Some Applications of Nunokawa’s Lemma



The purpose of the present paper is to prove a geometric property for analytic functions p in the open unit disk with \(p(0)=1\) by using Nunokawa’s result, which is a generalized form of well-known Jack’s lemma. This property concerns a boundary behavior of the functions p. As the applications of the main result, we obtain a few corollaries where several sufficient conditions for p to be of the real part greater than a given number \(\beta \ (0\le \beta <1)\) are also investigated.


Analytic functions Convex functions Starlike functions Univalent functions 

Mathematics Subject Classification

30C45 30C80 



The authors would like to express their thanks to the editor, Professor Rosihan M. Ali and the reviewers for many valuable advices regarding a previous version of this paper. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) and funded by the Ministry of Education, Science and Technology (No. 2011-0007037).


  1. 1.
    Jack, I.S.: Functions starlike and convex of order \(\alpha \). J. Lond. Math. Soc. 3, 469–474 (1971)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Nunokawa, M.: On properties of non-carathéodory functions. Proc. Jpn. Acad. Ser. A 68(6), 152–153 (1992)CrossRefMATHGoogle Scholar
  3. 3.
    Nunokawa, M.: On the order of strongly starlikeness of strongly convex functions. Proc. Jpn. Acad. Ser. A 69(7), 234–237 (1993)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Nunokawa, M., Sokół, J.: On the order of strongly starlikeness of convex functions of order alpha. Mediterr. J. Math. 11(2), 329–335 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Nunokawa, M., Sokół, J.: Remarks on some starlike functions. J. Inequal. Appl. 2013(593), 8 (2013)MathSciNetMATHGoogle Scholar
  6. 6.
    Tuneski, N., Aceska, R.: On the linear combination of the representations of starlikeness and convexity. Glasnik Matematicki 39(59), 265–272 (2004)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2016

Authors and Affiliations

  1. 1.University of GunmaChibaJapan
  2. 2.Department of MathematicsUniversity of RzeszówRzeszówPoland
  3. 3.Department of Applied Mathematics, College of Natural SciencesPukyong National UniversityBusanKorea

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