Sharp Bounds of the Hermitian Toeplitz Determinants for Some Classes of Close-to-Convex Functions

Theorem 1 ([11]) Let F be a subclass of A such that { f ∈F : a2 = 0} ̸= ∅and A2(F) := max{|a2| : f ∈F} exists. Then 1 −A2 2(F) ≤det T2,1( f ) ≤1. Both inequalities are sharp. We next define the classes of close-to-convex functions considered in this paper. First denote by S∗the subclass of S consisting of the starlike functions, i.e., f ∈S∗if and only if f ∈A and Re zf ′(z) f (z) > 0, z ∈D. A function f ∈A is called close-to-convex if there exist g ∈S∗and δ ∈R such that Re eiδzf ′(z) g(z) > 0,

Lemma 1 If p ∈P is of the form (5), then |cn| ≤2, n ∈N. (6) 123

Theorem 2 If f ∈F1, then −5 4 ≤det T2,1( f ) ≤1. Both inequalities are sharp. We now find the upper and lower bounds of det T3,1( f ) in the class F1, first noting that det T3,1( f ) =  1 a2 a3 a2 1 a2 a3 a2 1  = 2 Re

Theorem 3 If f ∈F1, then −1 ≤det T3,1( f ) ≤1. (14) Both inequalities are sharp.

Theorem 4 If f ∈F2, then −5 4 ≤det T2,1( f ) ≤1. Both inequalities are sharp. Now we estimate det T3,1( f ) for functions in the class F2.

Theorem 5 If f ∈F2, then det T3,1( f ) ≤11 9 . (43) The inequality is sharp.

Theorem 3, the inequality (15) holds with the function F(x, y) := 2x2y −2x2 −y2 + 1, (x, y) ∈[0, 3/2] × [0, 5/3]. Repeating argumentation in the proof of Theorem 3, we see that the function F does not have any relative maxima in (0, 3/2) × (0, 5/3). We consider F on the boundary of [0, 3/2] × [0, 5/3]. (1) On the side x = 0, F(0, y) = 1 −y2 ≤1, 0 ≤y ≤5 3. (2) On the side x = 3/2, F 3 2, y 

Theorem 6 If f ∈F2, then det T3,1( f ) ≥1 44