Certain classes of analytic functions of complex order and type beta. (English) Zbl 0764.30009

M. K. Aouf and M. A. Nasr [J. Nat. Sci. Math., 25, No. 1, 1- 12 (1985; Zbl 0596.30017)] introduced the class of starlike functions of order \((1-b)\), (\(b\neq 0\), complex). That is \(f(z)\in S(1-b)\) iff \(f(z)/z\neq 0\) in the unit disc \(U\) and \[ \text{Re}\left\{1+ {1\over b} \left({{zf'(z)} \over {f(z)}}-1\right)\right\}>0, \qquad z\in 0. \] Also the related class \(C(b)\), convex functions of order \(b\), \(g(z)\in C(b)\) iff \(zg'(z)\in S(1-b)\) was studied by the authors and others in earlier papers.
In the present paper, for fixed \(\beta\), \(0<\beta\leq 1\), the authors refine the definition (definition 3) to subclasses starlike of order \(1- b\), type \(\beta\). These are defined by the requirement, \(f(z)\in S(1- b,\beta)\) iff \[ \left| {{{{zf'(z)} \over {f(z)}}-1} \over {2\beta \bigl({{zf'(z)} \over {f(z)}}-1+b\bigr)- \bigl({{zf'(z)} \over {f(z)}}- 1\bigr)}} \right| <1, \qquad \text{for all }z\in U. \] Functions are convex of order \(b\) and type \(\beta\), \(g(z)\in C(b,\beta)\) iff \(zg'(z)\in S(1-b,\beta)\). The authors prove representation theorems, coefficients bounds and radii of starlikeness theorems for these classes. For special values of b and \(\beta\) many known results follow.
Reviewer: D.Shaffer


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


Zbl 0596.30017