The Sharp Bound of the Fifth Coefficient of Strongly Starlike Functions with Real Coefficients

  • Oh Sang Kwon
  • Adam Lecko
  • Young Jae SimEmail author
  • Barbara Śmiarowska


In this paper, we found the sharp bound of the fifth coefficient of strongly starlike functions of order \(\alpha \) with real coefficients for all \(\alpha \in (0,1]\).


Strongly starlike functions Carathéodory functions Schwarz functions Coefficient estimates 

Mathematics Subject Classification

30C45 30C50 



The authors thank the referees for their careful reading and useful comments. This work was partially supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP; Ministry of Science, ICT and Future Planning) (No. NRF-2017R1C1B5076778).


  1. 1.
    Ali, R.M.: Coefficients of the inverse of strongly starlike functions. Bull. Malays. Math. Sci. Soc. 26(2), 63–71 (2003)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ali, R.M., Singh, V.: On the fourth and fifth coefficients of strongly starlike functions. Results Math. 29, 197–202 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brannan, D.A., Kirwan, W.E.: On some classes of bounded univalent functions. J. Lond. Math. Soc. 2(1), 431–443 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cho, N.E., Kowalczyk, B., Kwon, O.S., Lecko, A., Sim, Y.J.: Some coefficient inequalities related to the Hankel determinant for strongly starlike functions of order alpha. J. Math. Inequal. 11(2), 429–439 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Goodman, A.W.: Univalent Functions. Mariner, Tampa (1983)zbMATHGoogle Scholar
  6. 6.
    Kwon, O.S., Lecko, A., Sim, Y.J.: On the fourth coefficient of functions in the Carathéodory class. Comput. Methods Funct. Theory.
  7. 7.
    Lecko, A.: Strongly starlike and spirallike functions. Ann. Polon. Math. 85(2), 165–192 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lecko, A.: Some methods in the theory of univalent functions. Oficyna Wydawnicza Poltechniki Rzeszowskiej (2005)Google Scholar
  9. 9.
    Lecko, A., Sim, Y.J.: A note on the fourth coefficient of strongly starlike functions. Results Math. 71, 1185–1189 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ma, W., Minda, D.: An internal geometric characterization of strongly starlike functions. Ann. Univ. Mariae Curie Skłodowska Sect. A 20, 89–97 (1991)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Ma, W., Owa, S.: Strongly starlike functions. Panam. Math. J. 3(2), 49–60 (1993)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Pommerenke, C.: Univalent Functions. Vandenhoeck & Ruprecht, Göttingen (1975)zbMATHGoogle Scholar
  13. 13.
    Prokhorov, D.V., Szynal, J.: Inverse coefficients for \((\alpha,\beta )\)-convex functions. Ann. Univ. Mariae Curie Skłodowska Sect. A 35, 125–143 (1981)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Sokół, J.: Coefficient estimates in a class of strongly starlike functions. Kyungpook Math. J. 49(2), 349–353 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Stankiewicz, J.: Quelques problémes extrémaux dans les classes des fonctions \(\alpha \)-angulairement étoilées. Ann. Univ. Mariae Curie Sklodowska Sect. A 20, 59–75 (1966)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Stankiewicz, J.: On a family of starlike functions. Ann. Univ. Mariae Curie Sklodowska Sect. A 22–24, 175–181 (1968/1970)Google Scholar
  17. 17.
    Sugawa, T.: A self-duality of strong starlikeness. Kodai Math. J. 28, 382–389 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2018

Authors and Affiliations

  1. 1.Department of MathematicsKyungsung UniversityBusanKorea
  2. 2.Department of Complex AnalysisUniversity of Warmia and Mazury in OlsztynOlsztynPoland

Personalised recommendations