Abstract
The aim of this particular article is at studying a holomorphic function f defined on the open-unit disc D = z ∈ ℂ : z < 1 for which the below subordination relation holds z f ′ z / f z ≺ q 0 z = 1 + tan h z . The class of such functions is denoted by S tan h ∗ . The radius constants of such functions are estimated to conform to the classes of starlike and convex functions of order β and Janowski starlike functions, as well as the classes of starlike functions associated with some familiar functions.
Highlights
To completely comprehend the mathematical concepts used throughout our key observations, some of the essential literature of the geometric function theory must be described and analyzed here
Let us begin with the symbol An which describes the family of holomorphic functions in a subset D of the complex plan C having the following series expansion f ðzÞ = z + an+1zn+1 + an+2zn+2+⋯: ð1Þ
Let the family of all univalent functions be denoted by S and is a subset of the class A1 = A: we define that the subordination between the function belongs to the class A
Summary
To completely comprehend the mathematical concepts used throughout our key observations, some of the essential literature of the geometric function theory must be described and analyzed here. Three significant subfamilies of S, which are well studied and have nice geometric interpretations, are the families of starlike S∗ðξÞ, convex KðξÞ, and strongly starlike S S∗ðζÞ functions of order ξð0 ≤ ξ < 1Þ and ζð0 < ζ ≤ 1Þ, respectively. These families are defined as follows: SS∗ðζÞ. (viii) By considering the function φðzÞ = 1 + sin h−1z, we get the recently examined family S∗ρ ≔ S∗ð1 + sin h−1zÞ introduced by Kumar and Arora [14] They discussed relationships of this class with the already known classes. We discuss the S∗tanh radius for some families of A, whose functions have been expressed as a ratio between two functions
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