Document Type: Research Paper

Authors

Department of Mathematics, Payame Noor University, I. R. of Iran.

Abstract

Assume that $\mathbb{D}$ is the open unit disk. Applying Ozaki's conditions, we consider two classes of locally univalent, which denote by $\mathcal{G}(\alpha)$ and $\mathcal{F}(\mu)$ as follows \begin{equation*}  \mathcal{G}(\alpha):=\left\{f\in \mathcal{A}:\mathfrak{Re}\left( 1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\right) <1+\frac{\alpha }{2},\quad 0<\alpha\leq1\right\}, \end{equation*} and \begin{equation*}  \mathcal{F}(\alpha):=\left\{f\in \mathcal{A}:\mathfrak{Re}\left( 1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\right) >\frac{1 }{2}-\mu,\quad -1/2<\mu\leq 1\right\}, \end{equation*} respectively, where $z \in \mathbb{D}$. In this paper, we study the mapping properties of this classes under general integral operator. We also, obtain some conditions for integral operator to be convex or starlike function.

Keywords

Main Subjects

[1] D. Breaz, M. Darus, and N. Breaz, Recent Studies on Univalent Integral Operators, Editure Aeternitas, Alba Iulia, 2010.

[2] D. Breaz, S. Owa, and N. Breaz,  Some properties for general integral operators,Scientific Journal, 3 (2014) 9-14.

[3] D. Bshouty and A. Lyzzaik,  Close-to-convexity criteria for planar harmonic mappings, Complex Anal. Oper. Theory, 5 (2011) 767-774.

[4] J.G. Clunie and T. Sheil-Small,  Harmonic Univalent Functions, Ann. Acad. Sci. Fenn. Ser. A. I., 1984.

[5] P. Duren,  Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics. 156, Cambridge University Press, Cambridge, 2004.

[6] P. Duren,  Univalent Functions (Grundlehren der mathematischen Wissenschaften 259), Springer, Berlin, 1983.

[7] A.W. Goodman, Univalent Functions, Vols. 1-2, Mariner, Tampa, Florida, 1983.

[8] P.T. Mocanu, Injective conditions in the complex plane, Complex Anal. Oper. Theory, {5} (2011) 759-786.

[9] M. Obradovi'c, S. Ponnusamy, and K.-J. Wirths, Cofficient charactrizations and sections for some univalent functions, Siberian Mathematical Journal, 54 (2013) 679-696.

[10] S. Ozaki, On the theory of multivalent functions, Sci. Rep. Tokyo Bunrika Daigaku, {4} (1941) 45-86.

[11] J.A. Pfaltgraff, M.O. Reade, and T. Umezawa, Sufficient conditions for univalence, Ann. Fac. Sci. de Kinshasa, Zaire; Sec. Math. Phys., 2 (1976) 94-100.

[12] S. Ponnusamy, S.K. Sahoo, and H. Yanagihara, Radius of convexity of partial sums of functions in the close-to-convex family, Nonlinear Analysis, 95 (2014) 219-228.

[13] M.S. Robertson, On the theory of univalent functions, Ann. Math., {37} (1936) 374-408.

[14] T. Umezawa, Analytic functions convex in one direction, J. Math. Soc. Jpn., {4} (1952) 194-202.