Abstract: The article introduces a generalization of the subclass $$C'$$ class of close-to-convex functions, which is defined by the condition of positivity of the functional $$ Re(zf'(z)g(z)/)$$, where $$g(z)$$ is an arbitrary convex function. The generalization of this class is related to both the membership of the values of the functional $$zf'(z)g(z)/$$ region of a special type contained in the right half-plane relative to the imaginary axis, and using starlike functions $$g(z)$$ of arbitrary order. In this case, functions with gaps in the expansion in a series are considered. In a generalized subclass of the class of close-to-convex functions, theorems of distortion and rotation, as well as the radius of convexity of this class, are established. All results are exact. The breadth of the introduced class of close-to-convex functions allows us to obtain a generalization of a number of results both for various subclasses of the class of close-to-convex functions, including the entire class of close-to-convex functions, and for the class of close-to-starlike functions and its subclasses. In particular, new results are obtained for classes of functions convex in the direction of the imaginary axis or in the positive direction of the real axis, as well as for typically real functions.It is also shown that using starlike functions of order 1/2 instead of convex functions leads to the same distortion, rotation, and convexity radius theorems as for the classical subclass $$C'$$ close-to-convex functions.

Keywords: univalent functions, starlike functions, close-to-convex functions, close-to-starlike functions, radii of convexity, radii of starlikeness, distortion theorems, growth theorems

Pages: 431-442