## 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

### MSC:

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

### Keywords:

starlike functions; convex functions

Zbl 0596.30017