# zbMATH — the first resource for mathematics

Certain classes of multivalent functions defined with higher-order derivatives. (English) Zbl 1414.30013
Summary: In this paper we derive some properties of multivalent functions belonging to the classes $$R_{p,q}(\alpha)$$, $$B_{p,q}(\alpha)$$, and $$M_{p,q}(\alpha)$$. The results obtained generalize the related works of some authors, and many other new results are obtained.
##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
Full Text:
##### References:
 [1] Aouf MK. On a class of p-valent starlike functions of order α . Int J Math Math Sci 1987; 10: 733-744 · Zbl 0624.30021 [2] Aouf MK. A generalization of multivalent functions with negative coefficients. J Korean Math Soc 1988; 25: 53-66. · Zbl 0648.30015 [3] Aouf MK. Certain classes of multivalent functions with negative coefficients defined by using a differential operator. J Math Appl 2008; 30: 5-21. · Zbl 1159.30304 [4] Aouf MK. Certain subclasses of p-valent starlike functions defined by using a differential operator. Appl Math Comput 2008; 206: 867-875. · Zbl 1155.30305 [5] Aouf MK. Some families of p-valent functions with negative coefficients. Acta Math Univ Comenian (NS) 2009; 78: 121-135. · Zbl 1199.30042 [6] Aouf MK. Bounded p-valent Robertson functions defined by using a differential operator. J Franklin Inst 2010; 347: 1972-1941. · Zbl 1207.30013 [7] Aouf MK. Some inclusion relationships associated with Dizok-Srivastava operator. Appl Math Comput 2010; 216: 431-437. · Zbl 1186.30008 [8] Bulboacă T. Differential Subordinations and Superordinations. New Results. Cluj-Napoca, Romania: House of Scientific Book Publications, 2005. [9] Fukui S, Ren F, Owa S, Nunokawa M. On certain multivalent functions. Bull Fac Edu Wakayama Univ Nat Sci 1989; 38: 5-8. [10] Jack IS. Functions starlike and convex of order α . J Lond Math Soc 1971; 2: 469-474. · Zbl 0224.30026 [11] Lee SK, Owa S. A subclass of p-valently close to convex functions of order α . Appl Math Lett 1992; 5: 3-6. · Zbl 0773.30010 [12] McCarty CP. Functions with real part greater than α . P Am Math Soc 1972; 35: 211-216. · Zbl 0258.30014 [13] McCarty CP. Two radius of convexity problems. P Am Math Soc 1974; 42: 153-160. · Zbl 0286.30005 [14] Miller SS. Distortion properties of alpha-starlike functions. P Am Math Soc 1973; 38: 311-318. · Zbl 0264.30014 [15] Miller SS, Mocanu PT. Differential Subordinations. Theory and Applications. Series on Monographs and Textbooks in Pure and Applied Mathematics, No. 255. New York, NY, USA: Marcel Dekker, 2000. [16] Miller SS, Mocanu PT, Reade MO. The order of starlikeness of alpha-convex functions. Mathematica (Cluj) 1978; 20: 25-30. · Zbl 0398.30008 [17] Mocanu PT. Une propriété de convexité generaliseé dans la théorie de la représentation conforme. Mathematica (Cluj) 1969; 11: 127-133 (in French). · Zbl 0195.36401 [18] Mocanu PT, Reade MO. On generalized convexity in conformal mappings. Rev Roum Math Pures Appl 1971; 46: 1541-1544. · Zbl 0224.30025 [19] Nunokawa M. On the theory of multivalent functions. Tsukuba J Math 1987; 11: 273-286. · Zbl 0639.30014 [20] Owa S. On certain classes of p-valent functions with negative coefficients. Bull Belg Math Soc Simon Stevin 1985; 59: 385-402. · Zbl 0593.30018 [21] Owa S. Some properties of certain multivalently functions. Math Nachr 1992; 155: 167-185. · Zbl 0771.30012 [22] Owa S, Aouf MK, Nasr MA. Note on certain subclass of close-to-convex functions of order α . Int J Math Math Sci 1990; 13: 189-192. · Zbl 0704.30018 [23] Owa S, Ma W, Liu L. On a class of analytic functions satisfying Re(f′(z)) > α . Bull Korean Math Soc 1988; 25: 211-224. 726 AOUF et al./Turk J Math [24] Owa S, Ren F. On a class of p-valently α -convex functions. Math Nachr 1990; 146: 17-21. · Zbl 0724.30014 [25] Saitoh H. Some properties of certain analytic functions. Topics in Univalent Functions and Its Applications 1990; 714: 160-167. [26] Saitoh H. Some properties of certain multivalent functions. Tsukuba J Math 1991; 15: 105-111. · Zbl 0737.30009 [27] Saitoh H. On certain class of multivalent functions. Math Japon 1992; 37: 871-875. · Zbl 0761.30009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.