Applications of extremal theorem and radius equation for a class of analytic functions

Abstract

Abstract A linear operator D 𝒫 k f(z) : S → S is introduced, where S denotes the class of univalent analytic functions in open unit disk and 𝒫 k is an arbitrary monotonically increasing function in k. We study a new class 𝒯𝒫 k (α 1, α 2, α 3, β) of analytic functions related to D 𝒫 k f(z). The main object of this paper is to give the radius equation between the class 𝒯𝒫 k (α 1, α 2, α 3,β) and the close-to-convex class of order α (0 ≤ α < 1). Also, we apply the extremal theorem to maximize |f (𝒳)(z)| over 𝒯𝒫 k (α 1, α 2, α 3, β), where 𝒳 = {0, 1, 2, …, k}, all the results are sharp.