EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 3, 2024, 1818-1830 ISSN 1307-5543 – ejpam.com Published by New York Business Global Hankel and Toeplitz Determinants of Logarithmic Coefficients of Inverse Functions for the Subclass of Starlike Functions with Respect to Symmetric Conjugate Points Nur Hazwani Aqilah Abdul Wahid1,∗, Adawiyah Tumiran1, Timilehin Gideon Shaba2 1 School of Mathematical Sciences, College of Computing, Informatics and Mathematics, Universiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia 2 Department of Mathematics, Landmark University, Omu-Aran 251103, Nigeria Abstract. This paper focuses on finding the upper bounds of the second Hankel and Toeplitz determinants, whose entries are logarithmic coefficients of inverse functions for a new subclass of starlike functions with respect to symmetric conjugate points associated with the exponential function defined by subordination. Results on initial Taylor coefficients and logarithmic coefficients of inverse functions for a new subclass are also presented. This study may inspire others to focus further to the coefficient functional problems associated with the inverse functions of various classes of univalent functions. 2020 Mathematics Subject Classifications: 30C45, 30C50 Key Words and Phrases: Univalent functions, starlike functions, symmetric conjugate points, exponential function, inverse functions, coefficient estimates, logarithmic inverse coefficients, Han- kel determinant, Toeplitz determinant, subordination 1. Introduction Let A denote the class of functions defined on the unit disk E = {z ∈ C : |z| < 1} which is normalized by the conditions f (0) = 0 and f ′ (0)− 1 = 0. The Taylor series of a function f (z) in A has the form f (z) = z + ∞∑ n=2 anz n, z ∈ E. (1) ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5250 Email addresses: hazwaniaqilah@uitm.edu.my (N. H. A. A. Wahid), adawiyahtumiran08@gmail.com (A. Tumiran), shabatimilehin@gmail.com (G. S. Timilehin) https://www.ejpam.com 1818 © 2024 EJPAM All rights reserved. N. H. A. A. Wahid, A. Tumiran, T. G. Shaba / Eur. J. Pure Appl. Math, 17 (3) (2024), 1818-1830 1819 The subclass of A that consists of analytic and univalent functions in the open unit disk E is denoted by S. On the other hand, it is well known that for each function f (z) ∈ S, there is an inverse function f−1 (w) in the form of f−1 (w) = w + ∞∑ n=2 Anw n, |w| < r0 (f) , r0 (f) ≥ 1 4 , (2) where particularly A2 = −a2, (3) A3 = −a3 + 2a2 2, (4) and A4 = −a4 + 5a2a3 − 5a2 3. (5) Let P denote the class of functions with a positive real part in E. A function p (z) in P has the form p (z) = 1 + ∞∑ n=1 pnz n, z ∈ E, (6) that is analytic in E and satisfying the condition Re (p (z)) > 0. It is known that p(z) ∈ P ⇔ p(z) = 1 + υ (z) 1− υ (z) , where υ (z) is a Schwarz function. Let H denotes the class of Schwarz functions υ (z) which are analytic in E given by υ (z) = ∞∑ k=1 bkz k, z ∈ E and satisfying υ (0) = 0 and |υ (z)| < 1. We assume that g1 (z) and g2 (z) are two analytic functions in E, and the symbol ≺ is a subordination. We say that the function g1 (z) is subordinate to another function g2 (z), denoted g1 (z) ≺ g2 (z) , if there exists a Schwarz function υ (z) ∈ H such that g1 (z) = g2 (υ (z)) for all z ∈ E. Furthermore, if g1 (z) is univalent in E, then we have the following equivalence: g1 (z) ≺ g2 (z) ⇔ g1 (0) = g2 (0) and g1 (E) = g2 (E) . The topic concerning Taylor coefficients in geometric function theory has stimulated more research into the Hankel and Toeplitz determinants for numerous classes of univalent functions. Because the upper bounds of both determinants for the classes of univalent functions are unknown in general and therefore remain an open problem. There is a close N. H. A. A. Wahid, A. Tumiran, T. G. Shaba / Eur. J. Pure Appl. Math, 17 (3) (2024), 1818-1830 1820 relationship between Toeplitz determinants and Hankel determinants. Constant entries are found along the diagonal of Toeplitz matrices, and along the reverse diagonal of Hankel matrices. The Hankel determinant Hq,n (f) and Toeplitz determinant Tq,n (f) , n, q ≥ 1 whose elements are Taylor coefficients an, n ≥ 2 of a function f (z) ∈ S are defined, respectively, by Pommerenke [15, 16] and Thomas and Halim [23, 24] as follows: Hq,n (f) = ∣∣∣∣∣∣∣∣∣ an an+1 · · · an+q−1 an+1 an+2 · · · an+q ... ... . . . ... an+q−1 an+q · · · an+2q+2 ∣∣∣∣∣∣∣∣∣ , a1 = 1 (7) and Tq,n (f) = ∣∣∣∣∣∣∣∣∣ an an+1 ... an+q−1 an+1 an ... an+q−2 · · · · · · ... · · · an+q−1 an+q−2 ... an ∣∣∣∣∣∣∣∣∣ . (8) The Hankel determinant is a valuable tool in the study of singularities. This is especially essential when investigating power series with integral coefficients [4, 5]. Meanwhile, the Toeplitz determinant has several applications in mathematics, both pure and applied. They appear in algebra, signal processing, partial differential equations, and time series analysis. [28] provides a good description of the applications of Toeplitz matrices across a wide spectrum of pure and applied mathematics. Furthermore, a recent study has focused on the Hankel and Toeplitz determinants, which involve the use of logarithmic coefficients, but in the direction of inverse functions for some classes of univalent functions, for instance, [2, 11, 12, 18] may provide further insight into this. The idea was that the classic concept of Hankel and Toeplitz determinants is generalized by replacing the entries with the logarithmic coefficients of inverse functions belonging to the classes of univalent functions. The Hankel determinant Hq,n ( Γf−1 ) and Toeplitz determinant Tq,n ( Γf−1 ) , n, q ≥ 1 whose elements are logarithmic coefficients of inverse functions belonging to the class S are defined, respectively, as follows [2, 11, 12, 18]: Hq,n ( Γf−1 ) = ∣∣∣∣∣∣∣∣∣ Γn Γn+1 ... Γn+q−1 Γn+1 Γn+2 ... Γn+q · · · · · · ... · · · Γn+q−1 Γn+q ... Γn+2q−2 ∣∣∣∣∣∣∣∣∣ (9) and Tq,n ( Γf−1 ) = ∣∣∣∣∣∣∣∣∣ Γn Γn+1 ... Γn+q−1 Γn+1 Γn ... Γn+q−2 · · · · · · ... · · · Γn+q−1 Γn+q−2 ... Γn ∣∣∣∣∣∣∣∣∣ . (10) N. H. A. A. Wahid, A. Tumiran, T. G. Shaba / Eur. J. Pure Appl. Math, 17 (3) (2024), 1818-1830 1821 The logarithmic coefficients of inverse functions Γn are defined in the series form of log f−1 (w) w = 2 ∞∑ n=1 Γnw n, |w| < 1 4 , where particularly Γ1 = −1 2 a2, (11) Γ2 = −1 2 ( a3 − 3 2 a2 2 ) , (12) Γ3 = −1 2 ( a4 − 4a2a3 + 10 3 a2 3 ) , (13) and Γ4 = −1 2 ( a5 − 5a2a4 + 15a2 2a3 − 5 2 a3 2 − 35 4 a2 4 ) . (14) We now introduce the subclass of starlike functions with respect to symmetric conju- gate points associated with the exponential function as follows: Definition 1. Let SSC ∗ (ez) be the class of functions defined by zf ′ (z) h (z) ≺ ϕ (z) , z ∈ E, where ϕ (z) = ez, is an analytic univalent function and h (z) = f(z)−f(−z) 2 . Remark 1. Changing the function ϕ (z) in Definition 1 gives us more subclasses of starlike functions with respect to symmetric conjugate points: (i) For ϕ (z) = 1+z 1−z , which has been introduced and studied in [8]. (ii) For ϕ (z) = 1+Az 1+Bz , − 1 ≤ B < A ≤ 1, which has been introduced and studied in [14]. (iii) For ϕ (z) = 1+Az 1+Bz , − 1 ≤ B < A ≤ 1 and considering the tilted factor eiα, |α| < π 2 , which has been defined and investigated in [25]. (iv) For ϕ (z) = 1 + sin z, which has been defined and studied in [26]. It is observed that there have been few studies on Hankel and Toeplitz determinants, whose entries are logarithmic coefficients of inverse functions for the subclass of univalent func- tions, particularly starlike functions with respect to other points, i.e., symmetric points, conjugate points, and symmetric conjugate points. We can refer the reader to [8, 9], who were among the early researchers who investigated these subclasses. Some researchers, including [14, 20, 21, 25–27], and references therein, have also carried out comprehensive studies related to these subclasses, which may provide diverse insights. Thus, inspired by the ideas of [11, 12, 18], in this paper, we aim to estimate the up- per bounds of the initial Taylor coefficients |an| , n = 2, 3, 4, 5, logarithmic coefficients of inverse functions |Γn| , n = 1, 2, 3, 4, and the second order Hankel and Toeplitz determi- nants whose entries are logarithmic coefficients of inverse functions belonging to the new subclass SSC ∗ (ez) , i.e., ∣∣H2,1 ( Γf−1 )∣∣ , ∣∣H2,2 ( Γf−1 )∣∣ , ∣∣T2,1 ( Γf−1 )∣∣ , and ∣∣T2,2 ( Γf−1 )∣∣ . N. H. A. A. Wahid, A. Tumiran, T. G. Shaba / Eur. J. Pure Appl. Math, 17 (3) (2024), 1818-1830 1822 2. Preliminary results In this section, we present certain lemmas that are essential to verify our main findings. Lemma 1. ([6]) For a function p (z) ∈ P of the form (6), the sharp inequality |pn| ⩽ 2 holds for each n ⩾ 1. Equality holds for the function p (z) = 1+z 1−z . Lemma 2. ([7]) Let p (z) ∈ P be a function of the form (6) and µ ∈ C. Then |pn − µpkpn−k| ⩽ 2max {1, |2µ− 1|} , 1 ⩽ k ⩽ n− 1. If |2µ− 1| ⩾ 1, then the inequality is sharp for the function p (z) = 1+z 1−z or its rotations. If |2µ− 1| < 1, then the inequality is sharp for the function p (z) = 1+zn 1−zn or its rotations. Lemma 3. ([10]) Let p (z) ∈ P be a function of the form (6) and α, β, γ ∈ ℜ. Then∣∣αp13 − βp1p2 + γp3 ∣∣ ⩽ 2 |α|+ 2 |β − 2α|+ 2 |α− β + γ| . 3. Main results This section is devoted to the proof of our main results. We will now determine the coefficient estimates for functions belonging to SSC ∗ (ez), followed by logarithmic coeffi- cients of inverse functions and the second Hankel and Toeplitz determinants of logarithmic coefficients of inverse functions for the new subclass SSC ∗ (ez), as follows: 3.1. Coefficient estimates Theorem 1. Let f (z) ∈ SSC ∗ (ez) . Then |a2| ≤ 1 2 , |a3| ≤ 1 2 , |a4| ≤ 25 96 , and |a5| ≤ 7 24 . Proof. If f (z) ∈ SSC ∗ (ez) and is the form of (1), then according to subordination relationship, there exists a Schwarz function υ (z) such that zf ′ (z) h (z) = eυ(z), (15) where h (z) = f(z)−f(−z) 2 . N. H. A. A. Wahid, A. Tumiran, T. G. Shaba / Eur. J. Pure Appl. Math, 17 (3) (2024), 1818-1830 1823 Define a function p (z) = 1 + υ (z) 1− υ (z) = 1 + ∞∑ n=1 pnz n ∈ P. This leads to υ (z) = p (z)− 1 p (z) + 1 . Hence, from the right-hand side of (15), we obtain eυ(z) = 1 + 1 2p1z + ( p2 2 − p12 8 ) z2 + ( p3 2 − p1p2 4 + p13 48 ) z3 + ( p4 2 − p1p3 4 − p22 8 + p12p2 16 + p14 384 ) z4 + · · · . On the other hand, since f (z) is in the form of (1), this gives zf ′ (z) = z + 2a2z 2 + 3a3z 3 + 4a4z 4 + 5a5z 5 + · · · and h (z) = z + a3z 3 + a5z 5 + · · · . Further, we have from (15) that z + 2a2z 2 + 3a3z 3 + 4a4z 4 + 5a5z 5 + · · · = ( z + a3z 3 + a5z 5 + · · · ) 1 + 1 2p1z + ( p2 2 − p12 8 ) z2 + ( p3 2 − p1p2 4 + p13 48 ) z3 + ( p4 2 − p1p3 4 − p22 8 + p12p2 16 + p14 384 ) z4 + · · ·  . (16) Now, equating the coefficients of zn, n = 1, 2, 3, 4, on both sides of (16) yields a2 = p1 4 , (17) a3 = 1 16 ( 4p2 − p1 2 ) , (18) a4 = 1 6144 ( 768p3 − 192p1p2 − 16p1 3 ) , (19) and a5 = 1 384 ( p1 4 − 24p1p3 + 48p4 ) . (20) Using Lemma 1 in (17), we get |a2| ≤ 1 2 . Applying Lemma 2 in (18) and Lemma 3 in (19), respectively, implies |a3| = 1 16 ∣∣4p2 − p1 2 ∣∣ ≤ 1 4 [ 2max { 1, ∣∣∣∣2(1 4 ) − 1 ∣∣∣∣}] = 1 2 N. H. A. A. Wahid, A. Tumiran, T. G. Shaba / Eur. J. Pure Appl. Math, 17 (3) (2024), 1818-1830 1824 and |a4| = 1 6144 ∣∣− ( 16p1 3 − (−192p1p2) + (−768p3) )∣∣ ≤ 1 6144 [2 |16|+ 2 |−192− 2 (16)|+ 2 |16− (−192) + (−768)|] = 25 96 . Rearranging the terms and taking modulus on both sides of (20), we can rewrite it as |a5| = 1 384 ∣∣48 (p4 − νp1p3) + p1 4 ∣∣ , where ν = 1 2 . Consequently, by applying Lemma 1 and Lemma 2 as well as the triangle inequality, we obtain |a5| ≤ 7 24 . This completes the proof of Theorem 1. 3.2. Logarithmic coefficients of inverse functions for SSC ∗ (ez) Theorem 2. Let f (z) ∈ SSC ∗ (ez) . Then |Γ1| ≤ 1 4 , |Γ2| ≤ 1 4 , |Γ3| ≤ 41 192 , and |Γ4| ≤ 197 256 . Proof. Putting (17)-(20) in (11)-(14), we obtain Γ1 = −p1 8 , (21) Γ2 = − 1 64 ( 8p2 − 5p1 2 ) , (22) Γ3 = − 1 768 ( 43p1 3 − 108p1p2 + 48p3 ) , (23) and Γ4 = − 1 256 ( −99 8 p1 4 − 28p1p3 + 16p4 + 45p1 2p2 − 20p2 2 ) . (24) The upper bounds of |Γ1| , |Γ2|, and |Γ3| follow from applying Lemma 1, Lemma 2, and Lemma 3, respectively. N. H. A. A. Wahid, A. Tumiran, T. G. Shaba / Eur. J. Pure Appl. Math, 17 (3) (2024), 1818-1830 1825 On the other hand, we write (24) as |Γ4| = 1 256 ∣∣p1 (αp13 − βp1p2 + γp3 ) + 16 ( p4 − µp2 2 )∣∣ , (25) where α = 99 8 , β = 45, γ = 28, and µ = 5 4 . Hence, implementing Lemma 2 and Lemma 3, we get the desired bound of |Γ4|. This completes the proof of Theorem 2. 3.3. Hankel determinant of logarithmic coefficients of inverse functions for SSC ∗ (ez) Theorem 3. Let f (z) ∈ SSC ∗ (ez) . Then∣∣H2,1 ( Γf−1 )∣∣ ≤ 95 768 . Proof. Using (21)-(23), we can establish H2,1 ( Γf−1 ) = Γ1Γ3 − Γ2 2 = p1 4096 ( 86 3 p1 3 − 72p1p2 + 32p3 ) − 1 4096 ( 64p2 2 − 80p1 2 + 25p1 4 ) = − 1 4096 ( −8p1 2p2 − 11 3 p1 4 − 32p1p3 + 64p2 2 ) . (26) Taking modulus and rearranging the terms in (26), it becomes∣∣H2,1 ( Γf−1 )∣∣ = 1 4096 ∣∣−p1 ( χp1 3 − λp1p2 + ηp3 ) + 64p2 2 ∣∣ , (27) where χ = 11 3 , λ = −8, and η = 32. By Lemma 3, we get∣∣χp13 − λp1p2 + ηp3 ∣∣ ≤ 2 ∣∣∣∣113 ∣∣∣∣+ 2 ∣∣∣∣−8− 2 ( 11 3 )∣∣∣∣+ 2 ∣∣∣∣113 − (−8) + 32 ∣∣∣∣ = 376 3 . Thus, from (27), in view of the triangle inequality as well as Lemma 1, we get the desired inequality. This completes the proof of Theorem 3. Theorem 4. Let f (z) ∈ SSC ∗ (ez) . Then∣∣H2,2 ( Γf−1 )∣∣ ≤ 7691 36864 . Proof. In view of (22)-(24), we obtain H2,2 ( Γf−1 ) = Γ2Γ4 − Γ3 2 = 1 131072 ( −2592p1 4p2 − 1792p1p2p3 + 1024p2p4 + 3680p1 2p2 2 −1280p2 3 + 495p1 6 + 1120p1 3p3 − 640p1 2p4 ) − 1 589824 ( 11664p1 2p2 2 − 9288p1 4p2 − 10368p1p2p3 +1849p1 6 + 4128p1 3p3 + 2304p3 2 ) = 1 1179648 ( −4752p1 4p2 + 4608p1p2p3 + 9216p2p4 + 9792p1 2p2 2 −11520p2 3 + 757p1 6 + 1824p1 3p3 − 5760p1 2p4 − 4608p3 2 ) . (28) N. H. A. A. Wahid, A. Tumiran, T. G. Shaba / Eur. J. Pure Appl. Math, 17 (3) (2024), 1818-1830 1826 Further, we can write (28) in the following expression:∣∣H2,2 ( Γf−1 )∣∣ = 1 1179648 ∣∣∣∣ p1 3 ( 757p1 3 − 4752p1p2 + 1824p3 ) − 4608p3 (p3 − p1p2) +9216p4 ( p2 − 5 8p1 2 ) − 11520p2 2 ( p2 − 17 20p1 2 ) ∣∣∣∣ . (29) Hence, by Lemma 2 and Lemma 3, we obtain that∣∣757p13 − 4752p1p2 + 1824p3 ∣∣ ≤ 12332, |p3 − p1p2| ≤ 2,∣∣∣∣p2 − 5 8 p1 2 ∣∣∣∣ ≤ 2, and ∣∣∣∣p2 − 17 20 p1 2 ∣∣∣∣ ≤ 2. Thus, from (29), making use of Lemma 1 and the triangle inequality yields the desired bound. This completes the proof of Theorem 4. 3.4. Toeplitz determinant of logarithmic coefficients of inverse functions for SSC ∗ (ez) Theorem 5. Let f (z) ∈ SSC ∗ (ez) . Then∣∣T2,1 ( Γf−1 )∣∣ ≤ 9 32 . Proof. It follows from (21) and (22) that T2,1 ( Γf−1 ) = Γ1 2 − Γ2 2 = 1 4096 ( 64p1 2 − 64p2 2 + 80p1 2p2 − 25p1 4 ) . (30) According to Lemma 2, we write∣∣T2,1 ( Γf−1 )∣∣ = 1 4096 ∣∣∣∣64p12 − 64p2 2 + 80p1 2 ( p2 − 5 16 p1 2 )∣∣∣∣ . (31) From (31), we find that∣∣∣∣p2 − 5 16 p1 2 ∣∣∣∣ ≤ 2max { 1, ∣∣∣∣2( 5 16 ) − 1 ∣∣∣∣} = 2. Hence, applying Lemma 1 and triangle inequality implies∣∣T2,1 ( Γf−1 )∣∣ ≤ 9 32 . This completes the proof of Theorem 5. N. H. A. A. Wahid, A. Tumiran, T. G. Shaba / Eur. J. Pure Appl. Math, 17 (3) (2024), 1818-1830 1827 Theorem 6. Let f (z) ∈ SSC ∗ (ez) . Then∣∣T2,2 ( Γf−1 )∣∣ ≤ 7165 9216 . Proof. Making use of (22) and (23), and after some calculations and simplifications, we obtain T2,2 ( Γf−1 ) = Γ2 2 − Γ3 2 = 1 4096 ( 64p2 2 − 80p1 2p2 − 86 3 p1 3p3 + 72p1p2p3 − 16p3 2 −81p1 2p2 2 + 129 2 p1 4p2 + 25p1 4 − 1849 144 p1 6 ) . (32) Considering (32) can be expressed as ∣∣T2,2 ( Γf−1 )∣∣ = 1 4096 ∣∣∣∣ 64p2 ( p2 − 5 4p1 2 ) + p3 ( −86 3 p1 3 + 72p1p2 − 16p3 ) −81p1 2p2 ( p2 − 43 54p1 2 ) + 25p1 4 − 1849 144 p1 6 ∣∣∣∣ . (33) Applying Lemma 2 and Lemma 3, from (33), we find that∣∣∣∣p2 − 5 4 p1 2 ∣∣∣∣ ≤ 2max { 1, ∣∣∣∣2(5 4 ) − 1 ∣∣∣∣} = 3, ∣∣∣∣p2 − 43 54 p1 2 ∣∣∣∣ ≤ 2max { 1, ∣∣∣∣2(43 54 ) − 1 ∣∣∣∣} = 2, and∣∣∣∣−86 3 p1 3 + 72p1p2 − 16p3 ∣∣∣∣ ≤ 2 ∣∣∣∣−86 3 ∣∣∣∣+2 ∣∣∣∣−72− 2 ( −86 3 )∣∣∣∣+2 ∣∣∣∣−86 3 − (−72) + (−16) ∣∣∣∣ = 424 3 . Hence, applying Lemma 1 and in view of the triangle inequality, (33) implies∣∣T2,2 ( Γf−1 )∣∣ ≤ 7165 9216 . This completes the proof of Theorem 6. 4. Conclusion Recent studies have provided strong motivation to find the upper bounds related to the Hankel and Toeplitz determinants whose entries are logarithmic coefficients of inverse functions for a new subclass SSC ∗ (ez). This paper specifically presents ∣∣H2,1 ( Γf−1 )∣∣ ,∣∣H2,2 ( Γf−1 )∣∣ , ∣∣T2,1 ( Γf−1 )∣∣ , and ∣∣T2,2 ( Γf−1 )∣∣ which also include estimates on initial Tay- lor coefficients |an| , n = 2, 3, 4, 5 and logarithmic coefficients of inverse functions |Γn|, n = 1, 2, 3, 4, which extends the existing knowledge in the field of geometric function the- ory. It appears that we may determine the upper bounds associated with the coefficient problems by using the lemma from the preliminary section. The obtained results of this study will prompt readers to further investigate other properties, such as Fekete-Szegö REFERENCES 1828 functional [25], Zalcman inequality [11, 13, 19], as well as the higher-order Hankel and Toeplitz determinants [1, 3, 10, 17, 22]. 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