# Third Hankel determinant and Zalcman functional for a class of starlike functions with respect to symmetric points related with sine function

Volume 25, Issue 1, pp 29--36
Publication Date: May 04, 2021 Submission Date: January 21, 2021 Revision Date: April 03, 2021 Accteptance Date: April 07, 2021
• 875 Views

### Authors

Muhammad Ghaffar Khan - Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat, Pakistan. Bakhtiar Ahmad - AhmadGovt. Degree College Mardan, 23200 Mardan, Pakistan. Gangadharan Murugusundaramoorthy - Department of Mathematics, School of Advanced Sciences, Vellore Institute Technology University Vellore - 632014, India. Wali Khan Mashwani - Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat, Pakistan. Sibel Yalçin - Department of Mathematics, Faculty of Arts and Sciences, Bursa Uludag University, 16059, Bursa, Turkey. Timilehin Gideon Shaba - Department of Mathematics, Physical Sciences, University of Ilorin, Nigeria. Zabidin Salleh - Department of Mathematics, Faculty of Ocean Engineering Technology and Informatics, University Malaysia Terengganu, 21030 Kuala Nerus, Terenggunu, Malaysia.

### Abstract

In this article we define a class of starlike functions with respect to symmetric points in the domain of sine function. Also, we investigate coefficients bounds and upper bounds for the third order Hankel determinant for this defined class. We also evaluate the Zalcman functional $|a_{3}^{2}-a_{5}|$. Specializing the parameters, we improve Zalcman functional for the class of starlike functions.

### Share and Cite

##### ISRP Style

Muhammad Ghaffar Khan, Bakhtiar Ahmad, Gangadharan Murugusundaramoorthy, Wali Khan Mashwani, Sibel Yalçin, Timilehin Gideon Shaba, Zabidin Salleh, Third Hankel determinant and Zalcman functional for a class of starlike functions with respect to symmetric points related with sine function, Journal of Mathematics and Computer Science, 25 (2022), no. 1, 29--36

##### AMA Style

Khan Muhammad Ghaffar, Ahmad Bakhtiar, Murugusundaramoorthy Gangadharan, Mashwani Wali Khan, Yalçin Sibel, Shaba Timilehin Gideon, Salleh Zabidin, Third Hankel determinant and Zalcman functional for a class of starlike functions with respect to symmetric points related with sine function. J Math Comput SCI-JM. (2022); 25(1):29--36

##### Chicago/Turabian Style

Khan, Muhammad Ghaffar, Ahmad, Bakhtiar, Murugusundaramoorthy, Gangadharan, Mashwani, Wali Khan, Yalçin, Sibel, Shaba, Timilehin Gideon, Salleh, Zabidin. "Third Hankel determinant and Zalcman functional for a class of starlike functions with respect to symmetric points related with sine function." Journal of Mathematics and Computer Science, 25, no. 1 (2022): 29--36

### Keywords

• Analytic functions
• subordinations
• sine function
• Hankel determinant
• Zalcman functional

•  30C45
•  30C50

### References

• [1] M. Arif, M. Raza, H. Tang, S. Hussain, H. Khan, Hankel determinant of order three for familiar subsets of analytic functions related with sine function, Open Math., 17 (2019), 1615--1630

• [2] K. O. Babalola, On H3 (1) Hankel determinant for some classes of univalent functions, Inequal. Theory Appl., 6 (2007), 1--7

• [3] D. Bansal, J. Sokol, Zalcman conjecture for some subclass of analytic functions, J. Fract. Calc. Appl., 8 (2017), 1--5

• [4] J. E. Brown, A. Tsao, On the Zalcman conjecture for starlike and typically real functions, Math. Z., 191 (1986), 467--474

• [5] N. E. Cho, B. Kowalczyk, O. S. Kwon , A. Lecko, J. Sim, Some coefficient inequalities related to the Hankel determinant for strongly starlike functions of order alpha, J. Math. Inequal., 11 (2017), 429--439

• [6] N. E. Cho, V. Kumar, S. S. Kumar, V. Ravichandran, Radius problems for starlike functions associated with the Sine function, Bull. Iranian Math. Soc., 45 (2019), 213--232

• [7] H. O. Guney, G. Murugusundaramoorthy, H. M. Srivastava, The second hankel determinant for a certain class of bi-close-to-convex function, Results Math.,, 74 (2019), 13 pages

• [8] W. K. Hayman, On the second Hankel determinant of mean univalent functions, Proc. London Math. Soc, 18 (1968), 77--94

• [9] A. Jangteng, S. A. Halim, M. Darus, Coefficient inequality for a function whose derivative has a positive real part, J. Inequal. Pure Appl. Math., 7 (2006), 1--5

• [10] F. R. Keogh, E. P. Merkes, Coefficient inequality for certain subclasses of analytic functions, Proc. Amer. Math. Soc., 20 (1969), 8--12

• [11] M. G. Khan, B. Ahmad, G. Muraugusundaramoorthy, R. Chinram, W. K. Mashwani, Applications of Modified Sigmoid Functions to a Class of Starlike Functions, J. Funct. Spaces, 2020 (2020), 8 pages

• [12] M. G. Khan, B. Ahmad, J. Sokol, Z. Muhammad, W. K. Mashwani, R. Chinram, P. Petchkaew, Coefficient problems in a class of functions with bounded turning associated with Sine function, Eur. J. Pure Appl. Math., 14 (2021), 53--64

• [13] A. Lecko, Y. J. Sim, B. Smiarowska, The sharp bound of the Hankel determinant of the third kind for starlike functions of order 1/2, Complex Anal. Oper. Theory, 13 (2019), 2231--2238

• [14] R. J. Libera, E. J. Złotkiewicz, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc., 87 (1983), 251--257

• [15] W. C. Ma, The Zalcman conjecture for close-to-convex functions, Proc. Amer. Math. Soc., 104 (1988), 741--744

• [16] S. Mahmood, M. Jabeen, S. N. Malik, H. M. Srivastava, R. Manzoor, S. M. J. Riaz, Some coefficient inequalities of q-starlike functions associated with conic domain defined by q-derivative, J. Funct. Spaces, 2018 (2018), 1--13

• [17] S. Mahmood, I. Khan, H. M. Srivastava, S. N. Malik, Inclusion relations for certain families of integral operators associated with conic regions, J. Inequal. Appl., 59 (2019), 11 pages

• [18] S. Mahmood, H. M. Srivastava, N. Khan, Q. Z. Ahmad, B. Khan, I. Ali, Upper bound of the third Hankel determinant for a subclass of q-starlike functions, Symmetry, 11 (2019), 1--13

• [19] S. Mahmood, H. M. Srivastava, S. N. Malik, Some subclasses of uniformly univalent functions with respect to symmetric points, Symmetry, 11 (2019), 14 pages

• [20] J. W. Noonan, D. K. Thomas, On the second Hankel determinant of areally mean p-valent functions, Trans. Amer. Math. Soc., 223 (1976), 337--346

• [21] H. Orhan, N. Magesh, J. Yamini, Bounds for the second Hankel determinant of certain bi-univalent functions, Turkish J. Math., 40 (2016), 679--687

• [22] C. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J. London Math. Soc., 41 (1966), 111--122

• [23] C. Pommerenke, On the Hankel determinants of univalent functions, Mathematika, 14 (1967), 108--112

• [24] C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Gottingen (1975)

• [25] R. K. Raina, J. Sokoł, On coefficient estimates for a certain class of starlike functions, Hacet. J. Math. Stat., 44 (2015), 1427--1433

• [26] V. Ravichandran, S. Verma, Bound for the fifth coefficient of certain starlike functions, C. R. Math. Acad. Sci. Paris, 353 (2015), 505--510

• [27] M. Raza, S. N. Malik, Upper bound of third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli, J. Inequal. Appl., 2013 (2013), 8 pages

• [28] K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan, 11 (1959), 72--75

• [29] M. Shafiq, H. M. Srivastava, N. Khan, Q. Z. Ahmad, M. Darus, S. Kiran, An upper bound of the third Hankel determinant for a subclass of q-starlike functions associated with k-Fibonacci numbers, Symmetry, 12 (2020), 1--17

• [30] L. Shi, M. G. Khan, B. Ahmad, Some geometric properties of a family of analytic functions involving a generalized q-operator, Symmetry,, 12 (2020), 11 pages

• [31] H. M. Srivastava, Q. Z. Ahmad, N. Khan, N. Khan, B. Khan, Hankel and Toeplitz determinants for a subclass of q-starlike functions associated with a general conic domain, Mathematics, 7 (2019), 15 pages

• [32] H. M. Srivastava, Q. Z. Ahmad, M. Darus, N. Khan, B. Khan, N. Zaman, H. H. Shah, Upper bound of the third Hankel determinant for a subclass of close-to-convex functions associated with the lemniscate of Bernoulli, Mathematics, 7 (2019), 10 pages

• [33] H. M. Srivastava, B. Khan, N. Khan, Q. Z. Ahmad, Coefficient inequalities for q-starlike functions associated with the Janowski functions, Hokkaido Math. J., 48 (2019), 407--425

• [34] H. M. Srivastava, B. Khan, N. Khan, M. Tahir, S. Ahmad, N. Khan, Upper bound of the third Hankel determinant for a subclass of q-starlike functions associated with the q-exponential function, Bull. Sci. Math., 167 (2021), 16 pages

• [35] P. Zaprawa, Third Hankel determinants for subclasses of univalent functions, Mediterr. J. Math., 14 (2017), 10 pages