In this paper, univalence criteria of certain integral operator defined by a generalised derivative

operator is obtained

Eghbiq, Abdussalam

Maslina Darus,

UKM Journal Article Repository

Journal of Quality Measurement and Analysis              JQMA 12(1-2), 2016, 107-114 Jurnal Pengukuran Kualiti dan Analisis        UNIVALENCE CRITERIA OF CERTAIN INTEGRAL OPERATOR (Kriterium Univalen bagi Pengoperasi Kamiran Tertentu) ABDUSSALAM EGHBIQ & MASLINA DARUS  ABSTRACT In this paper, univalence criteria of certain integral operator defined by a generalised derivative operator is obtained. Keywords: analytic function;  integral operator; derivative operator  ABSTRAK Dalam makalah ini, kriterium univalen bagi pengoperasi kamiran tertentu yang ditakrifkan oleh pengoperasi terbitan teritlak diperoleh. Kata kunci: fungsi analisis; pengoperasi kamiran; pengoperasi terbitan  1. Introduction and preliminaries Let S be the class of normalised analytic and univalent in a unit disc               Also let  be the class of functions analytic in             and let        be the subclasses of  consisting of functions of the form                               . Let A  be the subclasses of H  consisting of functions of the form                                  (1) Let )(nA  denote the class of functions )(zf  of the form                                                       (2) which are analytic in the open unit disc       For the function Af   given by (1), we define a new generalised derivative operator                        as follows:                                                               ,    (3) where                                                          111( , ) , 21 !kjj nk nc n k kn k      . Abdussalam Eghbiq & Maslina Darus  108  Here                  can also be written in terms of convolution as follows: If ,...2,1,0m  then                                                                                                                                                 =                   ,  where                                                                  is the Ruscheweyh derivative operator and                  is given by (3). If ,...,2,1m  then                                                                                                                                                                                                Note that                  =                       and                         .       By specialising the parameters of                   in (3), we get the following derivative and integral operators: a)                                                              introduced by Ruscheweyh (1975). b)                                                            introduced by Salagean (1983). c)                                               (Al-Oboudi 2004). d)                                                     (Al-Shaqsi & Darus 2009).  e)                                                        (Darus & Al-Shaqsi 2008).   f) The derivative operator introduced by Catas (2009):                                                                 . g)                                     introduced by Uralegaddi and Somanatha (1992). h)                                       ,  studied by Flett (1972). i)                                         , introduced by Cho and Kim (1983). j)                                                                 ,        introduced by Mustafa and Darus (2011).      Here, we introduce a new general integral operator by using generalised derivative operator given by (3).  Univalence criteria of certain integral operator   109      For       and               ,         and                 are complex numbers, we define a family of integral operator by                                                                        ,                             (4) where                        and              defined by (3), which generalises many integral operators. In fact, if we choose suitable values of parameters, we get the following interesting operators. For example a) For                            ,   reduces to         of Breaz et al (2009)  b) Let                    . Then it   reduces to        of Breaz and Breaz (2002). In this paper, we discuss the univalence properties of the new general integral operator. 2. Preliminary Result      To discuss our problems, we need the following results, called the Schwarz lemmas due to Pascu (1987). Lemma 1 (Pascu 1987).  Let      and   be a complex number with        .  If    satisfies                                , then for all      the integral operator                            is in the class  .        Pescar (1996) obtained univalence criterion of univalence and is given in the folowing lemma.  Lemma 2 (Pescar 1996). Let       , and      where          and             . If                                   , then for all      the function Abdussalam Eghbiq & Maslina Darus  110                           is analytic and univalent in   .          On the other hand, the following result due to Ozaki and Nunokawa (1972) is useful in studying the univalence of integral operators for certain subclass of  .  Theorem 1 (Ozaki and Nunokawa 1972).  Let       satisfies the following inequality                   , for all    . Then the function   is univalent in  .        Next theorem provides the univalence conditions for the functions                     given by (4) as follows:  Theorem 2. Let the functions                       , for               , satisfy the conditions                    ,    and                             . If     and   are complex numbers such that                           and                             ,  then the functions                    is univalent.  Proof. Since               by (3), we get                                                      Let   F   be defined by                                      . Then                                                      , and we have             .  Also, a simple computation yields                                                               . By differentiating the above equality, we get  Univalence criteria of certain integral operator   111                                                                , (5) and from (5), we have                                                                                                                                                                                           . (6)    We have Uz  and                         ,    ,              , then by Lemma 2, we obtain:                         . We apply this result in inequality (6) to obtain:                                                                                                                                                                                                                                    =                      =                    .  We have                                                                                                                                                d                                                                                                                             and obtain                                                                    . So, from there, we have                                    . Hence the theorem is proved.                                                                                                      Corollary 1. Let the functions                        , for               , satisfy the condition                       ,     and                             . If      and   are complex numbers such that                   and                      ,   then the functions                     is univalent. Abdussalam Eghbiq & Maslina Darus  112  Proof. Take              in Theorem 2.                                                                              Corollary 2. Let the functions                       , for             , satisfy the condition       and                      . If      and   are complex numbers such that                                                ,  then the functions                     is univalent.  Proof. Put     in Theorem 2.                                                                                                 We will need the following definition for further results.  Definition 1 (Nunokawa and Obradovi`c 1989).  For some real   with      ,  we define a subclass      of   consisting of all functions   which satisfy               , for all     . Singh (2000) has shown that if       , then   satisfies                      ,  for all       .    (7) Theorem 3. Let the functions                       , for             , satisfy the condition                                                  . If      and   are complex numbers such that                     and                              ,      (8) then the functions                     is univalent.  Proof.   We know from the proof of Theorem 2 that                                                                 and                                                                                                                                                                                          where                                                                                 , so  Univalence criteria of certain integral operator   113                                                    . Using (7) we have                                    . Applying Lemma 2, we obtain                     is belong to  .                                      Theorem 4. Let the functions                       , for            , satisfy the condition (7)   and                               . If      and   are complex numbers such that                     , then the functions                     is univalent.  Proof .  We know from the proof of theorem 3 that                                                                  . So, by the imposed conditions, we find that                                                                                                                                                                                                                                                                                                                                                                                                                                                                            By applying Lemma 2, we prove that                     .                                                 Corollary 3. Let the functions                       , for          , satisfy the condition                      ,     and                             . If     and   are complex numbers such that                   , and                      , then the functions                     is univalent.  Proof.    By taking                in Theorem 4.                                                                 Corollary 4. Let the functions                         , for              , satisfy the condition     and                      . If      and   are complex numbers such that                    and                   , thenthe functions                    is univalent.  Proof.  Put     in Theorem 4.                                                                                               Abdussalam Eghbiq & Maslina Darus  114 Acknowledgements  The work here is supported by MOHE grant FRGS/1/2016/STG06/UKM/01/1.  References Al-Oboudi F.M. 2004. On univalent functions defined by a generalized Salagean operator. International Journal of Mathematics and Mathematical Sciences 27:1429-1436. Al-Shaqsi K. & Darus M. 2009. On univalent functions with respect to k-symmetric points defined by a generalized Ruscheweyh derivatives operator. Journal of Analysis and Applications 7(1):53-61. Breaz D & Breaz N. 2002. Two integral operators. Studia Universitatis Babes-Bolyai 47(3):13-19.  Breaz D, Breaz N & Srivastava H.M. 2009. An extension of the univalent condition for a family of operators. Applied Mathematics Letters 22:41-44.  Catas A. 2009. On a certain differential sandwich theorem associated with a new generalized derivative operator. General Mathematics 17(4):83-95. Cho N.E & Kim T.H 2003.Multiplier transformations and strongly close-to-convex functions. Bulletin of the Korean Mathematical Society 40(3):399-410. Darus  M.  &  Al-Shaqsi  K.  2008. Differential sandwich theorems with generalised derivative  operator. Inter. J. Comput. Math. Sci. 2(2): 75-78. Flett T.M. 1972. The dual of an inequality of Hardy and Littlewood and some related Inequalities. Journal of Mathematical Analysis and Applications Soc. 38:746-765. Mustafa N.M & Darus M. 2011 Differential subordination and superordination for a new linear derivative operator. International Journal of Pure and Applied Mathematics. 70(6):825-835. Nehari Z. 1975. Comformal Mapping. New York: Dover Publications  Nunokawa M& Obradovi`c M. 1989. One criterion for univalency. Proceedings of the American Mathematical Society 106:1035-1037. Ozaki S &Nunokawa M. 1972. The Schwarzian derivative and univalent functions. Proceedings of the American Mathematical Society 33: 392-394. Pascu N. 1987. An improvement of Beckers univalence criterion. Proceedings of the Commemorative Session Simion Stoilow, Brasov: 43-48. Pescar V. 1996. A new generalization of Ahlfors’s and Becker’s criterion of univalence. Bulletin of the Malaysian Mathematical Sciences Society 19:53-54. Ruscheweyh S. 1975. New criteria for univalent functions. Proc. Amer. Math. Soc. 49:109-115. Salagean G.S. 1983. Subclasses of univalent functions. Lecture Notes in Mathematics 1013. Springer-Verlag: 362-372. Singh V. 2000. On a class of univalent functions. International Journal of Mathematics and Mathematical Sciences 12:855-857. Uralegaddi B.A & Somanatha C. 1992. Certain classes of univalent functions. In Current Topics in Analaytic Function Theory. Eds. Srivastava H.M. & Owa S., pp. 371-374. Singapore: World Scientific Publishing.   School of Mathematical Sciences Faculty of Science and Technology University Kebangsaan Malaysia 43600 UKM Bangi Selangor DE MALAYSIA E-mail:    eghbiq@gmail.com, maslina@ukm.edu.my*          *Corresponding author        

Univalence criteria of certain integral operator

Abstract

Similar works

Full text

Available Versions

UKM Journal Article Repository