Abstract
For an analytic function $f$ defined on the unit disk $|z|<1$, let $Δ(r,f)$ denote the area of the image of the subdisk $|z|<r$ under $f$, where $0<r\le 1$. In 1990, Yamashita conjectured that $Δ(r,z/f)\le πr^2$ for convex functions $f$ and it was finally settled in 2013 by Obradović and et. al.. In this paper, we consider a class of analytic functions in the unit disk satisfying the subordination relation $zf'(z)/f(z)\prec (1+(1-2β)αz)/(1-αz)$ for $0\le β<1$ and $0<α\le 1$. We prove Yamashita's
Results & Lemmas (7)
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Theorem 1.3.
Theorem 1.3. (Main Theorem) Let 0 < α ≤1 and 0 ≤β < 1. If f(z) ∈S(α, β) and z/f(z) is a non-vanishing analytic function in D, then we have…
Theorem 1.3. (Main Theorem) Let 0 < α ≤1 and 0 ≤β < 1. If f(z) ∈S(α, β) and z/f(z) is a non-vanishing analytic function in D, then we have the maximal area max f∈S(α,β) ∆ ρ, z f = 4πα2(1 −β)2ρ2 2F1(2β −1, 2β −1; 2; α2ρ2), |z| < ρ =: Aα,β(ρ) for all ρ, 0 < ρ ≤1. The maximum is attained only by the rotations of the function kα,β(z) defined by (1.2). This generalizes the main results which are discussed in [11] and [21].
Lemma 2.1.
Lemma 2.1. Let 0 < α ≤1, 0 ≤β < 1, and f(z) ∈S(α, β). If g(z) is a non-vanishing analytic function in D of the form (2.1), then it…
Lemma 2.1. Let 0 < α ≤1, 0 ≤β < 1, and f(z) ∈S(α, β). If g(z) is a non-vanishing analytic function in D of the form (2.1), then it necessarily satisfies the coefficient inequality ∞ X k=1 k2 −(k −2(1 −β))2α2 |bk|2 ≤4(1 −β)2α2.
Lemma 2.2.
Lemma 2.2. Let 0 < α ≤1, 0 ≤β < 1, and f(z) ∈S(α, β). For |z| < ρ ≤1 suppose that z f(z) = 1 + ∞ X k=1 bkzk and (1 + αz)2−2β = 1 + ∞ X k=1…
Lemma 2.2. Let 0 < α ≤1, 0 ≤β < 1, and f(z) ∈S(α, β). For |z| < ρ ≤1 suppose that z f(z) = 1 + ∞ X k=1 bkzk and (1 + αz)2−2β = 1 + ∞ X k=1 (−1)kckzk.
Lemma 2.3.
Lemma 2.3. Let 0 < α ≤1 and 0 ≤β < 1. Then f(z) ∈S(α, β) if and only if F defined by F(z) = z (f(z)/z) 1 1−β ∈S(α), z ∈D.
Lemma 2.3. Let 0 < α ≤1 and 0 ≤β < 1. Then f(z) ∈S(α, β) if and only if F defined by F(z) = z (f(z)/z) 1 1−β ∈S(α), z ∈D.
Theorem 3.1.
Theorem 3.1. Let for 0 < α ≤1, f ∈S(α) and z/f(z) be a non-vanishing analytic function in D. Then we have max f∈S(α) ∆ ρ, z f =…
Theorem 3.1. Let for 0 < α ≤1, f ∈S(α) and z/f(z) be a non-vanishing analytic function in D. Then we have max f∈S(α) ∆ ρ, z f = 2πα2ρ2(2 + α2ρ2) =: Aα(ρ) for all ρ, 0 < ρ ≤1. The maximum is attained only by the rotation of the function kα(z) defined by (1.3). We now collect the values of Aα(1) in the form of a table (see Table 2) for several values of α and affix geometrical pictures of the images of the unit disk under the extremal functions gα(z) = z/kα(z) = (1 −αz)2 for the corresponding α v
Corollary 3.2.
Corollary 3.2. [11, Theorem 2] We have max f∈St(1/2) ∆ ρ, z f = πρ2 for 0 < ρ ≤1, where the maximum is attained only by the rotation of…
Corollary 3.2. [11, Theorem 2] We have max f∈St(1/2) ∆ ρ, z f = πρ2 for 0 < ρ ≤1, where the maximum is attained only by the rotation of the Koebe function k(z) defined by (1.3). Moreover, if we choose α = 1 in Theorem 1.3, we get
Corollary 3.3.
Corollary 3.3. [11, Theorem 3] Let f ∈St(β) for some 0 ≤β < 1. Then we have max f∈St(β) ∆ ρ, z f = 4π(1 −β)2ρ2 2F1(2β −1, 2β −1; 2; ρ2)…
Corollary 3.3. [11, Theorem 3] Let f ∈St(β) for some 0 ≤β < 1. Then we have max f∈St(β) ∆ ρ, z f = 4π(1 −β)2ρ2 2F1(2β −1, 2β −1; 2; ρ2) for 0 < ρ ≤1, where the maximum is attained only by the rotation of the function kβ(z) defined by (1.3). 4. Concluding Remark For −1 ≤B < A ≤1, the Janowski class S∗(A, B) is defined by the subordination relation S∗(A, B) :=
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