Some subclasses of close-to-convex mappings associated with conic regions
In this paper, we introduce and investigate several new subclasses of close-to-convex functions and their related mappings. Various interesting properties including sufficiency criteria, coefficient estimates, arc-length problem and radius of convexity are investigated.
On certain p-valent close-to-convex functions of order $$\beta $$β and type $$\alpha $$α
A new class $$K_p(\alpha ,\beta )$$Kp(α,β) consisting of the functions which are p-valent close-to-convex of order $$\beta $$β and type $$\alpha $$α is introduced. The object of the present paper is to derive some sufficient conditions for functions to be in the class $$K_p(\alpha ,\beta )$$Kp(α,β).
On Hilbert's thirteenth problem and related questions
Hilbert's thirteenth problem involves the study of solutions of algebraic equations. The object is to obtain a complexity estimate for an algebraic function. As of now, the problem remains open. There are only a few partial algebraic results in this connection, but at the same time the problem has stimulated a series of studies in the theory of functions with their subsequent applications. The most brilliant result in this cycle is Kolmogorov's theorem on superpositions of continuous functions.