The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
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ESTIMATE OF THIRD ORDER HANKEL DETERMINANT FOR A CERTAIN SUBCLASS OF ANALYTIC FUNCTIONS ASSOCIATED WITH CARDIOID DOMAIN

  • Received : 2022.08.06
  • Accepted : 2022.11.02
  • Published : 2022.11.30
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Abstract

The present paper deals with the upper bound of third order Hankel determinant for a certain subclass of analytic functions associated with Cardioid domain in the open unit disc E = {z ∈ ℂ : |z| < 1}. The results proved here generalize the results of several earlier works.

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References

  1. S. Altinkaya & S. Yalcin: Third Hankel determinant for Bazilevic functions. Adv. Math., Scientific Journal 5 (2016), no. 2, 91-96.
  2. M. Arif, M. Raza, H. Tang, S. Hussain & H. Khan: Hankel determinant of order three for familiar subsets of ananlytic functions related with sine function. Open Math. 17(2019), 1615-1630. https://doi.org/10.1515/math-2019-0132
  3. K.O. Babalola: On H3(1) Hankel determinant for some classes of univalent functions. Ineq. Th. Appl. 6 (2010), 1-7.
  4. L. Bieberbach: Uber die koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln. Sitzungsberichte Preussische Akademie der Wissenschaften 138 (1916), 940-955.
  5. R. Bucur, D. Breaz & L. Georgescu: Third Hankel determinant for a class of analytic functions with respect to symmetric points. Acta Univ. Apulensis 42 (2015), 79-86.
  6. L. De-Branges: A proof of the Bieberbach conjecture. Acta Math. 154 (1985), 137-152. https://doi.org/10.1007/BF02392821
  7. R. Ehrenborg: The Hankel determinant of exponential polynomials. Amer. Math. Monthly 107 (2000), 557-560. https://doi.org/10.1080/00029890.2000.12005236
  8. M. Fekete and G. Szego: Eine Bemer Kung uber ungerade schlichte Functionen, J. Lond. Math. Soc., 8(1933), 85-89.
  9. A. Janteng, S.A. Halim & M. Darus: Hankel determinant for starlike and convex functions. Int. J. Math. Anal. 1 (2007), no. 13, 619-625.
  10. S.R. Keogh & E.P. Merkes: A coefficienit inequality for certain subclasses of analytic functions. Proc. Amer. Math. Soc. 20 (1969), 8-12. https://doi.org/10.1090/S0002-9939-1969-0232926-9
  11. J.W. Layman: The Hankel transform and some of its properties. J. Int. Seq. 4 (2001), 1-11.
  12. R.J. Libera & E.J. Zlotkiewiez: Early coefficients of the inverse of a regular convex function. Proc. Amer. Math. Soc., 85 (1982), 225-230. https://doi.org/10.1090/S0002-9939-1982-0652447-5
  13. R. J. Libera & E.J. Zlotkiewiez: Coefficient bounds for the inverse of a function with derivative in P. Proc. Amer. Math. Soc. 87 (1983), 251-257.
  14. W. Ma: Generalized Zalcman conjecture for starlike and typically real functions. J. Math. Anal. Appl. 234 (1999), 328-329. https://doi.org/10.1006/jmaa.1999.6378
  15. T.H. MacGregor: Functions whose derivative has a positive real part. Trans. Amer. Math. Soc. 104 (1962), 532-537. https://doi.org/10.1090/S0002-9947-1962-0140674-7
  16. T.H. MacGregor: The radius of univalence of certain analytic functions. Proc. Amer. Math. Soc. 14 (1963), 514-520. https://doi.org/10.1090/S0002-9939-1963-0148891-3
  17. S.N. Malik, M. Raza, J. Sokol & S. Zainab: Analytic functions associated with cardioid domain. Turkish J. Math. 44 (2020), 1127-1136. https://doi.org/10.3906/mat-2003-96
  18. S.N. Malik, M. Raza, Q. Xin, J. Sokol, R. Manzoor & S. Zainab: On convex functions associated with symmetric cardioid domain. Symmetry 13 (2021), 1-12.
  19. B.S. Mehrok & Gagandeep Singh: Estimate of second Hankel determinant for certain classes of analytic functions. Scientia Magna 8 (2012), no. 3, 85-94.
  20. G. Murugusundramurthi & N. Magesh: Coefficient inequalities for certain classes of analytic functions associated with Hankel determinant. Bull. Math. Anal. Appl. 1 (2009), no. 3, 85-89.
  21. J.W. Noonan & D.K. Thomas: On the second Hankel determinant of a really mean p-valent functions. Trans. Amer. Math. Soc. 223 (1976), no. 2, 337-346.
  22. K.I. Noor: Hankel determinant problem for the class of functions with bounded boundary rotation. Rev. Roum. Math. Pures Et Appl. 28 (1983), no. 8, 731-739.
  23. Ch. Pommerenke: Univalent functions. Math. Lehrbucher, vandenhoeck and Ruprecht, Gottingen, 1975.
  24. M. Raza, S. Mushtaq, S.N. Malik & J. Sokol: Coefficient inequalities for analytic functions associated with cardioid domain. Hacettepe J. Math. Stat. 49 (2020), no. 6, 2017-2027. https://doi.org/10.15672/hujms.595068
  25. M.O. Reade: On close-to-convex univalent functions. Michigan Math. J. 3 (1955-56), 59-62. https://doi.org/10.1307/mmj/1031710535
  26. G. Shanmugam, B. Adolf Stephen & K.O. Babalola: Third Hankel determinant for 𝛼 starlike functions. Gulf J. Math. 2 (2014), no. 2, 107-113.
  27. K. Sharma, N.K. Jain & V. Ravichandran: Starlike functions associated with a cardioid. Afrika Matematika 27 (2016), 923-939. https://doi.org/10.1007/s13370-015-0387-7
  28. L. Shi, I. Ali, M. Arif, N.E. Cho, S. Hussain & H. Raza: A study of Third hankel determinant problem for certain subfamilies of analytic functions involving cardioid domain. Mathematics 7 (2019), 1-15.
  29. G. Singh: Hankel determinant for a new subclass of analytic functions. Scientia Magna 8 (2012), no. 4, 61-65.
  30. G. Singh & G. Singh: On third Hankel determinant for a subclass of analytic functions. Open Sci. J. Math. Appl. 3 (2015), no. 6, 172-175.