Starlike functions associated with the generalized Koebe function

Abstract In this paper we study some properties of functions f which are analytic and normalized (i.e. $$f(0)=0=f'(0)-1$$ f ( 0 ) = ′ - 1 ) such that satisfy the following subordination relation $$\begin{aligned} \left( \frac{zf'(z)}{f(z)}-1\right) \prec \frac{z}{(1-pz)(1-qz)}, \end{aligned}$$ z ≺ p q , where $$(p,q) \in [-1,1] \times [-1,1]$$ ∈ [ ] × . These types starlike related to generalized Koebe function. Some features are: radius starlikeness order $$\gamma [0,1)$$ γ , image $$f\left( \{z:|z|<r\}\right) $$ { : | < r } $$r\in (0,1)$$ convexity, estimation initial logarithmic coefficients, Fekete–Szegö problem.