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Journal articles on the topic 'Second Hankel determinant'

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Published: 3 June 2025

Last updated: 31 July 2025

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1

Rathi, Sidik, Wan Muhammad Fakhrorrazzi Wan Ibrahim, and Muhammad Kamal Rafiq Mat Razali. "Second Hankel Determinants for a Subclass of Close-to-Convex Function Related to Certain Generalized Keobe Function." Mathematical Sciences and Informatics Journal 4, no. 1 (2023): 124–30. http://dx.doi.org/10.24191/mij.v4i2.23132.

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For ages, researchers have conducted numerous studies exploring every aspect of problems related to univalent functions. Most of the research has been concentrated on investigating the diverse properties of univalent functions. Notably, finding the upper bound of Hankel determinants has become an intriguing problem among researchers in this field. The aim of this paper is to solve on the second Hankel determinant problem for the class k z β ( ) C of close-to-convex functions related with the certain generalized starlike functions. We first give the definition of the class k z β ( ) C and use c

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2

Rehman,, H. U., K. A. Mashrafi,, and J. Salah,. "Estimating the Second Order Hankel Determinant for the Subclass of Bi-Close-to-Convex Function of Complex Order." Malaysian Journal of Mathematical Sciences 18, no. 1 (2024): 91–105. http://dx.doi.org/10.47836/mjms.18.1.06.

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The article aims to estimate the coefficient bounds for the second Hankel determinant by using the class of bi-close-to-convex functions of a complex order in the open unit disk. Making the direct application of Carathéodory function along with the closely related properties of starlike functions, we obtain the upper bound for the second Hankel determinant via certain subclass of bi-close-to-convex functions of complex order. The study discusses the maximization of the second Hankel determinant in both conventional graph and analytic methods. Moreover, we explore and modify some results on the

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3

Shu Huey, Kiu, Aini Janteng, Jaludin Janteng, and Andy Liew Pik Hern. "Second Hankel Determinant of Bi-univalent Functions." Malaysian Journal of Fundamental and Applied Sciences 19, no. 2 (2023): 269–79. http://dx.doi.org/10.11113/mjfas.v19n2.2807.

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Let be the class of functions which are analytic in the open unit disk and having the form . Denote to be the class for all functions in that are univalent in . Then, let denote the class of bi-univalent functions in . In this paper, we obtain the second Hankel determinant for certain classes of analytic bi-univalent function which are defined by subordinations in the open unit disk . In particular, we determine the initial coefficients and and obtained the upper bound for the functional of functions in the classes of analytic bi-univalent function which are defined by subordinations in .

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4

Sun, Yue-Juan, Muhammad Arif, Lei Shi, and Muhammad Imran Faisal. "Some Further Coefficient Bounds on a New Subclass of Analytic Functions." Mathematics 11, no. 12 (2023): 2784. http://dx.doi.org/10.3390/math11122784.

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The coefficient problem is an essential topic in the theory of univalent functions theory. In the present paper, we consider a new subclass SQ of analytic functions with f′(z) subordinated to 1/(1−z)2 in the open unit disk. This class was introduced and studied by Răducanu. Our main aim is to give the sharp upper bounds of the second Hankel determinant H2,3f and the third Hankel determinant H3,1f for f∈SQ. This may help to understand more properties of functions in this class and inspire further investigations on higher Hankel determinants for this or other popular sub-classes of univalent fun

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5

Khan, Mohammad Faisal, Jongsuk Ro, and Muhammad Ghaffar Khan. "Sharp estimate for starlikeness related to a tangent domain." AIMS Mathematics 9, no. 8 (2024): 20721–41. http://dx.doi.org/10.3934/math.20241007.

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In the recent years, the study of the Hankel determinant problems have been widely investigated by many researchers. We were essentially motivated by the recent research going on with the Hankel determinant and other coefficient bounds problems. In this research article, we first considered the subclass of analytic starlike functions connected with the domain of the tangent function. We then derived the initial four sharp coefficient bounds, the sharp Fekete-Szegö inequality, and the sharp second and third order Hankel determinant for the defined class. Also, we derived sharp estimates like sh

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6

Hamzat, Jamiu Olusegun, and Abiola Ayobami Oni. "Hankel Determinant for Certain Subclasses of Analytic Function." Asian Research Journal of Mathematics 5, no. 2 (2017): 1–10. https://doi.org/10.9734/ARJOM/2017/33817.

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Let <em>S</em> denote the class of analytic functions normalized and univalent in the open unit disk. <em>U</em>={Z:|Z|<1}. The prime focus of the present paper is to obtain sharp upper bounds for the functional |<em>a</em><sub>α+2</sub>- <em>μa<sup>2</sup><sub>α</sub></em><sub>+1</sub>| and a<sub>α+1</sub>a<sub>α+3</sub> - <em>a<sup>2</sup></em><sub>α+2</sub>| for functions belonging to the class <em>T<sub>n</sub><sup>α</sup></em>( <em>a,c,β,λ,l</em>).

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7

Rathi, Sidik, Wan Muhammad Fakhrorrazzi Wan Ibrahim, and Muhammad Kamal Rafiq Mat Razali. "Second Hankel Determinants for a Subclass of Close-to-Convex Function Related to Certain Generalized Keobe Function." Mathematical Sciences and Informatics Journal 4, no. 1 (2023): 124–30. http://dx.doi.org/10.24191/mij.v4i2.23123.

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Abstract:

For ages, researchers have conducted numerous studies exploring every aspect of problems related to univalent functions. Most of the research has been concentrated on investigating the diverse properties of univalent functions. Notably, finding the upper bound of Hankel determinants has become an intriguing problem among researchers in this field. The aim of this paper is to solve the second Hankel determinant problem for the class of close-to-convex functions related with the certain generalized starlike functions. Initiating with the definition of the class, specific preliminary lemmas was p

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8

Alsarari, Fuad, Latha Satyanarayana, and Maslina Darus. "The Second Hankel Determinant for k-symmetrical Functions." Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2(102) (November 2023): 3–10. http://dx.doi.org/10.56415/basm.y2023.i2.p3.

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9

Sokól, Janusz, and Derek K. Thomas. "The second Hankel determinant for alpha-convex functions." Lithuanian Mathematical Journal 58, no. 2 (2018): 212–18. http://dx.doi.org/10.1007/s10986-018-9397-0.

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10

Răducanu, Dorina, and Paweł Zaprawa. "Second Hankel determinant for close-to-convex functions." Comptes Rendus Mathematique 355, no. 10 (2017): 1063–71. http://dx.doi.org/10.1016/j.crma.2017.09.006.

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11

Liu, Ming-Sheng, Jun-Feng Xu, and Ming Yang. "Upper Bound of Second Hankel Determinant for Certain Subclasses of Analytic Functions." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/603180.

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In this present investigation, we first give a survey of the work done so far in this area of Hankel determinant for univalent functions. Then the upper bounds of the second Hankel determinant|a2a4−a32|for functions belonging to the subclassesS(α,β),K(α,β),Ss∗(α,β), andKs(α,β)of analytic functions are studied. Some of the results, presented in this paper, would extend the corresponding results of earlier authors.

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12

Sunthrayuth, Pongsakorn, Ibtisam Aldawish, Muhammad Arif, Muhammad Abbas, and Sheza El-Deeb. "Estimation of the Second-Order Hankel Determinant of Logarithmic Coefficients for Two Subclasses of Starlike Functions." Symmetry 14, no. 10 (2022): 2039. http://dx.doi.org/10.3390/sym14102039.

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In our present study, two subclasses of starlike functions which are symmetric about the origin are considered. These two classes are defined with the use of the sigmoid function and the trigonometric function, respectively. We estimate the first four initial logarithmic coefficients, the Zalcman functional, the Fekete–Szegö functional, and the bounds of second-order Hankel determinants with logarithmic coefficients for the first class Sseg* and improve the obtained estimate of the existing second-order Hankel determinant of logarithmic coefficients for the second class Ssin*. All the bounds t

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13

Illafe, Mohamed, Maisarah Haji Mohd, Feras Yousef, and Shamani Supramaniam. "Bounds for the Second Hankel Determinant of a General Subclass of Bi-Univalent Functions." International Journal of Mathematical, Engineering and Management Sciences 9, no. 5 (2024): 1226–39. http://dx.doi.org/10.33889/ijmems.2024.9.5.065.

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The Hankel determinant, which plays a significant role in the theory of univalent functions, is investigated here in the context of bi-univalent analytic functions. Specifically, this paper is dedicated to deriving an upper-bound estimate for the second-order Hankel determinant for a general subclass of bi-univalent analytic functions that incorporate Gegenbauer polynomials within the unit disk. Through the careful specialization of parameters in our main result, we unveil several novel findings.

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14

Örnek, B. N. "Estimates for analytic functions concerned with Hankel determinant." Ukrains’kyi Matematychnyi Zhurnal 73, no. 9 (2021): 1205–16. http://dx.doi.org/10.37863/umzh.v73i9.907.

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UDC 517.5 We give an upper bound of Hankel determinant of the first order for the classes of an analytic function. In addition, an evaluation with the Hankel determinant from below will be given for the second angular derivative of analytic function. For new inequalities, the results of Jack's lemma and Hankel determinant were used. Moreover, in a class of analytic functions on the unit disc, assuming the existence of an angular limit on the boundary point, the estimations below of the modulus of angular derivative have been obtained.

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15

Abbas, Muhammad, Reem K. Alhefthi, Daniele Ritelli, and Muhammad Arif. "Sharp Second-Order Hankel Determinants Bounds for Alpha-Convex Functions Connected with Modified Sigmoid Functions." Axioms 13, no. 12 (2024): 844. https://doi.org/10.3390/axioms13120844.

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The study of the Hankel determinant generated by the Maclaurin series of holomorphic functions belonging to particular classes of normalized univalent functions is one of the most significant problems in geometric function theory. Our goal in this study is first to define a family of alpha-convex functions associated with modified sigmoid functions and then to investigate sharp bounds of initial coefficients, Fekete-Szegö inequality, and second-order Hankel determinants. Moreover, we also examine the logarithmic and inverse coefficients of functions within a defined family regarding recent iss

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16

MODUPE, FATUNSIN LOLADE, and OPOOLA TIMOTHY OLOYEDE. "Hankel Determinant of Second Kind for a New Subclass of Analytic Functions Involving Chebyshev Polynomials." Asian Journal of Mathematics and Computer Research 32, no. 3 (2025): 40–50. https://doi.org/10.56557/ajomcor/2025/v32i39268.

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In this paper, a new subclass S∗(μ, β, ℘,Un(t)) of univalent functions is defined by using Opoola differential operator involving subordination principle. The upper bounds to the second Hankel determinant denoted by H2(2) for the subclass S∗(μ, β, ℘,Un(t)) is established in connection with Chebyshev polynomials of the second kind and Toeplitz determinants.

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17

Shi, Lei, Muhammad Arif, Khalil Ullah, Naseer Alreshidi, and Meshal Shutaywi. "On Sharp Estimate of Third Hankel Determinant for a Subclass of Starlike Functions." Fractal and Fractional 6, no. 8 (2022): 437. http://dx.doi.org/10.3390/fractalfract6080437.

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In our present investigation, a subclass of starlike function Sn−1,L* connected with a domain bounded by an epicycloid with n−1 cusps was considered. The main work is to investigate some coefficient inequalities, and second and third Hankel determinants for functions belonging to this class. In particular, we calculate the sharp bounds of the third Hankel determinant for f∈S4L* with zf′(z)f(z) bounded by a four-leaf shaped domain under the unit disk D.

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18

Rasheed Olawale Ayinla and Ayotunde Olajide Lasode. "Some coefficient properties of a certain family of regular functions associated with lemniscate of Bernoulli and Opoola differential operator." Malaya Journal of Matematik 12, no. 02 (2024): 218–28. http://dx.doi.org/10.26637/mjm1202/007.

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Abstract. In this exploration, we introduce a certain family of regular (or analytic) functions in association with the righthalf of the Lemniscate of Bernoulli and the well-known Opoola differential operator. For the regular function \(f\) studied in this work, some estimates for the early coefficients, Fekete-Szegö functionals and second and third Hankel determinants are established. Another established result is the sharp upper estimate of the third Hankel determinant for the inverse function \(f^{-1}\) of \(f\).

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19

Khan, Muhammmad Ghaffar, Wali Khan Mashwani, Lei Shi, Serkan Araci, Bakhtiar Ahmad, and Bilal Khan. "Hankel inequalities for bounded turning functions in the domain of cosine Hyperbolic function." AIMS Mathematics 8, no. 9 (2023): 21993–2008. http://dx.doi.org/10.3934/math.20231121.

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<abstract><p>In the present article, we define and investigate a new subfamily of holomorphic functions connected with the cosine hyperbolic function with bounded turning. Further some interesting results like sharp coefficients bounds, sharp Fekete-Szegö estimate, sharp $ 2^{nd} $ Hankel determinant and non-sharp $ 3^{rd} $ order Hankel determinant. Moreover, the same estimates have been investigated for 2-fold, 3-fold symmetric functions, the first four initial sharp bounds of logarithmic coefficient and sharp second Hankel determinant of logarithmic coefficients fort his defined

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20

Tang, Huo, Adeel Ahmad, Akhter Rasheed, Asad Ali, Saqib Hussain, and Saima Noor. "Sharp Coefficient and Hankel Problems Related to a Symmetric Domain." Symmetry 15, no. 10 (2023): 1865. http://dx.doi.org/10.3390/sym15101865.

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In the current article, we utilize the concept of subordination to establish a new subclass of analytic functions associated with a bounded domain that is symmetric about the real axis. By applying the convolution technique, we derive the necessary and sufficient condition, the radius of convexity for this recently introduced class. Furthermore, we prove the sharp upper bounds for the second-order Hankel determinants |H2,1ξ|,|H2,2ξ| and third-order Hankel determinant |H3,1ξ| for the functions ξ belonging to the newly defined class.

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21

Alahmade, Ayman, Zeeshan Mujahid, Ferdous M. O. Tawfiq, Bilal Khan, Nazar Khan, and Fairouz Tchier. "Third Hankel Determinant for Subclasses of Analytic and m-Fold Symmetric Functions Involving Cardioid Domain and Sine Function." Symmetry 15, no. 11 (2023): 2039. http://dx.doi.org/10.3390/sym15112039.

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In this research, we define a few subclasses of analytic functions which are connected to sine functions and the cardioid domain in the unit disk. We investigate initial coefficient bounds, the Fekete–Szego problem and second and third Hankel determinants for the functions f belonging to these newly defined classes. We also define the class of m-fold symmetric functions related with the sine function and then investigate the bounds of the third Hankel determinant for twofold symmetric and threefold symmetric functions.

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22

Derbal, Antisar Ali, and Mona Saleh Showia. "Second Hankel Determinant for Certain Class of Functions Defined by Differential Operator." International Science and Technology Journal 34, no. 1 (2024): 1–14. http://dx.doi.org/10.62341/amsh2017.

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The objective of this paper is to obtain an upper bound of second Hankel determinant for a class of functions M_(α,β,λ,δ)^k (φ) defined in the open unit disc U by using the differential operator〖 D〗_(α,β,λ,δ)^k f(z). In addition, we give particular values to the parameters A,B and k to study special cases of the results of this article. The class M_(α,β,λ,δ)^k (φ) and the differential operator〖 D〗_(α,β,λ,δ)^k f(z). Key words: differential operator, Second Hankel determinant, Starlike functions, Subordination property

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23

Orhan, Halit, Evrim Toklu, and Ekrem Kadıoğlu. "Second Hankel determinant problem for k-bi-starlike functions." Filomat 31, no. 12 (2017): 3897–904. http://dx.doi.org/10.2298/fil1712897o.

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In this paper we introduce and study some properties of k-bi-starlike functions defined by making use of the S?l?gean derivative operator. Upper bounds on the second Hankel determinant for k-bi-starlike functions are investigated. Relevant connections of the results presented here with various well-known results are briefly indicated.

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24

ÇAĞLAR, Murat, Erhan DENİZ, and Hari Mohan SRIVASTAVA. "Second Hankel determinant for certain subclasses ofbi-univalent functions." TURKISH JOURNAL OF MATHEMATICS 41 (2017): 694–706. http://dx.doi.org/10.3906/mat-1602-25.

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25

G, Shanmugam. "Second Hankel Determinant for Certain Classes of Analytic Functions." Bonfring International Journal of Data Mining 2, no. 2 (2012): 57–60. http://dx.doi.org/10.9756/bijdm.1368.

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26

Alarifi, Najla, Rosihan Ali, and V. Ravichandran. "On the second hankel determinant for the kth-root transform of analytic functions." Filomat 31, no. 2 (2017): 227–45. http://dx.doi.org/10.2298/fil1702227a.

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Let f be a normalized analytic function in the open unit disk of the complex plane satisfying zf'(z)/f(z) is subordinate to a given analytic function ?. A sharp bound is obtained for the second Hankel determinant of the kth-root transform z[f(zk)/zk]1/k. Best bounds for the Hankel determinant are also derived for the kth-root transform of several other classes, which include the class of ?-convex functions and ?-logarithmically convex functions. These bounds are expressed in terms of the coefficients of the given function ?, and thus connect with earlier known results for particular choices of

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27

Khan, Muhammad Ghaffar, Nak Eun Cho, Timilehin Gideon Shaba, Bakhtiar Ahmad, and Wali Khan Mashwani. "Coefficient functionals for a class of bounded turning functions related to modified sigmoid function." AIMS Mathematics 7, no. 2 (2022): 3133–49. http://dx.doi.org/10.3934/math.2022173.

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<abstract><p>The main objective of the present article is to define the class of bounded turning functions associated with modified sigmoid function. Also we investigate and determine sharp results for the estimates of four initial coefficients, Fekete-Szegö functional, the second-order Hankel determinant, Zalcman conjucture and Krushkal inequality. Furthermore, we evaluate bounds of the third and fourth-order Hankel determinants for the class and for the 2-fold and 3-fold symmetric functions.</p></abstract>

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28

Khan, Muhammmad Ghaffar, Wali Khan Mashwani, Jong-Suk Ro, and Bakhtiar Ahmad. "Problems concerning sharp coefficient functionals of bounded turning functions." AIMS Mathematics 8, no. 11 (2023): 27396–413. http://dx.doi.org/10.3934/math.20231402.

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<abstract><p>The work presented in this article has been motivated by the recent research going on the Hankel determinant bounds and their related consequences, as well as the techniques used previously by many different authors. We aim to establish a new subfamily of holomorphic functions connected with the hyperbolic tangent function with bounded boundary rotation. We investigate the sharp estimate of the third Hankel determinant for this newly defined family of functions. Moreover, for the defined functions family, the Krushkal inequality, the first four initial sharp bounds of

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29

Sümer Eker, Sevtap, Bilal Şeker, Bilal Çekiç, and Mugur Acu. "Sharp Bounds for the Second Hankel Determinant of Logarithmic Coefficients for Strongly Starlike and Strongly Convex Functions." Axioms 11, no. 8 (2022): 369. http://dx.doi.org/10.3390/axioms11080369.

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The logarithmic coefficients are very essential in the problems of univalent functions theory. The importance of the logarithmic coefficients is due to the fact that the bounds on logarithmic coefficients of f can transfer to the Taylor coefficients of univalent functions themselves or to their powers, via the Lebedev–Milin inequalities; therefore, it is interesting to investigate the Hankel determinant whose entries are logarithmic coefficients. The main purpose of this paper is to obtain the sharp bounds for the second Hankel determinant of logarithmic coefficients of strongly starlike funct

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30

Deniz, Erhan, and Levent Budak. "Second hankel determinat for certain analytic functions satisfying subordinate condition." Mathematica Slovaca 68, no. 2 (2018): 463–71. http://dx.doi.org/10.1515/ms-2017-0116.

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Abstract In this paper, we introduce and investigate the following subclass $$\begin{array}{} \displaystyle 1+\frac{1}{\gamma }\left( \frac{zf'(z)+\lambda z^{2}f''(z)}{\lambda zf'(z)+(1-\lambda )f(z)}-1\right) \prec \varphi (z)\qquad\left( 0\leq \lambda \leq 1,\gamma \in \mathbb{C} \smallsetminus \{0\}\right) \end{array} $$ of analytic functions, φ is an analytic function with positive real part in the unit disk 𝔻, satisfying φ (0) = 1, φ '(0) > 0, and φ (𝔻) is symmetric with respect to the real axis. We obtain the upper bound of the second Hankel determinant | a2a4− $\begin{array}{} a^2_3

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31

Han, Pengju, and Yang Chen. "Asymptotic relations for semi-classical Laguerre orthogonal polynomials and the associated Hankel determinants." Journal of Mathematical Physics 63, no. 7 (2022): 071506. http://dx.doi.org/10.1063/5.0072813.

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We study recurrence coefficients of semi-classical Laguerre orthogonal polynomials and the associated Hankel determinant generated by a semi-classical Laguerre weight [Formula: see text]. If t = 0, it is reduced to the classical Laguerre weight. For t > 0, this weight tends to zero faster than the classical Laguerre weight as x → ∞. In the finite n-dimensional case, we obtain two auxiliary quantities R n( t) and r n( t) by using the Ladder operator approach. We show that the Hankel determinant has an integral representation in terms of R n( t), where the quantity R n( t) is closely related

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32

Singh, Gagandeep, and Gurcharanjit Singh. "COEFFICIENT PROBLEMS FOR THE SUBCLASSES OF SAKAGUCHI TYPE FUNCTIONS ASSOCIATED WITH SINE FUNCTION." Jnanabha 51, no. 02 (2021): 237–43. http://dx.doi.org/10.58250/jnanabha.2021.51230.

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Abstract The estimation of the upper bound for certain coefficient relations of various subclasses of analytic functions is an active topic of research in Geometric function theory. In this paper, certain subclasses of Sakaguchi type functions are defined by subordinating to sine function in the open unit disc E = {z : |z| < 1} and some coefficient inequalities such as Fekete-Szegö inequality, Second Hankel determinant, Zalcman functional and third Hankel determinant are investigated.

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33

Krishna, D. Vamshee, and T. Ramreddy. "An upper bound to the nonlinear functional for certain subclasses of analytic functions associated with Hankel determinant." Asian-European Journal of Mathematics 07, no. 02 (2014): 1350042. http://dx.doi.org/10.1142/s1793557113500423.

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The objective of this paper is to obtain an upper bound to the second Hankel determinant [Formula: see text] for the functions belonging to strongly starlike and convex functions of order α(0 < α ≤ 1). Further, we introduce a subclass of analytic functions and obtain the same coefficient inequality for the functions in this class, using Toeplitz determinants.

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34

Palani, Jeyaraman Muthusamy, Parvatham Raman, and Aaisha Farzana Habibullah. "On coefficient functional and Bohr-radius for some classes of analytic functions." Ukrains’kyi Matematychnyi Zhurnal 77, no. 5 (2025): 368. https://doi.org/10.3842/umzh.v77i5.8397.

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UDC 517.5 We examine sharp bounds of coefficient functionals, such as Zalcman functional, second-order Hankel determinant, and third-order Toeplitz and Hermitian–Toeplitz determinants for the class of analytic functions. Growth estimates and bounds for the difference of successive coefficients are determined. Further, by using growth estimates, we obtained obtain the Bohr radius and the Bohr–Rogosinski phenomenon.

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35

Sim, Young Jae, and Paweł Zaprawa. "Third Hankel determinants for two classes of analytic functions with real coefficients." Forum Mathematicum 33, no. 4 (2021): 973–86. http://dx.doi.org/10.1515/forum-2021-0014.

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Abstract In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of H 2 , 2 {H_{2,2}} or H 3 , 1 {H_{3,1}} over different subclasses of 𝒮 {\mathcal{S}} . Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic functions with real coefficients. The first one is the class 𝒯 {\mathcal{T}} of typically real functions. The second object of our interest is 𝒦 ℝ ⁢ ( i ) {\mathcal{K}_{\mathbb{R}}(

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36

Mohamad, Daud, and Nur Hazwani Aqilah Abdul Wahid. "Hankel Determinant of Logarithmic Coefficients for Tilted Starlike Functions With Respect to Conjugate Points." International Journal of Analysis and Applications 21 (February 21, 2023): 10. http://dx.doi.org/10.28924/2291-8639-21-2023-10.

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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.

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37

Kowalczyk, Bogumiła, and Adam Lecko. "Second Hankel Determinant of Logarithmic Coefficients of Convex and Starlike Functions of Order Alpha." Bulletin of the Malaysian Mathematical Sciences Society 45, no. 2 (2021): 727–40. http://dx.doi.org/10.1007/s40840-021-01217-5.

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38

Al-Shbeil, Isra, Muhammad Imran Faisal, Muhammad Arif, Muhammad Abbas, and Reem K. Alhefthi. "Investigation of the Hankel Determinant Sharp Bounds for a Specific Analytic Function Linked to a Cardioid-Shaped Domain." Mathematics 11, no. 17 (2023): 3664. http://dx.doi.org/10.3390/math11173664.

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One of the challenging tasks in the study of function theory is how to obtain sharp estimates of coefficients that appear in the Taylor–Maclaurin series of analytic univalent functions, and for obtaining these bounds, researchers used the concepts of Carathéodory functions. Among these coefficient-related problems, the problem of the third-order Hankel determinant sharp bound is the most difficult one. The aim of the present study is to determine the sharp bound of the Hankel determinant of third order by using the methodology of the aforementioned Carathéodory function family. Further, we als

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Salehian, Safa, and Ahmad Motamednezhad. "Second Hankel determinant for certain subclass of bi-univalent functions." Filomat 35, no. 7 (2021): 2129–40. http://dx.doi.org/10.2298/fil2107129s.

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The main purpose of this paper is to obtain an upper bound for the second Hankel determinant for functions belonging to a subclass of bi-univalent functions in the open unit disk in the complex plane. Furthermore, the presented results in this work improve or generalize the recent works of other authors.

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P. H., Liew, Fuah K. H, and Janteng A. "Bounds for the second Hankel determinant of certain univalent functions." Global Journal of Pure and Applied Mathematics 12, no. 02 (2016): 1375. http://dx.doi.org/10.37622/gjpam/12.2.2016.1375-1385.

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AL-Ameedee, Sarah A., Waggas Galib Atshan, and Faez Ali AL-Maamori. "Second Hankel Determinant for Certain Subclasses of Bi-univalent functions." Journal of Physics: Conference Series 1664 (November 2020): 012044. http://dx.doi.org/10.1088/1742-6596/1664/1/012044.

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M. Patil, Sunita, and S. M. Khairnar. "Second Hankel Determinant for Analytic Functions Defined By Linear Operator." International Journal of Mathematics Trends and Technology 41, no. 3 (2017): 272–74. http://dx.doi.org/10.14445/22315373/ijmtt-v41p525.

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Marjono and D. K. Thomas. "The second Hankel determinant of functions convex in one direction." International Journal of Mathematical Analysis 10 (2016): 423–28. http://dx.doi.org/10.12988/ijma.2016.619.

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Altınkaya, Şahsene, and Sibel Yalçın. "Upper Bound of Second Hankel Determinant for Bi-Bazilevic̆ Functions." Mediterranean Journal of Mathematics 13, no. 6 (2016): 4081–90. http://dx.doi.org/10.1007/s00009-016-0733-5.

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Răducanu, Dorina. "Second Hankel determinant for a class of analytic functions defined by q-derivative operator." Analele Universitatii "Ovidius" Constanta - Seria Matematica 27, no. 2 (2019): 167–77. http://dx.doi.org/10.2478/auom-2019-0026.

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Khan, Muhammad Ghaffar, Sheza M. El-Deeb, Daniel Breaz, Wali Khan Mashwani, and Bakhtiar Ahmad. "Sufficiency criteria for a class of convex functions connected with tangent function." AIMS Mathematics 9, no. 7 (2024): 18608–24. http://dx.doi.org/10.3934/math.2024906.

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<abstract><p>The research here was motivated by a number of recent studies on Hankel inequalities and sharp bounds. In this article, we define a new subclass of holomorphic convex functions that are related to tangent functions. We then derive geometric properties like the necessary and sufficient conditions, radius of convexity, growth, and distortion estimates for our defined function class. Furthermore, the sharp coefficient bounds, sharp Fekete-Szegö inequality, sharp 2nd order Hankel determinant, and Krushkal inequalities are given. Moreover, we calculate the sharp coefficient

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Kund, S. N., and A. K. Mishra. "The second Hankel determinant for a class of analytic functions associated with the Carlson-Shaffer operator." Tamkang Journal of Mathematics 44, no. 1 (2013): 73–82. http://dx.doi.org/10.5556/j.tkjm.44.2013.963.

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In this paper a new class of analytic functions, associated with the Carlson-Shaffer operator, is investigated. The sharp estimate for the Second Hankel determinant and class preserving transforms are studied.

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Laxmi, K. Rajya, and R. Bharavi Sharma. "Second Hankel Determinants for Some Subclasses of Biunivalent Functions Associated with Pseudo-Starlike Functions." Journal of Complex Analysis 2017 (December 4, 2017): 1–9. http://dx.doi.org/10.1155/2017/6476391.

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We introduce second Hankel determinant of biunivalent analytic functions associated with λ-pseudo-starlike function in the open unit disc Δ subordinate to a starlike univalent function whose range is symmetric with respect to the real axis.

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Dhanalakshmi, K., D. Kavitha, and K. Anitha. "Certain Bounds for a Subclasses of Analytic Functions of Reciprocal Order." International Journal of Analysis and Applications 21 (December 21, 2023): 138. http://dx.doi.org/10.28924/2291-8639-21-2023-138.

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Rath, B., K. S. Kumar, and D. V. Krishna. "An exact estimate of the third Hankel determinants for functions inverse to convex functions." Matematychni Studii 60, no. 1 (2023): 34–39. http://dx.doi.org/10.30970/ms.60.1.34-39.

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Invesigation of bounds for Hankel determinat of analytic univalent functions is prominent intrest of many researcher from early twenth century to study geometric properties. Many authors obtained non sharp upper bound of third Hankel determinat for different subclasses of analytic univalent functions until Kwon et al. obtained exact estimation of the fourth coefficeient of Caratheodory class. Recently authors made use of an exact estimation of the fourth coefficient, well known second and third coefficient of Caratheodory class obtained sharp bound for the third Hankel determinant associated w

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