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Extensions of sufficient conditions for starlikeness and convexity of order $\alpha$. (English) Zbl 1216.30023
Summary: For analytic functions $f\left(z\right)$ in the open unit disk $𝕌$, there are many sufficient conditions for $f\left(z\right)$ to be starlike and convex in $𝕌$. 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 $f\left(z\right)$ to be starlike of order $\alpha$ and to be convex of order $\alpha$ are discussed.

##### MSC:
 30C45 Special classes of univalent and multivalent functions
##### 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 · doi:http://mistug.tubitak.gov.tr/bdyim/abs.php?dergi=mat&rak=0807-42 [4] Alexander, J. W.: Functions which map the interior of the unit circle upon simple regions, Ann. of math. 17, 12-22 (1915) · Zbl 45.0672.02 · doi:10.2307/2007212