Abstract
In this work, we estimate the bounds for the logarithmic coefficients \(\gamma _n\) of some well-known classes like \({\mathcal {F}}(c)\) for \(c\in (0,0.656]\cup \{2\}\). The best bound obtained for the logarithmic coefficient \(|\gamma _5|\) is sharp for \(c=2\). In a special case, it is important to note that we obtain the bounds for the logarithmic coefficients \(\gamma _n\) of the convex functions of order \(\alpha \) for some \(\alpha \). It is worthwhile mentioning that the given bounds would generalize some of the previous papers.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availability of data and materials
Not applicable.
References
Adegani, E.A., Cho, N.E., Jafari, M.: Logarithmic coefficients for univalent functions defined by subordination. Mathematics 7, Article ID 408, 1–12 (2019)
Adegani, E.A., Cho, N.E., Motamednezhad, A., Jafari, M.: Bi-univalent functions associated with Wright hypergeometric functions. J. Comput. Anal. Appl. 28, 261–271 (2020)
Ali, M.F., Vasudevarao, A.: On logarithmic coefficients of some close-to-convex functions. Proc. Am. Math. Soc. 146, 1131–1142 (2018)
Alimohammadi, D., Cho, N.E., Adegani, E.A., Motamednezhad, A.: Argument and coefficient estimates for certain analytic functions. Mathematics 8, Article ID 88, 1–13 (2020)
Bieberbach, L.: Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitkreises vermitteln. S.-B. Preuss. Akad. Wiss. 940–955, (1916)
Cho, N.E., Kowalczyk, B., Kwon, O.S., Lecko, A., Sim, Y.J.: On the third logarithmic coefficient in some subclasses of close-to-convex functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114, Article ID 52, 1–14 (2020)
De Branges, L.: A proof of the Bieberbach conjecture. Acta Math. 154, 137–152 (1985)
Duren, P.L.: Univalent Functions. Springer, Amsterdam (1983)
Duren, P.L., Leung, Y.J.: Logarithmic coeffcients of univalent functions. J. Anal. Math. 36, 36–43 (1979)
Ebadian, A., Bulboacă, T., Cho, N.E., Adegani, E.A.: Coefficient bounds and differential subordinations for analytic functions associated with starlike functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114, Article ID 128, 1–19 (2020)
Kargar, R.: On logarithmic coefficients of certain starlike functions related to the vertical strip. J. Anal. 27, 985–995 (2019)
Kargar, R., Ebadian, A., Sokół, J.: On Booth lemniscate and starlike functions. Anal. Math. Phys. 9, 143–154 (2019)
Kayumov, I.R.: On Brennan’s conjecture for a special class of functions. Math. Notes 78, 498–502 (2005)
Kumar, U.P., Vasudevarao, A.: Logarithmic coefficients for certain subclasses of close-to-convex functions. Monatsh. Math. 187, 543–563 (2018)
Ma, W.C., Minda, D.: A unified treatment of some special classes of univalent functions. In Proceedings of the Conference on Complex Analysis (Tianjin, 1992); Internat. Press, Cambridge, MA, USA, 157–169 (1992)
Milin, I.M.: Univalent functions and orthonormal systems. Am. Math. Soc. Transl. of Math. Monogr. 49, Proovidence, RI, (1977)
Milin, I.M.: On a property of the logarithmic coefficients of univalent functions. In: Metric Questions in the Theory of Functions, Naukova Dumka, Kiev 86–90 (1980) (in Russian)
Milin, I.M.: On a conjecture for the logarithmic coefficients of univalent functions. Zap. Nauch. Semin. Leningr. Otd. Mat. Inst. Steklova 125, 135–143 (1983) (in Russian)
Miller, S.S., Mocanu, P.T.: Univalent solutions of Briot–Bouquet differential equations. J. Differ. Equ. 56, 297–309 (1985)
Miller, S.S., Mocanu, P.T.: Differential Subordinations. Theory and Applications. Marcel Dekker Inc., New York (2000)
Obradović, M., Ponnusamy, S., Wirths, K.-J.: Logarithmic coeffcients and a coefficient conjecture for univalent functions. Monatsh. Math. 185, 489–501 (2018)
Ponnusamy, S., Sahoo, S.K., Yanagihara, H.: Radius of convexity of partial sums of functions in the close-to-convex family. Nonlinear Anal. 95, 219–228 (2014)
Ponnusamy, S., Sharma, N.L., Wirths, K.-J.: Logarithmic coefficients of the inverse of univalent functions. Results Math. 73, Article ID 160, 1–20 (2018)
Ponnusamy, S., Sharma, N.L., Wirths, K.-J.: Logarithmic coefficients problems in families related to starlike and convex functions. J. Aust. Math. Soc. 1–20, (2019)
Robertson, M.S.: A remark on the odd-schlicht functions. Bull. Am. Math. Soc. 42, 366–370 (1936)
Sokół, J.: A certain class of starlike functions. Comput. Math. Appl. 62, 611–619 (2011)
Sokół, J., Stankiewicz, J.: Radius of convexity of some subclasses of strongly starlike functions. Zeszyty Nauk. Politech. Rzeszowskiej Mat. 19, 101–105 (1996)
Umezawa, T.: Analytic functions convex in one direction. J. Math. Soc. Japan 4, 194–202 (1952)
Funding
Not applicable.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The authors declare that they have no conflict of interest.
Code availability
Not applicable.
Additional information
Communicated by V. Ravichandran.
Dedicated to the memory of Professor Gabriela Kohr (1967-2020)
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The fourth author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2019R1I1A3A01050861).
Rights and permissions
About this article
Cite this article
Alimohammadi, D., Analouei Adegani, E., Bulboacă, T. et al. Logarithmic Coefficients for Classes Related to Convex Functions. Bull. Malays. Math. Sci. Soc. 44, 2659–2673 (2021). https://doi.org/10.1007/s40840-021-01085-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-021-01085-z