Abstract
In this paper we investigate an interesting subclass $\mathcal{BS}(α)$ ($0\leq α<1$) of starlike functions in the unit disk $Δ$. The class $\mathcal{BS}(α)$ was introduced by Kargar et al. [R. Kargar, A. Ebadian and J. Sokół, {\it On Booth lemniscate and starlike functions}, Anal. Math. Phys. (2017) DOI: 10.1007/s13324-017-0187-3] which is strongly related to the Booth lemniscate. Some geometric properties of this class of analytic functions including, radius of starlikeness of order $γ$ ($0\leq
Results & Lemmas (12)
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Lemma 1.1. · radius
Lemma 1.1. (see [3]) Let Fα(z) be given by (1.3). Then for 0 ≤α < 1, we have (1.6) 1 α −1 < Re Fα(z) < 1 1 −α (z ∈∆). 2010 Mathematics…
Lemma 1.1. (see [3]) Let Fα(z) be given by (1.3). Then for 0 ≤α < 1, we have (1.6) 1 α −1 < Re {Fα(z)} < 1 1 −α (z ∈∆). 2010 Mathematics Subject Classification. 30C45. Key words and phrases. Booth lemniscate, radius of satarlikeness, starlike function, convex function, subordination. 1 arXiv:1802.03799v1 [math.CV] 11 Feb 2018
Lemma 1.2.
Lemma 1.2. (see [6]) Let F, G ∈H be any convex univalent functions in ∆. If f ≺F and g ≺G, then f ∗g ≺F ∗G in ∆. In this work, some…
Lemma 1.2. (see [6]) Let F, G ∈H be any convex univalent functions in ∆. If f ≺F and g ≺G, then f ∗g ≺F ∗G in ∆. In this work, some geometric properties of the class BS(α) are investigated. 2. Main results We start with the following lemma that gives the structural formula for the function of the considered class.
Lemma 2.1.
Lemma 2.1. The function f ∈A belongs to the class BS(α), 0 ≤α < 1, if and only if there exists an analytic function q, q(0) = 0 and q ≺Fα…
Lemma 2.1. The function f ∈A belongs to the class BS(α), 0 ≤α < 1, if and only if there exists an analytic function q, q(0) = 0 and q ≺Fα such that (2.1) f(z) = z exp Z z 0 q(t) t dt . The proof is easy. Putting q = Fα in Lemma 2.1 we obtain the function (2.2) ˜f(z) = z 1 + z√α 1 −z√α
Lemma 2.2.
Lemma 2.2. (Schwarz lemma) (see [1]) Let w be analytic in the unit disc ∆, with w(0) = 0 and |w(z)| < 1 in ∆. Then |w′(0)| ≤1 and |w(z)|…
Lemma 2.2. (Schwarz lemma) (see [1]) Let w be analytic in the unit disc ∆, with w(0) = 0 and |w(z)| < 1 in ∆. Then |w′(0)| ≤1 and |w(z)| ≤|z| in ∆. Strict inequality holds in both estimates unless w is a rotation of the disc: w(z) = eiθz.
Theorem 2.1.
Theorem 2.1. Let α ∈(0, 1) and γ ∈[0, 1) be given numbers. If f ∈BS(α), then f is starlike of order γ in the disc |z| < rs(α, γ) = √…
Theorem 2.1. Let α ∈(0, 1) and γ ∈[0, 1) be given numbers. If f ∈BS(α), then f is starlike of order γ in the disc |z| < rs(α, γ) = √ 1+4α(1−γ)−1 2α(1−γ) . The result is sharp.
Corollary 2.1.
Corollary 2.1. Let α ∈(0, 1). If f ∈BS(α) then f is starlike univalent in the disc |z| < rs(α) = √1+4α−1 2α. The result is sharp.
Corollary 2.1. Let α ∈(0, 1). If f ∈BS(α) then f is starlike univalent in the disc |z| < rs(α) = √1+4α−1 2α . The result is sharp.
Theorem 2.2.
Theorem 2.2. Let r ∈(0, 1] be the given number. If 0 ≤α < 1−r r2, then each function f ∈BS(α) maps a disc |z| < r onto a starlike domain.
Theorem 2.2. Let r ∈(0, 1] be the given number. If 0 ≤α < 1−r r2 , then each function f ∈BS(α) maps a disc |z| < r onto a starlike domain.
Theorem 2.3.
Theorem 2.3. Let n ≥2 be integer. If one of the following conditions holds (i) 1 α+n(1−α) < |c| < 1, (ii) n > 3−α 1−α and 1 α−2+n(1−α) <…
Theorem 2.3. Let n ≥2 be integer. If one of the following conditions holds (i) 1 α+n(1−α) < |c| < 1, (ii) n > 3−α 1−α and 1 α−2+n(1−α) < |c| < 1, (iii) n ≥2−α 1−α and |c| > 1,
Theorem 2.4.
Theorem 2.4. The function F(z) −1 is convex univalent in ∆.
Theorem 2.4. The function F(z) −1 is convex univalent in ∆.
Lemma 2.3.
Lemma 2.3. (see [4]) Suppose that Fα is given by (1.3). If 0 ≤α ≤3 −2 √ 2 ≈0.1715, then the curve Fα(eiϕ), ϕ ∈[0, 2π), is convex. If α ∈(3…
Lemma 2.3. (see [4]) Suppose that Fα is given by (1.3). If 0 ≤α ≤3 −2 √ 2 ≈0.1715, then the curve Fα(eiϕ), ϕ ∈[0, 2π), is convex. If α ∈(3 −2 √ 2, 1), then the curve Fα(eiϕ), ϕ ∈[0, 2π), is concave. Moreover, in both cases this curve is symmetric with respect to both axes.
Theorem 2.5.
Theorem 2.5. If a function f belongs to the class BS(α), 0 ≤α ≤3 −2 √ 2, then (2.6) f(z) z ≺F(z) (z ∈∆), where F(z) is given by (2.3).
Theorem 2.5. If a function f belongs to the class BS(α), 0 ≤α ≤3 −2 √ 2, then (2.6) f(z) z ≺F(z) (z ∈∆), where F(z) is given by (2.3).
Theorem 2.6.
Theorem 2.6. Let f ∈BS(α), 0 ≤α ≤3 −2 √ 2 and |z| = r < 1. Then (2.10) 1 −r√α 1 + √α 1 2√α ≤Re f(z) z ≤ 1 + r√α
Theorem 2.6. Let f ∈BS(α), 0 ≤α ≤3 −2 √ 2 and |z| = r < 1. Then (2.10) 1 −r√α 1 + √α 1 2√α ≤Re f(z) z ≤ 1 + r√α
Function classes studied:
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