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Some sufficient conditions for starlikeness and convexity. (English) Zbl 1195.30030
Summary: There are many results for sufficient conditions of functions $f\left(z\right)$ which are analytic in the open unit disc $𝕌$ to be starlike and convex in $𝕌$. In view of the results due to S. Ozaki, I. Ono and T. Umezawa [Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 5, 111–114 (1956; Zbl 0071.29501)], P.T. Mocanu [Rev. Roum. Math. Pures Appl. 33, No.1–2, 116–124 (1988; Zbl 0652.30004)], and M. Nunokawa [Proc. Japan Acad., Ser. A 69, No.7, 234–237 (1993; Zbl 0793.30007)], some sufficient conditions for starlikeness and convexity of $f\left(z\right)$ are discussed.

##### MSC:
 30C45 Special classes of univalent and multivalent functions
##### Keywords:
univalent function; starlike function; convex function