Abstract
We derive general formula for the fourth coefficient of the functions belonging to the Carathéodory class involving the parameters lying in the open unit disk. Further, we obtain sharp upper bounds of initial inverse coefficients for certain close-to-convex functions satisfying any one of the inequalities: \(\text {Re}((1-z)f^{'}(z))>0,\text {Re}((1-z^2)f^{'}(z))>0,\text {Re}((1-z+z^2)f^{'}(z))>0\) and \(\text {Re}((1-z)^2f^{'}(z))>0\).
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References
Ali, R.M.: Coefficients of the inverse of strongly starlike functions. Bull. Malays. Math. Sci. Soc. 2(26), 63–71 (2003)
Cho, N.E., Kowalczyk, B., Lecko, A.: Sharp bounds of some coefficient functionals over the class of functions convex in the direction of the imaginary axis. Bull. Aust. Math. Soc. 100, 86–96 (2019)
Choi, J.H., Kim, Y.C., Sugawa, T.: A general approach to the Fekete–Szegö problem. J. Math. Soc. Jpn 59, 707–727 (2007)
Grenander, U., Szegö, G.: Toeplitz Forms and their Applications, California Monographs in Mathematical Sciences. University of California Press, Berkeley (1958)
Jenkins, J.A.: On univalent functions with integral coefficients. Complex Var. Theory Appl. 9, 221–226 (1987)
Kaplan, W.: Close-to-convex schlicht functions. Mich. Math. J. 1, 169–185 (1952)
Kapoor, G.P., Mishra, A.K.: Coefficient estimates for inverses of starlike functions of positive order. J. Math. Anal. Appl. 329, 922–934 (2007)
Kumar, U.P., Vasudevarao, A.: Logarithmic coefficients for certain subclasses of close-to-convex functions. Monatsh. Math. 187, 543–563 (2018)
Kwon, O.S., Lecko, A., Sim, Y.J.: On the fourth coefficient of functions in the Carathéodory class. Comput. Methods Funct. Theory 18, 307–314 (2018)
Libera, R.J., Złotkiewicz, E.J.: Early coefficients of the inverse of a regular convex function. Proc. Amer. Math. Soc. 85, 225–230 (1982)
Libera, R.J., Złotkiewicz, E.J.: Coefficient bounds for the inverse of a function with derivative in \({mathcal P }\). II. Proc. Amer. Math. Soc. 92, 58–60 (1984)
Pommerenke, C.: Univalent Functions. Vandenhoeck & Ruprecht, Göttingen (1975)
Ponnusamy, S., Rasila, A., Kaliraj, A.S.: Harmonic close-to-convex functions and minimal surfaces. Complex Var. Elliptic Equ. 59, 986–1002 (2014)
Silverman, H.: Coefficient bounds for inverses of classes of starlike functions. Complex Var. Theory Appl. 12, 23–31 (1989)
Srivastava, H.M., Mishra, A.K., Kund, S.N.: Coefficient estimates for the inverses of starlike functions represented by symmetric gap series. Panamer. Math. J. 21, 105–123 (2011)
Xu, Q.H., Lv, C.B., Srivastava, H.M.: Coefficient estimates for the inverses of a certain general class of spirallike functions. Appl. Math. Comput. 219, 7000–7011 (2013)
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The work is supported by The Council of Scientific and Industrial Research(CSIR). Ref.No.:08/133(0018)/2017-EMR-I.
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The work is supported by The Council of Scientific and Industrial Research(CSIR). Ref.No.:08/133(0018)/2017-EMR-I.
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Goel, P., Kumar, S.S. On sharp bounds of certain close-to-convex functions. Afr. Mat. 35, 37 (2024). https://doi.org/10.1007/s13370-024-01176-7
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DOI: https://doi.org/10.1007/s13370-024-01176-7