Let p be an analytic function defined on the open unit disc with 01p. In this paper, we determine the conditions for  such that certain subordination properties hold for pz is subjected to certain geometric conditions involving the expressions

1',zpz 1'/zpzpz and 21'/zpzpz

of which each is subordinated to 1cz and the condition for

 is determined

Aini Janteng,

Liew, Andy Pik Hern

Rashidah Omar,

English

UKM Journal Article Repository

Journal of Quality Measurement and Analysis                                                          JQMA 16(1) 2020, 125-130 e-ISSN: 2600-8602                                                                                                                                        http://www.ukm.my/jqma                  The study reported in this paper was presented at the 27th National Symposium on Mathematical Sciences (SKSM27) at Hotel Tenera, Bangi, Selangor on 26 - 27 November 2019, organised by Department of Mathematics, Faculty of Science, Universiti Putra Malaysia.  FIRST ORDER DIFFERENTIAL SUBORDINATION ASSOCIATED WITH CASSINI CURVE (Subordinasi Pembeza Peringkat Pertama yang Bersekutu dengan Lengkung Cassini)    ANDY LIEW PIK HERN, RASHIDAH OMAR & AINI JANTENG*   ABSTRACT  Let p  be an analytic function defined on the open unit disc with  0 1p  . In this paper, we determine the conditions for   such that certain subordination properties hold for  p z  is subjected to certain geometric conditions involving the expressions  1 ' ,zp z    1 ' /zp z p z  and    21 ' /zp z p z  of which each is subordinated to 1 cz  and the condition for   is determined.  Keywords: analytic functions; univalent functions; differential subordination; Cassini curve   ABSTRAK  Andaikan p  fungsi analisis yang tertakrif pada cakera unit terbuka dengan  0 1p  . Di dalam makalah ini, syarat   ditentukan sedemikian sehingga sifat-sifat subordinasi adalah benar bagi  p z  tertakluk kepada syarat geometri mengandungi  1 ' ,zp z     1 ' /zp z p z  dan    21 ' /zp z p z  yang bersubordinasi kepada 1 .cz  Kata kunci: fungsi analisis; fungsi univalen; subordinasi pembezaan; lengkung Cassini                         1. Introduction  Let A  be the class of analytic functions in  and normalized by the condition    0 0 ' 0 1f f   . Let S  be the subset of A  denoting the class of univalent functions. Also, we denote by C  and S  respectively the class of convex and starlike functions. If an analytic function f  subordinates to function g  which also analytic, there must exists an analytic function w  so  0 0w   and   1w z   for 1z   and     f z g w z . When the function g  is univalent in D , then we say that    f z g z  and it is commensurate to    0 0f g  and    f D g D . In 1935, Goluzin (1935) proved that if the first order differential subordination    ' 'zp z zq z holds and  'zq z  is convex then the subordination    p z q z  holds and the function q  is the best dominant. After that, many researchers continued the study and established some generalizations of this first order differential subordination and the general theory was discussed in detail by Miller and Mocanu (1985). If  1 ' 1zp z z   hold, Nunokawa et al. (1989) proved the subordination   1p z z  also holds. Furthermore, there are lot of results obtained by other researchers, for example see Ali et al. (2007), Omar and Halim (2013), Omar et al. (2013), Cho et al. (2016), Ravichandran and Sharma (2015) and Sharma and Ravichandran (2016). Andy Liew Pik Hern, Rashidah Omar & Aini Janteng  126 A function of 1 z  is associated with the class called LS  introduced by Sokól and Stankiewics (1996). This class consists f A  so      : ' /w z zf z f z . It lies in the region that bounded by the right half of the lemniscate of Bernoulli which is 2 1 1w   . The shape of this lemniscate of Bernoulli resembles the symbol ∞. Besides, Aouf et al. (2011) generalized the idea and defined the class  cS q where   1cq z cz   for (0,1]c  as follow:       2': 1 ,czf zS q f A c z Df z                                               (1)   From this we may establish that     '1czf zf S q czf z    z D .                    (2)  Note that  1 LS q S  . We also denote by c  the set of all points in the right half-plane such that the product of the distances from each point to the focuses -1 and 1 is less than c :   2: Re 0, 1c w w c      ( ),        (3)  thus the boundary c  is the right loop of the Cassinian ovals    22 2 2 2 22 1x y x y c      and for 1c  ,  1 LS q S  . In Cho et al. (2019), the authors defined  sin : 1 sinz z    as      sin': : 1 sinzf zz f A zf z       .                    (4)  A function  sinf z  if and only if there exists an analytic function q , satisfies   0 1 sinq z q z  , z D . From this, we can get    01expz q tf z z dtt       .          (5)          In the paper by Ali et al. (2007), it showed the conditions of A, B, D and so that    1 ' / nzp z p z  ( 0,1,2n  ) are subordinated to 11DzEz implies  11Azp zBz. Likewise, researches in Omar and Halim (2013) found the values of   so that     1 ' / nzp z p z  ( 0,1,2n  ) subordinate to 11DzEz then   1p z z . Inspired by these researches, this First order differential subordination associated with Cassini curve   127 paper determines the conditions for   by considering the class  cS q and the function  sin z . 2. Preliminary Result  We investigate the subordination properties for the class  cS q associated with  sin z . We will need the following lemma to prove our results.  Lemma 2.1.  (Miller & Mocanu 2000) Let q  be univalent in D  and let   and   be analytic in a domain U  containing  q D  with   0w   when  w q D . Set       : 'Q z zq z q z  and       :h z q z Q z  . Suppose that  i. either h  is convex, or Q  is starlike univalent in D  and ii.     Re ' / 0zh z Q z   for z D . If p  is analytic in D , with    0 0p q ,  p D U  and                ' 'p z zp z p z q z zq z q z     , then    p z q z , and q  is best dominant.  The first result gives the bound for   so that  1 ' 1zp z cz   implies that the function p  is subordinate to the  function  sin z . 3.  Main Results  Theorem 3.1. Consider the analytic function p  D ,  0 1p   such that  1 ' 1zp z cz  . Then the following subordination result holds: if   2 log 1 1 1 1 log2sin 1c c          , then    sinp z z .  Proof.  Define the function                                               as follows:      21 1 log 1 1 log2 1Bq z cz cz                                          (6)  is analytic and it is the solution of  1 ' 1Bzq z cz   . First, denote   1w   ,  w   and we define         ' 'B B BQ z zq z q z zq z    where .  By considering 1 1cz  , let   1 1f z cz   , so we can get     1/2' 112 2 1zf z czf z cz . Then let    1/212 1h where 2 2| | | | | | 1, 1cz x yi x y        , so Andy Liew Pik Hern, Rashidah Omar & Aini Janteng  128     2 1 14 1 8 1x yihx yi x         . Thus,  2 1Re8h       and  1Re8h x      . Hence,   ' 1 1Re 02 8zf zf z       , then 1 1cz   is a starlike function, so Q  is also starlike. Function       Bh z q z Q z   satisfies     Re ' / 0zh z Q z   for z D .  Hence, from Lemma 2.1 we know that     1 ' 1 'Bzp z zq z    implies    Bp z q z . This concludes that    p z P z  for appropriate P . The result holds if    Bq z P z  holds. If    Bq z P z , then        1 1 1 1B BP q q P     . This gives a necessary condition for p P  to hold. So, by taking    sinP z z , the inequalities    sin1 1Bq     and    sin1 1Bq   reduce to 1   and 2  , where    12 log 1 1 1 1 log2sin 1c c          and   22 1 log 1 1 log2 1sin 1c c           respectively. The subordination    sinBq z z  holds if  1 2 1max ,     . The proof is complete.    When 1c  , we get the following corollary.   Corollary 3.2. If c=1 and   1 ' 1zp z z   where p  is analytic function p in D with  0 1p  . Then the following subordination result holds: If   2 1 log20.729325sin 1  , then    sinp z z   which is stated as Theorem 2.1 in Ahuja et al. (2018).             Next result gives bound on   so that    1 ' / 1zp z p z cz   implies p  is subordinate to  sin z  function.  Theorem 3.3. Suppose that p  be analytic function in D ,  0 1p   and    1 ' / 1zp z p z cz  . Then the following subordination result holds:      if    2 1 log 1 1 log 2 1log 1 sin 1c c         , then    sinp z z .   Proof. The function                                              defined by           2exp 1 log 1 1 log2 1Bq z cz cz                          is analytic and it is the solution of    1 ' / 1B Bzq z q z cz   . The function   1w   and   /w w   is defined and note that First order differential subordination associated with Cassini curve   129           : ' ' / 1 1B B B BQ z zq z q z zq z q z cz       is starlike in D . The function         : 1Bh z q z Q z Q z     satisfies     Re ' / 0zh z Q z   for z D . So, Lemma 2.1 is satisfied and the following subordination     ' '1 1BBzp z zq zp z q z                                  (7)  implies    Bp z q z . The, similar as the proof of Theorem 3.1, we obtain the desired results. The proof is complete.    Corollary 3.4.  For c = 1, the result reduces to Theorem 2.3 in Ahuja et al. (2018) and gives the following subordination:  if     2 2 1 log2 log 2 10.740256log 1 sin 1    , then    sinp z z .          Next,  a bound on   is determined such that    21 ' / 1zp z p z cz   implies p  is subordinate to  sin z  function.  Theorem 3.5.  Let    21 ' / 1zp z p z cz  , then the following subordination result holds:            if     2 1 sin 1 1 log 1 1 log 2 1sin 1c c       , then    sinp z z         where p  is an analytic function  in D  and  0 1p   .  Proof. The function  defined by      121 1 log 1 1 log 2 1Bq z cz cz                                         (8)  is analytic and is the solution of    21 ' / 1B Bzq z q z cz   . Similar as previous method, we define the function   1w   and   2/w w  , also           2: ' ' / 1 1B B B BQ z zq z q z zq z q z cz       is starlike in D , Q  is starlike function. The condition     Re ' / 0zh z Q z   for z D  is satisfied. Therefore, applying Lemma 2.1, we obtain     2 2' '1 1BBzp z zq zp z q z                                    (9)  implies    Bp z q z . Thence, similar as the proof in Theorem 3.1, we obtain the desired result. The proof is complete.  Andy Liew Pik Hern, Rashidah Omar & Aini Janteng  130 Corollary 3.6.  Let c =1 and making use Theorem 3, the following subordination result holds: if       2 1 sin 1 2 log 1 2 log 2 10.989098sin 1      , then    sinp z z  which is reduced to the result in Ahuja et al. (2018). Acknowledgments  Our thanks go to UMS Great grant GUG0269-2/2018 for financial support and all the anonymous reviewers. References  Ahuja O.P., Kumar S. & Ravichandran V. 2018. Application of first order differential subordination for functions with positive real part. Stud. Univ. Babeş-Bolyai Math 63: 303-311. Ali R.M., Ravichandran V. & Seenivasagan N. 2007. Sufficient conditions for Janowski starlikeness. Int. J. Math. Math. Sci. 2007: Article ID 62925. Aouf M.K., Dziok J. & Sokół J. 2011. On a subclass of strongly starlike functions. Appl. Math. Lett. 24: 27-32. Cho N.E., Kumar V., Kumar S. & Ravichandran V. 2019. Radius problems for starlike functions associated with the Sine function. Bull. Iran. Math. Soc. 45: 213-232. Cho N.E., Lee H.J., Park J.H. & Srivastava R. 2016. Some application of the first-order differential subordinations. Filomat 30(6): 1465-1474. Goluzin G.M. 1935. On the majorization principle in function theory. Dokl. Λkad. Nauk. SSSR 42: 647-650. Kumar S. & Ravichandran V. 2018. Subordinations for functions with positive real part. Complex Anal. Oper. Theory 12: 1179-1191. Miller S.S. & Mocanu P.T. 1985. On some classes of first-order differential subordinations. Michigan Math. J. 32(2): 185-195. Miller S.S. & Mocanu P.T. 2000. Differential Subordination. New York: Marcel Dekker. Nunokawa M., Obradović M. & Owa S. 1989. On criterion for univalency. Proc. Amer. Math. Soc. 106(4): 1035-1037. Omar R. & Halim S.A. 2013. Differential subordinations properties of Sokół-Stankiewicz starlike functions. Kyungpook Math. J. 53(3): 459-465. Omar R., Halim S.A. & Ibrahim R.W. 2013. Differential subordination properties of certain analytic functions. Int. J. of Math. 24(6), Article ID 1350044. Ravichandran V. & Sharma K. 2015. Sufficient conditions for starlikeness. J. Korean Math. Soc. 52(4): 727-749. Sharma K. & Ravichandran V. 2016. Applications of subordination theory to starlike functions. Bull. Iranian Math. Soc. 42(3): 761-777. Sokół J. & Stankiewicz J. 1996. Radius of convexity of some subclasses of strongly starlike functions. Zeszyty Nauk. Politech. Rzeszowskiej Mat. 19: 101-105.   Faculty of Science and Natural Resources Universiti Malaysia Sabah 88400 Kota Kinabalu Sabah, MALAYSIA E-mail:    andyliew1992@yahoo.com, aini_janteng@ums.edu.my*   Faculty of Computer and Mathematical Sciences Universiti Teknologi MARA Cawangan Sabah 88997 Kota Kinabalu Sabah, MALAYSIA                 Received: 9 March 2020 E-mail:   ashidah@hotmail.com                 Accepted: 17 July 2020   *Corresponding author       

First order differential subordination associated with Cassini curve

Let p be an analytic function defined on the open unit disc with p01. In this paper, we determine the conditions for such that certain subordination properties hold for p z is subjected to certain geometric conditions involving the expressions 1 ' , zp z1 ' /zp z p z and 2 1 ' / zp z p z of which each is subordinated to 1cz and the condition for is determined

Andy Liew Pik Hern

Rashidah Omar

Aini Janteng

UMS Institutional Repository

First order differential subordination associated with cassini curveABSTRACTLet p be an analytic function defined on the open unit disc with p01. In this paper, wedetermine the conditions for such that certain subordination properties hold for p z is subjectedto certain geometric conditions involving the expressions 1 ' , zp z1 ' /zp z p z and 2 1 ' / zp z pz of which each is subordinated to 1cz and the condition for is determined.

First order differential subordination associated with cassini curve

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First order differential subordination associated with Cassini curve

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