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.


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