Abstract
In this paper, by using a technique of the first-order differential subordination, we find several sufficient conditions for an analytic function p such that p(0)=1 to satisfy operatorname{Re}{ {mathrm{e}}^{{mathrm{i}}beta } p(z) } > gamma or | arg {p(z)-gamma } |<delta for all zin mathbb{D}, where beta in (-pi /2,pi /2), gamma in [0,cos beta ), delta in (0,1] and mathbb{D}:={zin mathbb{C}:|z|<1 }. The results obtained here will be applied to find some conditions for spirallike functions and strongly starlike functions in mathbb{D}.
Highlights
Introduction and definitionsFor real numbers β, γ, and δ satisfying –π/2 < β < π/2, 0 ≤ γ < cos β, and 0 < δ ≤ 1, define two domains γ (β) and γ (δ) in C by γ (β) = w ∈ C : Re e–iβ w > γ and (δ) = w ∈ C : arg(w – γ ) < π δ, γ respectively
By using a technique of the first-order differential subordination, we find several sufficient conditions for an analytic function p such that p(0) = 1 to satisfy Re{eiβ p(z)} > γ or | arg{p(z) – γ }| < δ for all z ∈ D, where β ∈ (–π /2, π /2), γ ∈ [0, cos β), δ ∈
The results obtained here will be applied to find some conditions for spirallike functions and strongly starlike functions in D
Summary
By using a technique of the first-order differential subordination, we find several sufficient conditions for an analytic function p such that p(0) = 1 to satisfy Re{eiβ p(z)} > γ or | arg{p(z) – γ }| < δ for all z ∈ D, where β ∈ (–π /2, π /2), γ ∈ [0, cos β), δ ∈ The results obtained here will be applied to find some conditions for spirallike functions and strongly starlike functions in D. As direct consequences of these results, we will obtain several sufficient conditions for spirallike functions or strongly starlike functions in D.
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