Abstract
Let h 1 z and h 2 z be two nonvanishing holomorphic functions in the open unit disc with h 1 0 = h 2 0 = 1 . For some holomorphic function q z , we consider the class consisting of normalized holomorphic functions f whose ratios f z / z q z and q z are subordinate to h 1 z and h 2 z , respectively. The majorization results are obtained for this class when h 1 z is chosen either h 1 z = cos z or h 1 z = 1 + sin z or h 1 z = 1 + z and h 2 z = 1 + sin z .
Highlights
In order to better explain the terminology included in our key observations, some of the essential relevant literature on geometric function theory needs to be provided and discussed here
We start with symbol A which represents the class of holomorphic functions in the region of open unit disc Ud = fz ∈ C : jzj < 1g, and if f ðzÞ is in A, it satisfies the relationship f ð0Þ = f ′ð0Þ − 1 = 0: the family S ⊂ A contains all univalent functions
Though function theory was started in 1851, in 1916, due to coefficient conjecture provided by Bieberbach [1], this field emerged as a good area of new research
Summary
In order to better explain the terminology included in our key observations, some of the essential relevant literature on geometric function theory needs to be provided and discussed here. They identified several subfamilies of a class S of univalent functions linked to various image domains. K ≕ : f ∈ S : R f ′ðzÞ > 0, ðz ∈ UdÞ;: In 1970, Roberston [3] established the idea of quasisubordination among holomorphic functions. H1ðzÞ = 1 + z, h2ðzÞ = 1 + sin z, and by applying the above-mentioned concepts, we consider the following classes:
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