Abstract
In the present exploration, the authors define and inspect a new class of functions that are regular in the unit disc D ≔ ς ∈ ℂ : ς < 1 , by using an adapted version of the interesting analytic formula offered by Robertson (unexploited) for starlike functions with respect to a boundary point by subordinating to an exponential function. Examples of some new subclasses are presented. Initial coefficient estimates are specified, and the familiar Fekete-Szegö inequality is obtained. Differential subordinations concerning these newly demarcated subclasses are also established.
Highlights
Introduction and Preliminary ResultsLet H be the class comprising of all holomorphic functions in the unit disc D ≔ fς ∈ C : jςj < 1g
Let A signify the subclass of H entailing of functions h ∈ A be of the form hðςÞ = ς + 〠 anςn, ς ∈ D, ð1Þ
≺ g if there exists ω ∈ H with ωð0Þ = 0 and ωðDÞ ⊂ D such that f ðςÞ = gðωðςÞÞ for every ς ∈ D: In precise, if g is univalent, f ≺ g if and only if f ð0Þ = gð0Þ and f ðDÞ ⊂ gðDÞ: Let P symbolize the class of functions p ∈ H with the normalization pð0Þ = 1, i.e., of the form pðςÞ = 1 + 〠 pnςn, ς ∈ D, ð3Þ
Summary
Let H be the class comprising of all holomorphic functions in the unit disc D ≔ fς ∈ C : jςj < 1g. ≺ g if there exists ω ∈ H with ωð0Þ = 0 and ωðDÞ ⊂ D such that f ðςÞ = gðωðςÞÞ for every ς ∈ D: In precise, if g is univalent, f ≺ g if and only if f ð0Þ = gð0Þ and f ðDÞ ⊂ gðDÞ: Let P symbolize the class of functions p ∈ H with the normalization pð0Þ = 1, i.e., of the form pðςÞ = 1 + 〠 pnςn, ς ∈ D, ð3Þ
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