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

Received: 18 July 2015     Accepted: 3 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

Published in Applied and Computational Mathematics (Volume 4, Issue 5)
DOI 10.11648/j.acm.20150405.12
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), 2015. 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.
Cite This Article
  • APA Style

    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://article.sciencepublishinggroup.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  - 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
    VL  - 4
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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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