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Extensions of sufficient conditions for starlikeness and convexity of order $$\alpha$$. (English) Zbl 1216.30023
Summary: For analytic functions $$f(z)$$ in the open unit disk $$\mathbb U$$, there are many sufficient conditions for $$f(z)$$ to be starlike and convex in $$\mathbb U$$. In view of the paper by P. T. Mocanu [Rev. Roum. Math. Pures Appl. 33, No. 1–2, 116–124 (1988; Zbl 0652.30004)], some extensions of sufficient conditions for $$f(z)$$ to be starlike of order $$\alpha$$ and to be convex of order $$\alpha$$ are discussed.

##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
Full Text:
##### References:
 [1] Mocanu, P.T., Some starlikeness conditions for analytic functions, Rev. roumaine math. pures appl., 33, 117-124, (1988) · Zbl 0652.30004 [2] Mocanu, P.T., Some simple criteria for starlikeness and convexity, Libertas math., 13, 27-40, (1993) · Zbl 0793.30008 [3] Nunokawa, M.; Owa, S.; Polatoglu, Y.; Caglar, M.; Duman, E.Y., Some sufficient conditions for starlikeness and convexity, Turkish J. math., 34, 333-337, (2010) · Zbl 1195.30030 [4] Alexander, J.W., Functions which map the interior of the unit circle upon simple regions, Ann. of math., 17, 12-22, (1915) · JFM 45.0672.02
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