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Abstract
Introducing a new method, we give sharp estimates
of the Hermitian Toeplitz determinants of third order for the
class S of functions univalent in the unit disc. The new approach
is also illustrated on some subclasses of the class S.
Key Words: univalent, Hermitian Toeplitz determinant of second order,
Hermitian Toeplitz determinant of third order, class U, convex functions
Mathematics Subject Classification 2010: 30C45, 30C50, 30C55
1
Results & Lemmas (7)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Theorem 1
Theorem 1 If f ∈S, then −3 ≤|T2,1(f)| ≤1 and −1 ≤|T3,1(f)| ≤8. All inequalities are sharp.
Theorem 1 If f ∈S, then −3 ≤|T2,1(f)| ≤1 and −1 ≤|T3,1(f)| ≤8. All inequalities are sharp.
Corollary 1
Corollary 1 and Corollary 3 in [4]). (ii) The same result as in Theorem 1 holds for the class U = U(1) since U ⊂S and both extremal…
Corollary 1 and Corollary 3 in [4]). (ii) The same result as in Theorem 1 holds for the class U = U(1) since U ⊂S and both extremal functions f1 and k belong to U.
Theorem 1
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Theorem 1 on other classes of univalent functions, it is enough to know the sharp estimates for |a2|, |a3| and |a3 −a2 2| and apply them on…
Theorem 1 on other classes of univalent functions, it is enough to know the sharp estimates for |a2|, |a3| and |a3 −a2 2| and apply them on |T2,1(f)| = 1 −|a2|2 (3) and on |T3,1(f)| ≤−|a3|2 + 2|a2|2|a3| −2|a2|2 + 1 =: ϕ(|a3|), (4) where ϕ(t) = −t2 + 2|a2|2t −2|a2|2 + 1 and t = |a3|. In the sense of Remark 2, for the class Us(λ), using the sharp estimates |a2| ≤1 + λ, |a3| ≤1 + λ + λ2 and |a3 −a2 2| ≤λ,
Theorem 2
Theorem 2 If f ∈U(λ), then −λ(2 + λ) ≤|T2,1(f)| ≤1, and if additionally f ∈Us(λ), then −λ2 ≤|T3,1(f)| ≤ 1, 0 ≤λ ≤λ0, λ2(1 + λ)(3 + λ), λ0…
Theorem 2 If f ∈U(λ), then −λ(2 + λ) ≤|T2,1(f)| ≤1, and if additionally f ∈Us(λ), then −λ2 ≤|T3,1(f)| ≤ 1, 0 ≤λ ≤λ0, λ2(1 + λ)(3 + λ), λ0 ≤λ ≤1, where λ0 = 0.44762 . . . is the positive real root of the equation λ2(1 + λ)(3 + λ) −1 = 0. All inequalities are sharp.
Corollary 1
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Corollary 1 If f ∈U ≡U(1), then −3 ≤|T2,1(f)| ≤1, and if f ∈Us ≡ Us(1), then −1 ≤|T3,1(f)| ≤8. All inequalities are sharp. We conclude with…
Corollary 1 If f ∈U ≡U(1), then −3 ≤|T2,1(f)| ≤1, and if f ∈Us ≡ Us(1), then −1 ≤|T3,1(f)| ≤8. All inequalities are sharp. We conclude with two more applictions of Remark 2.
Theorem 3
Theorem 3 If f ∈C:= C(0), then 0 ≤|T3,1(f)| ≤1. The estimate is sharp.
Theorem 3 If f ∈C := C(0), then 0 ≤|T3,1(f)| ≤1. The estimate is sharp.
Theorem 4
Theorem 4 If f ∈G:= G(1), then we have sharp estimates 1 2 ≤|T3,1(f)| ≤1.
Theorem 4 If f ∈G := G(1), then we have sharp estimates 1 2 ≤|T3,1(f)| ≤1.
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