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Extensions of sufficient conditions for starlikeness and convexity of order α for multivalent function. (English) Zbl 1254.30016
Summary: We obtain extensions of sufficient conditions for analytic functions f(z) in the open unit disk 𝕌 to be starlike and convex of order α. Our results unify and extend some starlikeness and convexity conditions for analytic functions discussed by P. T. Mocanu [Rev. Roum. Math. Pures Appl. 33, No. 1–2, 116–124 (1988; Zbl 0652.30004)] and N. Uyanık, M. Aydogan and S. Owa [Appl. Math. Lett. 24, No. 8, 1393–1399 (2011; Zbl 1216.30023)] and others.
30C45Special classes of univalent and multivalent functions
[1]Mocanu, P. T.: Some simple criteria for starlikeness and convexity, Libertas math. 13, 27-40 (1993) · Zbl 0793.30008
[2]Mocanu, P. T.; Oros, Gh.: A sufficient condition for starlikeness of order α, Int. J. Math. math. Sci. 28, No. 9, 557-560 (2001) · Zbl 1085.30501 · doi:10.1155/S0161171201011656 · doi:http://ijmms.hindawi.com/volume-28/S0161171201011656.html
[3]Uyanik, N.; Aydogan, M.; Owa, S.: Extension of sufficient conditions for starlikeness and convexity of order α, Appl. math. Lett. 24, No. 9, 1393-1399 (2011) · Zbl 1216.30023 · doi:10.1016/j.aml.2011.03.018
[4]Mocanu, P. T.: Some starlikeness conditions for analytic functions, Rev. roumaine math. Pures appl. 33, 117-124 (1988) · Zbl 0652.30004
[5]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
[6]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