Document Type : Research Paper


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


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.


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.