Abstract and Applied Analysis

A Note on the Class of Functions with Bounded Turning

Rabha W. Ibrahim and Adem Kılıçman

Full-text: Open access


We consider subclasses of functions with bounded turning for normalized analytic functions in the unit disk. The geometric representation is introduced, some subordination relations are suggested, and the upper bound of the pre-Schwarzian norm for these functions is computed. Moreover, by employing Jack's lemma, we obtain a convex class in the class of functions of bounded turning and relations with other classes are posed.

Article information

Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 820696, 10 pages.

First available in Project Euclid: 5 April 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Ibrahim, Rabha W.; Kılıçman, Adem. A Note on the Class of Functions with Bounded Turning. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 820696, 10 pages. doi:10.1155/2012/820696. https://projecteuclid.org/euclid.aaa/1365125169

Export citation


  • Y. C. Kim and T. Sugawa, “Norm estimates of the pre-Schwarzian derivatives for certain classes of univalent functions,” Proceedings of the Edinburgh Mathematical Society, vol. 49, no. 1, pp. 131–143, 2006.
  • R. Parvatham, S. Ponnusamy, and S. K. Sahoo, “Norm estimates for the Bernardi integral transforms of functions defined by subordination,” Hiroshima Mathematical Journal, vol. 38, no. 1, pp. 19–29, 2008.
  • R. W. Ibrahim and M. Darus, “General properties for Volterra-type operators in the unit disk,” ISRN Mathematical Analysis, vol. 2011, Article ID 149830, 11 pages, 2011.
  • M. Darus and R. W. Ibrahim, “Coefficient inequalities for concave Cesáro operator of non-concave analytic functions,” European Journal of Pure and Applied Mathematics, vol. 3, no. 6, pp. 1086–1092, 2010.
  • P. T. Mocanu, “On a subclass of starlike functions with bounded turning,” Revue Roumaine de Mathématiques Pures et Appliquées, vol. 55, no. 5, pp. 375–379, 2010.
  • N. Tuneski, “Convex functions and functions with bounded turning,” Tamsui Oxford Journal of Mathematical Sciences, vol. 26, no. 2, pp. 161–172, 2010.
  • M. Darus, R. W. Ibrahim, and I. H. Jebril, “Bounded turning for generalized integral operator,” International Journal of Open Problems in Complex Analysis, vol. 1, no. 1, pp. 1–7, 2009.
  • M. Darus and R. W. Ibrahim, “On Cesáro means of order $\mu $ for outer functions,” International Journal of Nonlinear Science, vol. 9, no. 4, pp. 455–460, 2010.
  • M. Darus and R. W. Ibrahim, “Partial sums of analytic functions of bounded turning with appli-cations,” Computational & Applied Mathematics, vol. 29, no. 1, pp. 81–88, 2010.
  • H. M. Srivastava, M. Darus, and R. W. Ibrahim, “Classes of analytic functions with fractional powers defined by means of a certain linear operator,” Integral Transforms and Special Functions, vol. 22, no. 1, pp. 17–28, 2011.
  • I. S. Jack, “Functions starlike and convex of order k,” Journal of the London Mathematical Society, vol. 3, pp. 469–474, 1971.