Inequalities for the Coefficients of Schwarz Functions

The relation between a considered family of analytic functions and the class P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}$$\end{document} of functions with a positive real part is one of the main tools used in solving various extremal problems, among others coefficient problems. Another approach can be useful in solving such tasks. This approach is to exploit the correspondence between a considered family and the family B0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}_0$$\end{document} of bounded analytic functions ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document} such that ω(0)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (0)=0$$\end{document}. Such functions appear in the well-known Schwarz lemma, so they are called Schwarz functions. In the literature, there are numerous coefficient functionals discussed for functions in P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}$$\end{document}. On the other hand, relative functionals for functions in B0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}_0$$\end{document} are not so commonly studied. Consequently, we do not know so much about coefficient inequalities for Schwarz functions. We shall fill the gap to some extent considering two types of functionals. The first one is a Zalcman-type functional cn-ckcn-k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{n}-c_{k}c_{n-k}$$\end{document}; the other one is the Hankel determinant cn-1cn+1-cn2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{n-1}c_{n+1}-c_{n}{}^2$$\end{document}. For these functionals, bounds with respect to a fixed first coefficient c1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_1$$\end{document} (or a few initial coefficients) are obtained. Some generalizations of these functionals are also given. All results are sharp.

written as a power series ω(z) = ∞ n=1 c n z n z ∈ D. (1) Denote by P the class of functions analytic in D, given by p(z) = 1 + ∞ n=1 p n z n (2) and having a positive real part. It is clear that if then p ∈ P if and only if ω ∈ B 0 .
This property makes it possible to discuss problems in B 0 considering the class P and vice versa. Further, we apply this property to establish a relation between the initial coefficients of ω ∈ B 0 and p ∈ P. From the Schwarz-Pick lemma, it follows that for ω ∈ B 0 of the form (1), This inequality can be easily improved as follows. For any μ ∈ C, Carlson in [2] obtained another generalization of the Schwarz-Pick lemma. In fact, he established some inequalities for the set B of functions bounded by 1 (the assumption ω(0) = 0 is not necessarily satisfied). Here, we adapt these inequalities for the class B 0 (for details, see [8]).
Although the majority of our results will be derived with the use of the theorems given above, in the proof of Theorem 20 we apply a different approach. We express the initial coefficients of a Schwarz function ω ∈ B 0 by the corresponding coefficients of a function with a positive real part p ∈ P.
Let p(z) and ω(z) be of the form (2) and (1), respectively. Comparing coefficients at powers of z in we obtain Consequently, we need the following lemma (see, [4]).
Throughout the paper, we assume that the first coefficient of ω ∈ B 0 is a nonnegative real number. Consequently, we assume that c 1 = c ∈ [0, 1]. This assumption does not restrict the generality of our consideration because for any ϕ ∈ R A suitable choice of ϕ makes c 1 being real and greater than or equal to 0.

Zalcman Functionals
For a given ω ∈ B 0 of the form (1), consider the functional (ω) = c n − c k c n−k . A related functional ( f ) = a n − a k a n−k+1 defined for an analytic function is called a general Zalcman functional. Its classical version 0 ( f ) = a 2n−1 − a n 2 appeared in the late 1960 s and was connected with the famous Zalcman conjecture for analytic univalent functions in S. Zalcman conjectured (see, [1]) that |a 2n−1 − a n 2 | ≤ (n − 1) 2 for f ∈ S and n ≥ 2. This conjecture was verified for S and n = 2, 3, 4, 5, 6 as well as for many other subclasses of S.
For a function p ∈ P of the form (2), an analogous functional is defined by ( p) = p n − p k p n−k . It was Livingston who proved in [5] that | p n − p k p n−k | ≤ 2 for p ∈ P and 0 ≤ k ≤ n.

Theorem 4
If ω ∈ B 0 is given by (1), then the following sharp inequality holds for all Equality holds for each ω(z) = z j , j ∈ N, 2 ≤ j ≤ n.
Consequently, we have

Proof of Theorem 4 Applying Theorem 2 for
This results in

Theorem 6
If ω ∈ B 0 is given by (1), then the following sharp inequalities hold for all n ∈ N and μ ∈ R n j=2 and 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 λ n = 1, λ n−k = −c k and λ i = 0 for all i = k where an integer k is chosen to satisfy 2 ≤ k < n, leads to This results in

Theorem 7
If ω ∈ B 0 is given by (1), then the following sharp inequality holds for all n, k ∈ N such that 2 ≤ k < n. Equality holds for each Consequently,

Corollary 8
If ω ∈ B 0 is given by (1), then 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 ∈ B 0 is given by (1), then is true for all n ∈ N, n ≥ 3.

Remark 10
A slight modification of the proof of Theorem 7 yields and Interestingly, this inequality can be improved by applying Carlson's theorem. We can write This means that holds for all ω ∈ B 0 given by (1) and all integers m ≥ 2. From Corollary 9, we know that for ω ∈ B 0 given by (1) This inequality can be slightly improved if we apply Carlson's theorem once again.

Theorem 11
If ω ∈ B 0 is given by (1) and c = c 1 ∈ [0, 1], then the following inequality holds Inequality (24) is sharp for c = 0 and c ∈ [ 2 3 , 1]. In the first case, the extremal function is ω(z) = z 3 . In the other, the extremal function is given by Comparing two bounds for |c 3 − c 1 c 2 |, i.e., the bound in (24) and √ 1 − c 2 which follows from (19), we can see that the first bound is better and the equality in [0, 1] holds only for c = 0, c = √ 2/2 and c = 1.

Proof of Theorem 11
From the triangle inequality and Theorem 1, where the set of variability of (|c 1 |, For a fixed x ∈ [0, 1], the function h(·, y) is increasing for y < 1 2 x(1 + x) and decreasing for y > 1 2 x(1 + x). Hence, This results in (24).
In the same way, we can prove what follows.

Hankel Determinants
For a given analytic function f of the form (12), we define the second Hankel determinant as In recent years, the second Hankel determinant has been widely discussed for various subclasses of S as well as for some subclasses of non-univalent functions. The research mainly focused on H 2 (2) (for numerous result, see [7]). It is worth noting that the sharp bound of a 2 a 4 − a 3 2 for the whole class S is still not known. In this section, we derive the sharp bounds of H 2 (n) for ω ∈ B 0 and n ∈ N.

with equality for the function defined by (25)
Proof From the triangle inequality and from (3),

Remark 15
The same bound can be obtained replacing |c 1 c 3 − c 2 2 | by |c 1 c 3 + c 2 2 |. In this case, the bound is not sharp for all c ∈ [0, 1], but for c = 0 and c = 1. The extremal functions are ω(z) = z 2 and ω(z) = z, respectively.
A slight modification of the above proof leads to a more general version of Theorem 14.
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 c n−1 c n+1 − c 2 n . Applying (7), we are able to obtain the following general result.

Theorem 18
If ω ∈ B 0 is given by (1) and c = c 1 ∈ [0, 1], then for all n ∈ N, n ≥ 3, Proof Inequality (7) applied with N = n + 1 implies Consequently, By omitting the last two components which are non-positive, we have which results in (31) with the equality for ω(z) = z and ω(z) = z n .
Furthermore, we can generalize the inequality in (31) in two directions. For the first one, let k, m, n be integers greater than 1 and k < n, m < n and let N = min{k, m}. Then, The equality holds for ω(z) = z j , j = 1, 2, . . . , N or j = n.
To obtain the other generalization, we discuss two cases. Assume that μ ∈ [0, 1]. Hence, Let now μ ≥ 1. We have Finally, observe that Theorem 18 is still valid if we replace c n−1 c n+1 −c 2 n by c n−1 c n+1 + c 2 n . Combining information given above, we have
For the rotation of a function given by (36), there is It is clear that for n ≥ 3 and we have This suggests that the exact bound of |c n−1 c n+1 −c 2 n | for all n ≥ 3 is equal to (1−c 2 ) 2 . Now, we shall estimate the sum At the beginning, we prove the following lemma.

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 j=1 |c j | 2 ≤ 1.
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Declarations
Conflict of interest Author has no relevant financial or non-financial interests to disclose.
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