On a certain subclass of strongly starlike functions

Lemma 2.1. [1, Lemma 2.1] For n ∈N, let Bn(α1, α2, c) be defined by (2.6). Then for |α1+α2| < 2, we have |Bn(α1, α2, c)| ≤|Bn(α1, α2, 1)|. As a result of Lemma 2.1, we get the following:

Corollary 2.1. Let An be defined as in (2.4) for n = 1, 2, 3, . . ., and |α1 + α2| < 2 for α1, α2 ∈ (0, 1]. Then |An| ≤λ |Bn(α1, α2, 1)|, where (2.7) λ = (α1 + α2) cos πθ 2  with θ = α2 −α1 α2 + α1 . The main purpose of this paper is to study the class S∗(α1, α,2 ), which is provided below:

Lemma 2.2. (See [17]) Let λ and γ be complex numbers with λ ̸= 0 and let q be convex univalent in D with Re{λq(z) + γ} ≥0. If p is analytic in D with p(0) = q(0), then p(z) + zp′(z) λp(z) + γ ≺q(z) ⇒p(z) ≺q(z).

Lemma 2.3. (See [28]) Let F, H ∈H be any convex univalent functions in D. If f ≺F and g ≺H, then (2.10) f(z) ∗g(z) ≺F(z) ∗H(z), where “*” denotes the well-known Hadamard product. In the following, we recall some other useful lemmas.

Lemma 2.4 (See [25]). Let q(z) = P∞ n=1 Qnzn be analytic and univalent in D, and suppose that q maps D onto a convex domain. If p(z) = P∞ n=1 Pnzn is analytic in D and satisfies the following subordination: p(z) ≺q(z), then |Pn| ≤|Q1|, (n = 1, 2, 3, . . .).

Lemma 2.5. [2, Lemma 1] If w(z) = P∞ k=1 wkzk is a Schwarz function, then |w2 −tw2 1| ≤           

Lemma 2.6. (See [19]) If w(z) = P∞ n=1 wnzn is a Schwarz function, then for any complex numbers ρ and τ the following sharp estimate holds: |w3 + ρw1w2 + τw3 1| ≤H(ρ, τ),

Lemma 2.7. If the function f ∈S satisfies the following subordination, 1 + zf ′′(z) f ′(z) ≺G(z), then f belongs to the class S∗(α1, α2), where G is given by (2.2).

Lemma 2.2. □ 3. Main Results Determining the sharp coefficient bounds for strongly starlike functions appears to pose a challenging classical problem. To date, the estimation of the coefficients of these functions remains an open problem. There are several known results for specific subclasses of strongly starlike functions, but there is no general solution for the entire class. One of the main challenges in this problem is that the definition of strongly starlike functions is quite broad, and t

Theorem 3.1. Let the function f ∈A belong to the class SS∗(β), where β ∈(0, 1]. Then the following sharp inequalities hold: |a2| ≤2β, |a3| ≤    β, 0 < β ≤1/3; 3β2, 1/3 ≤β ≤1, and |a4| ≤  

Theorem 3.2. Let f be of the form (2.1) belonging to the class S∗(α1, α2), where 0 < α1, α2 ≤1. Then |a2| ≤λ and for n ≥3, (3.1) |an| ≤ λ n −1 n−1 Y k=2  1 + λ k −1  ,

Theorem 3.3. If the function f of the form (2.1) is a strongly starlike function of order β, then |an| ≤      2β, n = 2; 2β n−1 Qn−1 k=2  1 +

Theorem 3.4. Consider a function f ∈A, which takes the form (2.1) and belongs to the class S∗(α1, α2), where α1, α2 ∈(0, 1]. Let λ be defined by (2.7), and γ = (3α−1)/2, where α is given by (2.3). Then |a2| ≤λ, |a3| ≤      λ 2 , 0 < α ≤1/3; λ 2

Theorem 3.5. If the function f ∈A belongs to the class S∗(α1, α2), then (3.14) log f(z) z  ≺ Z z 0 G(t) −1 t dt, where the function G is defined by (2.2). Moreover, (3.15) eG(z) :=

Theorem 3.6. Let eG be of the form (3.15) and 0 < α1, α2 ≤1. If f(z) ∈S∗(α1, α2), then r exp eG(−r) ≤|f(z)| ≤r exp eG(r), for each r = |z| < 1. Both inequalities are sharp.

Theorem 3.7. Let f ∈S∗(α1, α2) and the coefficients of log(f(z)/z) be given by (3.13) and 0 < α1, α2 ≤1. Then (3.19) |γn| ≤λ 2n, (n ≥1), where λ is given by (2.7). The result is sharp.

Corollary 3.1. Let f be a strongly starlike function of order β, where 0 < β ≤1. Then the logarithmic coefficients of f satisfy the following sharp inequality: |γn| ≤1 nβ, (n ≥1). In particular, taking β = 1 gives us an estimate of the logarithmic coefficients of starlike func- tions.

Theorem 3.8. Suppose that f belongs to the class A. If f ∈S∗(α1, α2), then Re zf ′(z) f(z)  ≥ 1 −(1 + 2 cos(πθ/2)) r 1 −r (α1+α2)/2 , 0 ≤|z| = r ≤ 1 1 + 2 cos(πθ/2), and Re

Corollary 3.2. Let f be a strongly starlike function of order β, where 0 < β ≤1. Then 1 −3r 1 −r β ≤Re zf ′(z) f(z)  ≤ 1 + r 1 −r β , (|z| = r ≤1/3). In particular, if f is a strongly starlike function of order β, and |z| = 1/3, then

Definition 2.1. Let 0 < α1, α2 ≤1. A function f ∈S belongs to the class S∗(α1, α2), if f satisfies the following two-sided inequality (2.8) −πα1 2 < arg zf ′(z) f(z)  < πα2