Abstract:
The article introduces and investigates a special subclass of close-to-convex functions in a unit disk which is defined in terms of starlike functions of the order 1/2. Functions of this class, as well as starlike functions of the order 1/2, may have omissions in the decomposition.
Some close subclasses have been actively studied in the works of Gao C.-Y., Zhou S.-Q., Kowalczyk J., Les-Bomba E., Prajapat J.K., and other authors published in the last decade. The class of functions introduced in this paper is characterized by the fact that the range of values of the functional used is contained in the right half of the generalized Bernoulli lemniscate with a nodal point 0 and a given angle between the tangents at the nodal point, which allows us to consider many special cases. For example, a disk of arbitrary radius with a point 0 at the border, an angle with a vertex at point 0 of a given size, and others. For the introduced class of functions and its subclasses, exact distortion theorems and convexity radii are found, and extreme functions are given. Both new results and generalizations of previously known results are obtained.
To solve the extreme problems under consideration, the article provides accurate estimates of the logarithmic derivative in the class of analytical functions, the codomain of values of which are contained in the right half of the generalized Bernoulli lemniscate. These estimates are given both for the case of a standard decomposition of a function in a series, and in the presence of omissions in the decomposition and summarize the well-known results of Goel R.M. and Shaffer D.B.
Keywords:starlike functions, close-to-convex functions, distortion estimates, radii of convexity.
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