## Abstract and Applied Analysis

### General Univalence Criterion Associated with the $n$th Derivative

#### Abstract

For normalized analytic functions $f(z)$ with $f(z)\ne 0$ for $0<|z|<1$, we introduce a univalencecriterion defined by sharp inequality associated with the $n$th derivative of $z/f(z)$, where $n\in \{3,4,5,\dots \}$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 307526, 9 pages.

Dates
First available in Project Euclid: 5 April 2013

https://projecteuclid.org/euclid.aaa/1365168178

Digital Object Identifier
doi:10.1155/2012/307526

Mathematical Reviews number (MathSciNet)
MR2947662

Zentralblatt MATH identifier
1254.30011

#### Citation

Al-Refai, Oqlah; Darus, Maslina. General Univalence Criterion Associated with the $n$ th Derivative. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 307526, 9 pages. doi:10.1155/2012/307526. https://projecteuclid.org/euclid.aaa/1365168178

#### References

• L. A. Aksentev, “Sufficient conditions for univalence of regular functions,” Izvestija Vysših Učebnyh Zavedeniĭ Matematika, vol. 1958, no. 3(4), pp. 3–7, 1958 (Russian).
• M. Obradović and S. Ponnusamy, “Coefficient characterization for certain classes of univalent functions,” Bulletin of the Belgian Mathematical Society, vol. 16, no. 2, pp. 251–263, 2009.
• M. Obradović and S. Ponnusamy, “New criteria and distortion theorems for univalent functions,” Complex Variables, vol. 44, no. 3, pp. 173–191, 2001.
• S. Ozaki and M. Nunokawa, “The Schwarzian derivative and univalent functions,” Proceedings of the American Mathematical Society, vol. 33, pp. 392–394, 1972.
• M. Obradović, N. N. Pascu, and I. Radomir, “A class of univalent functions,” Mathematica Japonica, vol. 44, no. 3, pp. 565–568, 1996.
• M. Nunokawa, M. Obradović, and S. Owa, “One criterion for univalency,” Proceedings of the American Mathematical Society, vol. 106, no. 4, pp. 1035–1037, 1989.
• D. Yang and J. Liu, “On a class of univalent functions,” International Journal of Mathematics and Mathematical Sciences, vol. 22, no. 3, pp. 605–610, 1999.
• D. Breaz and N. Breaz, “Univalence conditions for integral operators on $S(\alpha )$-class,” Libertas Mathematica, vol. 24, pp. 211–214, 2004.
• D. Breaz and N. Breaz, “Operatori integrali pe clasa T$_{2}$,” in Proceedings of the 6th Annual Conference of the Romanian Society of Mathematical Sciences, pp. 348–352, Sibiu, Romania, 2003.
• V. Pescar, “Univalence conditions for certain integral operators,” Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 4, article 147, 2006.