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Some Applications of Nunokawa’s Lemma

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Abstract

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.

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Acknowledgments

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).

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Authors and Affiliations

  1. University of Gunma, Hoshikuki-cho 798-8, Chuou-Ward, Chiba, 260-0808, Japan

    Mamoru Nunokawa

  2. Department of Mathematics, University of Rzeszów, ul. Rejtana 16A, 35-310, Rzeszów, Poland

    Janusz Sokół

  3. Department of Applied Mathematics, College of Natural Sciences, Pukyong National University, Busan, 608-737, Korea

    Nak Eun Cho

Authors
  1. Mamoru Nunokawa
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  2. Janusz Sokół
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  3. Nak Eun Cho
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Corresponding author

Correspondence to Nak Eun Cho.

Additional information

Communicated by V. Ravichandran.

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Nunokawa, M., Sokół, J. & Cho, N.E. Some Applications of Nunokawa’s Lemma. Bull. Malays. Math. Sci. Soc. 40, 1791–1800 (2017). https://doi.org/10.1007/s40840-016-0398-z

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  • DOI: https://doi.org/10.1007/s40840-016-0398-z

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