Applied and Computational Mathematics

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On a Subclass of Close-to-Convex Functions Associated with Fixed Second Coefficient

Received: 18 July 2015    Accepted: 03 August 2015    Published: 19 August 2015
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Abstract

We consider a subclass of univalent functions f (z) for which there corresponds a convex function g(z) of order α such that Re(zf'(z) / g(z)) ≥ β. We investigate the influence of the second coefficient of g(z) on this class. We also prove distortion, covering, and radius of convexity theorems

DOI 10.11648/j.acm.20150405.12
Published in Applied and Computational Mathematics (Volume 4, Issue 5, October 2015)
Page(s) 342-345
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Analytic Function, Univalent Function, Convex Function of Order α, Close-to-Convexity, Fixed Second Coefficient, Radius of Convexity

References
[1] O.P. Ahuja, “The influence of second coefficient on spirallike and Robertson functions”, Yokohama Math. J. 34(1-2) (1986) 3 - 1.
[2] H.S. Al-Amiri, “On close-to-star functions of order a”, Proc. Amer. Math. Soc. 29 (1971) 103 - 108.
[3] V.V. Anh, “Starlike functions with a fixed coefficient”, Bult. Austral. Math. Soc. 39(1) (1989) 145 - 158.
[4] P.L. Duren, “Univalent functions”, Springer-Verlag, N.Y. Berlin, Heidelberg, Tokyo, 1983.
[5] M. Finkelstein, “Growth estimates of convex functions”, Proc. Amer. Math. Soc. 18 (1967), 412 - 418.
[6] R.M. Goel, “The radius of convexity and starlikeness for certain classes of analytic functions with fixed coefficient”, Ann. Univ. Mariar Euric Sklodowska Sect. A, 25 (1971) 33 - 39.
[7] A.W. Goodman, “Univalent functions”, Vol. I, II, Mariner Tampa, Florida, 1983.
[8] T.H. Gronwall, “On the distortion in conformal mapping when the second coefficient in the mapping function has an assigned value”, Prof. Nat. Acad. Proc. 6 (1920) 300 - 302.
[9] W.K. Hayman, “Multivalent functions”, Cambridge University Press, 1958.
[10] W. Kaplan, “Close-to-convex functions”, Mich. Math. J. 1 (1952) 169 - 185.
[11] R.J. Libera, “Some radius of convexity problems”, Duke Math. J. 31 (1964) 143 - 158.
[12] A.E. Livingston, “On the radius of univalence of certain analytic functions”, Proc. Amer. Math. Soc. 17 (1965) 352 - 357.
[13] K.I. Noor, “Radius problem for a subclass of close-to-convex univalent functions”, Int. J. Math. Sci. 14(4) (1992) 719 - 726.
[14] M.S. Robertson, “On the theory of univalent functions”, Ann. Math. 37 (1936) 374 - 408.
[15] C. Selvaraj, “A subclass of close-to-convex functions”, Southeast Asian Bull. Math., 28 (2004) 113 - 123.
[16] C. Selvaraj and N. Vasanthi, “A certain subclass of close-to-convex functions defined in the unit disk”, Far East J. Math. Sci. 24(2) (2010) 241 - 253.
[17] H. Silverman, “On a close-to-convex functions”, Proc. Amer. Math. Soc. 36(2) (1972) 477 - 484.
Author Information
  • Department of Mathematics, Presidency College (Autonomous), Chennai, India

  • Department of Mathematics, Tagore Engineering College, Vandalur, Chennai, India

  • Department of Mathematics, Presidency College (Autonomous), Chennai, India

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    Selvaraj Chellian, Stelin Simpson, Logu Sivalingam. (2015). On a Subclass of Close-to-Convex Functions Associated with Fixed Second Coefficient. Applied and Computational Mathematics, 4(5), 342-345. https://doi.org/10.11648/j.acm.20150405.12

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    ACS Style

    Selvaraj Chellian; Stelin Simpson; Logu Sivalingam. On a Subclass of Close-to-Convex Functions Associated with Fixed Second Coefficient. Appl. Comput. Math. 2015, 4(5), 342-345. doi: 10.11648/j.acm.20150405.12

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    AMA Style

    Selvaraj Chellian, Stelin Simpson, Logu Sivalingam. On a Subclass of Close-to-Convex Functions Associated with Fixed Second Coefficient. Appl Comput Math. 2015;4(5):342-345. doi: 10.11648/j.acm.20150405.12

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  • @article{10.11648/j.acm.20150405.12,
      author = {Selvaraj Chellian and Stelin Simpson and Logu Sivalingam},
      title = {On a Subclass of Close-to-Convex Functions Associated with Fixed Second Coefficient},
      journal = {Applied and Computational Mathematics},
      volume = {4},
      number = {5},
      pages = {342-345},
      doi = {10.11648/j.acm.20150405.12},
      url = {https://doi.org/10.11648/j.acm.20150405.12},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.acm.20150405.12},
      abstract = {We consider a subclass of univalent functions f (z) for which there corresponds a convex function g(z) of order α such that Re(zf'(z) / g(z)) ≥ β. We investigate the influence of the second coefficient of g(z) on this class. We also prove distortion, covering, and radius of convexity theorems},
     year = {2015}
    }
    

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    AU  - Stelin Simpson
    AU  - Logu Sivalingam
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    AB  - We consider a subclass of univalent functions f (z) for which there corresponds a convex function g(z) of order α such that Re(zf'(z) / g(z)) ≥ β. We investigate the influence of the second coefficient of g(z) on this class. We also prove distortion, covering, and radius of convexity theorems
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