Int. J. Anal. Appl. (2023), 21:10 Hankel Determinant of Logarithmic Coefficients for Tilted Starlike Functions With Respect to Conjugate Points Daud Mohamad, Nur Hazwani Aqilah Abdul Wahid∗ School of Mathematical Sciences, College of Computing, Informatics and Media, Universiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia ∗Corresponding author: hazwaniaqilah@uitm.edu.my Abstract. The growth of the Hankel determinant whose elements are logarithmic coefficients for different subclasses of univalent functions has recently attracted considerable interest. In this paper, we obtain the bounds for the first four initial logarithmic coefficients for the subclass of starlike functions with respect to conjugate points in an open unit disk. Furthermore, we determine the upper bounds of the second Hankel determinant of logarithmic coefficients for this subclass. We also present some new consequences of our results. 1. Introduction Let A be the class of analytic functions f (z) in an open unit disk E = {z ∈C : |z| < 1} which satisfy f (0) = f ′ (0) − 1 = 0 and has the series representation f (z) = z + ∞∑ n=2 anz n, z ∈ E. (1.1) We also denote by S the subclass of A consisting of univalent functions in E. Let P be the class of analytic functions p (z) defined in E which satisfy Re p (z) > 0 and has the series representation p (z) = 1 + ∞∑ n=1 pnz n, z ∈ E. (1.2) This class is also known as the class of Carathéodory functions. Received: Jan. 7, 2023. 2020 Mathematics Subject Classification. 30C45, 30C50. Key words and phrases. univalent functions; starlike functions with respect to conjugate points; logarithmic coefficient; Hankel determinant of logarithmic coefficients; subordination. https://doi.org/10.28924/2291-8639-21-2023-10 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-10 2 Int. J. Anal. Appl. (2023), 21:10 We denote H as the class of Schwarz functions ω (z) defined in E which satisfy ω (0) = 0 and |ω (z)| < 1, and has the series representation ω (z) = ∞∑ k=1 bkz k, z ∈ E. (1.3) If there exists a Schwarz function ω (z) ∈ H such that f (z) = g (ω (z)) for all z ∈ E, then the analytic function f (z) is subordinate to another analytic function g (z) and is symbolically written as f (z) ≺ g (z) . Furthermore, if g (z) is univalent in E, then f (z) ≺ g (z) ⇔ f (0) = g (0) and f (E) = g (E) . In [1], El-Ashwah and Thomas introduced the class of functions that are starlike with respect to conjugate points. The class is denoted by SC ∗ which satisfies Re { 2zf ′ (z) f (z) + f (z) } > 0, z ∈ E. (1.4) In [2], Halim introduced the class SC ∗ (δ) consisting of functions of the form (1.1) and satisfying Re { 2zf ′ (z) f (z) + f (z) } > δ,0 6 δ < 1, z ∈ E. (1.5) By means of subordination, Dahhar and Janteng [3] introduced the class SC ∗ (A,B) consisting of functions of the form (1.1) and satisfying 2zf ′ (z) f (z) + f (z) ≺ 1 + Az 1 + Bz , − 1 6 B < A 6 1, z ∈ E. (1.6) From (1.6), it follows that f (z) ∈ SC∗ (A,B) if and only if 2zf ′ (z) f (z) + f (z) = 1 + Aω (z) 1 + Bω (z) , ω (z) ∈ H. (1.7) In [4], Wahid et al. introduced the class SC ∗ (α,δ) consisting of functions of the form (1.1) and satisfying Re { eiα zf ′ (z) g (z) } > δ, 0 6 δ < 1, |α| < π 2 , z ∈ E, (1.8) where g (z) = f (z)+f (z) 2 . The functions in the class SC ∗ (α,δ) are known as tilted starlike func- tions with respect to conjugate points of order δ. In terms of subordination, they defined the class SC ∗ (α,δ,A,B) which satisfies{ eiα zf ′ (z) g (z) −δ − i sin α } 1 tαδ ≺ 1 + Az 1 + Bz , − 1 6 B < A 6 1, z ∈ E, (1.9) where tαδ = cos α−δ. From (1.9), it follows that f (z) ∈ SC∗ (α,δ,A,B) if and only if{ eiα zf ′ (z) g (z) −δ − i sin α } 1 tαδ = 1 + Aω (z) 1 + Bω (z) , ω (z) ∈ H. (1.10) Remark 1.1. It is observed that by choosing the specific values of the parameters α, δ, A and B in the class SC ∗ (α,δ,A,B) leads to the following classes: Int. J. Anal. Appl. (2023), 21:10 3 (a) If we let α = δ = 0,A = 1 and B = −1, then the class SC∗ (α,δ,A,B) reduces to the class SC∗ given in (1.4). (b) If we let α = 0,A = 1 and B = −1, then the class SC∗ (α,δ,A,B) reduces to the class SC∗ (δ) given in (1.5). (c) If we replace α = δ = 0, then the class SC ∗ (α,δ,A,B) reduces to the class SC ∗ (A,B) given in (1.6). (d) If we replace A = 1 and B = −1, then the class SC∗ (α,δ,A,B) reduces to the class SC∗ (α,δ) given in (1.8). Aside from that, many interesting results especially related to coefficients of f (z) ∈ S for various subclasses of starlike functions with respect to symmetric points, symmetric conjugate points and conjugate points were obtained by several authors. We may point interested readers to recent advances in these subclasses and their further results which point in a different direction than the current study, see for example, coefficient estimates [3,5,6], Fekete-Szegö inequality [7], Hankel determinant [8–12], Toeplitz determinant [13] and Zalcman coefficient functional [8,14]. The logarithmic coefficients γn, n > 1 for a function f (z) ∈ S of the form (1.1) play an important role in Milin’s conjecture [15,16], Brennan’s conjecture [17] and can also be used to find estimations for the coefficients of an inverse function. It is given in the series representation Ff (z) := log ( f (z) z ) = 2 ∞∑ n=1 γnz n, z ∈ E. (1.11) By differentiating (1.11) and comparing the coefficients of zn, the logarithmic coefficients γn, n = 1, 2, 3, 4 are given as follows: γ1 = 1 2 a2, (1.12) γ2 = 1 2 ( a3 − 1 2 a2 2 ) , (1.13) γ3 = 1 2 ( a4 −a2a3 + 1 3 a2 3 ) (1.14) and γ4 = 1 2 ( a5 −a2a4 + a22a3 − 1 2 a3 2 − 1 4 a2 4 ) . (1.15) The growth of the inequalities problems related to the upper bound of the Hankel determinant has been studied for different subclasses of A. The Hankel determinant was defined by Pommerenke [18,19] for a function f (z) ∈ S of the form (1.1) which is given by Hq,n (f ) = ∣∣∣∣∣∣∣∣∣∣∣ an an+1 ... an+q−1 an+1 an+2 ... an+q · · · · · · ... · · · an+q−1 an+q ... an+2q−2 ∣∣∣∣∣∣∣∣∣∣∣ , (1.16) 4 Int. J. Anal. Appl. (2023), 21:10 where n, q ∈ N. It is very useful for example in the theory of singularities [20] and in the study of power series with integral coefficients. Recently, Kowalczyk and Lecko [21,22] proposed the study of the Hankel determinant whose ele- ments are logarithmic coefficients of f (z) ∈ S which is given by Hq,n (Ff /2) = ∣∣∣∣∣∣∣∣∣∣∣ γn γn+1 ... γn+q−1 γn+1 γn+2 ... γn+q · · · · · · ... · · · γn+q−1 γn+q ... γn+2q−2 ∣∣∣∣∣∣∣∣∣∣∣ . (1.17) The results of Hq,n (Ff /2) broaden the knowledge of logarithmic coefficients for different subclasses of S. In particular, for values of q = 2, n = 1 and q = 2, n = 2, respectively, we have H2,1 (Ff /2) = γ1γ3 −γ22 (1.18) and H2,2 (Ff /2) = γ2γ4 −γ32. (1.19) The problem of finding the upper bounds of |γn| and |Hq,n (Ff /2)| has been considered for some subclasses of univalent functions. Some significant contributions have been obtained recently to these problems; see for instance [8, 23–31]. However, as far as we know, no one has used the coefficients of logarithmic functions to obtain the bound for the second Hankel determinant for the classes SC ∗, SC ∗ (δ), SC ∗ (A,B) and SC ∗ (α,δ). Thus, in this paper, we continue the research dealing with the logarithmic coefficients and the Hankel determinant of logarithmic coefficients for the class SC ∗ (α,δ,A,B) introduced in (1.9). Our main aim is to obtain the upper bounds of the logarithmic coefficients |γn| , n = 1, 2, 3, 4 and the second Hankel determinants of logarithmic coefficients, i.e., |H2,1 (Ff /2)|and |H2,2 (Ff /2)|. Furthermore, we give several new consequences of our results based on the special choices of the involved parameters. 2. Preliminary results In this section, we present some lemmas which will be used to prove our main results. Lemma 2.1. ( [15]) Let p (z) ∈ P of the form p (z) = 1 + ∞∑ n=1 pnz n. Then |pn|6 2, n > 1. The inequality is sharp for the function p (z) = 1+z 1−z . Lemma 2.2. ( [32]) Let p (z) ∈ P of the form p (z) = 1 + ∞∑ n=1 pnz n and µ ∈C. Then |pn −µpkpn−k|6 2max{1, |2µ− 1|} , 1 6 k 6 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+z n 1−zn or its rotations. Int. J. Anal. Appl. (2023), 21:10 5 Lemma 2.3. ( [33]) Let p (z) ∈ P of the form p (z) = 1 + ∞∑ n=1 pnz n. Then ∣∣Ic13 −Xc1c2 + V c3∣∣ 6 2 |I| + 2 |X − 2I| + 2 |I −X + V | , where I, X and V are real numbers. 3. Main results In this section, we find the estimate for initial logarithmic coefficients for functions belonging to the class SC ∗ (α,δ,A,B) . Furthermore, we obtain the upper bounds of the second Hankel determinant of logarithmic coefficients for the case of q = 2 and n = 1, and q = 2 and n = 2 for functions from the class SC ∗ (α,δ,A,B) . Theorem 3.1. If f (z) ∈ SC∗ (α,δ,A,B) and has the series representation (1.1), then |γ1|6 T 2 , |γ2|6 T 4 , |γ3|6 T 6 and |γ4|6 T (1 + 2Υ) 8 , where T = (A−B)tαδ, tαδ = cos α−δ and Υ = 1 + B. Proof. Let f (z) = z + ∞∑ n=2 anz n ∈ SC∗ (α,δ,A,B). The coefficients an, n = 2, 3, 4, 5 are given by [14] a2 = k1ξ 2 , (3.1) a3 = ξ 8 [ 2k2 + k1 2 (ξ− Υ) ] , (3.2) a4 = ξ 48 [ 8k3 + k1k2 (6ξ− 8Υ) + k13 ( ξ2 − 3Υξ + 2Υ2 )] (3.3) and a5 = ξ 384 [ 48k4 + k1k3 (32ξ− 48Υ) + k22 (12ξ− 24Υ) + k14 ( ξ3 − 6Υξ2 + 11Υ2ξ− 6Υ3 ) +k1 2k2 ( 12ξ2 − 44Υξ + 36Υ2 )] , (3.4) where ξ = Te−iα, T = (A−B)tαδ, tαδ = cos α−δ and Υ = 1 + B. Using (3.1)−(3.4), from (1.12)−(1.15), respectively, we obtain γ1 = k1ξ 4 , (3.5) γ2 = 1 2 [ ξ 8 ( 2k2 + k1 2 (ξ− Υ) ) − k1 2ξ2 8 ] = ξ 16 ( 2k2 −k12Υ ) , (3.6) 6 Int. J. Anal. Appl. (2023), 21:10 γ3 = 1 2 [ ξ 48 ( 8k3 + k1k2 (6ξ− 8Υ) + k13 ( ξ2 − 3Υξ + 2Υ2 )) − k1ξ 2 16 ( 2k2 + k1 2 (ξ− Υ) ) + k1 3ξ3 24 ] = ξ 48 ( k1 3Υ2 − 4k1k2Υ + 4k3 ) (3.7) and γ4 = 1 2 [ ξ 384 ( 48k4 + k1k3 (32ξ− 48Υ) + k22 (12ξ− 24Υ) + k14 ( ξ3 − 6Υξ2 + 11Υ2ξ− 6Υ3 ) +k1 2k2 ( 12ξ2 − 44Υξ + 36Υ2 )) − k1ξ 2 96 ( 8k3 + k1k2 (6ξ− 8Υ) + k13 ( ξ2 − 3Υξ + 2Υ2 )) + k1 2ξ3 32 ( 2k2 + k1 2 (ξ− Υ) ) − ξ2 128 ( 4k2 2 + 4k1 2k2 (ξ− Υ) + k14(ξ− Υ)2 ) − k1 4ξ4 64 ] = ξ 128 ( 8k4 − 4k22Υ −k14Υ3 + 6k12k2Υ2 − 8k1k3Υ ) . (3.8) For γ1, implementing Lemma 2.1 in (3.5), we obtain |γ1|6 T 2 . For γ2, γ3 and γ4, we can write (3.6)−(3.8), respectively, as γ2 = ξ 8 ( k2 −µk12 ) , (3.9) γ3 = ξ 48 ( Ik1 3 −Xk1k2 + V k3 ) (3.10) and γ4 = ξ 128 ( 8 ( k4 −µk22 ) −k1 ( I∗k1 3 −X∗k1k2 + V ∗k3 )) , (3.11) where µ = Υ 2 , I = Υ2, X = 4Υ, V = 4, I∗ = Υ3, X∗ = 6Υ2 and V ∗ = 8Υ. Implementing Lemma 2.2 in (3.9), Lemma 2.3 in (3.10) and both Lemma 2.2 and Lemma 2.3 in (3.11), and application of triangle inequality, respectively, we get |γ2|6 T 4 , |γ3|6 T 6 and |γ4|6 T (1 + 2Υ) 8 . This completes the proof. � Int. J. Anal. Appl. (2023), 21:10 7 Theorem 3.2. If f (z) ∈ SC∗ (α,δ,A,B) and has the series representation (1.1), then |H2,1 (Ff /2)|6 7T 2 48 , where T = (A−B)tαδ, tαδ = cos α−δ and Υ = 1 + B. Proof. Using (3.5)−(3.7), from (1.18), we have H2,1 (Ff /2) = k1ξ 2 192 ( k1 3Υ2 − 4k1k2Υ + 4k3 ) − ξ2 256 ( 4k2 2 − 4k12k2Υ + k14Υ2 ) . (3.12) Hence, simplifying (3.12), we can write it as H2,1 (Ff /2) = ξ2 768 ( k1 ( Ik1 3 −Xk1k2 + V ∗∗k3 ) − 12k22 ) , (3.13) where I = Υ2, X = 4Υ and V ∗∗ = 16. Thus, applying Lemma 2.1 and Lemma 2.3, and by triangle inequality implies that |H2,1 (Ff /2)|6 7T 2 48 . This completes the proof. � Theorem 3.3. If f (z) ∈ SC∗ (α,δ,A,B) and has the series representation (1.1), then |H2,2 (Ff /2)|6 T 2 ( 2Υ2 + 9Υ + 17 ) 288 , where T = (A−B)tαδ, tαδ = cos α−δ and Υ = 1 + B. Proof. Substituting (3.9)−(3.11) in (1.19) and after simplification, we get H2,2 (Ff /2) = ξ2 2048 ( 2k2 −k12Υ )( 8k4 − 4k22Υ −k14Υ3 + 6k12k2Υ2 − 8k1k3Υ ) − ξ2 2304 ( k1 3Υ2 − 4k1k2Υ + 4k3 )2 = ξ2 18432 ( k1 6Υ4 − 8k14k2Υ3 + 8k13k3Υ2 + 144k2k4 − 72k12k4Υ − 72k23Υ +16k1 2k2 2Υ2 −128k32 + 112k1k2k3Υ ) . (3.14) By rearranging the terms in (3.14), we may write H2,2 (Ff /2) = ξ2 18432 [ k1 3Υ2 ( Ik1 3 −V ∗k1k2 + X∗k3 ) + 144k4 ( k2 −µk12 ) − 72k22Υ ( k2 −νk12 ) −128k3 (k3 −ηk1k2)] , (3.15) where I = Υ2, X∗ = 8, V ∗ = 8Υ, µ = Υ 2 , ν = 2Υ 9 and η = 7Υ 8 . Thus, implementing Lemma 2.2 and Lemma 2.3, and by triangle inequality, (3.15) yields |H2,2 (Ff /2)|6 T 2 ( 2Υ2 + 9Υ + 17 ) 288 . This completes the proof. � 8 Int. J. Anal. Appl. (2023), 21:10 Upon choosing the specific values of the parameters α, δ, A and B in Theorem 3.1, Theorem 3.2 and Theorem 3.3, respectively, we get the following consequences: Corollary 3.1. (a) Let SC ∗ (0, 0, 1,−1) ≡ SC∗. Then we have |γ1|6 1, |γ2|6 1 2 , |γ3|6 1 3 and |γ4|6 1 4 . (b) Let SC ∗ (0,δ, 1,−1) ≡ SC∗ (δ) . Then we have |γ1|6 (1 −δ) , |γ2|6 (1 −δ) 2 , |γ3|6 (1 −δ) 3 and |γ4|6 (1 −δ) 4 . (c) Let SC ∗ (0, 0,A,B) ≡ SC∗ (A,B) . Then we have |γ1|6 (A−B) 2 , |γ2|6 (A−B) 4 , |γ3|6 (A−B) 6 and |γ4|6 (A−B) (1 + 2Υ) 8 . (d) Let SC ∗ (α,δ, 1,−1) ≡ SC∗ (α,δ). Then we have |γ1|6 tαδ, |γ2|6 tαδ 2 , |γ3|6 tαδ 3 and |γ4|6 tαδ 4 . Corollary 3.2. (a) Let SC ∗ (0, 0, 1,−1) ≡ SC∗. Then we have |H2,1 (Ff /2)|6 7 12 . (b) Let SC ∗ (0,δ, 1,−1) ≡ SC∗ (δ) . Then we have |H2,1 (Ff /2)|6 7(1 −δ)2 12 . (c) Let SC ∗ (0, 0,A,B) ≡ SC∗ (A,B) . Then we have |H2,1 (Ff /2)|6 7(A−B)2 48 . (d) Let SC ∗ (α,δ, 1,−1) ≡ SC∗ (α,δ). Then we have |H2,1 (Ff /2)|6 7tαδ 2 12 . Corollary 3.3. (a) Let SC ∗ (0, 0, 1,−1) ≡ SC∗. Then we have |H2,2 (Ff /2)|6 17 72 . (b) Let SC ∗ (0,δ, 1,−1) ≡ SC∗ (δ) . Then we have |H2,2 (Ff /2)|6 17(1 −δ)2 72 . (c) Let SC ∗ (0, 0,A,B) ≡ SC∗ (A,B) . Then we have |H2,2 (Ff /2)|6 (A−B)2 ( 2Υ2 + 9Υ + 17 ) 288 . Int. J. Anal. Appl. (2023), 21:10 9 (d) Let SC ∗ (α,δ, 1,−1) ≡ SC∗ (α,δ). Then we have |H2,2 (Ff /2)|6 17tαδ 2 72 . 4. Conclusion In this paper, we have obtained the upper bounds of the initial logarithmic coefficients and the second Hankel determinant of logarithmic coefficients for functions from the class SC ∗ (α,δ,A,B). It is shown in corollaries that the obtained results lead to new results for some existing subclasses, i.e., SC ∗, SC ∗ (δ) , SC ∗ (A,B) and SC ∗ (α,δ). Corollary 3.1(a) also coincides with the inequality |γn| 6 1n, n > 1 that holds for the well-known class of starlike functions S ∗. 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