Ma-Minda φ-classes studied in this paper:
Abstract
We say that a class $\mathcal{F}$ consisting of analytic functions $f(z)=\sum_{n=0}^{\infty} a_{n}z^{n}$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}: |z|<1\}$ satisfies a Bohr phenomenon if there exists $r_{f} \in (0,1)$ such that $$
\sum_{n=1}^{\infty} |a_{n}z^{n}|\leq d(f(0),\partial f(\mathbb{D})) $$ for every function $f \in \mathcal{F}$ and $|z|=r\leq r_{f}$, where $d$ is the Euclidean distance. The largest radius $r_{f}$ is the Bohr radius for the class $\mathcal{F}$. In this paper,
Results & Lemmas (17)
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Lemma 1.10.
Lemma 1.10. [32] Let f ∈S∗(φ). Then zf ′(z)/f(z) ≺zh′(z)/h(z) and f(z)/z ≺ h(z)/z.
Lemma 1.10. [32] Let f ∈S∗(φ). Then zf ′(z)/f(z) ≺zh′(z)/h(z) and f(z)/z ≺ h(z)/z.
Lemma 1.11.
Lemma 1.11. [32] Assume f ∈S∗(φ) and |z| = r < 1. Then (1.12) −h(−r) ≤|f(z)| ≤h(r). Equality holds for some z ̸= 0 if, and only if, f is a…
Lemma 1.11. [32] Assume f ∈S∗(φ) and |z| = r < 1. Then (1.12) −h(−r) ≤|f(z)| ≤h(r). Equality holds for some z ̸= 0 if, and only if, f is a rotation of h. It has been pointed out in [32] that −h(−r) is increasing in (0, 1) and bounded by 1 because each f ∈S∗(φ) is normalized by f(0) = f ′(0) −1 = 0. Therefore, limr→1 −h(−r) exists and denote it by −h(−1). The following subordination and growth theorem for the class C(φ) have been established in [32].
Lemma 1.13.
Lemma 1.13. [32] Let f ∈C(φ). Then zf ′′(z)/f ′(z) ≺zk′′(z)/k′(z) and f ′(z) ≺ k′(z).
Lemma 1.13. [32] Let f ∈C(φ). Then zf ′′(z)/f ′(z) ≺zk′′(z)/k′(z) and f ′(z) ≺ k′(z).
Lemma 1.14.
Lemma 1.14. [32] Assume f ∈C(φ) and |z| = r < 1. Then (1.15) −k(−r) ≤|f(z)| ≤k(r). Equality holds for some z ̸= 0 if, and only if, f is a…
Lemma 1.14. [32] Assume f ∈C(φ) and |z| = r < 1. Then (1.15) −k(−r) ≤|f(z)| ≤k(r). Equality holds for some z ̸= 0 if, and only if, f is a rotation of k. It is justified in [32] that −k(−r) is increasing in (0, 1) and bounded by 1 beacuse each f ∈C(φ) is normalized by f(0) = f ′(0) −1 = 0. Therefore, limr→1 −k(−r) exists and denote it by −k(−1). Ma and Minda [32] have introduced the analytic and univalent function φ with certain conditions, one of which is φ′(0) > 0. Recently, Kumar and Banga [29]
Lemma 1.18.
Lemma 1.18. [45] A function G is in Gα if, and only if, there exists a function s ∈S∗(α) such that G(z) = (1 −z)2(1−α) s(z) z. The regions…
Lemma 1.18. [45] A function G is in Gα if, and only if, there exists a function s ∈S∗(α) such that G(z) = (1 −z)2(1−α) s(z) z . The regions of variability for the class Gα has been studied extensively by Pon- nusamy et al. [38]. The folowing growth theorem for the class Gα has been estab- lished by Silverman and Silvia [45].
Lemma 1.19.
Lemma 1.19. [45] If g ∈Gα then (1.20) 1 −r 1 + r 2(1−α) ≤|g(z)| ≤ 1 + r 1 −r 2(1−α) for |z| = r. Equality holds for g(z) = ((1 −z)/(1 +…
Lemma 1.19. [45] If g ∈Gα then (1.20) 1 −r 1 + r 2(1−α) ≤|g(z)| ≤ 1 + r 1 −r 2(1−α) for |z| = r. Equality holds for g(z) = ((1 −z)/(1 + z))2(1−α) at z = r and z = −r.
Lemma 1.22.
Lemma 1.22. [13] Let f and g be anlytic in D with Taylor expansions (1.1) and (1.21) respectively and g ≺f, then (1.23) ∞ X n=0 |cn|rn ≤ ∞…
Lemma 1.22. [13] Let f and g be anlytic in D with Taylor expansions (1.1) and (1.21) respectively and g ≺f, then (1.23) ∞ X n=0 |cn|rn ≤ ∞ X n=0 |an|rn for z| = r ≤1/3. The following coefficients bounds for the class S∗(α) are required to obtain the Bohr radius for the class Gα.
Lemma 1.24.
Lemma 1.24. [21] Let f ∈S∗(α) be given by (1.15). Then |an| ≤ 1 (n −1)! n Y k=0 (k −2α) for n ≥2. The equalities in above estimates are…
Lemma 1.24. [21] Let f ∈S∗(α) be given by (1.15). Then |an| ≤ 1 (n −1)! n Y k=0 (k −2α) for n ≥2. The equalities in above estimates are attained for f(z) = z/(1 −z)2(1−α) for z ∈D. In this paper, we establish the Bohr phenomenon for the classes S∗(φ) and C(φ), where all the coefficients of associated Ma-Minda function φ are positive such that φ ∈H2, the Hardy class of analytic functions in D. As a consequence, we obtain several important corollaries for particular choices of φ. We also obtain the
Theorem 2.1.
Theorem 2.1. Let f ∈S∗(φ) be given by (1.5) and φ(z) be given by (1.6) with all Bn > 0, n ≥1 such that φ ∈H2, the Hardy class of analytic…
Theorem 2.1. Let f ∈S∗(φ) be given by (1.5) and φ(z) be given by (1.6) with all Bn > 0, n ≥1 such that φ ∈H2, the Hardy class of analytic functions in D. Then (2.2) |z| + ∞ X n=0 |an||z|n ≤d(f(0), ∂f(D)) for |z| = r ≤min{rf, 1/3}, where rf is the smallest positive root of (2.3) h(r) + h(−1) = 0 in (0, 1) and h(z) is defined in (1.9).
Corollary 2.4.
Corollary 2.4. For φ(z) = α + (1 −α)ez, we have S∗ α,e:= S∗(α + (1 −α)ez). Let f ∈S∗ α,e be given by (1.5) with 0 ≤α < 0.05284. Then the…
Corollary 2.4. For φ(z) = α + (1 −α)ez, we have S∗ α,e := S∗(α + (1 −α)ez). Let f ∈S∗ α,e be given by (1.5) with 0 ≤α < 0.05284. Then the inequality (2.2) is satisfied for |z| = r ≤rf, where 0 < rf < 1/3. The radius rf is the best possible. In particular, for α = 0 in Corollary 2.4, we obtain the sharp Bohr radius for the class S∗ e.
Corollary 2.5.
Corollary 2.5. For φ(z) = 1 + 4z/3 + 2z2/3, S∗(φ) reduces to S∗ C. For f ∈S∗ C of the form (1.5), the inequality (2.2) is satisfied for |z|…
Corollary 2.5. For φ(z) = 1 + 4z/3 + 2z2/3, S∗(φ) reduces to S∗ C. For f ∈S∗ C of the form (1.5), the inequality (2.2) is satisfied for |z| = r ≤rf, where 0 < rf < 1/3. The radius rf is the best possible.
Corollary 2.6.
Corollary 2.6. Let φ be the rational functon such that φ(z) = 1+(z/k) ((k + z)/(k −z)), where k = √ 2 + 1. Then S∗(φ) reduces to the class…
Corollary 2.6. Let φ be the rational functon such that φ(z) = 1+(z/k) ((k + z)/(k −z)), where k = √ 2 + 1. Then S∗(φ) reduces to the class S∗ R. Then the ineuality (2.2) is satisfied for |z| = r ≤1/3 for the class S∗ R.
Corollary 2.7. · radius
Corollary 2.7. For φ(z) = (1 + Az)/(1 + Bz), the class S∗(φ) reduces to the Janowski starlike class S∗[A, B]. Let f ∈S∗[A, B] be given by…
Corollary 2.7. For φ(z) = (1 + Az)/(1 + Bz), the class S∗(φ) reduces to the Janowski starlike class S∗[A, B]. Let f ∈S∗[A, B] be given by (1.5) with −1 ≤B < (1 −3k)/(1 + 3k) < 0 and 0 ≤A ≤1, where k = B/(B −A). Then the ineuality (2.2) is satisfied for |z| = r ≤rf, where 0 < rf < 1/3. The radius rf is the best possible.
Corollary 2.8.
Corollary 2.8. For φ(z) = 1 + z/(1 −αz2), the class S∗(φ) reduces to BS∗(α). Let f ∈BS∗(α) be given by (1.5) with 0 ≤α < 1. Then the…
Corollary 2.8. For φ(z) = 1 + z/(1 −αz2), the class S∗(φ) reduces to BS∗(α). Let f ∈BS∗(α) be given by (1.5) with 0 ≤α < 1. Then the inequality (2.2) is satisfied for |z| = r ≤rf, where 0 < rf < 1/3. The constant rf cannot be improved.
Corollary 2.9.
Corollary 2.9. Let φ = (1 + sz)2, then S∗(φ) reduces to the class ST L(s). Let f ∈ST L(s) be given by (1.5) with 0.444981 < s ≤1/ √ 2. Then…
Corollary 2.9. Let φ = (1 + sz)2, then S∗(φ) reduces to the class ST L(s). Let f ∈ST L(s) be given by (1.5) with 0.444981 < s ≤1/ √ 2. Then the inequality (2.2) satisfied for |z| = r ≤rf, where 0 < rf < 1/3. The radius rf is the best posible. By using Lemmas 1.14 and 1.22, we establish Bhor phenomenon for the class C(φ).
Theorem 2.10.
Theorem 2.10. Let f ∈C(φ) be given by (1.5) and φ(z) be given by (1.6) with all Bn > 0, n ≥1 such that φ ∈H2, the Hardy class of analytic…
Theorem 2.10. Let f ∈C(φ) be given by (1.5) and φ(z) be given by (1.6) with all Bn > 0, n ≥1 such that φ ∈H2, the Hardy class of analytic functions in D. Then (2.11) |z| + ∞ X n=0 |an||z|n ≤d(f(0), ∂f(D)) for |z| = r ≤min{rf, 1/3}, where rf is the smallest positive root of (2.12) k(r) + k(−1) = 0 in (0, 1) and k(z) is defined in (1.9). Let G ∈Gα with the power series representation (2.13) G(z) = 1 +
Theorem 2.14.
Theorem 2.14. Let G ∈Gα be given by (2.13). Then (2.15) ∞ X n=1 |dn||z|n ≤d(G(0), ∂G(D)) for |z| = r ≤rf, where rf = (21/2(1−α)…
Theorem 2.14. Let G ∈Gα be given by (2.13). Then (2.15) ∞ X n=1 |dn||z|n ≤d(G(0), ∂G(D)) for |z| = r ≤rf, where rf = (21/2(1−α) −1)/(21/2(1−α) + 1). The radius rf is the best possible. 3. Proof of the main results
Function classes studied:
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