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New properties for starlike and convex functions of complex order. (English) Zbl 1197.30006

Summary: For functions \(f(z)\) which are starlike of complex order \(b\) \((b \neq 0)\) in the open unit disk \(\mathbb U\) introduced by M. A. Nasr and M. K. Aouf [J. Nat. Sci. Math. 25, No. 1, 1–12 (1985; Zbl 0596.30017)], in consideration of its properties, some sufficient conditions, necessary conditions and distortion theorems with coefficient inequalities of \(f(z)\) as the improvement of the well-known result due to H. Silverman [Proc. Am. Math. Soc. 51, 109–116 (1975; Zbl 0311.30007)] are discussed. Furthermore, some new coefficient inequalities including some new interesting corollaries are discussed.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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