## Abstract and Applied Analysis

### Differential Subordinations for Nonanalytic Functions

#### Abstract

In the paper by Mocanu (1980), Mocanu has obtained sufficient conditions for a function in the classes ${C}^{\mathrm{1}}(U)$, respectively, and ${C}^{\mathrm{2}}(U)$ to be univalent and to map $U$ onto a domain which is starlike (with respect to origin), respectively, and convex. Those conditions are similar to those in the analytic case. In the paper by Mocanu (1981), Mocanu has obtained sufficient conditions of univalency for complex functions in the class ${C}^{\mathrm{1}}$ which are also similar to those in the analytic case. Having those papers as inspiration, we try to introduce the notion of subordination for nonanalytic functions of classes ${C}^{\mathrm{1}}$ and ${C}^{\mathrm{2}}$ following the classical theory of differential subordination for analytic functions introduced by Miller and Mocanu in their papers (1978 and 1981) and developed in their book (2000). Let $\mathrm{\Omega }$ be any set in the complex plane $\mathbb{C}$, let $p$ be a nonanalytic function in the unit disc $U$, $p\in {C}^{\mathrm{2}}(U),$ and let $\psi (r,s,t;z):{\mathbb{C}}^{\mathrm{3}}{\times}U\to \mathbb{C}$. In this paper, we consider the problem of determining properties of the function $p$, nonanalytic in the unit disc $U$, such that $p$ satisfies the differential subordination $\psi (p(z),Dp(z),{D}^{\mathrm{2}}p(z)-Dp(z);z)\subset \mathrm{\Omega }{\Rightarrow}p(U)\subset \mathrm{\Delta }$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 251265, 9 pages.

Dates
First available in Project Euclid: 6 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412605987

Digital Object Identifier
doi:10.1155/2014/251265

Mathematical Reviews number (MathSciNet)
MR3226182

Zentralblatt MATH identifier
07022003

#### Citation

Oros, Georgia Irina; Oros, Gheorghe. Differential Subordinations for Nonanalytic Functions. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 251265, 9 pages. doi:10.1155/2014/251265. https://projecteuclid.org/euclid.aaa/1412605987

#### References

• P. T. Mocanu, “Starlikeness and convexity for non-analytic functions in the unit disc,” Mathematica, vol. 22, no. 1, pp. 77–83, 1980.
• P. T. Mocanu, “Sufficient conditions of univalency for complex functions in the class C1,” Revue d'Analyse Numerique et de Theorie de l'Approximation, vol. 10, no. 1, pp. 75–79, 1981.
• S. S. Miller and P. T. Mocanu, “Second order differential inequalities in the complex plane,” Journal of Mathematical Analysis and Applications, vol. 65, pp. 298–305, 1978.
• S. S. Miller and P. T. Mocanu, “Differential subordinations and univalent functions,” Michigan Mathematical Journal, vol. 28, pp. 157–171, 1981.
• S. S. Miller and P. T. Mocanu, Differential Subordination. Theory and Applications, Marcel Dekker, New York, NY, USA, 2000. \endinput