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 http://dx.doi.org/10.22436/jmcs.025.01.04

Publication Date: May 04, 2021 Submission Date: January 21, 2021 Revision Date: April 03, 2021 Accteptance Date: April 07, 2021

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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.

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Keywords

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

MSC

•  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
  • View Article


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


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


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


• [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
  • View Article


• [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
  • View Article


• [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
  • View Article


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


• [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
  • View Article


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


• [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
  • View Article


• [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
  • View Article


• [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
  • View Article


• [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
  • View Article


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


• [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
  • View Article


• [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
  • View Article


• [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
  • View Article


• [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
  • View Article


• [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
  • View Article


• [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
  • View Article


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


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


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


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


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


• [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
  • View Article


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


• [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
  • View Article


• [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
  • View Article


• [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
  • View Article


• [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
  • View Article


• [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
  • View Article


• [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
  • View Article


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