Hermitian Toeplitz determinants of the second and third-order for classes of close-to-star functions

Theorem 1 Let F be a subclass of A such that F(2) ̸= ∅and A2(F) exists. Then 1 −A2 2(F) ≤T2,1( f ) ≤1. Both inequalities are sharp. Let S∗be the subclass of S of starlike functions ([1]), i.e., f ∈S∗if f ∈A and Re zf ′(z) f (z) > 0, z ∈D. A function f ∈A is called close-to-star if there exist g ∈S∗and β ∈R such that Re eiβ f (z) g(z) > 0, z ∈D. (2) The class CST of all close-to-star functions which was introduced by Reade [36] bear the

Lemma 1 If p ∈P is of the form (7), then |cn| ≤2, n ∈N. (8) Moreover c1 = 2ζ1 (9) and c2 = 2ζ 2 1 + 2(1 −|ζ1|2)ζ2 (10) for some ζi ∈D, i ∈{1, 2}. For ζ1 ∈T, there is a unique function p ∈P with c1 as in (9), namely, p(z) = 1 + ζ1z 1 −ζ1z ,

Theorem 2 If f ∈ST (i), then −3 ≤T2,1( f ) ≤1. Both inequalities are sharp. 123

Theorem 3 If f ∈ST (i), then T3,1( f ) ≤8. (15) The inequality is sharp.

Theorem 4 If f ∈ST (i), then T3,1( f ) ≥−9. (17) The inequality is sharp.

Theorem 5 If f ∈ST (1), then −15 ≤T2,1( f ) ≤1. Both inequalities are sharp. We next compute the upper and lower bounds of T3,1( f ).

Theorem 6 If f ∈ST (1), then T3,1( f ) ≤176. (27) The inequality is sharp.

Theorem 7 If f ∈ST (1), then T3,1( f ) ≥−442 + 208 √ 3 + 20  28 + 10 √ 3 −21  84 + 30 √ 3 9(−1 + 3 √ 3)