International Journal of Analysis and Applications Volume 19, Number 3 (2021), 477-493 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-19-2021-477 Received February 28th, 2021; accepted April 7th, 2021; published May 4th, 2021. 2010 Mathematics Subject Classification. 30C45, 30C50. Key words and phrases. coefficient bounds; univalent functions; starlike functions; Toeplitz determinants ©2021 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 477 BOUNDS ON TOEPLITZ DETERMINANT FOR STARLIKE FUNCTIONS WITH RESPECT TO CONJUGATE POINTS DAUD MOHAMAD, NUR HAZWANI AQILAH ABDUL WAHID* Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia *Corresponding author: hazwaniaqilah@tmsk.uitm.edu.my ABSTRACT. This paper is concerned with the estimate of the upper bounds of the Toeplitz determinants  2 3T and  3 3T for functions belonging to the subclass of starlike functions with respect to conjugate points. The results presented would extend the results for some existing subclasses in the literature. 1. INTRODUCTION Let  be the class of functions  f z which are analytic in an open unit disk  : 1E z z  and having the power series expansion (1.1)   2 n n n f z z a z     in .E Let S be the class of functions  f z  and univalent in .E Let P be the class of functions  p z of the form https://doi.org/10.28924/2291-8639-19-2021-477 Int. J. Anal. Appl. 19 (3) (2021) 478 (1.2)   1 1 nn n p z p z     that is analytic in E and satisfying the condition  Re 0, .p z z E  Functions in P are called Carathéodory functions. It is well known that if   ,p z P then a Schwarz function  z exists with  0 0,    1,z  z E such that [1]       1 . 1 z p z z      For two functions  F z and  G z analytic in ,E we say that the function  F z is subordinate to  G z and we write it as    F z G z if there exists a Schwarz function  z which is analytic in E with  0 0,    1,z  such that     .F z G z Further, if  G z is univalent in ,E then        0 0F z G z F G  and    F E G E (see Miller and Mocanu [2, 3] for details). Let *S denote the class of starlike functions in .S It is known that   *f z S if and only if (1.3)     Re 0, . zf z z E f z          El-Ashwah and Thomas [4] defined the following class: (1.4)         * 2: Re 0, .C zf z S f z z E f z f z                    Functions in the class * CS are called starlike functions with respect to conjugate points. Halim [5] defined the following class: (1.5)           * 2: Re , 0 <1, .C zf z S f z z E f z f z                         In terms of subordination, Dahhar and Janteng [6] generalized the class *CS and it is denoted by  * , .CS A B This class is defined as follows: Int. J. Anal. Appl. 19 (3) (2021) 479 (1.6)           * 2 1, : , 1 1, . 1 C zf z Az S A B f z B A z E Bzf z f z                  Wahid et al. [7] introduced the subclass of tilted starlike functions with respect to conjugate points of order ,  * , , ,CS A B  and it is given by (1.7)         * 1 1, , , : sin , , 1 i C zf z Az S A B f z e i z E g z t Bz                          where       , 2 f z f z g z   cos 0,t     0 <1, 2    and 1 1.B A    In particular,  * *0 ,C CS S  * *1, 1C CS S  and   * *0,0,1, 1 .C CS S  Toeplitz matrices are one of the well-studied classes of structured matrices. The concept of Toeplitz matrices led to the development of the studies related to Toeplitz determinants, Toeplitz kernel, Toeplitz operators, and q-deformed Toeplitz matrices [8]. In a recent investigation, the Toeplitz determinant has been studied by [9-18], and they succeeded in estimating the coefficient bounds for Toeplitz determinant   ,qT n , 1n q  for the first few values of n and q over some subclasses of . The Toeplitz determinant  ,qT n , 1n q  of functions  f z of the form (1.1), is defined by Thomas and Halim [9]   1 1 1 2 1 1 2 , 1. n n n q n n n q q n q n q n a a a a a a T n a a a a                   However, apart from these works, there was no study of finding estimates for  2 3T and  3 3T for the subclasses introduced by El-Ashwah and Thomas [4], Halim [5], Dahhar and Janteng [6], and Wahid et al. [7]. In fact, as far as we are concerned, no bound for  3 3T was obtained for the class of univalent functions and its subclasses in the existing literature. Therefore, in this paper, we obtain the upper bounds for the Toeplitz determinant for  * , , ,CS A B  as defined in (1.7) for the case of 3,n 2q  and 3,n 3q  namely Int. J. Anal. Appl. 19 (3) (2021) 480 (1.8)   3 42 4 3 3 a a T a a  and (1.9)   3 4 5 3 4 3 4 5 4 3 3 . a a a T a a a a a a  We also give some results for the subclasses introduced by El-Ashwah and Thomas [4], Halim [5], and Dahhar and Janteng [6]. We shall state the following lemmas to prove our main results. 2. PRELIMINARY RESULTS Lemma 2.1. [19] For a function  p z P of the form (1.2), the sharp inequality 2np  holds for each 1.n  Equality holds for the function   1 . 1 z p z z    Lemma 2.2. [20] Let  p z P of the form (1.2) and . Then  2max 1, 2 1 , 1 1.n k n kp p p k n       If 2 1 1,   then the inequality is sharp for the function   1 1 z p z z    or its rotations. If 2 1 1,   then the inequality is sharp for the function   1 1 n n z p z z    or its rotations. 3. MAIN RESULTS Theorem 3.1. If the function  f z given by (1.1) belongs to the class  * , , , ,CS A B  then                    2 4 2 3 3 2 2 4 2 2 2 2 2 2 3 2 3 3 2 2 3 832 64 1 3 3 11 12 2 4 2304 8 72 144 72 144 144 8 128 128 72 192 96 16 16 32 24 88 32 24 12 16 92 72 T T                                                                   where ,iTe     ,T A B t  cost    and 1 .B  Int. J. Anal. Appl. 19 (3) (2021) 481 Proof. From (1.7), since    * , , , ,Cf z S A B  according to subordination relationship, so there exists a Schwarz function  z such that (3.1)         11 sin , 1 i zf z A ze i g z t B z                  where       , 2 f z f z g z   cos .t    Define a function       1 1 1 . 1 n n n z h z k z z           We have  h z P and (3.2)       1 . 1 h z z h z     Using (3.2), from (3.1), we have (3.3)              1 1 1 1 i i i e B T h z e B Tzf z e g z B h z B                   where   .T A B t  Using the series expansion in (3.3), we get (3.4)                 2 3 4 2 3 4 2 3 4 2 3 2 3 4 1 2 3 2 3 4 2 3 4 2 3 4 2 3 2 3 4 1 2 3 1 2 3 4 1 2 3 4 1 1 1 1 . i i i i e B z a z a z a z e B z a z a z a z k z k z k z e B T z a z a z a z e B T z a z a z a z k z k z k z                                                Equating the coefficients of 3z and 4z respectively in the expansion of (3.4) and for simplicity, we take iTe   and 1 ,B  give us (3.5) 2 2 2 2 1 1 3 2 8 k k k a      Int. J. Anal. Appl. 19 (3) (2021) 482 and (3.6) 2 3 3 3 2 3 2 3 1 2 1 2 1 1 1 4 8 6 8 3 2 . 48 k k k k k k k k a               Squaring (3.5) and (3.6), respectively, we get (3.7) 2 2 4 2 2 2 2 4 3 2 3 4 4 2 2 1 1 2 1 1 2 1 3 4 4 2 4 64 k k k k k k k k a               and 2 2 2 6 4 6 3 6 2 2 6 4 6 2 3 6 3 2 6 4 4 3 1 1 1 1 1 1 1 3 2 3 2 3 3 2 3 2 1 2 3 1 2 3 1 3 1 3 1 3 1 3 1 3 2 2 2 2 2 2 2 2 4 3 4 1 2 1 2 1 2 1 2 1 2 64 3 2 3 11 12 4 2304 96 128 16 24 16 24 16 36 96 64 12 34 a k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k                                                  2 4 2 3 41 2 1 2 4 3 4 2 2 4 3 1 2 1 2 1 2 36 16 6 36 16 . k k k k k k k k k k                  (3.8) From the equations (1.8), (3.7), and (3.8), yield    2 2 2 4 3 2 2 6 4 6 3 6 2 2 6 4 6 2 3 6 3 2 6 4 3 1 1 1 1 1 1 1 3 2 3 2 3 3 2 3 2 1 2 3 1 2 3 1 3 1 3 1 3 1 3 1 3 2 2 2 2 2 2 2 2 4 3 1 2 1 2 1 2 1 2 3 64 3 2 3 11 12 4 2304 96 128 16 24 16 24 16 36 96 64 12 T a a k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k                                                          4 2 4 2 3 4 1 2 1 2 1 2 4 3 4 2 2 4 3 1 2 1 2 1 2 2 2 4 2 4 2 4 2 2 1 1 2 1 1 2 1 2 2 2 6 4 2 3 2 3 2 4 3 2 1 4 2 1 34 36 16 6 36 16 144 36 144 72 144 36 64 144 1 3 3 11 2 12 4 2304 36 72 36 k k k k k k k k k k k k k k k k k k k k k k k k                                                                               2 2 1 2 1 2 3 2 2 2 2 3 2 2 2 1 2 1 3 4 3 2 2 2 3 3 1 2 144 144 96 128 36 96 64 16 16 24 24 16 12 6 34 36 36 16 16 . k k k k k k k k k k k                                                         Further, by suitably arranging the terms yield Int. J. Anal. Appl. 19 (3) (2021) 483             2 2 2 6 4 2 3 2 3 2 4 2 3 2 1 2 2 2 2 1 2 1 2 2 1 2 3 1 2 3 2 2 2 1 3 64 144 1 3 3 11 2 12 4 2304 36 36 72 144 144 144 144 36 96 64 96 128 96 128 16 16 24 16 T k k k k k k k k k k k k                                                                                                    2 3 2 2 3 3 1 2 2 2 2 2 2 2 2 2 2 6 4 2 3 2 3 2 4 3 2 1 2 2 1 2 1 24 12 6 36 34 36 16 16 16 16 24 16 24 16 16 24 16 24 64 144 1 3 3 11 2 12 4 2304 144 144 k k k k k k k k k                                                                                              1 2 3 1 2 3 2 2 2 1 3 1 2 96 128 16 16 24 16 24 k k k k k k k k k                      (3.9) where 2 236 36 72 , 144 144             2 236 96 64 96 128             and 3 3 2 2 2 2 3 3 2 2 2 2 12 6 36 34 36 16 16 . 16 16 24 16 24                                  Consequently, by the triangle inequality, from (3.9), we get              2 2 2 6 4 2 3 3 2 2 4 2 3 2 1 2 2 1 2 1 1 2 3 1 2 3 2 2 2 1 3 1 2 3 64 144 1 3 3 11 12 2 4 2304 144 144 96 128 16 16 24 16 24 . T T k k k k k k k k k k k k k k k                                                (3.10) By Lemma 2.2, Int. J. Anal. Appl. 19 (3) (2021) 484 (3.11)  22 1 2 2 2max 1, 2 1 72 144 72 144 144 2max 1, , 144 144 k k                         (3.12)  3 1 2 2 2 2max 1, 2 1 128 128 72 192 96 2max 1, 96 128 k k k                         and (3.13)         3 1 2 2 3 2 3 3 2 2 2 2 2 2 2 2 2 2 2max 1, 2 1 16 32 24 88 32 24 12 2 max 1, 16 16 24 16 24 16 92 72 . 16 16 24 16 24 k k k                                                          By making use of Lemma 2.1 together with (3.11)-(3.13), we find that (3.14)  2 22 2 1 2 1 72 144 72 144 144 144 144 8 144 144 , 144 144 k k k                          (3.15)  2 2 1 2 3 1 2 128 128 72 192 96 96 128 8 96 128 96 128 k k k k k                       and                       3 2 2 2 1 3 1 2 2 3 2 3 2 2 2 2 2 2 3 2 2 2 2 2 16 16 24 16 24 16 32 24 88 32 16 16 16 24 16 24 16 16 24 16 24 24 12 16 92 72 . 16 16 24 16 24 k k k k                                                                  (3.16) Again by applying Lemma 2.1 along with (3.14)-(3.16), from (3.10), we obtain Int. J. Anal. Appl. 19 (3) (2021) 485                    2 4 2 3 3 2 2 4 2 2 2 2 2 2 3 2 3 3 2 2 3 832 64 1 3 3 11 12 2 4 2304 8 72 144 72 144 144 8 128 128 72 192 96 16 16 32 24 88 32 24 12 16 92 72 . T T                                                                   The result is sharp for the function given by     1 1 sin . 1 i zf z ze i g z t z                This completes the proof of Theorem 3.1. Remark 3.1. For 0,  0 ,  1A  and 1,B   Theorem 3.1 yields  2 3 25.T  This inequality coincides with the result obtained by Ali et al. [14] for *.S Theorem 3.2. If the function  f z given by (1.1) belongs to the class  * , , , ,CS A B  then                   3 2 2 2 3 3 2 2 2 4 2 3 2 3 2 3 4 4 3 8 12 36 22 44 12 12 2 384 16 12 6 8 4 48 48 4 24 12 192 2304 2048 8 144 144 8 288 576 288 576 576 9216 64 24 5 3 88 35 96 51 32 18 4 144 288 288 T T T                                                                                         2 2 3 2 2 2 3 3 2 8 360 792 576 448 736 16 108 96 32 328 576 48 92 32 256 16                                where ,iTe     ,T A B t  cost    and 1 .B  Proof. Upon simplification of (1.9), the determinant  3 3T can be written as     2 23 3 5 3 4 3 53 2T a a a a a a    and by using the triangle inequality, we get   2 23 3 5 3 4 3 53 2 .T a a a a a a    Now, equating the coefficient of 5z in the expansion of (3.4) and for simplicity, we take iTe   and 1 ,B  give us (3.17) 2 2 2 2 4 4 4 3 4 1 3 1 3 2 2 1 1 5 4 2 2 4 3 2 3 2 2 2 2 1 1 1 2 1 2 1 2 48 32 48 12 24 6 384 11 6 12 44 36 . 384 k k k k k k k k k a k k k k k k k k                                Int. J. Anal. Appl. 19 (3) (2021) 486 From the equations (3.5) and (3.17), we obtain   2 2 2 2 2 2 2 4 4 3 5 2 1 1 4 1 3 1 3 2 2 1 4 3 4 2 2 4 3 2 3 2 2 2 2 1 1 1 1 2 1 2 1 2 4 3 2 2 3 2 2 2 1 1 2 1 96 48 48 48 32 48 12 24 384 6 11 6 12 44 36 1 6 11 6 36 44 12 384 a a k k k k k k k k k k k k k k k k k k k k k k k                                                                             2 21 2 1 3 2 448 48 24 12 48 32 96 48 .k k k k k k                  (3.18) Further, by suitably arranging the terms, we get             2 2 2 2 3 2 2 3 3 5 1 2 1 2 2 1 2 4 1 3 2 3 2 2 3 2 2 2 2 1 2 1 2 2 4 1 3 1 36 44 12 6 11 6 384 48 48 24 12 8 6 6 4 96 6 11 6 36 44 12 384 36 44 12 6 4 48 6 a a k k k k k k k k k k k k k k k                                                                                                2 2 1 2 2 2 2 2 2 1 2 1 4 1 3 2 2 1 2 2 48 48 24 12 96 36 44 12 48 384 48 48 24 12 96 k k k k k k k k k k k k                                             (3.19) where 3 2 2 3 2 2 6 11 6 36 44 12                    and 6 4 . 6      Consequently, by the triangle inequality, from (3.19), we get (3.20)   2 2 2 2 3 5 1 2 1 4 1 3 2 2 2 1 2 36 44 12 48 384 96 48 48 24 12 . T a a k k k k k k k k k                         By making use of Lemma 2.1 and Lemma 2.2, we find that Int. J. Anal. Appl. 19 (3) (2021) 487 (3.21)     2 2 2 2 2 2 1 2 1 3 2 2 2 3 2 2 36 44 12 8 36 44 12 12 36 22 44 12 12 2 36 44 12 k k k                                         and (3.22) 4 1 3 12 6 8 48 96 . 6 k k k        Again by applying Lemma 2.1 along with (3.21) and (3.22), from (3.20) yields (3.23)      3 2 2 2 3 3 5 8 12 36 22 44 12 12 2 384 16 12 6 8 4 48 48 4 24 12 192 . T a a                                  In view of (3.5), (3.7), (3.8), and (3.17), we have       2 2 3 4 3 5 2 2 4 2 2 2 2 1 1 2 2 2 6 4 3 2 2 4 2 3 3 2 4 3 1 3 2 2 2 2 1 2 3 1 3 2 2 1 2 2 576 144 144 288 576 576 9216 512 8 24 16 24 88 96 32 9216 768 1024 128 192 128 192 128 a a a a k k k k k k k k k k k k k                                                                           2 2 4 3 2 2 3 1 2 2 3 2 2 3 3 2 4 2 1 2 3 6 4 3 2 2 3 4 3 2 2 1 1 3 2 2 1 4 1 288 768 512 96 272 288 128 48 288 128 288 72 144 192 288 9216 3 21 51 51 18 96 240 144 144 144 k k k k k k k k k k k k k k                                                                                      2 2 2 2 4 3 2 2 3 1 2 2 2 2 2 2 3 1 4 6 4 2 3 2 3 2 4 3 4 1 2 4 2 2 1 2 1 2 4 108 372 288 42 204 306 144 576 512 144 144 9216 24 5 3 88 35 96 32 51 18 576 576 144 144 288 288 k k k k k k k k k k k k k k                                                                                     3 2 2 2 2 2 1 2 3 1 2 3 2 2 2 1 3 4 3 2 2 2 3 3 1 2 72 144 576 736 180 396 224 32 192 48 128 16 54 48 68 288 18 16 128 . k k k k k k k k k                                                     Int. J. Anal. Appl. 19 (3) (2021) 488 By suitably arranging the terms, we get             2 2 3 4 3 5 2 2 2 2 2 3 1 4 6 4 2 3 2 3 2 4 3 4 1 2 2 2 2 2 1 2 1 2 4 2 2 576 512 144 144 9216 24 5 3 88 35 96 32 51 18 144 144 288 72 144 576 576 288 576 576 288 a a a a k k k k k k k k k k k                                                                                          2 2 1 2 3 1 2 3 2 2 2 1 3 2 2 2 3 3 3 1 2 2 2 2 180 396 224 576 736 576 736 32 192 48 128 16 54 48 68 288 18 16 128 32 192 48 128 16 k k k k k k k k k                                                                          and further yields                    2 2 3 4 3 5 2 2 2 2 2 3 1 4 6 4 2 3 2 3 2 4 3 4 1 2 2 2 1 2 1 2 4 2 1 2 3 1 2 3 2 2 2 1 2 576 512 144 144 9216 24 5 3 88 35 96 32 51 18 576 576 288 576 736 32 192 48 128 16 a a a a k k k k k k k k k k k k k k k k k                                                                        3 1 2k k k    (3.24) where 2 2144 144 288 , 576 576             72 144 , 288       2 2180 396 224 576 736             Int. J. Anal. Appl. 19 (3) (2021) 489 and           3 2 2 2 3 3 2 2 2 . 54 48 68 288 18 16 128 32 192 48 128 16                                Consequently, by the triangle inequality, from (3.24), we obtain               2 2 3 4 3 5 2 2 2 2 2 3 1 4 6 4 2 3 2 3 2 4 3 4 1 2 2 2 1 2 1 2 4 2 1 2 3 1 2 3 2 2 2 1 3 1 2 2 576 512 144 144 9216 24 5 3 88 35 96 32 51 18 576 576 288 576 736 32 192 48 128 16 . a a a a T k k k k k k k k k k k k k k k k k k k k                                                             (3.25) By Lemma 2.2, (3.26)     2 2 1 2 2 2max 1, 2 1 288 576 288 576 576 2max 1, , 576 576 k k                          (3.27)  24 2 2max 1, 2 1 144 288 288 2max 1, , 288 k k                  (3.28)     3 1 2 2 2 2max 1, 2 1 360 792 576 448 736 2max 1, 576 736 k k k                          and             3 1 2 3 2 2 2 3 3 2 2 2 2 2max 1, 2 1 108 96 328 576 32 92 32 48 256 16 2max 1, . 32 192 48 128 16 k k k                                             (3.29) Hence, applying Lemma 2.1 together with (3.26)-(3.29), we find that Int. J. Anal. Appl. 19 (3) (2021) 490 (3.30)  2 22 2 1 2 1 288 576 288 576 576 576 576 8 576 576 , 576 576 k k k                          (3.31) 2 2 4 2 1152 144 288 288 288 , 288 k k k         (3.32)  2 2 1 2 3 1 2 360 792 576 448 736 576 736 8 576 736 576 736 k k k k k                          and                   3 2 2 2 1 3 1 2 2 2 2 3 2 2 2 3 3 2 2 2 2 32 192 48 128 16 16 32 192 48 128 16 108 96 328 576 32 92 32 48 256 16 . 32 192 48 128 16 k k k k                                                             (3.33) Again by applying Lemma 2.1 along with (3.30)-(3.33), from (3.25) yields                  2 2 3 4 3 5 2 2 2 4 2 3 2 3 2 3 4 4 2 2 3 2 2 2 2304 2048 8 144 144 8 288 576 288 576 576 9216 64 24 5 3 88 35 96 51 32 18 4 144 288 288 8 360 792 576 448 736 16 108 96 32 328 576 48 92 a a a a T                                                                           2 3 3 232 256 16 .      (3.34) Finally, from (3.23) and (3.34), we obtain                   3 2 2 2 3 3 2 2 2 4 2 3 2 3 2 3 4 4 3 8 12 36 22 44 12 12 2 384 16 12 6 8 4 48 48 4 24 12 192 2304 2048 8 144 144 8 288 576 288 576 576 9216 64 24 5 3 88 35 96 51 32 18 4 144 288 288 T T T                                                                                         2 2 3 2 2 2 3 3 2 8 360 792 576 448 736 16 108 96 32 328 576 48 92 32 256 16 .                                Int. J. Anal. Appl. 19 (3) (2021) 491 This completes the proof of Theorem 3.2. By putting the specific values for the parameters , , A and B in Theorem 3.1 and Theorem 3.2, we obtain the coefficient bounds for the Toeplitz determinants for the subclasses introduced by El-Ashwah and Thomas [4], Halim [5], and Dahhar and Janteng [6], respectively as follows. Corollary 3.1. For  * 0,0,1, 1 ,Cf S  we obtain  2 3 25T  and  3 3 240.T  Corollary 3.2. For  * 0, ,1, 1 ,Cf S   we obtain                    2 4 2 2 2 3 2 4 1 3 832 64 16 1 8 288 1 288 1 2304 8 288 1 192 1 16 192 1 64 1 T                           and                                      2 3 3 2 2 4 2 3 2 2 1 3 8 48 1 16 1 16 6 16 1 4 96 1 4 24 1 192 384 4 1 2304 2048 8 288 1 8 1152 1 1152 1 9216 64 80 1 4 288 1 288 8 1440 1 1152 1 16 864 1 128 1 . T                                                         Corollary 3.3. For  * 0,0, , ,Cf S A B we obtain                                    2 4 32 2 23 2 4 22 22 2 3 2 3 3 22 3 832 64 1 3 3 11 2304 12 2 4 8 72 144 72 144 144 8 128 128 72 192 96 16 16 32 24 88 32 24 12 16 92 72 A B T A B A B A B A B A B A B A B A B A B A B A B                                                                    Int. J. Anal. Appl. 19 (3) (2021) 492 and                                      2 33 2 2 3 2 22 4 3 22 2 3 3 3 8 12 36 22 44 12 12 2 384 16 12 6 8 4 48 48 4 24 12 192 2304 2048 8 144 144 9216 8 288 576 288 576 576 64 24 5 3 88 35 96 51 A B T A B A B A B A B A B A B A B A B A B A B A B A B A B                                                                                          4 4 2 32 22 2 3 3 2 32 18 4 144 288 288 8 360 792 576 448 736 16 108 96 32 328 576 48 92 32 256 16 . A B A B A B A B A B A B A B                                            It is observed that the result of  2 3T for *S and * CS are shown to be equivalent. 4. CONCLUSION In this paper, we have obtained the coefficient bounds for  2 3T and  3 3T for the subclass of tilted starlike functions with respect to conjugate points of order ,  * , , , .CS A B  The results obtained can be reduced to the results for some existing subclasses in the literature by considering specific values for the parameters , , A and .B Acknowledgements: The authors wish to thank the anonymous referees for their careful reading. Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publication of this paper. REFERENCES [1] I. Graham, Geometric function theory in one and higher dimensions, CRC Press, New York, 2003. [2] S. S. Miller, P. T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Anal. Appl. 65(2) (1978), 289-305. [3] S. S. Miller, P. T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J. 28(2) (1981), 157-172. Int. J. Anal. Appl. 19 (3) (2021) 493 [4] R. M. El-Ashwah, D. K. Thomas, Some subclasses of close-to-convex functions, J. Ramanujan Math. Soc. 2(1) (1987), 85-100. [5] S. Halim, Functions starlike with respect to other points, Int. J. Math. Math. Sci. 14(3) (1991), 451-456. [6] S. A. F. M. Dahhar, A. Janteng, A subclass of starlike functions with respect to conjugate points, Int. Math. Forum, 4(28) (2009), 1373-1377. [7] N. H. A. A. Wahid, D. Mohamad, S. Cik Soh, On a subclass of tilted starlike functions with respect to conjugate points, Menemui Mat. (Discover. Math.) 37(1) (2015), 1-6. [8] K. Ye, L. H. Lim, Every matrix is a product of Toeplitz matrices, Found. Comput. Math. 16(3) (2016), 577-598. [9] D. K. Thomas and S. A. Halim, Toeplitz matrices whose elements are the coefficients of starlike and close-to-convex functions, Bull. Malaysian Math. Sci. Soc. 40(4) (2016), 1781-1790. [10] V. Radhika, S. Sivasubramanian, G. Murugusundaramoorthy, J. M. Jahangiri, Toeplitz matrices whose elements are the coefficients of functions with bounded boundary rotation, J. Complex Anal. 2016 (2016), Art. ID 4960704. [11] S. Sivasubramanian, M. Govindaraj, G. Murugusundaramoorthy, Toeplitz matrices whose elements are the coefficients of analytic functions belonging to certain conic domains, Int. J. Pure Appl. Math. 109(10) (2016), 39-49. [12] C. Ramachandran, D. Kavitha, Toeplitz determinant for some subclasses of analytic functions, Glob. J. Pure Appl. Math. 13(2) (2017), 785-793. [13] N. Magesh, Ş. Altınkaya, S. Yalçın, Construction of Toeplitz matrices whose elements are the coefficients of univalent functions associated with q-derivative operator, ArXiv:1708.03600 [Math]. (2017). [14] M. F. Ali, D. K. Thomas, A. Vasudevarao, Toeplitz determinants whose elements are the coefficients of analytic and univalent functions, Bull. Aust. Math. Soc. 97(2) (2018), 253-264. [15] V. Radhika, J. M. Jahangiri, S. Sivasubramanian, G. Murugusundaramoorthy, Toeplitz matrices whose elements are coefficients of Bazilevič functions, Open Math. 16(1) (2018), 1161-1169. [16] H. M. Srivastava, Q. Z. Ahmad, N. Khan, B. Khan, Hankel and Toeplitz determinants for a subclass of q-starlike functions associated with a general conic domain, Mathematics, 7(2) (2019), 181. [17] H. Y. Zhang, R. Srivastava, H. Tang, Third-order Hankel and Toeplitz determinants for starlike functions connected with the sine function, Mathematics, 7(5) (2019), 404. [18] S. N. Al-Khafaji, A. Al-Fayadh, A. H. Hussain, S. A. Abbas, Toeplitz Determinant whose Its Entries are the Coefficients for Class of Non-Bazilevic Functions, J. Phys.: Conf. Ser. 1660 (2020), 012091. [19] P. L. Duren, Univalent Functions vol. 259, Springer, New York-Berlin–Heidelberg–Tokyo, 1983. [20] I. Efraimidis, A generalization of Livingston's coefficient inequalities for functions with positive real part, J. Math. Anal. Appl. 435(1) (2016), 369-379.