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Abstract
Estimates on the initial coefficients are obtained for normalized analytic functions $f$ in the open unit disk with $f$ and its inverse $g=f^{-1}$ satisfying the conditions that $zf'(z)/f(z)$ and $zg'(z)/g(z)$ are both subordinate to a starlike univalent function whose range is symmetric with respect to the real axis. Several related classes of functions are also considered, and connections to earlier known results are made.
Results & Lemmas (7)
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Theorem 2.1.
Theorem 2.1. If f ∈Hσ(ϕ) is given by f(z) = z + ∞ X n=2 anzn, (2.2) then |a2| ≤ B1 √B1 p |3B2 1 −4B2 + 4B1| and
Theorem 2.1. If f ∈Hσ(ϕ) is given by f(z) = z + ∞ X n=2 anzn, (2.2) then |a2| ≤ B1 √B1 p |3B2 1 −4B2 + 4B1| and
Theorem 2.2.
Theorem 2.2. Let f given by (2.2) be in the class ST σ(α, ϕ). Then |a2| ≤ B1 √B1 p |B2 1(1 + 4α) + (B1 −B2)(1 + 2α)2|, (2.15) and |a3| ≤B1…
Theorem 2.2. Let f given by (2.2) be in the class ST σ(α, ϕ). Then |a2| ≤ B1 √B1 p |B2 1(1 + 4α) + (B1 −B2)(1 + 2α)2| , (2.15) and |a3| ≤B1 + |B2 −B1| (1 + 4α) . (2.16)
Corollary 2.1.
Corollary 2.1. Let f given by (2.2) be in the class ST σ(ϕ). Then |a2| ≤ B1 √B1 p |B2 1 + B1 −B2| and |a3| ≤B1 + |B2 −B1|.
Corollary 2.1. Let f given by (2.2) be in the class ST σ(ϕ). Then |a2| ≤ B1 √B1 p |B2 1 + B1 −B2| and |a3| ≤B1 + |B2 −B1|.
Theorem 3.1
Theorem 3.1]. Next, a function f ∈σ belongs to the class Mσ(α, ϕ), α ≥0, if the following subordinations hold: (1 −α)zf ′(z) f(z) + α 1 +…
Theorem 3.1]. Next, a function f ∈σ belongs to the class Mσ(α, ϕ), α ≥0, if the following subordinations hold: (1 −α)zf ′(z) f(z) + α 1 + zf ′′(z) f ′(z) ≺ϕ(z) and (1 −α)wg′(w) g(w) + α 1 + wg′′(w)
Theorem 2.3.
Theorem 2.3. Let f given by (2.1) be in the class Mσ(α, ϕ). Then |a2| ≤ B1 √B1 p (1 + α)|B2 1 + (1 + α)(B1 −B2)| (2.23) and |a3| ≤B1 + |B2…
Theorem 2.3. Let f given by (2.1) be in the class Mσ(α, ϕ). Then |a2| ≤ B1 √B1 p (1 + α)|B2 1 + (1 + α)(B1 −B2)| (2.23) and |a3| ≤B1 + |B2 −B1| 1 + α . (2.24)
Corollary 2.2.
Corollary 2.2. Let f given by (2.1) be in the class CVσ(ϕ). Then |a2| ≤ B1 √B1 p 2|B2 1 + 2B1 −2B2| and |a3| ≤1 2(B1 + |B2 −B1|).
Corollary 2.2. Let f given by (2.1) be in the class CVσ(ϕ). Then |a2| ≤ B1 √B1 p 2|B2 1 + 2B1 −2B2| and |a3| ≤1 2(B1 + |B2 −B1|).
Theorem 2.4.
Theorem 2.4. Let f given by (2.1) be in the class Lσ(α, ϕ). Then |a2| ≤ 2B1 √B1 p |2(α2 −3α + 4)B2 1 + 4(α −2)2(B1 −B2)| (2.32) and |a3| ≤…
Theorem 2.4. Let f given by (2.1) be in the class Lσ(α, ϕ). Then |a2| ≤ 2B1 √B1 p |2(α2 −3α + 4)B2 1 + 4(α −2)2(B1 −B2)| (2.32) and |a3| ≤ 2(3 −2α) B1 + |B1 −B2| |(3 −2α)(α2 −3α + 4)|
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