Inequalities for the Coefficients of Schwarz Functions

Theorem 1 ([2]) Let ω(z) = ∞ n=1 cnzn be in B0. Then, |c2n| ≤1 −|c1|2 −|c2|2 −. . . −|cn|2 , n = 1, 2, . . . (5) and |c2n+1| ≤1 −|c1|2 −|c2|2 −. . . −|cn|2 −|cn+1|2 1 + |c1| , n = 1, 2, . . . . (6) 123

Theorem 2 ([9]) For a function ω analytic in  with the power series expansion (1), the following conditions are equivalent: 1. ω ∈B0 2. for all positive integers N and for all λ j ∈C, j = 1, 2, . . . , N we have N  j=1  N  k= j ck+1−jλk  2 ≤

Lemma 3 If p ∈P is of the form (2) with p1 ≥0, then 2p2 = p2 1 + (4 −p2 1)x, (9) 4p3 = p3 1 + (4 −p2 1)p1x(2 −x) + 2(4 −p2 1)(1 −|x|2)y (10) and 8p4 =p4 1 + (4 −p2 1)x

Theorem 4 If ω ∈B0 is given by (1), then the following sharp inequality holds for all n ∈N n  j=2 c j −c1c j−1 2 ≤1. (14) Equality holds for each ω(z) = z j, j ∈N, 2 ≤j ≤n. Consequently, we have

Corollary 5 If ω ∈B0 is given by (1), then (13) is true for all n ∈N, n ≥2.

Theorem 6 If ω ∈B0 is given by (1), then the following sharp inequalities hold for all n ∈N and μ ∈R n  j=2 c j −μc1c j−1 2 ≤max{1, μ2} (15) and |cn −μc1cn−1| ≤max{1, |μ|}. (16) Equalities hold for each ω(z) = z j, j ∈N, 2 ≤j ≤n. Observe that (15) is a generalization of (4). The application of the same method as in the proof of Theorem 4, but with the choice of λn = 1, λn−k = −ck and λi = 0 for all i ̸= k where an integer k is chosen

Theorem 7 If ω ∈B0 is given by (1), then the following sharp inequality n  j=k+1 c j −ckc j−k 2 ≤1 − k−1  j=1 |c j|2 (17) holds for all n, k ∈N such that 2 ≤k < n. Equality holds for each ω(z) = z j, j ∈N \ {k}, j ≤n. Consequently,

Corollary 8 If ω ∈B0 is given by (1), then |cn −ckcn−k|2 ≤1 − k−1  j=1 |c j|2. (18) is true for all n, k ∈N and 2 ≤k < n. Taking k = n −1 results in the following improvement in (13).

Corollary 9 If ω ∈B0 is given by (1), then |cn −cn−1c1|2 ≤1 − n−2  j=1 |c j|2. (19) is true for all n ∈N, n ≥3.

Theorem 11 If ω ∈B0 is given by (1) and c = c1 ∈[0, 1], then the following inequality holds |c3 −c1c2| ≤  1 4(1 + c)(2 −c)2, c ∈[0, 2 3] 2c(1 −c2), c ∈[ 2 3, 1]. (24) Inequality (24) is sharp for c = 0 and c ∈[ 2 3, 1]. In the first case, the extremal function is ω(z) = z3. In the other, the extremal function is given by

Theorem 12 If ω ∈B0 is given by (1) and c = c1 ∈[0, 1], μ ∈R, then the following inequality holds |c3 −μc1c2| ≤  1 4(1 + c)[4(1 −c) + μ2c2], c ∈[0, 2 2+|μ|] (1 + |μ|)c(1 −c2), c ∈[ 2 2+|μ|, 1]. (26) In particular, if μ = 2, then

Theorem 13 If ω ∈B0 is given by (1) and c = c1 ∈[0, 1], then the following inequality holds |c3 −2c1c2| ≤  1 + c3, c ∈[0, 1 2] 3c(1 −c2), c ∈[ 1 2, 1]. (27) 3 Hankel Determinants For a given analytic function f of the form (12), we define the second Hankel determinant as H2(n) =

Theorem 14 If ω ∈B0 is given by (1) and c = c1 ∈[0, 1], then the following sharp inequality holds c1c3 −c2 2  ≤1 −c2 (28) with equality for the function defined by (25)

Theorem 16 If ω ∈B0 is given by (1) and c = c1 ∈[0, 1], μ ∈R, then c1c3 −μc2 2  ≤  c(1 −c2), |μ| ≤ c 1+c (1 −c2)[c2 + |μ|(1 −c2)], |μ| ≥ c 1+c. (29) The result is sharp.

Corollary 17 If ω ∈B0 is given by (1) and c = c1 ∈[0, 1], then 2c1c3 −c2 2  ≤1 −c4. (30) The result is sharp. In Theorem 16, so consequently in Corollary 17, the equality holds for a function given by (25). Now, let us turn to the estimation of cn−1cn+1 −c2 n . Applying (7), we are able to obtain the following general result.

Theorem 18 If ω ∈B0 is given by (1) and c = c1 ∈[0, 1], then for all n ∈N, n ≥3, cn−1cn+1 −c2 n  ≤1 −c2. (31)

Theorem 19 If ω ∈B0 is given by (1) and c = c1 ∈[0, 1], then for all n ∈N, n ≥3, μ ∈R, cn−1cn+1 −μc2 n  ≤  1 −|μ|c2 |μ| ≤1 |μ| −c2 |μ| ≥1. (34) Although Theorem 14 is sharp for all c ∈[0, 1], the equality in Theorem 18 holds only for c = 0 and c = 1. We shall find the sharp estimate for all c ∈[0, 1] also for case n = 3.

Theorem 20 If ω ∈B0 is given by (1) and c = c1 ∈[0, 1], then c2c4 −c2 3  ≤(1 −c2)2. (35) Equality holds for rotations ε−1ω(εz) of ω(z) = z(c + z2) 1 + cz2 = cz + (1 −c2)z3 −c(1 −c2)z5 + . . . (36) where |ε| = 1. 123

Lemma 21 If ω ∈B0 is given by (1), then for all n ∈N, n ≥4, cn−2cn −c2 n−1  2 + cn−3cn−1 −c2 n−2  2 ≤|cn−2|2 + |cn−1|2. (42)

Theorem 22 If ω ∈B0 is given by (1), then ∞  n=2 cn−1cn+1 −c2 n  2 ≤ ∞  n=2 |cn|2. (44) 123

Corollary 23 If ω ∈B0 is given by (1), then ∞  n=2 cn−1cn+1 −c2 n  2 ≤1 −|c1|2. (45) 4 Conclusions From the results proved in two previous sections, we can observe that for Schwarz functions given by (1) we have three similar inequalities valid for all integers n ≥2. The first one is the inequality n