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Abstract
In this paper, using the Mittag-Leffler-type Borel distribution, the authors introduce a new class of bi-Bazilevic functions de-
fined in the open unit disc associated with Legendre polynomials, we find estimates for the general Taylor-Maclaurin coefficients
of the functions in the subclass introduced, and the Fekete-Szego problem is solved.
Results & Lemmas (4)
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Lemma 2.1
Lemma 2.1 ([10]). If w(z) = c1z + c2z2 + c3z3 + · · ·, c1 ̸= 0 is analytic and satisfies |w(z)| < 1 on the unit disk ∆, then for each 0 < r…
Lemma 2.1 ([10]). If w(z) = c1z + c2z2 + c3z3 + · · · , c1 ̸= 0 is analytic and satisfies |w(z)| < 1 on the unit disk ∆, then for each 0 < r < 1, |w ′(0)| < 1 and |w(reiδ)| < 1 unless w(z) = reiδ for some real number δ.
Theorem 2.2.
Theorem 2.2. Let f(z) as assumed in (1.1) and f ∈Mλ,α,β Σ (δ, σ; x). Then |a2| ≦ x √ 2x cos2 δ q x2 cos δ (σ −1)(σ + 2)φ2 2 + 2(σ + 2)φ3…
Theorem 2.2. Let f(z) as assumed in (1.1) and f ∈Mλ,α,β Σ (δ, σ; x). Then |a2| ≦ x √ 2x cos2 δ q x2 cos δ (σ −1)(σ + 2)φ2 2 + 2(σ + 2)φ3 −(3x2 −1) (σ + 1)2 eiδφ2
Theorem 3.1.
Theorem 3.1. Let f is fixed as in (1.1) and f ∈Mλ,α,β Σ (δ, σ; x), then |a3 −µa2 2| ⩽ |x| cos δ (σ+2) φ3, if 0 < |h(µ)| < 1 2(σ+2) φ3,…
Theorem 3.1. Let f is fixed as in (1.1) and f ∈Mλ,α,β Σ (δ, σ; x), then |a3 −µa2 2| ⩽ |x| cos δ (σ+2) φ3 , if 0 < |h(µ)| < 1 2(σ+2) φ3 , 2|xh(µ)| cos δ, if |h(µ)| ⩾
Corollary 3.2.
Corollary 3.2. Let f be assumed in (1.1) and f ∈Mλ,α,β Σ (δ, σ; x), then |a3 −a2 2| ⩽ |x| cos δ (σ + 2) φ3, where δ ∈ −π 2, π 2 , σ ⩾0,…
Corollary 3.2. Let f be assumed in (1.1) and f ∈Mλ,α,β Σ (δ, σ; x), then |a3 −a2 2| ⩽ |x| cos δ (σ + 2) φ3 , where δ ∈ −π 2 , π 2 , σ ⩾0, α, β ∈C, ℜ(α) > 0, ℜ(β) > 0, 0 < λ ⩽1 and the coefficients φk are given by (1.5).
Function classes studied:
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