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Ma-Minda φ-classes studied in this paper:
Abstract
In this article we define a class of starlike functions with respect to symmetric points in the domain of sine function. Also,
we investigate coefficients bounds and upper bounds for the third order Hankel determinant for this defined class. We also
evaluate the Zalcman functional |a2
3 −a5|. Specializing the parameters, we improve Zalcman functional for the class of starlike
functions.
Results & Lemmas (10)
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Lemma 1.1.
Lemma 1.1. If h ∈P is expressed in series expansion (1.2), then |pn| ⩽2 for n ⩾1, (1.4) pi+j −µpipj ⩽2 for 0 ⩽µ ⩽1, (1.5) and for complex…
Lemma 1.1. If h ∈P is expressed in series expansion (1.2), then |pn| ⩽2 for n ⩾1, (1.4) pi+j −µpipj ⩽2 for 0 ⩽µ ⩽1, (1.5) and for complex number ξ, we have p2 −ξp2 1 ⩽2 max {1, |2ξ −1|} . (1.6) where the inequalities (1.4) and (1.5), are taken from [24] and (1.6) is obtained in [10].
Lemma 1.2
Lemma 1.2 ([1]). Let h ∈P has power series (1.2), then αp3 1 −βp1p2 + γp3 ⩽2 |α| + 2 |β −2α| + 2 |α −β + γ|.
Lemma 1.2 ([1]). Let h ∈P has power series (1.2), then αp3 1 −βp1p2 + γp3 ⩽2 |α| + 2 |β −2α| + 2 |α −β + γ| .
Lemma 1.3
Lemma 1.3 ([26]). Let m, n, l and a satisfy the inequalities 0 < m < 1, 0 < r < 1, and 8r (1 −r) h (mn −2l)2 + (m (r + m) −n)2i + m (1 −m)…
Lemma 1.3 ([26]). Let m, n, l and a satisfy the inequalities 0 < m < 1, 0 < r < 1, and 8r (1 −r) h (mn −2l)2 + (m (r + m) −n)2i + m (1 −m) (n −2rm)2 ⩽4m2 (1 −m)2 r (1 −r) . If h(z) ∈P and has power series (1.2), then lp4 1 + rp2 2 + 2mp1p3 −3 2np2 1p2 −p4 ⩽2. 2. Coefficients estimates and Fekete-Szeg¨o inequality In this section we evaluate the coefficients estimates for the class S∗ s (sin) . Further we evaluate Fekete-
Theorem 2.1.
Theorem 2.1. If f(z) is of the form (1.1) and belongs to S∗ s (Ψ), then |a2| ⩽1 2, |a3| ⩽1 2, |a4| ⩽1 4, |a5| ⩽3 4. (2.1)
Theorem 2.1. If f(z) is of the form (1.1) and belongs to S∗ s (Ψ), then |a2| ⩽1 2, |a3| ⩽1 2, |a4| ⩽1 4, |a5| ⩽3 4. (2.1)
Theorem 2.2.
Theorem 2.2. If f(z) is of the form (1.1) and belongs to S∗ s (Ψ), then for any complex number ξ a3 −ξa2 2 ⩽1 2 max
Theorem 2.2. If f(z) is of the form (1.1) and belongs to S∗ s (Ψ), then for any complex number ξ a3 −ξa2 2 ⩽1 2 max
Corollary 2.3.
Corollary 2.3. If f ∈S∗ s (Ψ) and ξ = 1, then a3 −a2 2 ⩽1 2. (2.8) 3. Hankel determinants Now we obtain other important results on the…
Corollary 2.3. If f ∈S∗ s (Ψ) and ξ = 1, then a3 −a2 2 ⩽1 2. (2.8) 3. Hankel determinants Now we obtain other important results on the basis of which we will evaluate the third Hankel for this class.
Theorem 3.1.
Theorem 3.1. If f is of the form (1.1) and belongs to S∗ s (Ψ), then |a2a3 −a4| ⩽1 4. (3.1)
Theorem 3.1. If f is of the form (1.1) and belongs to S∗ s (Ψ), then |a2a3 −a4| ⩽1 4. (3.1)
Theorem 3.2.
Theorem 3.2. If f(z) is of the form (1.1) and belongs to S∗ s (Ψ), then a2a4 −a2 3 ⩽11 16. (3.2)
Theorem 3.2. If f(z) is of the form (1.1) and belongs to S∗ s (Ψ), then a2a4 −a2 3 ⩽11 16. (3.2)
Theorem 3.3.
Theorem 3.3. If f(z) = z + a2z2 + a3z3 + · · · belongs to S∗ s (Ψ), then |H3,1 (f)| ⩽25 32 ≃0.781 25.
Theorem 3.3. If f(z) = z + a2z2 + a3z3 + · · · belongs to S∗ s (Ψ), then |H3,1 (f)| ⩽25 32 ≃0.781 25.
Theorem 4.1.
Theorem 4.1. If f(z) = z + a2z2 + a3z3 + · · · belongs to S∗ s (Ψ), then a2 3 −a5 ⩽1 4 ≃0.25.
Theorem 4.1. If f(z) = z + a2z2 + a3z3 + · · · belongs to S∗ s (Ψ), then a2 3 −a5 ⩽1 4 ≃0.25.
Function classes studied:
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