Toeplitz determinants whose elements are the coefficients of univalent functions

Lemma 1.1. [1, p. 41] For a function p ∈P of the form (1.3), the sharp inequality |cn| ≤2 holds for each n ≥1. Equality holds for the function p(z) = (1+z)/(1−z).

Lemma 1.2. [2, Theorem 1] Let p ∈P be of the form (1.3) and µ ∈C. Then |cn −µckcn−k| ≤2 max{1, |2µ −1|}, 1 ≤k ≤n −1. If |2µ −1| ≥1 then the inequality is sharp for the function p(z) = (1 + z)/(1 −z) or its rotations. If |2µ −1| < 1 then the inequality is sharp for the function p(z) = (1 + zn)/(1 −zn) or its rotations. 2. Main Results

Theorem 2.1. Let f ∈S be of the form (1.1). Then (i) |T2(n)| = |a2 n −a2 n+1| ≤2n2 + 2n + 1 for n ≥2, (ii) |T3(1)| ≤24. Both inequalities are sharp.

Theorem 2.2. Let f ∈S∗be of the form (1.1). Then |T3(2)| ≤84. The inequality is sharp.

Lemma 2.1. [3, Theorem 3.1] Let g ∈S∗and be of the form g(z) = z +P∞ n=2 bnzn. Then |b2b4 −b2 3| ≤1, and the inequality is sharp for the Koebe function k(z) = z/(1 −z)2, or its rotations.

Lemma 2.2. [4, Lemma 3] Let g ∈S∗be of the form g(z) = z + P∞ n=2 bnzn. Then for any λ ∈C, |b3 −λb2 2| ≤max{1, |3 −4λ|}. The inequality is sharp for k(z) = z/(1 −z)2, or its rotations if |3 −4λ| ≥1, and for (k(z2))1/2, or its rotations if |3 −4λ| < 1.

Lemma 2.3. [5, Theorem 2.2] Let g ∈S∗be of the form g(z) = z + P∞ n=2 bnzn. Then |λbnbm −bn+m−1| ≤λnm −(n + m −1) for λ ≥2(n + m −1) nm , where n, m = 2, 3, . . .. The inequality is sharp for the Koebe function k(z) = z/(1 − z)2, or its rotations.

Lemma 2.4. Let f ∈K be of the form (1.1). Then |a2a4 −2a2 3| ≤21/2.

Theorem 2.3. Let f ∈K be of the form (1.1). Then |T3(2)| ≤86.

Theorem 2.4. Let f ∈C be of the form (1.1). Then (i) |T2(n)| ≤2 for n ≥2. (ii) |T3(1)| ≤4. (iii) |T3(2)| ≤4. All the inequalities are sharp.

Theorem 2.5. Let f ∈R be of the form (1.1). Then (i) |T2(n)| ≤4 n2 + 4 (n + 1)2 for n ≥2. (ii) |T3(1)| ≤35 9 . (iii) |T3(2)| ≤7 3. The inequalities in (i) and (ii) are sharp.

Theorem 2.6. Let J : [α, β] →Rn be continuous. Suppose that there exists a positive integer k, such that for each non-zero −→p in Rn the number of solutions of any equation ⟨−−→ J(t), −→p ⟩= const, α ≤t ≤β is not greater than k. Then, for every µ ∈P[α,β] such that Jµ belongs to the boundary of the convex hull of J([α, β]), the following statements are true: (1) if k = 2m, then (a) |supp(µ)| ≤m, or (b) |supp(µ)| = m + 1 and {α, β} ⊂supp(µ). (2) if k = 2m + 1, then (a) |supp(µ)| ≤m, or (b) |supp(µ

Lemma 2.5. The boundary of A2,3 consists of points (a2, a3) that correspond to the functions F(z, 1, t, 0) = k(z, t) or F(z, α, 1, −1) with 0 ≤α ≤1 and −1 ≤t ≤1 where F(z, α, t1, t2) is defined by (2.21). In a similar way, one can obtain the following:

Lemma 2.6. The boundary of A3,4 consists of points (a3, a4) that correspond to the functions F(z, α, t, −1) or F(z, α, t, 1) with 0 ≤α ≤1 and −1 ≤t ≤1 where F(z, α, t1, t2) is defined by (2.21). Before we proceed further, we give some example of typically real functions.

Lemma 2.7. If f ∈T then T2(n) attains its extreme values on the boundary of An,n+1.

Lemma 2.8. If f ∈T then T3(1) attains its extreme values on the boundary of A2,3. Since all coefficients of f ∈T are real, we look for the lower and the upper bounds of Tq(n) instead of the bound of |Tq(n)|. The proof of the following theorem is obvious.

Theorem 2.7. For every function f ∈T of the form (1.1), we have −(n + 1)2 ≤ T2(n) ≤n2. In particular (i) if n is odd then max{T2(n) : f ∈T } = n2 and equality attained for the function F(z, 1/2, 1, −1). (ii) if n is even then min{T2(n) : f ∈T } = −(n + 1)2 and equality attained for the function F(z, 1/2, 1, −1).

Theorem 2.8. For f ∈T , max{T2(2) : f ∈T } = 5 4.

Corollary 2.1. For f ∈T , we have the sharp inequality −9 ≤T2(2) ≤5 4.

Theorem 2.9. For f ∈T , we have min{T2(3) : f ∈T } = −7.

Corollary 2.2. For f ∈T , we have the sharp inequality −7 ≤T2(3) ≤9.

Theorem 2.10. For f ∈T , we have max{T3(1) : f ∈T } = 8, and min{T3(1) : f ∈ T } = −8.