Let $$D\subset \mathbb {C}$$ D ⊂ C and $$0\in D$$ 0 ∈ D . A set D is circularly symmetric if, for each $$\varrho \in \mathbb {R}^+$$ ϱ ∈ R + , a set $$D\cap \{\zeta \in \mathbb {C}:|\zeta |=\varrho \}$$ D ∩ { ζ ∈ C : | ζ | = ϱ } is one of three forms: an empty set, a whole circle, a curve symmetric with respect to the real axis containing $$\varrho $$ ϱ . A function f analytic in the unit disk $$\Delta \equiv \{\zeta \in \mathbb {C}:|\zeta |<1\}$$ Δ ≡ { ζ ∈ C : | ζ | < 1 } and satisfying the normalization condition $$f(0)=f^{\prime }(0)-1=0$$ f ( 0 ) = f ′ ( 0 ) - 1 = 0 is circularly symmetric, if $$f(\Delta )$$ f ( Δ ) is a circularly symmetric set. The class of all such functions is denoted by X. In this paper, we focus on the subclass $$X^{\prime }$$ X ′ consisting of functions in X which are locally univalent. We obtain the results concerned with omitted values of $$f\in X^{\prime }$$ f ∈ X ′ and some covering and distortion theorems. For functions in $$X^{\prime }$$ X ′ we also find the upper estimate of the n-th coefficient, as well as the region of variability of the second and the third coefficients. Furthermore, we derive the radii of starlikeness, convexity and univalence for $$X^{\prime }$$ X ′ .
