Ma-Minda φ-classes studied in this paper:
Abstract
Let $\mathcal{A}$ be the set of all analytic functions $f$ defined in the open unit disk $\mathbb{D}$ and satisfying $f(0)=f'(0)-1=0$. In this paper, we consider the function $\varphi_{\scriptscriptstyle {Ne}}(z):=1+z-z^3/3$, which maps the unit circle $\{z:|z|=1\}$ onto a $2$-cusped curve called nephroid given by $\left((u-1)^2+v^2-\frac{4}{9}\right)^3-\frac{4 v^2}{3}=0$, and the function class $\mathcal{S}^*_{Ne}$ defined as \begin{align*} \mathcal{S}^*_{Ne}:=\left\{f\in\mathcal{A}:\frac{zf'(z
Results & Lemmas (23)
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Lemma 1.1
Lemma 1.1 ([20, Theorem 3.4h, p. 132]). Let q: D →C be univalent, and let λ and ϑ be analytic in a domain Ω⊇q(D) with λ(ξ) ̸= 0 whenever ξ…
Lemma 1.1 ([20, Theorem 3.4h, p. 132]). Let q : D →C be univalent, and let λ and ϑ be analytic in a domain Ω⊇q(D) with λ(ξ) ̸= 0 whenever ξ ∈q(D). Define Θ(z) := zq′(z) λ(q(z)) and h(z) := ϑ(q(z)) + Θ(z), z ∈D. Suppose that either (i) h(z) is convex, or (ii) Θ(z) is starlike. In addition, assume that
Theorem 2.1.
Theorem 2.1. Let p ∈H satisfies p(0) = 1, and let ϕL(z):= √1 + z, where the branch of the square root is chosen in order that ϕL(0) = 1.…
Theorem 2.1. Let p ∈H satisfies p(0) = 1, and let ϕL(z) := √1 + z, where the branch of the square root is chosen in order that ϕL(0) = 1. Then each of the following subordi- nations imply p(z) ≺ϕNe(z) := 1 + z −z3/3. (a) 1 + βzp′(z) ≺ϕL(z) for β ≥3(1 −log 2) ≈0.920558. (b) 1 + β zp′(z) p(z) ≺ϕL(z) for β ≥ 2( √ 2+log 2−1−log(1+ √ 2)) log(5/3)
Corollary 2.1.
Corollary 2.1. Let f ∈A, and let G(z):= 1 −zf ′(z) f(z) + zf ′′(z) f ′(z), z ∈D. (2.3) Then each of the following is sufficient to imply f…
Corollary 2.1. Let f ∈A, and let G(z) := 1 −zf ′(z) f(z) + zf ′′(z) f ′(z) , z ∈D. (2.3) Then each of the following is sufficient to imply f ∈S∗ Ne. (a) 1 + βG(z) zf′(z) f(z) ≺ϕL(z) for β ≥3(1 −log 2),
Theorem 2.1.
Theorem 2.1.
Theorem 2.1.
Theorem 2.2.
Theorem 2.2. Let ϕRL(z):= √ 2 −( √ 2 −1) s 1 −z 1 + 2( √ 2 −1)z and g0(z):= r 2
Theorem 2.2. Let ϕRL(z) := √ 2 −( √ 2 −1) s 1 −z 1 + 2( √ 2 −1)z and g0(z) := r 2
Lemma 1.1
Lemma 1.1 as Θ = ϕRL −1 and h = 1 + Θ = ϕRL. Again, the function h(z) = ϕRL(z)
Lemma 1.1 as Θ = ϕRL −1 and h = 1 + Θ = ϕRL. Again, the function h(z) = ϕRL(z)
Corollary 2.2.
Corollary 2.2. Let f ∈A and let G(z) be given by (2.3). Then each of the following conditions imply f ∈S∗ Ne. (a) 1 + βG(z) zf′(z) f(z) …
Corollary 2.2. Let f ∈A and let G(z) be given by (2.3). Then each of the following conditions imply f ∈S∗ Ne. (a) 1 + βG(z) zf′(z) f(z) ≺ϕRL(z) for β ≥− 3 2( √ 2−1) log 1
Theorem 2.3.
Theorem 2.3. Let ϕSG(z):= 2/(1 + e−z), and let ℓ(z) = Z z 0 et −1 t (et + 1) dt. Then, for p ∈H with p(0) = 1, each of the following…
Theorem 2.3. Let ϕSG(z) := 2/(1 + e−z), and let ℓ(z) = Z z 0 et −1 t (et + 1) dt. Then, for p ∈H with p(0) = 1, each of the following differential subordinations is suffi- cient for the subordination p ≺ϕNe: (a) 1 + βzp′ ≺ϕSG(z) for β ≥3ℓ(1)/2 ≈0.730333, (b) 1 + βzp′/p ≺ϕSG(z) for β ≥ℓ(1)/log(5/3) ≈0.953141, (c) 1 + βzp′/p2 ≺ϕSG(z) for β ≥5ℓ(1)/2 ≈1.21722. The bounds on β can not be improved further.
Theorem 2.4.
Theorem 2.4. Let ϕe(z):= ez be the exponential function, and let p(z) be analytic such that p(0) = 1. If any one of the following…
Theorem 2.4. Let ϕe(z) := ez be the exponential function, and let p(z) be analytic such that p(0) = 1. If any one of the following differential subordinations hold true, then p ≺ϕNe. Each estimate on β is sharp. (a) 1 + βzp′ ≺ϕe(z) for β ≥P∞ n=1 3 2n(n!) ≈1.97685, (b) 1 + βzp′/p ≺ϕe(z) for β ≥ P∞ n=1 1 n(n!) log(5/3) ≈2.57995,
Theorem 3.1.
Theorem 3.1. Let p ∈H with p(0) = 1, and let ϕ (z) for β ≥ 3( √ 2−log(1+ √ 2)+log 2) 2 ≈1.83898, (b) 1 + β zp′(z) p(z)
Theorem 3.1. Let p ∈H with p(0) = 1, and let ϕ$(z) := z + √ 1 + z2. If any of the following differential subordinations hold true, then p ≺ϕNe. (a) 1 + βzp′(z) ≺ϕ$(z) for β ≥ 3( √ 2−log(1+ √ 2)+log 2) 2 ≈1.83898, (b) 1 + β zp′(z) p(z)
Lemma 1.1
Lemma 1.1, we have 1 + βzp′/p ≺1 + βzˆq′ β/ˆqβ =⇒p ≺ˆqβ. To prove p ≺ϕNe, it only remains to show that ˆqβ ≺ϕNe. The later subordination is…
Lemma 1.1, we have 1 + βzp′/p ≺1 + βzˆq′ β/ˆqβ =⇒p ≺ˆqβ. To prove p ≺ϕNe, it only remains to show that ˆqβ ≺ϕNe. The later subordination is true if, and only if, 1/3 < ˆqβ(−1) < ˆqβ(1) < 5/3. This condition on further simplification shows that ˆqβ ≺ϕNe provided β ≥max − √ 2 −2 + log 2 −log 1 +
Corollary 3.1.
Corollary 3.1. Let f ∈A and let G(z) be defined by (2.3). Then each of the following conditions sufficiently implies that f is a member of S∗…
Corollary 3.1. Let f ∈A and let G(z) be defined by (2.3). Then each of the following conditions sufficiently implies that f is a member of S∗ Ne. (a) 1 + βG(z)(zf ′/f) ≺ϕ$(z) for β ≥ 3( √ 2−log(1+ √ 2)+log 2) 2 , (b) 1 + βG(z) ≺ϕ$(z) for β ≥ √ 2+log(2)−log(1+ √
Theorem 3.2.
Theorem 3.2. Let ϕC(z):= 1 + 4z/3 + 2z2/3. Then, for p ∈H with p(0) = 1, each of the following subordinations is sufficient to imply that p…
Theorem 3.2. Let ϕC(z) := 1 + 4z/3 + 2z2/3. Then, for p ∈H with p(0) = 1, each of the following subordinations is sufficient to imply that p ≺ϕNe. Moreover, each estimate on β is sharp. (a) 1 + βzp′ ≺ϕC(z) for β ≥5/2, (b) 1 + βzp′/p ≺ϕC(z) for β ≥5/3 log(5/3) ≈3.26269, (c) 1 + βzp′/p2 ≺ϕC(z) for β ≥25/6.
Theorem 3.3.
Theorem 3.3. Let p ∈H with p(0) = 1, and let ϕ0(z):= 1 + z k k + z k −z !, k = √ 2 + 1. If any one of the following differential…
Theorem 3.3. Let p ∈H with p(0) = 1, and let ϕ0(z) := 1 + z k k + z k −z ! , k = √ 2 + 1. If any one of the following differential subordinations hold true, then p ≺ϕNe. Each of the respective bounds on β is best possible.
Corollary 3.2.
Corollary 3.2. Let f ∈A and G(z) be given by (2.3). If any one of the following inequalities: (a)
Corollary 3.2. Let f ∈A and G(z) be given by (2.3). If any one of the following inequalities: (a)
Theorem 3.4.
Theorem 3.4. Let ϕS(z):= 1 + sin z, and let p ∈H satisfies p(0) = 1. Each of the following subordinations imply p ≺ϕNe: (a) 1 + βzp′ ≺ϕS(z)…
Theorem 3.4. Let ϕS(z) := 1 + sin z, and let p ∈H satisfies p(0) = 1. Each of the following subordinations imply p ≺ϕNe: (a) 1 + βzp′ ≺ϕS(z) for β ≥3 2 P∞ n=0 (−1)n (2n+1)!×(2n+1) ≈1.41912, (b) 1 + βzp′/p ≺ϕS(z) for β ≥ P∞ n=0 (−1)n (2n+1)!×(2n+1) log(5/3) ≈1.85207,
Theorem 4.1.
Theorem 4.1. Let −1 < B < A ≤1, B ̸= 0, and let p ∈H satisfies p(0) = 1. Then each of the following differential subordinations sufficiently…
Theorem 4.1. Let −1 < B < A ≤1, B ̸= 0, and let p ∈H satisfies p(0) = 1. Then each of the following differential subordinations sufficiently ensures the subordination p ≺ϕNe. Moreover, the respective estimates on the real β are best possible. (a) 1 + βzp′(z) ≺1+Az 1+Bz for β ≥max {β1, β2}, where β1 = A −B 2B log(1 −B)−3 and β2 = A −B 2B log(1 + B)3. (b) 1 + β zp′(z) p(z) ≺1+Az
Theorem 2.1
Theorem 2.1(b) leads to the desired subordination. (c): Considering the function ˜qβ: D →C given by ˜qβ(z) =
Theorem 2.1(b) leads to the desired subordination. (c): Considering the function ˜qβ : D →C given by ˜qβ(z) =
Corollary 4.1.
Corollary 4.1. Let f ∈A and G(z) be defined as in (2.3). If any one of the following conditions hold true, then f ∈S∗ Ne. (a)
Corollary 4.1. Let f ∈A and G(z) be defined as in (2.3). If any one of the following conditions hold true, then f ∈S∗ Ne. (a)
Lemma 5.1
Lemma 5.1 (K¨ustner [15, Theorem 1 (a)]). If 0 < a ≤b ≤c, then 1 − ab b + c ≤σ (zF(a, b; c; z)) ≤1 −ab 2c. In this section, we use Lemma…
Lemma 5.1 (K¨ustner [15, Theorem 1 (a)]). If 0 < a ≤b ≤c, then 1 − ab b + c ≤σ (zF(a, b; c; z)) ≤1 −ab 2c. In this section, we use Lemma 5.1 along with Lemma 1.1 to find sharp bounds on β so that the first-order differential subordination p(z) + βzp′(z) ≺ √ 1 + z, or , 1 + z implies the subordination p ≺ϕNe.
Theorem 5.1.
Theorem 5.1. Let p ∈H satisfies p(0) = 1, and let p(z) + βzp′(z) ≺ϕL(z) = √ 1 + z, β > 0. If β ≥βL, then p ≺ϕNe, where βL is the unique root…
Theorem 5.1. Let p ∈H satisfies p(0) = 1, and let p(z) + βzp′(z) ≺ϕL(z) = √ 1 + z, β > 0. If β ≥βL, then p ≺ϕNe, where βL is the unique root of 3 Γ(−1 2) ∞ X j=0 Γ(−1 2 + j) j! (1 + jβ) −1 = 0.
Corollary 5.1.
Corollary 5.1. Let G(z) be defined as in (2.3), and let f ∈A satisfies (1 + β G(z)) zf ′(z) f(z) ≺ϕL(z). Then f ∈S∗ Ne for β ≥βL.
Corollary 5.1. Let G(z) be defined as in (2.3), and let f ∈A satisfies (1 + β G(z)) zf ′(z) f(z) ≺ϕL(z). Then f ∈S∗ Ne for β ≥βL.
Theorem 5.2.
Theorem 5.2. Let p(z) + βzp′(z) ≺1 + z, where p ∈H satisfies p(0) = 1 and β > 0. Then p ≺ϕNe for β ≥1/2. The estimate on β is sharp.
Theorem 5.2. Let p(z) + βzp′(z) ≺1 + z, where p ∈H satisfies p(0) = 1 and β > 0. Then p ≺ϕNe for β ≥1/2. The estimate on β is sharp.
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