Certain subclass of analytic functions. (English) Zbl 1090.30012

Darus, Maslina

J. Inst. Math. Comput. Sci., Math. Ser. 16, No. 2, 89-93 (2003).

Summary: Let \(f\) be analytic in \(D=\{z:|z|<1\}\) with \(f(0)=f'(0)-1=0\). Suppose \(\lambda \geq 0\) and \(\lambda+\mu>0\). For \(0<\beta\leq 1\), the largest \(\alpha(\beta, \lambda,\mu)\) is found such that \[ \lambda \left(1+\frac{zf''(z)}{f'(z)}\right)+\mu\frac{zf'(z)}{f(z)}\prec \left(\frac {1+z}{1-z}\right)^{\alpha(\beta,\lambda, \mu)}\Rightarrow\frac{zf'(z)} {f(z)}\prec\left(\frac{1+z}{1-z}\right)^\beta. \] The result solves the inclusion problem for certain subclass of analytic functions defined in a sector.

MSC:

30C45

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