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Abstract
Let $f$ be analytic in the unit disk $\mathbb D$ and normalized so that $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper, we give upper bounds of the Hankel determinant of second order for the classes of starlike functions of order $α$, Ozaki close-to-convex functions and two other classes of analytic functions. Some of the estimates are sharp.
Results & Lemmas (4)
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Theorem 1.
Theorem 1. Let f(z) = z+a2z2+a3z3+· · · belongs to the class S⋆(α), 0 ≤α < 1. Then we have the next sharp estimation: |H2(2)| = |a2a4 −a2…
Theorem 1. Let f(z) = z+a2z2+a3z3+· · · belongs to the class S⋆(α), 0 ≤α < 1. Then we have the next sharp estimation: |H2(2)| = |a2a4 −a2 3| ≤(1 −α)2.
Theorem 2.
Theorem 2. Let f(z) = z+a2z2+a3z3+· · · belongs to the class C(α), −1 2 ≤α < 1. Then we have the next estimations: |H2(2)| ≤ …
Theorem 2. Let f(z) = z+a2z2+a3z3+· · · belongs to the class C(α), −1 2 ≤α < 1. Then we have the next estimations: |H2(2)| ≤ (1−α)2(5α+6) 48(1+α) , −1 2 ≤α ≤0 (1−α)2(17α2−36α+36) 144(α2−2α+2) ,
Theorem 3.
Theorem 3. Let f(z) = z +a2z2 +a3z3 +· · · belongs to the class G(α), 0 < α ≤1. Then we have the next estimation: |H2(2)| ≤α2 144 17 4 − α…
Theorem 3. Let f(z) = z +a2z2 +a3z3 +· · · belongs to the class G(α), 0 < α ≤1. Then we have the next estimation: |H2(2)| ≤α2 144 17 4 − α 4 + α2 .
Theorem 4.
Theorem 4. Let f(z) = z + a2z2 + a3z3 + · · · belongs to the class S⋆(q). Then we have the next sharp estimation: |H2(2)| ≤1 4.
Theorem 4. Let f(z) = z + a2z2 + a3z3 + · · · belongs to the class S⋆(q). Then we have the next sharp estimation: |H2(2)| ≤1 4.
Function classes studied:
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