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Abstract
The sharp bounds for the fourth-order Hermitian Toeplitz determinant over the class
of convex functions are computed.
Results & Lemmas (3)
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Lemma 1
Lemma 1 If p ∈P is of the form (5) with c1 ≥0, then 2c2 = c2 1 + (4 −c2 1)ζ (6) and 4c3 = c3 1 + (4 −c2 1)c1ζ(2 −ζ) + 2(4 −c2 1)(1 −|ζ|2)η…
Lemma 1 If p ∈P is of the form (5) with c1 ≥0, then 2c2 = c2 1 + (4 −c2 1)ζ (6) and 4c3 = c3 1 + (4 −c2 1)c1ζ(2 −ζ) + 2(4 −c2 1)(1 −|ζ|2)η (7) for some ζ, η ∈D := {z ∈C : |z| ≤1} 2 Main result In [4] the sharp bounds for the Hermitian-Toeplitz determinants of the second and third-order for the class of convex functions of order α were computed. In particular
Theorem 1
Theorem 1 If f ∈Sc, then 0 ≤|T3,1( f )| ≤1. (8) Both inequalities are sharp with equalities attained by f (z) = z 1 −z, z ∈D, (9) and by…
Theorem 1 If f ∈Sc, then 0 ≤|T3,1( f )| ≤1. (8) Both inequalities are sharp with equalities attained by f (z) = z 1 −z , z ∈D, (9) and by the identity, respectively.
Theorem 2
Theorem 2 If f ∈Sc, then 0 ≤|T4,1( f )| ≤1. (12) Both inequalities are sharp with equalities attained by the function (9) and the identity,…
Theorem 2 If f ∈Sc, then 0 ≤|T4,1( f )| ≤1. (12) Both inequalities are sharp with equalities attained by the function (9) and the identity, respectively.
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