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A Study of Sharp Coefficient Bounds for a New Subfamily of Starlike Functions

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Nonlinear Analysis, Geometry and Applications

Part of the book series: Trends in Mathematics ((TM))

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Abstract

The main contribution of this article includes derivations of sharp inequalities involving the Taylor-Maclaurin coefficients for functions belonging to the class \(\mathcal {S}_{\tanh }^{\ast }\) of starlike functions in \(\mathbb {D}\). In particular, the bounds of the first three Taylor-Maclaurin coefficients, the estimates of the Fekete-Szegö type functional, and the estimates of the second- and third-order Hankel determinants are the main problems that are proposed to be studied here.

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References

  1. A. Alotaibi, M. Arif, M.A. Alghamdi, Starlikness associated with cosine hyperbolic function. Mathematics 8, Article ID 1118, 1–16 (2020)

    Google Scholar 

  2. Ş. Altınkaya, S. Yalçın, Third Hankel determinant for Bazilevič functions. Adv. Math. Sci. J. 5, 91–96 (2016)

    MATH  Google Scholar 

  3. Ş. Altınkaya, S. Yalçın, Upper bound of second Hankel determinant for bi-Bazilevic functions. Mediterr. J. Math. 13, 4081–4090 (2016)

    Article  MathSciNet  Google Scholar 

  4. 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, 1615–1630 (2019)

    Article  MathSciNet  Google Scholar 

  5. M. Arif, S. Umar, M. Raza, T. Bulboaca, M.U. Farooq, H. Khan, On fourth Hankel determinant for functions associated with Bernoulli’s lemniscate. Hacet. J. Math. Stat. 49, 1777–1780 (2020)

    MathSciNet  MATH  Google Scholar 

  6. K.O. Babalola, On H 3(1) Hankel determinant for some classes of univalent functions. Inequal. Theory Appl. 6, 1–7 (2010)

    Google Scholar 

  7. K. Bano, M. Raza, Starlike functions associated with cosine function. Bull. Iran. Math. Soc. 20 pp. (2020). https://doi.org/10.1007/s41980-020-00456-9

  8. D. Bansal, Upper bound of second Hankel determinant for a new class of analytic functions. Appl. Math. Lett. 26, 103–107 (2013)

    Article  MathSciNet  Google Scholar 

  9. D. Bansal, S. Maharana, J.K. Prajapat, Third order Hankel determinant for certain univalent functions. J. Korean Math. Soc. 52, 1139–1148 (2015)

    Article  MathSciNet  Google Scholar 

  10. O. Barukab, M. Arif, M. Abbas, S.K. Khan, Sharp bounds of the coefficient results for the family of bounded turning functions associated with petal shaped domain. J. Funct. Spaces 2021, Article ID 5535629, 1–9 (2021)

    Google Scholar 

  11. L. Bieberbach, Über dié koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln. Sitzungsber. Preussische Akad. Wiss. 138, 940–955 (1916)

    MATH  Google Scholar 

  12. M. Çaglar, E. Deniz, H.M. Srivastava, Second Hankel determinant for certain subclasses of bi-univalent functions. Turk. J. Math. 41, 694–706 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. N.E. Cho, B. Kowalczyk, O.S. Kwon, A. Lecko, Y.J. Sim, The bounds of some determinants for starlike functions of order alpha. Bull. Malaysian Math. Sci. Soc. 41, 523–535 (2018)

    Article  MathSciNet  Google Scholar 

  15. 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, 213–232 (2019)

    Article  MathSciNet  Google Scholar 

  16. L. De Branges, A proof of the Bieberbach conjecture. Acta Math. 154, 137–152 (1985)

    Article  MathSciNet  Google Scholar 

  17. A. Ebadian, T. Bulboacă, N.E. Cho, E.A. Adegani, Coefficient bounds and differential subordinations for analytic functions associated with starlike functions. Rev. Real Acad. Cienc. Exact. Fís. Natur. Ser. A. Mat. 114, Article ID 128, 1–19 (2020)

    Google Scholar 

  18. H.Ö. Güney, G. Murugusundaramoorthy, H.M. Srivastava, The second Hankel determinant for a certain class of bi-close-to-convex functions. Results Math. 74, Article ID 93, 1–13 (2019)

    Google Scholar 

  19. W.K. Hayman, On second Hankel determinant of mean univalent functions. Proc. Lond. Math. Soc. Ser. 3 18, 77–94 (1968)

    Article  MathSciNet  Google Scholar 

  20. W. Janowski, Extremal problems for a family of functions with positive real part and for some related families. Ann. Polon. Math. 23, 159–177 (1970)

    Article  MathSciNet  Google Scholar 

  21. A. Janteng, S.A. Halim, M. Darus, Coefficient inequality for a function whose derivative has a positive real part. J. Inequal. Pure Appl. Math. 7(2), Article ID 50, 1–5 (2006)

    Google Scholar 

  22. A. Janteng, S.A. Halim, M. Darus, Hankel determinant for starlike and convex functions. Int. J. Math. Anal. 1, 619–625 (2007)

    MathSciNet  MATH  Google Scholar 

  23. S. Kanas, E.A. Adegani, A. Zireh, An unified approach to second Hankel determinant of bi-subordinate functions. Mediterr. J. Math. 14, Article ID 233, 1–12 (2017)

    Google Scholar 

  24. F. Keough, E. Merkes, A coefficient inequality for certain subclasses of analytic functions. Proc. Am. Math. Soc. 20, 8–12 (1969)

    Article  Google Scholar 

  25. B. Kowalczyk, A. Lecko, Y.J. Sim, The sharp bound of the Hankel determinant of the third kind for convex functions. Bull. Aust. Math. Soc. 97, 435–445 (2018)

    Article  MathSciNet  Google Scholar 

  26. D.V. Krishna, B. Venkateswarlu, T. RamReddy, Third Hankel determinant for bounded turning functions of order alpha. J. Nigerian Math. Soc. 34, 121–127 (2015)

    Article  MathSciNet  Google Scholar 

  27. S.S. Kumar, K. Arora, Starlike functions associated with a petal shaped domain. Preprint, arXiv:2010.10072

    Google Scholar 

  28. O.S. Kwon, A. Lecko, Y.J. Sim, On the fourth coefficient of functions in the Carathéodory class. Comput. Methods Funct. Theory 18, 307–314 (2018)

    Article  MathSciNet  Google Scholar 

  29. O.S. Kwon, A. Lecko, Y.J. Sim, The bound of the Hankel determinant of the third kind for starlike functions. Bull. Malaysian Math. Sci. Soc. 42, 767–780 (2019)

    Article  MathSciNet  Google Scholar 

  30. A. Lecko, Y.J. Sim, B. Śmiarowska, The sharp bound of the Hankel determinant of the third kind for starlike functions of order \(\frac {1}{2}\). Complex Anal. Oper. Theory 13, 2231–2238 (2019)

    Google Scholar 

  31. S.K. Lee, V. Ravichandran, S. Supramaniam, Bounds for the second Hankel determinant of certain univalent functions. J. Inequal. Appl. 2013, Article ID 281, 1–17 (2013)

    Google Scholar 

  32. R.J. Libera, E.J. Złotkiewicz, Early coefficients of the inverse of a regular convex function. Proc. Am. Math. Soc. 85, 225–230 (1982)

    Article  MathSciNet  Google Scholar 

  33. M.-S. Liu, J.-F. Xu, M. Yang, Upper bound of second Hankel determinant for certain subclasses of analytic functions. Abstr. Appl. Anal. 2014, Article ID 603180, 1–10 (2014)

    Google Scholar 

  34. W.C. Ma, D. Minda, A unified treatment of some special classes of univalent functions, in Proceedings of the Conference on Complex Analysis, ed. by Z. Li, F. Ren, L. Yang, S. Zhang (Tianjin, People’s Republic of China, June 19–22, 1992). Conference Proceedings and Lecture Notes in Analysis, vol. I (International Press, Cambridge, 1994), pp. 157–169

    Google Scholar 

  35. 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, Article ID 347, 1–11 (2019)

    Google Scholar 

  36. R. Mendiratta, S. Nagpal, V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function. Bull. Malaysian Math. Sci. Soc. 38, 365–386 (2015)

    Article  MathSciNet  Google Scholar 

  37. M. Obradović, N. Tuneski, Hankel determinants of second and third order for the class S of univalent functions. Math Slovaca 71, 649–654 (2021)

    Article  MathSciNet  Google Scholar 

  38. Ch. Pommerenke, On the coefficients and Hankel determinants of univalent functions. J. Lond. Math. Soc. (Ser. 1) 41, 111–122 (1966)

    Google Scholar 

  39. Ch. Pommerenke, On the Hankel determinants of univalent functions. Mathematika 14, 108–112 (1967)

    Article  MathSciNet  Google Scholar 

  40. Ch. Pommerenke, Univalent Function (Vanderhoeck & Ruprecht, Göttingen, 1975)

    Google Scholar 

  41. 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, Article ID 412, 1–8 (2013)

    Google Scholar 

  42. 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, Article ID 1043, 1–17 (2020)

    Google Scholar 

  43. K. Sharma, N.K. Jain, V. Ravichandran, Starlike functions associated with a cardioid. Afr. Mat. 27, 923–939 (2016)

    Article  MathSciNet  Google Scholar 

  44. G. Shanmugam, B.A. Stephen, K.O. Babalola, Third Hankel determinant for α-starlike functions. Gulf J. Math. 2, 107113 (2014)

    Article  Google Scholar 

  45. L. Shi, I. Ali, M. Arif, N.E. Cho, S. Hussain, H. Khan, A study of third Hankel determinant problem for certain subfamilies of analytic functions involving cardioid domain. Mathematics 7, Article ID 418, 1–15 (2019)

    Google Scholar 

  46. L. Shi, H.M. Srivastava, M. Arif, S. Hussain, H. Khan, An investigation of the third Hankel determinant problem for certain subfamilies of univalent functions involving the exponential function. Symmetry 11, Article ID 598, 1–14 (2019)

    Google Scholar 

  47. J. Sokół, J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions. Zeszyty Nauk. Politech. Rzeszowskiej Mat. Fizyka 19, 101–105 (1996)

    MathSciNet  MATH  Google Scholar 

  48. H.M. Srivastava, Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis. Iran. J. Sci. Technol. Trans. A Sci. 44, 327–344 (2020)

    Article  MathSciNet  Google Scholar 

  49. H.M. Srivastava, Some parametric and argument variations of the operators of fractional calculus and related special functions and integral transformations. J. Nonlinear Convex Anal. 22, 1501–1520 (2021)

    MathSciNet  Google Scholar 

  50. H.M. Srivastava, S. Altınkaya, S. Yalcın, Hankel determinant for a subclass of bi-univalent functions defined by using a symmetric q-derivative operator. Filomat 32, 503–516 (2018)

    Article  MathSciNet  Google Scholar 

  51. 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, Article ID 848, 1–10 (2019)

    Google Scholar 

  52. H.M. Srivastava, Q.Z. Ahmad, N. Khan, B. Khan, Hankel and Toeplitz determinants for a subclass of q-starlike functions associated with a general conic domain. Mathematics 7, Article ID 181, 1–15 (2019)

    Google Scholar 

  53. H.M. Srivastava, N. Raza, E.S.A. AbuJarad, G. Srivastava, M.H. AbuJarad, Fekete-Szegö inequality for classes of (p, q)-starlike and (p, q)-convex functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. A Mat. 113, 3563–3584 (2019)

    Article  MathSciNet  Google Scholar 

  54. H.M. Srivastava, N. Khan, M. Darus, S. Khan, Q.Z. Ahmad, S. Hussain, Fekete-Szegö type problems and their applications for a subclass of q-starlike functions with respect to symmetrical points. Mathematics 8, Article ID 842, 1–18 (2020)

    Google Scholar 

  55. H.M. Srivastava, G. Kaur, G. Singh, Estimates of the fourth Hankel determinant for a class of analytic functions with bounded turnings involving cardioid domains. J. Nonlinear Convex Anal. 22, 511–526 (2021)

    MathSciNet  Google Scholar 

  56. 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, Article ID 102942, 1–16 (2021)

    Google Scholar 

  57. P. Zaprawa, Third Hankel determinants for subclasses of univalent functions. Mediterr. J. Math. 14, Article ID 19, 1–10 (2017). https://doi.org/10.1007/s00009-016-0829-y

  58. P. Zaprawa, M. Obradović, N. Tuneski, Third Hankel determinant for univalent starlike functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. A Mat. 115, Article ID 49, 1–6 (2021)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Informatics, University of Agadez, Agadez, Niger

    Sama Arjika

  2. International Chair in Mathematical Physics and Applications, University of Abomey-Calavi, Godomey, Benin

    Sama Arjika

  3. Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Pakistan

    Khalin Ullah & Muhammad Arif

  4. Department of Mathematics and Statistics, University of Victoria, British Columbia, VIC, Canada

    Hari Mohan Srivastava

  5. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

    Hari Mohan Srivastava

  6. Department of Mathematics and Informatics, Azerbaijan University, Baku, Azerbaijan

    Hari Mohan Srivastava

  7. Institute of Space Technology, University of Islamabad, Islamabad, Pakistan

    Ayesha Rafiq

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  1. Sama Arjika
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  2. Khalin Ullah
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  3. Hari Mohan Srivastava
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  4. Ayesha Rafiq
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  5. Muhammad Arif
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Editor information

Editors and Affiliations

  1. Department of Mathematics of Decision, University Cheikh Anta Diop of Dakar, Dakar, Senegal

    Diaraf Seck

  2. Department of Mathematics and Computer Science, Felix Houphouet Boigny University, Abidjan, Côte d'Ivoire

    Kinvi Kangni

  3. Mathematics Laboratory, Ecole Normale Supérieure de Libreville, Libreville, Gabon

    Philibert Nang

  4. University Assane Seck of Ziguinchor, Ziguinchor, Senegal

    Marie Salomon Sambou

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Arjika, S., Ullah, K., Srivastava, H.M., Rafiq, A., Arif, M. (2022). A Study of Sharp Coefficient Bounds for a New Subfamily of Starlike Functions. In: Seck, D., Kangni, K., Nang, P., Salomon Sambou, M. (eds) Nonlinear Analysis, Geometry and Applications. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-04616-2_15

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