An improvement of Ozaki's condition

We improve the Ozaki's condition for univalence of convex functions of order @a. The modified Nunokawa's Lemma is the key in the argumentation and it may have a lot of applications in the solutions of the other problems. Some new conditions for starlikeness and strongly starlikeness of order alpha are also proved in this work.

[1]  Mamoru Nunokawa,et al.  On the theory of multivalent functions , 1987 .

[2]  Saibah Siregar,et al.  The starlikeness of analytic functions of Koebe type , 2011, Math. Comput. Model..

[3]  梅沢 敏夫.  On the theory of multivalent functions , 1955 .

[4]  Toshio Umezawa,et al.  ON THE THEORY OF UNIVALENT FUNCTIONS , 1955 .

[5]  Mamoru Nunokawa,et al.  On the order of strongly starlikeness of strongly convex functions , 1993 .

[6]  Janusz Sokól,et al.  Starlikeness of Hadamard product of certain analytic functions , 2007, Appl. Math. Comput..

[7]  Hari M. Srivastava,et al.  Subordination properties of certain classes of multivalently analytic functions , 2010, Math. Comput. Model..

[8]  Zhi-Hong Liu,et al.  Some criteria for meromorphic multivalent starlike functions , 2011, Appl. Math. Comput..

[9]  T. Umezawa.  Analytic functions convex in one direction. , 1952 .

[10]  E. Deniz,et al.  ON SOME SUFFICIENT CONDITIONS FOR UNIVALENCE , 2010 .

[11]  J. Sokół,et al.  On some sufficient conditions for univalence and starlikeness , 2012 .

[12]  Mamoru Nunokawa,et al.  On properties of non-Carathéodory functions , 1992 .

[13]  Maslina Darus,et al.  On the starlikeness of certain class of analytic functions , 2011, Math. Comput. Model..

[15]  D. Brannan,et al.  On Some Classes of Bounded Univalent Functions , 1969 .

[16]  Janusz Sokól.  On sufficient condition to be in a certain subclass of starlike functions defined by subordination , 2007, Appl. Math. Comput..

[17]  Sanford S. Miller,et al.  Differential Subordinations: Theory and Applications , 2000 .

[18]  N. Seenivasagan.  Sufficient Conditions For Univalence , 2022 .