Abstract
A starlike function $f$ is characterized by the quantity $zf'(z)/f(z)$ lying in the right half-plane. This paper deals with sharp bounds for certain symmetric Toeplitz determinants whose entries are the coefficients of the functions $f$ for which the quantity $zf'(z)/f(z)$ takes values in certain specific subset in the right half-plane. The results obtained include several new special cases and some known results.
Results & Lemmas (6)
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Theorem 2.1
Theorem 2.1 and Theorem 2.2 respectively give the sharp bound for T2(2) = a2 3 −a2 2 for functions f ∈S∗(ϕ) and f ∈K(ϕ).
Theorem 2.1 and Theorem 2.2 respectively give the sharp bound for T2(2) = a2 3 −a2 2 for functions f ∈S∗(ϕ) and f ∈K(ϕ).
Theorem 2.1.
Theorem 2.1. If f ∈S∗(ϕ) and ϕ(z) = 1 + B1z + B2z2 + · · · with 0 < B1 ≤|B2 + B2 1|, then the Toeplitz determinant T2(2) satisfies the sharp…
Theorem 2.1. If f ∈S∗(ϕ) and ϕ(z) = 1 + B1z + B2z2 + · · · with 0 < B1 ≤|B2 + B2 1|, then the Toeplitz determinant T2(2) satisfies the sharp bound: |T2(2)| ≤1 4(B2 + B2 1)2 + B2 1.
Theorem 2.2.
Theorem 2.2. If f ∈K(ϕ) and ϕ(z) = 1 + B1z + B2z2 + · · · with 0 < B1 ≤|B2 + B2 1|, then the Toeplitz determinant T2(2) satisfies the sharp…
Theorem 2.2. If f ∈K(ϕ) and ϕ(z) = 1 + B1z + B2z2 + · · · with 0 < B1 ≤|B2 + B2 1|, then the Toeplitz determinant T2(2) satisfies the sharp bound given by |T2(2)| ≤1 36 B2 1 + B2 2 + 1 4B2 1.
Theorem 2.3
Theorem 2.3 and Theorem 2.4 give the sharp bound for the Toeplitz determinant T3(1) for functions respectively in the classes S∗(ϕ) and…
Theorem 2.3 and Theorem 2.4 give the sharp bound for the Toeplitz determinant T3(1) for functions respectively in the classes S∗(ϕ) and K(ϕ).
Theorem 2.3.
Theorem 2.3. If f ∈S∗(ϕ) and ϕ(z) = 1 + B1z + B2z2 + · · ·, with B1 > 0 and B1 −B2 1 ≤B2 ≤3B2 1 −B1, then the Toeplitz determinant T3(1)…
Theorem 2.3. If f ∈S∗(ϕ) and ϕ(z) = 1 + B1z + B2z2 + · · · , with B1 > 0 and B1 −B2 1 ≤B2 ≤3B2 1 −B1, then the Toeplitz determinant T3(1) satisfies the sharp bound: |T3(1)| ≤1 + 2B2 1 + 1 4(B2 + B2 1)(3B2 1 −B2).
Theorem 2.4.
Theorem 2.4. If f ∈K(ϕ) and ϕ(z) = 1 + B1z + B2z2 + · · · with B1 > 0 and B1 −B2 1 ≤B2 ≤2B2 1 −B1, then the Toeplitz determinant T3(1)…
Theorem 2.4. If f ∈K(ϕ) and ϕ(z) = 1 + B1z + B2z2 + · · · with B1 > 0 and B1 −B2 1 ≤B2 ≤2B2 1 −B1, then the Toeplitz determinant T3(1) satisfies the sharp bound: |T3(1)| ≤1 + 1 2B2 1 + 1 36(B2 1 + B2)(2B2 1 −B2).
Definitions (1)
Def 1.1.
Definition 1.1. For an analytic univalent function ϕ with positive real part in D, ϕ(0) = 1, ϕ′(0) > 0 and ϕ′′(0) ∈R, the classes S∗(ϕ) and…
Definition 1.1. For an analytic univalent function ϕ with positive real part in D, ϕ(0) = 1, ϕ′(0) > 0 and ϕ′′(0) ∈R, the classes S∗(ϕ) and K(ϕ) are defined by S∗(ϕ) := f ∈S : zf ′(z) f(z) ≺ϕ(z) and K(ϕ) :=