SOME SUFFICIENT CONDITIONS FOR THE CLASS U
Skender Avdiji1, Nikola Tuneski2
1Faculty of Natural Sciences and Mathematics
State University of Tetovo
Ilinden b.b., 1200 Tetovo
Republic of MACEDONIA
2Faculty of Mechanical Engineering
Ss. Cyril and Methodius University in Skopje
Karpos II b.b., 1000 Skopje
Republic of MACEDONIA

Abstract

Let $\A$ be the class of functions $f$ that are analytic in the unit disk and normalized by $f(0)=f'(0)-1=0$. The classes $\widetilde{\mathcal{U}}(\mu)$ and $\mathcal{U}(\lambda,\mu)$ ($\mu>0$ and $0<\lambda\le1$) consist of functions $f$ from $\A$ that satisfy

\begin{displaymath}\real U(f,\mu;z) >0 \quad (z\in\mathbb{D}), \end{displaymath}

and respectively

\begin{displaymath}\left\vert U(f,\mu;z) -1\right\vert <\lambda \quad (z\in\mathbb{D}),\end{displaymath}

where

\begin{displaymath}U(f,\mu;z)=\left[\frac{z}{f(z)}\right]^{1+\mu}\cdot f'(z).\end{displaymath}

In this paper, using methods from the theory of first order differential subordinations, sufficient conditions (some of them sharp) are obtained in terms of the analytical representations of starlikeness and convexity that embed a function $f$ from $\A$ in the class $\mathcal{U}(\lambda,\mu)$ or $\widetilde{\mathcal{U}}(\mu)$.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 31
Issue: 2
Year: 2018

DOI: 10.12732/ijam.v31i2.10

Download Section



Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

  1. [1] L.A. Aksentév, Sufficient conditions for univalence of regular functions (In Russian), Izv. Vysš. Učebn. Zaved. Matematika, 4, No 3 (1958), 3–7.
  2. [2] P.L. Duren, Univalent Functions, Springer-Verlag, New York (1983).
  3. [3] R. Fournier, S. Ponnusamy, A class of locally univalent functions defined by a differential inequality, Complex Var. Elliptic Equ., 52, No 1 (2007), 1–8.
  4. [4] S.S. Miller, P.T. Mocanu, Differential Subordinations, Theory and Applications, Marcel Dekker, New York-Basel (2000). SOME SUFFICIENT CONDITIONS FOR THE CLASS U 297
  5. [5] S.S. Miller, P.T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J., 28, No 2 (1981), 157–172.
  6. [6] S.S. Miller, P.T. Mocanu, On some classes of first-order differential subordinations, Michigan Math. J., 32, No 2 (1985), 185–195.
  7. [7] M. Nunokawa, M. Obradovic, S. Owa, One criterion for univalency, Proc. Amer. Math. Soc., 106 (1989), 1035–1037.
  8. [8] M. Obradovic, S. Ponnusamy, Product of univalent functions, Math. Comput. Modelling, 57, No 3-4 (2013), 793–799.
  9. [9] M. Obradovic, S. Ponnusamy, Criteria for univalent functions in the unit disk, Arch. Math. (Basel), 100, No 2 (2013), 149–157.
  10. [10] M. Obradovic, S. Ponnusamy, On harmonic combination of univalent functions, Bull. Belg. Math. Soc. Simon Stevin, 19, No 3 (2012), 461–472.
  11. [11] M. Obradovic, S. Ponnusamy, New criteria and distortion theorems for univalent functions, Complex Variables Theory Appl., 44 (2001), 173–191.
  12. [12] M. Obradovic, S. Ponnusamy, V. Singh, P. Vasundhra, Univalency, starlikeness and convexity applied to certain classes of rational functions, Analysis (Munich), 22, No 3 (2002), 225–242.
  13. [13] M. Obradovic, S. Ponnusamy, Univalence and starlikeness of certain integral transforms defined by convolution of analytic functions, J. Math. Anal. Appl., 336 (2007), 758–767.
  14. [14] M. Obradovic, S. Ponnusamy, Coefficient characterization for certain classes of univalent functions, Bull. Belg. Math. Soc. (Simon Stevin), 16 (2009), 251–263.
  15. [15] S. Ozaki, M. Nunokawa, The Schwarzian derivative and univalent functions, Proc. Amer. Math. Soc., 33 (1972), 392–394.