Abstract
In this paper, we investigate the sharp bounds of the second Hankel determinant of Logarithmic coefficients for the starlike and convex functions with respect to symmetric points in the open unit disk.
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References
Ali, M.F., Allu, V.: On logarithmic coefficients of some close-to-convex functions. Proc. Amer. Math. Soc. 146, 1131–1142 (2017)
Ali, M.F., Allu, V.: Logarithmic coefficients of some close-to-convex functions. Bull. Aust. Math. Soc. 95, 228–237 (2017)
Allu, V., Lecko, A., Thomas, D. K.: Hankel, Toeplitz and Hermitian-Toeplitz Determinants for Ozaki Close-to-convex Functions, Mediterr. J. Math. (to appear)
Allu, V., Arora, V.: Second Hankel determinant of logarithmic coefficients of certain analytic functions, arXiv: 2110.05161
de Branges, L.: A proof of the Bieberbach conjecture. Acta Math. 154, 137–152 (1985)
Cho, N.E., Kowalczyk, B., Kwon, O., Lecko, A., Sim, Y.: On the third logarithmic coefficient in some subclasses of close-to-convex functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. 114, (2020). https://doi.org/10.1007/s13398-020-00786-7
Das, R.N., Singh, P.: On subclasses of schlicht mapping. Indian J. Pure Appl. Math. 8, 864–872 (1977)
Duren, P.L.: Univalent functions (Grundlehren der mathematischen Wissenschaften 259, New York, Berlin, Heidelberg, Tokyo), Springer-Verlag, (1983)
Efraimidis, I.: A generalization of Livingston’s coefficient inequalities for functions with positive real part. J. Math. Anal. Appl. 435, 369–379 (2016)
Elhosh, M.M.: On the logarithmic coefficients of close-to-convex functions. J. Aust. Math. Soc. A 60, 1–6 (1996)
Girela, D.: Logarithmic coefficients of univalent functions. Ann. Acad. Sci. Fenn. Math. 35(2), 337–350 (2010)
Kowalczyk, B., Lecko, A.: Second hankel determinant of logarithmic coefficients of convex and starlike functions, Bull. Aust. Math. Soc. https://doi.org/10.1017/S0004972721000836
Milin, I. M.: Univalent functions and orthonormal systems, Translations of Mathematical Monographs, Volume 49 (1977)
Nezhmetdinov, I.R., Ponnusamy, S.: On the class of univalent functions starlike with respect to N-symmetric points. Hokkaido Math. J. 31(1), 61–77 (2002)
Pommerenke, Ch.: On the coefficients and Hankel determinants of univalent functions. J. Lond. Math. Soc. 41, 111–122 (1966)
Pommerenke, Ch.: On the Hankel determinants of Univalent functions. Mathematika 14, 108–112 (1967)
Ponnusamy, S., Sharma, N.L., Wirths, K.J.: Logarithmic coefficients problems in families related to starlike and convex functions. J. Aust. Math. Soc. 109, 230–249 (2020)
Ponnusamy, S., Sugawa, T.: Sharp inequalities for logarithmic coefficients and their applications, Bulletin des Sciences Mathématiques 166, 23 pages, Article 102931 (2021)
Pranav Kumar, U., Vasudevarao, A.: Logarithmic coefficients for certain subclasses of close-to-convex functions. Monatsh. Math. 187, 543–563 (2018)
Sakaguchi, K.: On a certain univalent mapping. J. Math. Soc. Japan 11, 72–75 (1959)
Sim, Y.J., Lecko, A., Thomas, D.K.: The second Hankel determinant for strongly convex and Ozaki close-to-convex functions. Ann. Mat. Pura. Appl. 200, 2515–2533 (2021)
Thomas, D.K.: On the logarithmic coefficients of close-to-convex functions. Proc. Amer. Math. Soc. 144, 1681–1687 (2016)
Zaprawa, P.: Initial logarithmic coefficients for functions starlike with respect to symmetric points. Bol. Soc. Mat. Mex. 27, (2021). https://doi.org/10.1007/s40590-021-00370-y
Acknowledgements
The first author thanks SERB-CRG, the second author thanks IIT Bhubaneswar for providing Institute Post Doctoral Fellowship, and the third author’s research work is supported by CSIR-UGC.
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Allu, V., Arora, V. & Shaji, A. On the Second Hankel Determinant of Logarithmic Coefficients for Certain Univalent Functions. Mediterr. J. Math. 20, 81 (2023). https://doi.org/10.1007/s00009-023-02272-x
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DOI: https://doi.org/10.1007/s00009-023-02272-x
Keywords
- Univalent functions
- Logarithmic coefficients
- Hankel determinant
- Starlike functions
- Convex functions
- Symmetric points
- Schwarz function