Hankel Determinant of Logarithmic Coefficients for Tilted Starlike Functions With Respect to Conjugate Points

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DOI: https://doi.org/10.28924/2291-8639-21-2023-10
Published: Feb 21, 2023
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Daud Mohamad, Nur Hazwani Aqilah Abdul Wahid, Hankel Determinant of Logarithmic Coefficients for Tilted Starlike Functions With Respect to Conjugate Points, Int. J. Anal. Appl., 21 (2023), 10.

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Daud Mohamad, Nur Hazwani Aqilah Abdul Wahid

Abstract

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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References

  1. R.M. El-Ashwah, D.K. Thomas, Some Subclasses of Close-to-Convex Functions, J. Ramanujan Math. Soc. 2 (1987), 85-100.
  2. S. Abdul Halim, Functions Starlike With Respect to Other Points, Int. J. Math. Math. Sci. 14 (1991), 451–456. https://doi.org/10.1155/s0161171291000613.
  3. S.A.F.M. Dahhar, A. Janteng, A Subclass of Starlike Functions With Respect to Conjugate Points, Int. Math. Forum. 4 (2009), 1373-1377.
  4. N.H.A.A. Wahid, D. Mohamad, S.C. Soh, on a Subclass of Tilted Starlike Functions With Respect to Conjugate Points, Menemui Mat. (Discover. Math.) 37 (2015), 1-6.
  5. C. Selvaraj, N. Vasanthi, Subclasses of Analytic Functions With Respect to Symmetric and Conjugate Points, Tamkang J. Math. 42 (2011), 87-94. https://doi.org/10.5556/j.tkjm.42.2011.87-94.
  6. Q.H. Xu, G.P. Wu, Coefficient Estimate for a Subclass of Univalent Functions With Respect to Symmetric Point, Eur. J. Appl. Math. 3 (2010), 1055-1061.
  7. T.N. Shanmugam, C. Ramachandram, V. Ravichandran, Fekete-Szegö Problem for Subclasses of Starlike Functions With Respect to Symmetric Points, Bull. Korean Math. Soc. 43 (2006), 589–598. https://doi.org/10.4134/BKMS.2006.43.3.589.
  8. M. Ghaffar Khan, B. Ahmad, G. Murugusundaramoorthy, et al. Third Hankel Determinant and Zalcman Functional for a Class of Starlike Functions With Respect to Symmetric Points Related With Sine Function, J. Math. Computer Sci. 25 (2021), 29-36. https://doi.org/10.22436/jmcs.025.01.04.
  9. A.K. Mishra, J.K. Prajapat, S. Maharana, Bounds on Hankel Determinant for Starlike and Convex Functions With Respect to Symmetric Points, Cogent Math. 3 (2016), 1160557. https://doi.org/10.1080/23311835.2016.1160557.
  10. G. Singh, G. Singh, Coefficient Inequality for Subclasses of Starlike Functions With Respect to Conjugate Points, Int. J. Modern Math. Sci. 8 (2013), 48-56.
  11. G. Singh, Hankel Determinant for Analytic Functions With Respect to Other Points, Eng. Math. Lett. 2 (2013), 115-123.
  12. N.H.A.A. Wahid, D. Mohamad, S.C. Soh, Second Hankel Determinant for a Subclass of Tilted Starlike Functions With Respect to Conjugate Points, MATEMATIKA, 31 (2015), 111-119.
  13. N.H.A.A. Wahid, D. Mohamad, Toeplitz Determinant for a Subclass of Tilted Starlike Functions With Respect to Conjugate Points, Sains Malays. 50 (2021), 3745-3751. http://doi.org/10.17576/jsm-2021-5012-23.
  14. D. Mohamed, N.H.A.A. Wahid, Zalcman Coefficient Functional for Tilted Starlike Functions With Respect to Conjugate Points, J. Math. Computer Sci. 29 (2022), 40–51. https://doi.org/10.22436/jmcs.029.01.04.
  15. P.L. Duren, Univalent Functions, Springer-Verlag, New York, 1983.
  16. I.M. Milin, Univalent Functions and Orthonormal Systems, Izdat. "Nauka", Moscow, 1971 (in Russian); English transl. American Mathematical Society, Providence, (1977).
  17. I.P. Kayumov, On Brennan’s Conjecture for a Special Class of Functions, Math Notes. 78 (2005), 498–502. https://doi.org/10.1007/s11006-005-0149-1.
  18. C. Pommerenke, On the Coefficients and Hankel Determinants of Univalent Functions, J. Lond. Math. Soc. s1-41 (1966), 111–122. https://doi.org/10.1112/jlms/s1-41.1.111.
  19. C. Pommerenke, On the Hankel Determinants of Univalent Functions, Mathematika. 14 (1967), 108–112.
  20. P. Dienes, The Taylor Series: An Introduction to the Theory of Functions of a Complex Variable, Dover, Mineola, 1957.
  21. B. Kowalczyk, A. Lecko, Second Hankel Determinant of Logarithmic Coefficients of Convex and Starlike Functions, Bull. Aust. Math. Soc. 105 (2021), 458–467. https://doi.org/10.1017/s0004972721000836.
  22. B. Kowalczyk, A. Lecko, Second Hankel Determinant of Logarithmic Coefficients of Convex and Starlike Functions of Order Alpha, Bull. Malays. Math. Sci. Soc. 45 (2021), 727–740. https://doi.org/10.1007/s40840-021-01217-5.
  23. D. Alimohammadi, E. Analouei Adegani, T. Bulboaca, et al. Logarithmic Coefficient Bounds and Coefficient Conjectures for Classes Associated with Convex Functions, J. Funct. Spaces. 2021 (2021), 6690027. https://doi.org/10.1155/2021/6690027.
  24. V. Allu, V. Arora, A. Shaji, On the Second Hankel Determinant of Logarithmic Coefficients for Certain Univalent Functions, Mediterr. J. Math. 20 (2023), 81. https://doi.org/10.1007/s00009-023-02272-x.
  25. V. Allu, V. Arora, Second Hankel Determinant of Logarithmic Coefficients of Certain Analytic Functions, (2021). https://doi.org/10.48550/ARXIV.2110.05161.
  26. B. Khan, I. Aldawish, S. Araci, M.G. Khan, Third Hankel Determinant for the Logarithmic Coefficients of Starlike Functions Associated with Sine Function, Fractal Fract. 6 (2022), 261. https://doi.org/10.3390/fractalfract6050261.
  27. M. Obradovic, S. Ponnusamy, K.J. Wirths, Logarithmic Coefficients and a Coefficient Conjecture for Univalent Functions, Monatsh Math. 185 (2017), 489–501. https://doi.org/10.1007/s00605-017-1024-3.
  28. L. Shi, M. Arif, A. Rafiq, et al. Sharp Bounds of Hankel Determinant on Logarithmic Coefficients for Functions of Bounded Turning Associated with Petal-Shaped Domain, Mathematics. 10 (2022), 1939. https://doi.org/10.3390/math10111939.
  29. K. Tra.bka-Wie.cław, On Coefficient Problems for Functions Connected with the Sine Function, Symmetry. 13 (2021), 1179. https://doi.org/10.3390/sym13071179.
  30. Z. Ye, The Logarithmic Coefficients of Close-to-Convex Functions, Bull. Inst. Math. Acad. Sin. 3 (2008), 445–452.
  31. P. Zaprawa, Initial Logarithmic Coefficients for Functions Starlike With Respect to Symmetric Points, Bol. Soc. Mat. Mex. 27 (2021), 62. https://doi.org/10.1007/s40590-021-00370-y.
  32. I. Efraimidis, A Generalization of Livingston’s Coefficient Inequalities for Functions With Positive Real Part, J. Math. Anal. Appl. 435 (2016), 369–379. https://doi.org/10.1016/j.jmaa.2015.10.050.
  33. M. Arif, M. Raza, H. Tang, et al. Hankel Determinant of Order Three for Familiar Subsets of Analytic Functions Related With Sine Function, Open Math. 17 (2019), 1615–1630. https://doi.org/10.1515/math-2019-0132.