A STUDY ON APPLICATIONS OF SOME GENERALIZED LINEAR OPERATORS TO SUBCLASSES OF ANALYTIC FUNCTIONS SARA AREF SAAD. Dr. JAMAL MOHAMMED SHENAN

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1 AL-AZHAR UNIVERSITY-GAZA DEANSHIP OF GRADUATE STUDIES A STUDY ON APPLICATIONS OF SOME GENERALIZED LINEAR OPERATORS TO SUBCLASSES OF ANALYTIC FUNCTIONS by SARA AREF SAAD suervisor Dr JAMAL MOHAMMED SHENAN A THESIS Submitted in Partial Fulfillment of the Requirements of the Degree of Master of Science in Mathematics Deartment of Mathematics Faculty of Science Al-Ahar University Gaa Palestine June-04

2 Al Ahar University Gaa Deartment of Mathematics Faculty of Science Deanshi of Postgraduate Studies and Research A STUDY ON APPLICATIONS OF SOME GENERALIZED LINEAR OPERATORS TO SUBCLASSES OF ANALYTIC FUNCTIONS by SARA AREF SAAD suervisor Dr JAMAL MOHAMMED SHENAN This thesis was defended successfully on 4/ 6/ 04 Aroved by Committee members Signature Suervisor: Dr JAMAL M SHENAN Internal examiner: Dr ABDUL SHAKOR S TEIM External examiner: Dr GHAZI KHAMMASH II

3 بسم اهلل الرمحن الرحيم أ وح س )و ي لونك ع ن الر ق م م ن ال ا أ و م وت يت ل الر وح ا م إ ل ق م ن أم ر ر ) ل يل ع ل ي ب اإلسراء- 58 III

4 DEDICATION I dedicate this wor to my glorious suervisor who has illuminated my way I dedicate this wor also to the Palestinian martyrs who resonded to Allah orders I as well dedicate it to my symathetic arents I as Allah to bless them to my husband brothers and sisters wishing them all the best Finally I dedicate it to all my teachers and friends who have heled me to achieve this wor in which I indebted to all of them and I as allah to hel them to achieve their goals too SARA AREF SAAD IV

5 Declaration I declare that this whole wor submitted for the degree of Master is the result of my own wor excet where otherwise acnowledged in the text and that this wor (or any art of it) has not been submitted for another degree at any other university or institution Signature: Name: SARA AREF SAAD Date: 4/6/ 04 V

6 ACKNOWLEDGEMENT First of all I would lie to than Allah for guiding me through my education I also than my family members secially my symathetic father Dr Aref Saad and my comassionate mother they have always been suorting me and doing their best to mae me able to comlete my study thans to Allah that I have one of my arents wishes comes true More over I feel roud to exress my secial regards to my teacher Dr Jamal Shenan who has never disaointed me in any asect and he has delivered so much information to me I am really so grateful to him I also greatly areciate the hel of all of my teachers who gave me all the nowledge that I needed during my education eriod I would never forget my dear husband Eng Rami Khader and my devoted friend Lubna Elburai for the great hel they used to give me VI

7 ABSTRACT In this thesis we defined new subclasses of analytic and multivalent functions and meromorhic functions in terms of certain linear oerator and we study the roerties of these subclasses There are four chaters that constitute this thesis together with an extensive list of references In chater we resent a brief history of geometric function theory and fractional calculus In chater we introduce the subclass S g ac ( ; ) TS ( g ; ) of P -valent -uniformly convex and starlie functions defined by convolutions of certain linear oerators and we rove certain results such as coefficient estimates closure roerty extreme oints convolutions results and others In chater 3 we rove several inclusion relations associated with -neighborhood by maing use of the familiar concet of neighborhoods of analytic functions of the subclasses L ( a c ; b ) which consists of analytic functions of comlex order with negative coefficients defined by generalied Saitoh oerator L( a c ) In chater 4 we introduce the subclasses A ( A B n) and A ( A B n) of meromorhic multivalent functions defined by using a differential oerator n D f ( ) We obtain coefficient estimates distortion theorem radius of convexity closure and other Theorems for the class A ( A B n) VII

8 العربية باللغة الملخص في ىذه الرسالة تم تعريف مجموعو من الفصول الجزئية المعينة لمدوال التحميمية والمتعددة القطبية و الفصول الجزئية المعينة لمددوال المورومورفيدو باسدتددام بعدض المعدامال الدطيدة ومدن مدم تدم د ارسدة بعض دصائص ىذه الفصول و تتكون ىذه الرسدالة مدن بربدح وحددال منفصدمة الوحددة األولد ىدي مقدمدو وممددص عدن موضدوع الرسدالة يشدمل بعدض التعدارف والمفداىيم األساسدية فدي نظريدة الددوال اليندسدية وفدي الوحددة المانيدة تدم تعريدف صدف جديدد مدن الددوال التحميميدة و المعرفدة بد لدة المعامدل التفاضدمي الكسدر لسدايقو ومدن مدم تدم دا رسدة بعدض دصدائص ىدذا الصدف ممممدة بنظريدال معامدل التقددددي ارل نظريدددال الفصدددل نظريدددال نصدددف القطدددر لمددددوال النجميدددة والمحدبدددة نظريدددال المعامدددل التكدداممي نظريددال ا لتفدداف نظريددال اق ددا وبعددض النظريددال األدددر فددي الوحدددة المانيددة تددم تعريف صف آدر مدن الددوال التحميميدة المتعدددة القطبيدة و المعرفدة بد لدة المعامدل الدطدي لسديتوش وتم دارسة دصائص بدر ليذا الصف ممممة بنظريال معامل التقدي ارل نظريدال الجدوار نظريدال المجدداميح الجزئيددة نظريددال ا لتددوار ونظريددال بدددر وفددي الوحدددة األديددرة مددن الرسددالة تددم تعريددف مجموعدددو مدددن الفصدددول الجزئيدددة المعيندددة لمددددوال المورومورفيدددو والمتعدددددة القطبيدددة باسدددتددام بعددددض المعامال الدطية ومن مم تم د ارسة بعض دصائص ىذه الفصدول ممممدة بدبعض النظريدال اليامدة والمعروفة فدي نظريدة الددوال اليندسدية ممدل نظريدال معامدل التقددي ارل نظريدال نصدف القطدر لمددوال النجمية والمحدبة نظريال اق ا وبعض النظريال األدر VIII

9 CONTENTS CHAPTER Concets of Geometric Function Introduction Univalent functions 3 3 Multivalent functions and some of their subclasses 6 4 Meromorhic functions and some of their subclasses 7 5 On some secial functions 9 6 Fractional calculus 7 Subordination 9 3 CHAPTER Certain subclasses of uniformly starlie and convex functions defined by convolution with negative coefficients 4 Introduction 5 Coefficient estimates 7 3 Distortion Theorems for the class TS ( g ; ) 8 4 Extreme oints of the class TS ( g ; ) 0 5 Radii of close-to-convexity starlieness and convexity 6 A family of integral oerators 3 7 Convolution Results for the class Tac ( g ) Closure roerties CHAPTER 3 On certain roerties of neighborhoods of multivalent functions involving the generalied saitoh oerator 33 3 Introduction 3 Inclusion relationshis involving N n hfor the class L ( a c ; b ) 33 - neighborhood for the class L ( a c ; b ) 34 Subordination Results 35 Partial sums 36 Integral means inequalities IX

10 CHAPTER 4 Subclasses of meromorhically multivalent functions associated with a certain linear oerator 49 4 Introduction 50 4 Coefficient estimates for the class A ( A B n) 5 43 Distortion roerties for the class A ( A B n) The radii of meromorhically -valent starlieness and Convexity Closure Theorems for the class A ( A B n) Neighborhoods and artial sums for the class A ( A B n) References 63 X

11 List of Abbreviations Abbreviation Stands for N Set of natural numbers Z Set of integers R Set of real numbers C Comlex ane Subordination between two functions Hadamard roduct (or convolution) of two functions U Unit disc U unctured unit disc A The class of functions analytic in U S The class of all functions in A which are univalent in U K Koebe's constant S Class of starlie functions of order K T Class of convex functions of order The subclass of A of negative Coefficient T ( n ) The class of functions of the form (6) S The intersection of the classes S with T K The intersection of the classes K with T C C ( ) The intersection of the classes C with T R ( ) Class of closed to convex functions of order Class of re-starlie functions of order R ( ) The intersection of the classes R ( ) with T S Class of starlie functions of comlex order K Class of convex functions of comlex order S The class of functions analytic and -valent in U T Subclass of S of negative Coefficient XI

12 S ( ) Class of all P -valent starlie functions of order K ( ) Class of all P -valent convex functions of order S ( ) The intersection of the classes S ( ) with T K ( ) The intersection of the classes K ( ) with T ( ) Class of uniformly starlie functions of order S ( ) Class of uniformly convex functions of order K Class of functions which are analytic in the unctured unit dis U of negative Coefficient Subclass of ( ) The class of all meromorohic -valent starlie functions of order ( ) The class of all meromorohic -valent convex functions of order The intersection of the classes ( ) with The intersection of the classes ( ) with D D The fractional integral of order The derivative integral of order XII

13 CHAPTER Some reliminary concets of geometric function theory

14 Introduction Geometric Function theory is that branch of comlex analysis which deals and studies the geometric roerties of the analytic functions That is geometric function theory is an area of mathematics characteried by an intriguing marriage between geometry and analysis Its origins date from the 9th century but new alications arise continually The theory of univalent functions is one of the most beautiful subjects in geometric function theory which was initiated by Koebe [8] in 907 yet it remains an active field of current research A single valued function f ( ) is said to be univalent (or Schlicht) in a domain D C (the comlex lane) if it never taes on the same values twice that is if f ( ) f ( ) for D imlies that In other words f ( ) is one to one on D In this case the equation f ( ) has at most one root in D for any comlex number In the geometric function theory one studies the classes of analytic functions in a domain lying in the comlex lane C subject to various conditions In the resent thesis we study the classes of analytic functions which has a Taylor series exansion of the form defined in the unit disc U : f ( ) a () also we study the classes of functions which have the ower series reresentation f ( ) a () and analytic in the unctured unit disc : 0 0 U : U U with a simle ole at 0 with residue such functions are called meromorhic functions Let A denote the class of functions of the form () analytic in U Further let S denote the class of all functions in A which are univalent in U Each function f S 0 0 f 0 is normalied by the conditions f and The origin of the contemorary study of univalent functions can be traced bac to a aer of Koebe [8] in 907 on the uniformiation of algebraic curves In this aer Koebe showed that there is a constant K (now called the Koebe's constant) such that the boundary of the ma of U by any function f in the class S is always at a distance not less than K from the origin Koebe's wor was ursued by a number of eminent mathematicians (See eg Bieberbach [] [3] ) The "Area rincile" was first discovered by Gronwall [3] In 96 Bieherbach [] rediscovered the same and with the hel of it obtained the recise value of Koebe's constant vi K and the shar estimate a 4 for every f in S

15 He also observed that the Koebe function his results This led him to conjecture that for every f in S K is an extremal function for an n for n = 3 4 This was later nown as the Bieberbach conjecture This conjecture determined the course of research in univalent functions for nearly seven decades and finally in 985 it was roved in affirmative by de Branges [ 6 7] Univalent functions Erstwhile when the Bieberbach conjecture remained unsolved It was verified for some subclasses of univalent functions with extra geometric roerties namely the class of convex functions the class of starlie functions the class of close-to- convex functions etc Presently there are several elegant looing and readily alicable results on these subclasses Thus they occuy a central osition in the study of univalent functions indeendent of Bieberbach's conjecture In this section we give definitions and some basic results concerning subclasses of A related to univalent and multivalent functions [8] A set is said to be starlie with resect to 0 if the line segment joining 0 to every other oint lies entirely in If the function f ( ) mas a domain D C onto a domain that is starlie with resect to 0 then f ( ) is said to be starlie with resect to 0 A function f in S is said to be starlie if f mas D C onto a domain starlie with resect to the origin ie the line segment joining 0 to any oint of f ( D ) lies in f ( D ) Definition A function f ( ) in S is said to be starlie of order (0 ) if and only if ( ) Re f U f ( ) () we denote the class of such functions by S (0 ) We note that S S 0 where S 0 is the class of starlie function with resect to the origin in U ie f is in S 0 if and only if f ( ) Re 0 U f ( ) () The set is said to be convex if the line segment joining any two oints of lies t t for every and 0t entirely in Equivalently 3

16 Definition A function in f is said to be convex of order (0 ) if and only if f ( ) Re ( U) f ( ) (3) such class of functions will be denoted by K We note that K K 0 0 f where K is the class of convex functions with resect to the origin in U ie K(0) if and only if f ( ) Re 0 ( U) f ( ) (4) The classes S and K were first introduced by Robertson [46] and were studied subsequently by Schild [55] and others We note that K (0) S (0) S and K S S if and only if (0 ) According to Alexander [4] if f is analytic in U with and thus f f 0 0 and 0 f K if and only if f S f then K(0) if and only if f S 0 We assume T to denote the subclass of A consisting of functions of the form f ( ) a a 0 U (5) also let T ( n ) denotes the class of functions of the form f a 0; a n N n which are analytic and univalent in the unit dis U Denote by S and K resectively of the classes S and (6) the classes obtained by considering intersection K with T ie ; 0 ; 0 S S T K K T The subclasses S and K were introduced by Silverman [56] Definition 3 A function f ( ) belonging A is said to be close-to-convex denote by C [cf Goodman and Shaff ()] if there exists a convex function g such that 4

17 f ( ) Re 0 U g( ) (7) A function f ( ) belonging A is said to be close-to-convex of order (0 ) and is denoted by C if there exists a convex function g such that ( ) Re f U g( ) (8) We note that We denote by Let K S C S (0 ) C ( ) the class obtained by taing the intersection of C ( ) with T g ( ) b (9) be analytic function in U The Hadamard roduct (convolution) of the analytic functions f () defined by () and g () defined by (9) is denoted by f g ( ) and defined by ( f g )( ) b a (0) Definition 4 A function f ( ) in S is said to be re-starlie of order (0 ) [cf Silverman and Silvia (57)] if ( ) f ( ) S ( ) () We denote by R ( ) the class of all re-starlie of order Indeed R (0) K (0) K and R( ) S( ) We denote by R ( ) the class obtained by taing the intersection of R ( ) with T Definition 5 A function f T is said be starlie of comlex order denoted by S if it satisfies the inequality f Re f 0 U; C {0} () Furthermore a function f T is said be convex of comlex order denoted by K if it satisfies the inequality 5

18 f Re f 0 U; C {0} (3) The classes S and K Wiatrowsi [64] (see also[8 6]) were considered earlier by Nasr and Aouf [37] and 3 Multivalent functions and some of their subclasses A function f ( ) analytic in a domain D C is said to be the multivalent ( -valent) function [cf Haymann(4)] = in D if for every comlex number the equation f ( ) does not have more than roots in D and there exists a comlex number 0 such that the equation f ( ) 0 has exactly roots in D Let S denote the class of functions of the form f ( ) a (3) which are analytic and P -valent in the unit dis U And let T denote the subclass of S consisting of analytic and P -valent functions which can be exressed in the form Let f ( ) a ( a 0) (3) g ( ) b ( b 0) (33) then the modified Hadamard roduct ( f g )( ) of f ( ) and g( ) is defined by ( f g )( ) a b ( a 0andb 0) (34) Definition 6 A function f ( ) S is said to be P -valent starlie of order if and only if f ( ) Re ( U) (35) f ( ) for some ( 0 ) We denoted the class of all P -valent starlie functions of order by S ( ) Further a function f () from S is said to be convex of order if and only if 6

19 f ( ) Re ( U) f ( ) (36) for some ( 0 ) We denote the class of all -valent convex functions of order by K ( ) The classes S ( ) and K ( ) were first introduced was introduced by Patil and Thaare [43] see also [39] For 0 we obtain the class S( ) and K of -valent starlie and convex functions with resect to the origin [cf Goodman (8)] and for P = the class S and K ( ) are obtained We denote by S ( ) and K ( ) the classes obtained by taing intersections of the classes S ( ) and K ( ) with T resectively These classes are introduced by Owa [39] The classes S Silverman [56] ( ) S ( ) and K ( ) K ( ) were studied by Definition 7 A function f ( ) S is said to be uniformly starlie functions of order denoted by ( ) iff S f ( ) f ( ) Re f ( ) f ( ) (37) for some ( ) 0 and U and is said to be if uniformly convex of order denoted by ( ) if and only K f ( ) f ( ) Re f ( ) f ( ) (38) for some ( ) 0 and U The classes ( ) and ( ) were introduced and studied by Goodman [0] S K Ronning[47] and Minda and Ma[35] The class 0 K ( ) K ( ) 0 S ( ) S ( ) and 4 Meromorhic functions and some of their subclasses This section deals with meromorhic multivalent functions defined in the unctured disc U : 0 A function f ( ) is said to be meromorhic in a region D if it is analytic in D excet at a finite number of oles Let denote the class of functions of the form 7

20 f ( ) a 0 {3} (4) which are analytic in the unctured unit dis U : 0 U 0 ole of order at 0 Also let denote the subclass of which have the ower series reresentation 0 with a of meromorhic multivalent functions in U f ( ) a ( a 0) (4) A function f ( ) is said to be -valent meromorhically starlie of order if and only if f ( ) f ( ) Re ( U ) (43) for some ( 0 ) We denoted the class of all meromorohic -valent starlie functions of order by ( ) Further a function f () in is said to be meromorohic -valent convex of order if and only if f ( ) f ( ) Re ( U ) (44) for some ( 0 ) We denote the class of all meromorohic -valent convex functions of order by ( ) The classes ( ) and ( ) and various other subclasses of have been studied rather extensively by Aouf etal ([5] and [8]) Joshi and Srivastava [5] Kularni et al [7] Owa et al [4] and others For 0 we obtain the class ( ) of meromorohic -valent starlie and and convex functions with resect to the origin Denote by and resectively of the classes the classes obtained by considering intersection and ( ) with ie ; 0 ; 0 (45) For f ( ) given by (4) and g ( ) given by g ( ) b 0 8 {3} (46) the Hadamard roduct (or convolution) of f and g is denoted by f g ( ) and defined by f g ( ) a b (47) 0

21 5 On some secial functions In what follows we deict certain secial functions used in our study for fractional calculus and analytic functions Gauss in 8 introduced the hyergeometric series defined by ( a) ( b) F ( a b; c; ) (5) ( c)! 0 for a b c C ; c 0 ) and ( a ) = ( a) 0 ( a) a( a )( a )( a ) N (5) is the Pochhammer symbol The incomlete beta function ( a c; ) is defined by ( a) ( a c; ) (53) () c U ; a c R \ Z R 0 0 corresonding to the incomlete beta function ( a c; ) defined by (53) the function ( a c; ) is defined in terms of the Hadamard roduct (or convolution) by the following condition ( a c; ) ( a c; ) ( ) which can be easily written in the following form ( a c; ) 6 Fractional calculus ( ) ( c) 9 0!( a) (54) (55) The original question that led to means of fractional calculus was can the meaning of a n n derivative of integer order d y / dx n 0 be extended when n is any number fractional irrational or comlex? The affirmative answer exosed to the world the comlete theory of fractional calculus which is also named as the theory of oerators of differentiation and integration of arbitrary order and further in late 70s it is termed as differ integrals of fractional order or fractional differ integrals The earlier studies in this

22 direction seem to have been made in the beginning and middle of the 9 th century by Liouville [3] and Riemann[45] The theory of fractional calculus is mainly based uon the study of the well nown Riemann-Liouvill integral and derivative oerators defined as follows Definition 8 [40] (Fractional integral oerator) Let the function f be analytic in a simly connected region (of the -lane) containing the origin and let 0 then the fractional integral of order is defined by 0 f D f d 0 (6) where the multilicity of when 0 is removed by requiring log to be real Definition 9 [40] (Fractional derivative oerator) Let the function f be analytic in a simly connected region (of the -lane) containing the origin and let 0 then the fractional derivative of order is defined by 0 d f D f d 0 d (6) where the multilicity of when 0 is removed by requiringlog to be real Definition 0 [40] (Extended fractional derivative oerator) Under the hyotheses of Definition 9 the fractional derivative of order defined by n is n n d D f D f 0 ; n N 0 N 0 (63) n d More recently Owa and Srivastava [4] introduced and studied an oerator : A A0 given by where f D f 3 (64) D is the fractional derivative of f of order 0

23 Note that f f f f 0 The oerators D and have been studied in the literature rather extensively to investigate various subclasses of univalent and multivalent functions with negative Taylor coefficients[7 4] Definition ([5] [5]) (Saigo Fractional integral oerator) let >0 and R then the generalied fractional integral oerator of a function f ( ) is defined by 0 0 of order 0 t f ( ) ( t ) F ; ; f ( t ) dt (65) where the function f ( ) is analytic in simly-connected region of the -lane containing the origin and the multilicity of when t 0 - t is removed by requiring log - ( ) rovided further that for f ( ) ( ) 0 t to be real max 0 (66) Definition ([5] [5]) (Saigo Fractional derivative oerator) let 0 < and R then the generalied fractional derivative oerator of a function f ( ) defined by J 0 of order d t J 0 f ( ) ( t ) F ; ; f ( t ) dt d 0 n d n J 0 f ( ) ( n n ; n N ) n d (67) where the function f ( ) is analytic in a simly-connected region of the -lane containing the origin with the order as given in (66) and multilicity of ( ) removed by requiring log t to be real when ( t) 0 It follows from Definition and that I ) 0 f ( t D f () ( 0) (68) is J f () f () ( 0 ) (69) 0 D

24 Lemma (cf Srivastava et al [6] ) The (generalied) fractional integral I 0 and the (generalied) fractional derivative J of a ower function are given by 0 I 0 = ( ) ( ) ( ) ( ) ( 0; max{0 } ) (60) and J 0 = ( ) ( ) ( ) ( ) (0 ; max{0 } ) (6) Definition 3 [6] The fractional oerator convenience as U for f ( ) A is defined in terms of 0 J 0 and for 0 ( ) ( ) J 0 f ( ) 0 () ( ) U 0 f ( ) ( ) ( ) 0 f ( ) 0 () ( ) (6) Using Lemma for f ( ) defined by () we get U f () ( ) 0 ( ) a ( ) ( ) Note that and U f 0 ( ) U f For 0 0 ( ) also for = D f f (63) ( 0 ) (64) = D f f ( 0) (65) 00 U 0 f ( ) = f () (66) U 0 f ( ) = f () (67)

25 The fractional derivative oerator (6) is defined by U (0 ) for the function f ( ) given by 0 U 0 f ( ) ( ) ( ) ( ) J f 0 ( ) (68) ( ) () a ( ) ( ) = n (0 ; ; max{ } ) 7 Subordination Definition 4 [Lindelof (9)] An analytic function g is said to be subordinate to an analytic function f written g f if for some analytic function with ( ) g f (7) The subordination function f need not to be univalent In articular if g() is univalent in U the subordination is equivalent to f g and f ( u) g u (0) 0 (7) Not that f f S f (73) and f K // f / f (74) 3

26 CHAPTER Certain subclasses of uniformly starlie and convex functions defined by convolution with negative coefficients 4

27 In this chater we introduce the classes S ( g ; ) and TS ( g ; ) of P -valent -uniformly convex and starlie functions defined by convolutions of certain linear oerators We obtain coefficient estimates distortion theorems extreme oints the results of modified Hadamard roduct and radii of close-to-convexity starlieness and convexity for function belonging to the class TS ( g ; ) Also we obtain convolution results and closure roerty for function belonging to the class Tac ( g; ) TS ( g 0; ) Introduction Let f ( ) and g( ) defined by () and (9) resectively using the Hadamard roduct ( f g )( ) and the convolution of the oerator U 0 defined by (63) with the function ( a c; ) defined by (53 ) we define the oerator M as follows Definition For real numbers ; ; Ra ; 0 ; and c 0-- we define the oerator M : A A by M ( f g )( ) ( a c; ) U ( f g )( ) a c 0 = ( ) a b () where ( a) () ( ) ( ) ( c) ( ) ( ) () Note that if g( ) then 00 M f ( ) and M f '( ) Now maing use of the oerator subclass S ( g ; ) M of univalent analytic functions defined by () we introduced the following 5

28 Definition The class S ( g ; ) consists of functions f ( ) A of the form () and the functions g ( ) A of the form (5) that satisfy the analytic criterion: M ( f g )( ) Re ( ) M a c ( f g )( ) M a c ( f g )( ) (3) M ( f g )( ) ( ) M ( f g )( ) M ( f g )( ) a c a c ( ; ; R;0 ; ; 0; a c R \ Z ) Further we define the class TS a c a c ( g ; ) by TS ( g ; ) T S ( g ; ) also for 0 let 0 TS ( g 0; ) T ( g; ) a c a c It may be noted that several classes studied by different authors can be derived from the class Sac ( g ; ) and TS ( g ; ) by choosing different values of the arameters and for instance If g( ) ac and = 0 the class class of univalent - uniformly starlie function S ( g ; ) reduces to the If g( ) ac and =0 the class class of univalent - uniformly convex function S ( g ; ) reduces to the 3 TS g K which is the class studied by Khairnar and More [6] ( 0; ) ( ) 4 00 S ( g ; ) S ( g ; ) and the classes studied by Aouf et al[65] 00 TS ( g ; ) TS ( g ; ) which are 5 Putting g( ) ac 0 in the class Sac ( g ; ) we obtain the class defined by Murugusundaramoorthy and Magesh [36] 6

29 Coefficient Estimates Theorem A function f ( ) defined by () is in the class S ( g ; ) if ( ) ( ) ( ) ( ) b a () ( ; ; R;0 ; ; 0; a c R \ Z ) where b a 0 and ( ) is given by () Proof It suffices to show that a c a c M ( f g )( ) ( ) M ( f g )( ) M ( f g )( ) M ( f g )( ) Re ( ) M a c ( f g )( ) M a c ( f g )( ) Assume that the inequality () holds true Then we have M ( f g )( ) ( ) M ( f g )( ) M ( f g )( ) a c a c M ( f g )( ) Re ( ) M a c ( f g )( ) M a c ( f g )( ) M ( f g )( ) ( ) ( ) M ( f g )( ) M ( f g )( ) a c a c 0 ( ) ( )( ) ( ) b a ( ) ( ) b a The last inequality is bounded above by (- ) ( ) ( ) ( ) ( ) b a Theorem A necessary and sufficient condition for the function f ( ) defined by (5) to be in the class TS ( g ; ) if is that 7

30 ( ) ( ) ( ) ( ) ba () ( ; ; R;0 ; ; 0; a c R \ Z ) where ( ) is given by ()The result is shar Proof In view of Theorem we need only to rove the sufficient art let f ( ) TS ( g ; ) and -be real then in view of (5) and (3) we have 0 ( ) b a ( )( ) ( ) b a b a b a ( ) ( ) ( ) ( ) Allowing along the real axis we obtain the inequality () The equality in () is attained for the extremal function f ( ) ( ) ( ) ( ) ( ) b ( ) (3) This comletes the roof of Theorem Corollary let the function f ( ) defined by (5) be in the class TS ( g ; ) then a ( ) ( ) ( ) ( ) b ( ) (4) 3 Distortion Theorems Theorem 3 let the function f ( ) defined by (5) be in the class then for r we have TS ( g ; ) and where f r r (3) b f r r (3) b a () (33) c( )( ) 8

31 where b a 0 The equalities in (3) and (3) are attained for the function f ( ) given by Proof Since for b f ( ) b Using Theorem we have (34) ( ) ( ) ( ) ( ) b b a ( ) ( ) ( ) ( ) b a that is From (5) and (35)we have and b a (35) b f r r a r r b f r r a r r This comletes the roof of Theorem 3 Theorem 4 let the function f ( ) defined by (5) be in the class then for r we have TS ( g ; ) and b f r (36) b f r (37) where b a 0 and () is given by (33) The result is shar for the function f ( ) given by (3) Proof From Theorem and (35) we have a b and the remaining art of the roof is similar to the roof of Theorem 3 9

32 4 Extreme oints of the class TS Theorem 5 Let and f ( g ; ) f ( ) ( ) ( ) ( ) ( ) b Then f TS g if and only if ac ( ; ) f can be exressed in the form ( ) (4) where 0 and f f (4 ) Proof Let (4) holds then by (4) we have f ( ) ( ) ( ) ( ) b Now ( ) ( ) ( ) ( ) b ( ) ( ) ( ) ( ) b = f TS ( g ; ) So by Theorem Conversely suose that f TS ( g ; ) Since a ( ) ( ) ( ) ( ) b ( ) ( ) ( ) ( ) ( ) ab Setting ( ) and we see that f can be exressed in the form (4) This comletes the roof of the Theorem 0

33 Corollary The extreme oints of the class TS ( g ; ) f ( ) ( ) ( ) ( ) ( ) b 5 Radii of close-to-convexity starlieness and convexity are Theorem 6 let the function f ( ) defined by (5) be in the class then f ( ) is close-to-convex of order (0 ) in r where f and ( ) (43) TS ( ) ( ) ( ) ( ) b r inf The result is shar with the extremal function f ( ) given by (3) Proof We must show that f ( ) for r where r is given by (5) Indeed we find from (5) that ( g ; ) (5) f ( ) a Thus f ( ) if But by Theorem (5) will be true if a (5) ( ) ( ) ( ) ( ) b that is if ( ) ( ) ( ) ( ) b (53) Theorem 6 follows easily from (53) Theorem 7 let the function f ( ) defined by (5) be in the class then f ( ) is starlie of order (0 ) in r TS ( g ; )

34 where r ( ) ( ) ( ) ( ) b inf (54) The result is shar with the extremal function f ( ) given by (3) Proof We must show that f ( ) for r f ( ) where r is given by (54) Indeed we find from (5) that f ( ) f ( ) a a Thus f ( ) if f ( ) But by Theorem (55) will be true if a (55) ( ) ( ) ( ) ( ) b that is if ( ) ( ) ( ) ( ) b (56) Theorem 7 follows easily from (56) Theorem 8 let the function f ( ) defined by (5) be in the class then f ( ) is convex of order (0 ) in r3 where TS ( ) ( ) ( ) ( ) b r3 inf ( g ; ) (57)

35 The result is shar with the extremal function f ( ) given by (3) Proof We must show that f ( ) f ( ) for 3 r where r 3 is given by (57) Indeed we find from (5) that f ( ) f ( ) a a Thus f ( ) f ( ) if a (58) But by Theorem (58) will be true if ( ) ( ) ( ) ( ) b that is if ( ) ( ) ( ) ( ) b (59) Theorem 8 follows easily from (59) 6 A family of integral oerators Theorem 8 let the function f ( ) defined by (5) be in the class let be a real number such that Then the function F () defined by also belongs to TS 0 TS ( g ; ) F t f () t dt (6) ( g ; ) 3

36 roof Let f () defined by (5) be in class written by TS ( g ; ) then F () can be where Now d a a F ( ) d (6) (63) ( ) ( ) ( ) ( ) b d ( ) ( ) ( ) ( ) b a ( ) since f ( ) TS ( g ) Hence by Theorem F( ) TS ( g ; ) Theorem 9 Let be a real number such that F ( ) a belongs to the class TS ( g ; ) then the function f () a c defined by (6) is univalent in R where ( ) ( ) ( ) ( ) ( ) b R inf ( )( ) The result is shar Proof from (6) we have If (64) f ( ) a (65) to rove the result it sufficient to show that f in ( ) f a a Thus f if ( ) In the view of the Theorem (66) will be satisfied if R (66) 4

37 or Setting we tae R ( ) ( ) ( ) ( ) ( ) b a ( ) ( ) ( ) ( ) ( ) ( ) b ( )( ) (67) in (67) and simlify we get the result Sharness of the result follows if ( )( ) f ( ) ( ) ( ) ( ) ( ) ( ) b ( ) (68) 7 Convolution Results for the class T For the functions ( g; ) j j j f ( ) a a 0 j n (7) the modified Hadmard roduct (or convolution) is defined by f f f ( ) n a a a n a j 0 j n For ositive real numbers r r r n the generalied hadmard roduct is defined by r r r n f f f n ( r r rn ; ) a a a n n Or j j j j If we tae r r r n then we have f f f n (; ) j f ( r ; ) a (7) j f f f ( ) r It is also noted that if f f j n and j j f ; f ( ) r j 5 n j rj then

38 Theorem 0 let the function f ( ) defined by (5) be in the class and h( ) d be in the class T ( g; ) Also assume T ( g; ) that a( ) Then ( f h)( ) Tac ( g; ) where c( )( ) ( )( )( ) (73) () b ( )( ) where a( ) () (74) c( )( ) and the result is shar Proof To rove the theorem we need to show that ( ) ( ) ( ) b ad (75) ( ) where ( ) is defined by () and is defined in (73) Now f ( ) and Tac ( g; ) and h ( ) Tac ( g; ) and thus we have ( ) ( ) ( ) b a (76) ( ) ( ) ( ) ( ) b d (77) ( ) By alying Cauchy-Schwar inequality to (76) and (77) we get Hence (75) is true if ( ) ( ) ( ) ( ) ( ) b a d ( )( ) (78) 6

39 ( ) ( ) ( ) b ad ( ) ( ) ( ) ( ) ( ) ( ) b a d ( )( ) or equivalently ad ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (79) In view of (78) and (79) (75) is true if ( )( ) ( ) b ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) which simlifies to ( )( )( )( ) (70) ( ) b ( ) ( ) ( ) ( ) ( )( ) Under the stated conditions in the theorem we observe that the function ( ) is a decreasing function for ( ) and thus (70) is satisfied if is given by (73) Finally the result is shar for f ( ) and g( ) defined resectively by and f ( ) b (7) g ( ) b (7) This comletes the roof Theorem Let that function f ( ) and h ( ) defined as in Theorem 0 be in the class T ( g; ) Then ( f h)( ) Tac ( g; ) where 7

40 for () given by (74) ( )( ) b ( ) Proof Substituting in theorem 0 the result follows Theorem Let the function f ( ) defined by (5) be in the class let with h( ) d Proof Notice that d Then ( f h)( ) Tac ( g; ) ( ) ( ) ( ) b a d ( ) ( ) ( ) b a d ( ) ( ) ( ) b a ( ) Hence by Theorem ( f h)( ) T ( g; ) (73) T ( g; ) Theorem 3 Let f ( ) j n be defined by (7) belong to the class T a c ( g; j ) j for each j Then ; j f j T a c ( g ; ) where r j and min n j ( )( ) ( ) b ( ) ( j ) ( ) b j rj ( ) b (74) Proof since f ( ) T ( g; ) j n then by Theorem we have j a c j Hence ( ) ( ) ( ) b a j j ( j n) j (75) 8

41 Thus using Holder's inequality we get ( ) ( ) ( ) b a j j (76) j ( ) ( ) ( ) b j j rj a j (77) j j j since n ; rj j f j a j r j j We need to show r n ( ) ( ) ( ) b a rj j (78) j Now in view of (77) the inequality (78) holds if ( ) ( ) ( ) b ( ) ( j ) ( ) b j j which simlifies to (74) hence we get the result Theorem 4 Let f ( ) j n be defined by (7) belong to the class T a c ( g; j ) j for each j Then the function h ( ) defined by r j r j n h( ) n rj a j j r j j (79) with rj 0 for j n belongs to the class T min n j ( g; ) where ( )( ) (70) ( ) ( j ) j 9

42 Proof since f ( ) T ( g; ) j n then from (75) we have j a c j ( ) ( ) ( ) b a r r j j j j ( j n) j or n j r j n ( ) ( j ) ( ) b ra j j j j (7) To show that h ( ) belongs T a c ( g; j ) we need to show n j r j ( ) ( ) ( ) b r a n j j j (7) Now in view of (7) the inequality (7) holds if ( ) ( ) n ( ) ( j ) j which simlifies to (70) hence we get the result j 8 Closure roerties Theorem 5 Let the function f ( ) and h ( ) defined as in Theorem 0 be in the class T ( g; ) is in the class T then the function ( ) defined by ( g; ) (8) ( ) a d where b (8) and () given by (74) The result is shar 30

43 Proof In view of Theorem it is sufficient to show that Notice that f ( ) and h ( ) belong to ( ) ( ) ( ) b a d (83) T ( g; ) and so ( ) ( ) ( ) b ( ) ( ) ( ) b a a (84) and ( ) ( ) ( ) b ( ) ( ) ( ) b d d (85) Adding (84 ) and (85) we get Thus (83) will hold if That is if ( ) ( ) ( ) b a d (86) ( ) ( ) ( ) ( ) b( ) ( ) ( ) b ( ) (87) Notice that the function ( ) is a decreasing for ( ) and thus (87) is satisfied if is given by (8) Finally the result is shar for f ( ) and g() given by (7) and (7) resectively Theorem 6 Let f j ( ) be defined by (7) belong to the class r j the function h f is also in the classt r j ( g; ) T ( g; ) Then Proof since f ( ) T ( g; ) in view of Theorem we have j a c ( ) ( ) ( ) b a j (88) 3

44 Now r r h( ) f a e r r j j j j where e r a j r j Thus by (88) we have r ( ) ( ) ( ) b a j j r Hence by Theorem h( ) T ( g; ) 3

45 CHAPTER 3 On certain roerties of neighborhoods of multivalent functions involving the generalied Saitoh oerator 33

46 In this chater we introduce the generalied Saitoh oerator L( a c ) and using this oerator we define the new subclasses of analytic functions with negative coefficients M ( a c ; b ) and L ( a c ; b ) We rove several inclusion relations associated with the neighborhood for the class L ( a c ; b ) Secial cases of some of these inclusion relations are shown Some subordination results results of artial sum and integral means inequalities are also derived for the class L ( a c ; b ) 3 Introduction For f ( ) defined by (3) Saitoh [53] introduce a linear oerator: defined by where L ( a c): S S L ( a c) ( a c; ) f ( ) U (3) ( a) ( a c; ) (3) () c 0 and ( a ) is the Pochammer symbol The generalied saitoh oerator L( a c ) is defined as follows Let a R and C Rl Z 0 where 0 0 and 0 then for f ( ) defined by (3) the oerator L( a c ) : S S is defined by where It easy to show that for f ( ) S L( a c ) f ( ) ( a c; ) D f ( ) (33) Df ( ) ( ) f ( ) f ( ) ( 0 U) (34) ( a) L( a c ) f ( ) a () c Note that for 0 we obtain the saitoh oerator (3) In 004 Cho Kwon and Srivastava [5] introduced the linear oerator L ( a c): S S analogous to L ( a c ) defined by L ( a c) f ( ) ( a c; ) f ( ) U ; a c R \ Z ; (35) 34 0

47 where ( a c ; ) is the function defined in terms of the Hadamard roduct (or convolution) by the following condition: ( a c; ) ( a c; ) (36) ( ) We can easily find from (3) (35) and (36) and for the function f ( ) S that ( ) ( c) L a c f a (37) ( ) ( )!( a) It is easily verified from (37) that and L ( a c) f ( ) al ( a c) f ( ) ( a ) L ( a c) f ( ) (38) L ( a c) f ( ) ( ) L ( a c) f ( ) L ( a c) f ( ) (39) for f ( ) defined by (3) and in terms of the Hadamard roduct (or convolution) we define the oerator L ( a c ) as follows Definition 3 Let a R C Rl Z 0 and then for f ( ) defined by 0 (3) the oerator L ( a c ) : S S is defined by L ( a c ) f ( ) ( a c; ) D f ( ) U = ( ) ( c) a!( a) (30) Note that for 0 we obtain the Cho Kwon and Srivastava oerator (37) Also by secialiing the arameters and we obtain from (30) and f ( ) L( 0) f ( ) f ( ) L( 0) f ( ) (3) L a a f D f n (3) n n ( 0) ( ) ( ) where n D is the well-nown Ruschewehy derivative of order n 35

48 Definition 3 Following [ ] we define the -neighborhood of a function f ( ) T by ( ) : ( ) N f g g T g b and a b (33) In articular for the identity function h( ) we immediately N ( h) g : g T g ( ) b and ( ) b (34) Now maing use of the oerator L ( a c ) defined by (30) we introduced the following subclasses L ( a c ; b ) and M ( a c ; b ) of valent analytic function Definition 33 A function f ( ) S defined by (3) is said to be in the class M ( a c ; b ) if it is satisfies the following inequality : L ( a c ) f ( ) L ( a c ) f ( )) b (35) ( U ; N ; a c R / Z ; b C /{0}; ;0 ; 0; 0) 0 Further we define the class L ( a c ; b ) by L ( a c ; b ) T M ( a c ; b ) It may be noted that by secifying the arameters a b c and the classes L ( a c ; b ) and M ( a c ; b ) generalies and extends several classes of analytic and valent functions such that f ( ) f ( ) (i) L ( 0; b ) ( ) b ( U; N;0 ; 0; bc /{0}) (ii) f ( ) L ( 0; b0 ) b ( U; N;0 ; bc /{0}) 36

49 (iii) f ( ) L ( 0; b ) b ( U; N;0 ; bc /{0}) 3 Inclusion relationshis involving N h for the class L ( a c ; b ) In our investigation of the inclusion relations involving following Theorem N h we shall require the Theorem 3 A function f ( ) T defined by (3) belongs to the class L ( a c ; b ) if and only if a b (3) ( ) ( c)!( a) ( U ; N ; a c R / Z ; b C /{0};0 ; 0 0) 0 The result is shar Proof Let f ( ) T defined by (3) belong to class L ( a c ; b ) / L ( a c ) f ( ) ( L ( a c ) f ( )) ( ) ( ) ( c) a b!( a) or equivalently ( ) ( c) Re a b!( a) ( U ) (3) We now choose values of on the real axis and let through the real values then the inequality (3) immediately yields the desired condition (3) Conversely assume that the inequality (3) is hold true and let then I ( a c ) f ( ) ( I ( a c ) f ( )) ( ) ( ) ( c) a!( a) 37

50 c ( ) a b!( a) Hence we have f ( ) L ( a c ; b ) the equality in (3) is hold for the function!( a) b f ( ) ( )( )( ) ( c) ( ) (33) Theorem 3 If then a b ( ) c( )( )( ) ( b ) (34) L ( a c ; b ) N ( h) (35) Proof Let f ( ) L ( a c ; b ) then in view of Theorem 3 we have c( )( )( ) ( ) ( c) a ( ) a b a!( a) which immediately yields a b a (36) c( )( )( ) On the other hand we also find from (3) and (36) ( ) ( c) ( ) ( c) ( ) a ( ) a b a a!( )!( ) or c( )( ) c( )( ) ( ) a ( ) a a a ( ) ( c) a ( ) ( c) a b ( ) ( )!( a)!( a) that is c( )( ) c( )( ) ( ) a ( ) a b a or a c( )( ) c( )( ) ( ) a b ( ) a a a b b ( ) b ( ) ( ) ( ) 38

51 Hence a b ( ) ( ) a ( b ) (37) c( )( )( ) which by means of (3) comletes the roof of Theorem Corollary 3 If then ( ) b (38) ( ) L ( 0; b ) N ( h) (39) Proof Let a c 0 in Theorem 3 we obtain the result Corollary 3 If then b ( ) (30) L ( 0; b0 ) N ( h) (3) Proof Let a c 0 in Theorem 3 we obtain the result Corollary 33 If b (3) then L ( 0; b ) N ( h) (33) Proof Let a c 0 in Theorem 3 we obtain the result 33 - neighborhood for the class L ( a c ; b ) In this section we determine the neighborhood for the class L ( a c ; b ) which is define as follows A function f ( ) T is said to in the class L ( a c ; b ) if there exist a function g ( ) L ( a c ; b ) such that f ( ) g( ) ( U0 ) (33) 39

52 Theorem 33 Let g ( ) L ( a c ; b ) and then c ( )( )( ) (33) ( ) c( )( )( ) a b N ( g ) L ( a c ; b ) (333) Proof suose that f ( ) N ( g) we find that from (33) which immediately have a ( ) a b (334) b Next as g ( ) L ( a c ; b ) then we have from Theorem 3 so that ( N) (335) a b b (336) c( )( )( ) a b a b b b f ( ) g( ) c ( )( )( ) ( ) ( )( )( ) ( ) c a b rovided that is given by (33) Thus roves the theorem f L a c b ( ) ( ; ) This evidently Corollary 34 Let then g L b and ( ) ( 0; ) ( ) (337) ( ) ( ) b ( ) ( 0; ) N g L b (338) Proof Let a c 0 in Theorem 33 we obtain the result 40

53 Corollary 35 Let then g L b and ( ) ( 0; 0 ) ( ) b ( ) ( 0; 0 ) N g L b (339) Proof Let a c 0 in Theorem 33 we obtain the result Corollary 36 Let g L b and ( ) ( 0; ) then ( ) b ( ) ( 0; ) (330) N g L b (33) Proof Let a c 0 in Theorem 33 we obtain the result 34 Subordination Results Definition 34 A sequence { } 0 sequence if for any regular and convex function b of comlex numbers is called subordination factor with U c g ( ) 0 c bc g( ) ( U) 0 (34) In 96 Wilf [65] resented the following necessary and sufficient condition for a sequence to be subordination factor sequence: Lemma 3 [65] The sequence { } 0 b is subordination factor sequence if and only if Re 0 b 0 ( U) (34) Now we obtain the subordination result for the class L ( a c ; b ) 4

54 Theorem 34 Let f ( ) L ( a c ; b ) of the form (3) and 0 be regular and convex function in U then Moreover g ( ) c c c( )( )( ) ( )( )( ) c a b 0 ( f g )( ) g ( ) (343) ( U ; N ; a c R / Z ; b C /{0}; ;0 ; 0; 0) c( )( )( ) a b Re{ f ( )} c( )( )( ) and the subordinating result (343) is shar for the maximum factor c( )( )( ) ( )( )( ) c a b (344) (345) Proof Let f ( ) L ( a c ; b ) of the form (3) and 0 g ( ) c c be regular and convex function in U To show subordinating result (343) we need to show that c( )( )( ) ( )( )( ) c a b is a subordinating factor sequence with a which in view of lemma 3 is true if Since 0 (346) c( )( )( ) Re 0 ( U) (347) 0 c ( )( )( ) a b ( ) ( c) ( )( ) c 0 ( a)! a on using Theorem 3 we have for r Re c( )( )( ) c( )( )( ) a b 0 a ( ) 4

55 c( )( )( ) Re c( )( )( ) a b c( )( )( ) a b c( )( )( ) a c( )( )( ) a b c( )( )( ) c( )( )( ) a b a( )( c) ( ) ( ) a!( a) c( )( )( ) r c ( )( )( ) a b c ( )( )( ) a b a b r c( )( )( ) a b 0 c ( )( )( ) a b c ( )( )( ) a b which evidently roves of (347) and hence the subordination result (343) Taing g ( ) in the subordination result (343) we easily get the result (344) and for the function c( )( )( ) f ( ) L ( a c ; b ) it can be verified that subordination result in (343) c( )( )( ) a b c( )( )( ) c( )( )( ) a b is the maximum factor for the 35 Partial sums In this section we determine inequalities involving artial sums of artial sums of f ( ) T of the form (3) is defined as follows: f ( ) T where the n 0 ( ) and f n ( ) a ( a 0; n ) f (35) 43

56 Theorem 35 Let then and where (354) f ( ) T be defined by (3) belong to L ( a c ; b ) f ( ) Re f n( ) S n( a c ; b ) U n (35) f n( ) S n( a c ; b ) Re (353) f ( ) S n ( a c ; b ) ( ( n )) (( ( n )) ( ) ( c) S a c b n n n ( ; ) ( a) n() n b ( U ; N ; a c R / Z ; b C /{0}; ;0 ; 0 0) 0 Proof Let f ( ) T be defined by (3) belong to L ( a c ; b ) then from Theorem 3 and using ( ; ) S ( ; ) n a c b S n a c b (355) we get Set n a S n a S a n (356) f ( ) g( ) S n ( a c ; b ) (357) f n( ) S n( a c ; b ) = n ( ; ) n a S a c b a which is analytic in U and g (0) 0 If (357) holds we find that g g S n ( a c ; b ) a ( ) n ( ) n ( ; ) n a S a c b a 44

57 S n ( a c ; b ) a n a S n a c b a n ( ; ) which shows that Re( g ( )) 0 and from (356) we obtain the inequality (35) similarly if we ut f n( ) S n ( a c ; b ) g ( ) S n ( a c ; b ) f ( ) S n ( a c ; b ) and maing use ( 355) we find that g g n n a S ( a c ; b ) a S n ( a c ; b ) a ( ) n ( ) n ( ; ) n a S a c b a S n ( a c ; b ) a n a S n a c b a n ( ; ) which roved the inequality (353) (358) 36 Integral means inequalities Lemma 3 ( Littlewood [33]) Let f and g be analytic in the unit disc and suose that i g f then for 0 and re ( 0 r ) g ( ) d f ( ) d (36) 0 0 Theorem 36 Let f ( ) L ( a c ; b ) and suose that a j (36) if there exist analytic function w() given by ( ) ( c) b!( a) ( j )( j ) 45

58 ( j )( j ) ( ) ( c) w a (363) b!( a) j ( ) i then for re ( 0 r ) where f j( ) is given by j 0 0 L ( a c ) f ( ) d L ( a c ) f ( ) d 0 (364) b j!( a) f j( ) ( j )( j )( ) ( c) j j j j (365) Proof By virtue of (30) and (365) we have ( ) ( c) I ( a c ) f ( ) a!( a) (366) and then we must show that b I ( a c ) f j ( ) ( j) j ( ) ( c) b j 0!( a) ( j ) 0 (367) 0 a d d by Lemma (3) it is sufficient to show that ( ) ( c) b j a!( a) ( j ) (368) Setting ( ) ( c) b a w ( )!( a) ( j ) which readily yields (363) with w ( ) 0 Now we rove that the analytic function w satisfies w U using (36) we obtain ( j )( j ) ( ) ( c) w a b!( a) ( ) j ( j )( j ) ( ) ( c) a b!( a) 46

59 ( j )( j ) ( ) ( c) a b!( a) This comletes the roof of the Theorem Corollary 37 Let f L b and suose that ( ) ( ; ) a j (369) if there exist analytic function w() given by i then for re ( 0 r ) where f j( ) is given by b ( j )( j ) ( )( ) ( ) j j j w a (360) b 0 0 f ( ) d f j( ) d 0 (36) b f j( ) ( j )( j ) j (36) Proof Let a c in Theorem 36 we obtain the result Corollary 38 let f L b and suose that ( ) ( ; ) a j (363) b ( j) if there exist analytic function w() given by i then for re ( 0 r ) where f j( ) is given by ( ) ( ) j j w a (364) b 0 0 f ( ) d f j( ) d 0 (365) b f j( ) ( j) j (366) 47

60 Proof Let a c in Theorem 3 6 we obtain the result Corollary 39 Let f L b and suose that ( ) ( ; 0 ) a j (367) if there exist analytic function w() given by i then for re ( 0 r ) where f j( ) is given by b ( j) ( ) ( ) j j w a (368) b 0 0 f ( ) d f j( ) d 0 (369) b f j( ) ( j) j (360) Proof Let a c 0 in Theorem 36 we obtain the result 48

61 CHAPTER 4 Subclasses of meromorhically multivalent functions associated with a certain linear oerator 49

62 In this chater we introduce the subclasses A ( A B n) and A ( A B n) of meromorhic multivalent functions in the unctured unit dis U : 0 by using a differential oerator D n f ( ) We obtain coefficient estimates distortion theorem radius of convexity and closure Theorems for the class A ( A B n) The familiar concet of neighborhoods of analytic functions is also extended and alied to the functions considered here 4 Introduction The extended linear derivative oerator of Ruscheweyh tye R : is defined by the following convolution: R f ( ) f ( ) ; f ( ) In terms of binomial coefficient (4) can be written as R f ( ) a 0 ; f (4) (4) In articular when n n N it is easily observed from (4) and (4) that so that (4) becomes The linear oerator n ( n ) n ( f ( )) R f ( ) n N 0 N {0} (43) n! n n R f ( ) a 0 n R n N ; f 0 (44) defined by (4) is motivated essentially by familiar Ruscheweyh oerator D which has been used widely on the sace of analytic and univalent functions (see for details Rusheweyh [48] Raina and Srivastava [44] Yang[66] For the function f ( ) Aouf [6] define the following differential oerator 50

63 S f ( ) f ( ) 0 S f ( ) ( ) f ( ) f ( ) ( ) ( 0 ) a S f N 0 S f ( ) S ( D f ( )) S f ( ) S ( S f ( )) n n n n ( ) S f ( ) S f ( ) ( 0; n N ) (45) It can be easily seen that n n 0 0 (46) S f ( ) a ( n N N 0 ; N ) With the aid of the diferential oerator S n f ( ) and the Ruscheweyh derivative R f ( ) we defined the following differential oerator for the function f ( ) D f ( ) ( ) S f ( ) R f ( ) (47) n n n for n N 0 and 0 Let f () be given by (4) then by maing use of (44) and (46) (47) can be easily written as where for n N 0 and 0 D f ( ) Q ( n ) a (48) n K 0 K n n QK ( n ) ( ) n (49) With the aid of the differential oerator D n f ( ) we define the following subclasses of multivalent and meromorhic functions 5

64 Definition 4 A function f ( ) defined by (4) is said to be in the class A ( A B n) if it satisfies the following subordination condition: n n ( D f ( ) [ B ( A B )( P )] ( U ) D f ( ) B or equivalently if the following inequality holds true: (40) n ( D f ( ) D f ( ) n n ( D f ( ) B [ B ( A B )( P )] n D f ( ) ( U ) (4) Also let A ( A B n) = A ( A B n) (0 P; A B ;0 B ; N ; n N ; 0; 0) It may be noted that for suitable choice of A B n and the class A ( A B n) extends several classes of analytic and -valent meromorhic functions such that Aouf and Shammay[9] Srivastava et al [60] and Uralegaddi and Ganigi [63] It may be also noted that A ( 0) A ( ) 0 P;0 0 0 is the class of meromorhic -valent convex function of order ( 0 ) and tye (0 ) with negative coefficients 4 Coefficient estimates for the class A ( A B n) We first determine a necessary and sufficient condition for a function f ( ) of the form (4) to be in the class A ( A B n) Theorem 4 Let the function f ( ) defined by (4) then if and only if where f ( ) A ( A B n) QK ( n ) M K ( A B P) a ( B A)( P ) (4) 0 (0 P; A B 0; A B ;0 B ; N ; n N ; 0; 0) and Q ( n ) is given by (49) K M K ( A B P) ( B ) ( A ) ( B A) (4) 0 5

65 Proof Suose that the function f ( ) defined by (4) be in the class A ( A B n) then from (4) we have n n ( D f ( ) D f ( ) n n n ( ) ( ( )) [ ( )( )] ( ) B D f D f B A B P D f = QK ( n ) a 0 K 0 P ( B A )( P ) Q ( n )[ B ( B A ) A] a K ( U ) (43) Since Re{ } for any real value then (4 3) yield 0 Q ( n ) a choosing to be real and letting through K P( B A)( P ) Q ( n )[ B ( B A ) A] a K (4 4) K 0 which leads us immediate to the coefficient inequality (4 ) Next in order to rove the converse we assume that the inequality (4) holds true then we observe that n n ( D f ( ) D f ( ) n n n ( ) ( ( )) [ ( )( )] ( ) B D f D f B A B P D f K 0 QK ( n ) a 0 P ( B A )( P ) Q ( n )[ B ( B A ) A] a K ( U) (4 5) Hence by maximum modulus theorem we have the roof of Theorem f ( ) A ( A B n) This comletes 53