## Certain subclass of analytic functions.(English)Zbl 1090.30012

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.)