Radius Estimates and Convolution Properties for Analytic Functions

Abstract

For given real numbers $$\alpha $$ and $$\beta \;(\alpha< 1 <\beta )$$ , let $${\mathcal {P}}(\alpha , \beta )$$ be the class of analytic functions p with $$p(0)=1$$ satisfying $$\alpha< {{\mathrm{\mathfrak {R}}}}\{ p(z) \}< \beta $$ in the open unit disk $${\mathbb {D}}:= \left\{ z\in {\mathbb {C}}:|z|<1 \right\} $$ . For $$|z|=r<1$$ , the lower and upper bounds on the real and imaginary parts for the analytic functions $$p\in {\mathcal {P}}(\alpha , \beta )$$ are investigated. For such functions, the radii are found so that $$p(z)+zp'(z)/p(z)$$ and $$p(z)+zp'(z)$$ belong to the class $${\mathcal {P}}(\alpha , \beta )$$ . With the help of these estimates, the radius estimates of strongly, parabolic and lemniscate starlikeness for starlike functions associated with the vertical strip domain under consideration are obtained. Further, convolution and inclusion properties are also investigated. The results investigated in this paper extend and improve several existing results for starlike functions related to vertical strip domains.