Abstract and Applied Analysis

Certain Properties of a Class of Close-to-Convex Functions Related to Conic Domains

Wasim Ul-Haq and Shahid Mahmood

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We aim to define a new class of close-to-convex functions which is related to conic domains. Many interesting properties such as sufficiency criteria, inclusion results, and integral preserving properties are investigated here. Some interesting consequences of our results are also observed.

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Abstr. Appl. Anal., Volume 2013 (2013), Article ID 847287, 6 pages.

First available in Project Euclid: 27 February 2014

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Ul-Haq, Wasim; Mahmood, Shahid. Certain Properties of a Class of Close-to-Convex Functions Related to Conic Domains. Abstr. Appl. Anal. 2013 (2013), Article ID 847287, 6 pages. doi:10.1155/2013/847287. https://projecteuclid.org/euclid.aaa/1393511867

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  • A. W. Goodman, Univalent Functions. Vol. I, Polygonal Publishing House, Washington, DC, USA, 1983.
  • S. Kanas and A. Wisniowska, “Conic regions and $k$-uniform convexity,” Journal of Computational and Applied Mathematics, vol. 105, no. 1-2, pp. 327–336, 1999.
  • S. Kanas and A. Wisniowska, “Conic domains and starlike functions,” Revue Roumaine de Mathématiques Pures et Appliquées, vol. 45, no. 4, pp. 647–657, 2000.
  • M. Acu, “On a subclass of $n$-uniformly close to convex functions,” General Mathematics, vol. 14, no. 1, pp. 55–64, 2006.
  • E. Aqlan, J. M. Jahangiri, and S. R. Kulkarni, “New classes of $k$-uniformly convex and starlike functions,” Tamkang Journal of Mathematics, vol. 35, no. 3, pp. 1–7, 2004.
  • S. Kanas, “Alternative characterization of the class $k$-$UCV$ and related classes of univalent functions,” Serdica. Mathematical Journal, vol. 25, no. 4, pp. 341–350, 1999.
  • S. Kanas and H. M. Srivastava, “Linear operators associated with $k$-uniformly convex functions,” Integral Transforms and Special Functions, vol. 9, no. 2, pp. 121–132, 2000.
  • S. Shams, S. R. Kulkarni, and J. M. Jahangiri, “Classes of uni-formly starlike and convex functions,” International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 55, pp. 2959–2961, 2004.
  • H. M. Srivastava, S.-H. Li, and H. Tang, “Certain classes of $k$-uniformly close-to-convex functions and other related functions defined by using the Dziok-Srivastava operator,” Bulletin of Mathematical Analysis and Applications, vol. 1, no. 3, pp. 49–63, 2009.
  • K. I. Noor, “On a generalization of uniformly convex and related functions,” Computers & Mathematics with Applications, vol. 61, no. 1, pp. 117–125, 2011.
  • K. I. Noor and F. M. Al-Oboudi, “Alpha-quasi-convex univalent functions,” Caribbean Journal of Mathematics, vol. 3, pp. 1–8, 1984.
  • K. I. Noor and S. N. Malik, “On generalized bounded Mocanu variation associated with conic domain,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 844–852, 2012.
  • K. I. Noor, M. Arif, and W. Ul-Haq, “On $k$-uniformly close-to-convex functions of complex order,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 629–635, 2009.
  • K. G. Subramanian, T. V. Sudharsan, and H. Silverman, “On uni-formly close-to-convex functions and uniformly quasiconvex functions,” International Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 48, pp. 3053–3058, 2003.
  • R. J. Libera, “Some radius of convexity problems,” Duke Mathematical Journal, vol. 31, pp. 143–158, 1964.
  • K. I. Noor, “On quasi-convex functions and related topics,” International Journal of Mathematics and Mathematical Sciences, vol. 2, pp. 241–258, 1987.
  • S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, vol. 225 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000.
  • S. D. Bernardi, “Convex and starlike univalent functions,” Tran-sactions of the American Mathematical Society, vol. 135, pp. 429–446, 1969.
  • R. J. Libera, “Some classes of regular univalent functions,” Pro-ceedings of the American Mathematical Society, vol. 16, pp. 755–758, 1965.