A note on the class of functions with bounded turning. (English) Zbl 1237.30004

Summary: 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.


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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[1] 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. · Zbl 1112.30012 · doi:10.1017/S0013091504000306
[2] 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. · Zbl 1158.30009
[3] 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. · Zbl 1217.30015 · doi:10.5402/2011/149830
[4] 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. · Zbl 1213.30018
[5] 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. · Zbl 1240.30056
[6] N. Tuneski, “Convex functions and functions with bounded turning,” Tamsui Oxford Journal of Mathematical Sciences, vol. 26, no. 2, pp. 161-172, 2010. · Zbl 1211.30032
[7] 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.
[8] 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. · Zbl 1394.30002
[9] M. Darus and R. W. Ibrahim, “Partial sums of analytic functions of bounded turning with applications,” Computational & Applied Mathematics, vol. 29, no. 1, pp. 81-88, 2010. · Zbl 1186.30012 · doi:10.1590/S1807-03022010000100006
[10] 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. · Zbl 1207.30031 · doi:10.1080/10652469.2010.489796
[11] I. S. Jack, “Functions starlike and convex of order k,” Journal of the London Mathematical Society, vol. 3, pp. 469-474, 1971. · Zbl 0224.30026 · doi:10.1112/jlms/s2-3.3.469
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