat with elements of the subclass as follows f(z)= = + (1.12) AIJRSTEM ; 2013, AIJRSTEM All Rights Reserved Page 128

3 = and. f(z) = +, (1.13) =. From (1.13), follows that f(z) can be written in the convolution fm as f(z)= (f*l)(z), (1.14) l(z)= + (1.15) Again following the manner of H. Orhan et al. [10], we use the linear operat to define a subclass of the class as follows Definition 1.4. A function f is said to be in the subclass if it satisfies the condition <, (1.16) f some, > 0, (0 < 1), ( 0 < 1) and (z We consider another subclass of given by =. (1.17) In this paper a symmetric investigation of the subclasses and is presented. 2. PROPERTIES of THE SUBCLASS ] We start with the following theem, in terms of subdination, f a function to be in the subclass. Theem 2.1. A function f is in the subclass if and only if Let f (z (2.1). Then from Definition 1.4 we have <, -2 +1< (1- -2[1+ < + if, we have -2 that is <. (2.2) AIJRSTEM ; 2013, AIJRSTEM All Rights Reserved Page 129

4 The inequality (2.2) shows that the values region of the function F(z) = is contained in the disc centered at The function W = G(z) = and radius maps the unit disc U onto the disc W- < (2.3) Since G is univalent and F(0) = G(0), F(U) G(U), we obtain that F(z) G(z), that is Conversely, suppose that (z (2.4) But f, the function W = G(z )= maps the unit disc U onto the disc W- < (2.5) Hence. (2.6) After simple calculations, from (2.6) we obtain < Therefe, we complete the proof of Theem 2.1. In Theem 2.1, putting =1 gives the following collary Collary 2.2. A function f is in the subclass if and only if (z (2.7) Remark 2.3. Since, it follows that Re{ }> which shows that Using the subdination relationship f the subclass f the subclass and then f the subclass., we derive a structural fmula, first Theem 2.4. A function f belong to the subclass if and only if there exists a probability measure (x) on U such that f(z)= [ *[ exp( (z (2.8) In view of the subdination condition (2.7) and Remark 2.3, we obtain that if and only if From Limma 1.2, we have = (2.9) Integrating (2.9) with respect to z we get exp(, (2.10) AIJRSTEM ; 2013, AIJRSTEM All Rights Reserved Page 130

5 Now, from (1.16), (1.17) and (2.10) we obtain f(z)=[ + ]*(f*l)(z). (2.11) Hence f(z)=[ + ]* [ exp( (2.12) Therefe, we complete the proof of Theem 2.4. Using a result of Goluzin [5] (see also [12]), we obtain the following Theem 2.5. Let a function f belong to the subclass. Then (z Let f Then by Collary 2.2 we have The function is univalent and convex in U, then in view of Goluzin's result, we obtain log( d (z thus, there exists a function w log( such that which is equivalent to the required result. Hence, we complete the proof of Theem 2.5 Next, we obtain a structural fmula f the subclass. Theem 2.6. Let a function f belong to the subclass. Then f(z)= [ *[ exp( (z (2.13) Ω. Suppose that f then from Theem 2.1 follows that Taking integration of (2.16) with respect to z, we get (z (2.14) (z (2.15) (z (2.16) log( ) = (2.17) = (2.18) Now, from (1.16), (1.17) and (2.18) we get the result AIJRSTEM ; 2013, AIJRSTEM All Rights Reserved Page 131

6 f(z)= [ *[ exp( (z (2.19) Hence, we complete the proof of Theem 2.6. Theem 2.7. Let f be given as in (1.6). If f (0 <1) and (0 < 1) Then f. Proof; Since f(z)= +. We have M = z( )'+p - z( )' +p(2 = z( + )'+p( + - z( + )' +p(2 + =. F 0 < z = r < 1, we obtain M -, M -2p Since the above inequality holds f all r (0 < r <1, letting r 1, we have M - 2p Making use of (2.20), we obtain M<0, that is <. Consequently, f. Therefe, we complete the proof of Theem In this section we prove that the condition in (2.20) is both necessary and sufficient f a function to be in the subclass Theem 3.1. Let f Then f belongs to the subclass and only if (2.20) Proof; In view of Theem 2.7, we have to prove " only if " part. Assume that f(z) = +, ( z is in the subclass Then = f all z U. Since Re(z) z f all z, it follows that Re{. (3.1) We choose the values of z on the real axis such that (3.1) and letting z 1 through positive values, we obtain 2p (1- Hence, we complete the proof of Theem p (1- is real. Upon clearing the denominat in, AIJRSTEM ; 2013, AIJRSTEM All Rights Reserved Page 132

7 REFERENCES [1] Aouf M.K.and Hossen H.M., New criteria f meromphic p-valen tstarlike functions, Tsukuba J. Math., 17 (1993), pp [2] Dziok J., Srivastava H.M. Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput., (1999),103 pp [3] Dziok J., Srivastava H.M. Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transfms Spec. Funct., (2003),14 pp [4] Dziok J., Srivastava H.M. Some subclasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeometric function, Adv. Stud. Contemp. Math., (2002),5 pp [5] Goluzin G.M., On the majization principle in function they, Dokl. Akad. Nauk. SSSR, 42(1935), pp [6] Liu J.L, Srivastava H.M. Certain properties of the Dziok-Srivastava operat, Appl. Math. Comput., (2004),159 pp [7] Liu J.L., On subdinations f certain analytic functions associated with the Dziok-Srivastava linear operat, Taiwanse J. Math,(2009), 13 pp [8] Liu J.L., Srivastava H.M., Classes of meromphically multivalent functions associated with the generalized hypergeometric function, Math. Comput. Modelling, (2004),39 pp [9] Miller S.S. and Mocanu P.T., Differential subdinations they and applications, Dekker, New Yk, [10] Orhan H., R\u{a}ducanu D. and Deniz E., Subclasses of meromphically multivalent functions defined by a differential operat, arxiv: v1 [math.cv] 27 Aug [11] J. Patel J., Mishra A.K., Srivastava H.M. Classes of multivalent analytic functions involving the Dziok-Srivastava operat, Comput. Math. Appl.,(2007), 54, pp [12]. Pommerenke C., Univalent functions, Vanderhoeck and Ruprecht G ttingen, [13] Ramachandran C., Shanmugam T.N., Srivastava H.M. A. Swaminathan, A unified class of k-unifmly convex functions defined by the Dziok-Srivastava operat, Appl. Math. Comput.,(2007), 190 pp AIJRSTEM ; 2013, AIJRSTEM All Rights Reserved Page 133