Inequalities for the Coefficients of Schwarz Functions

Abstract

The relation between a considered family of analytic functions and the class ${\mathcal {P}}$ 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 ${\mathcal {B}}_0$ of bounded analytic functions $\omega$ such that $\omega (0)=0$. 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 ${\mathcal {P}}$. On the other hand, relative functionals for functions in ${\mathcal {B}}_0$ 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 $c_{n}-c_{k}c_{n-k}$; the other one is the Hankel determinant $c_{n-1}c_{n+1}-c_{n}{}^2$. For these functionals, bounds with respect to a fixed first coefficient $c_1$ (or a few initial coefficients) are obtained. Some generalizations of these functionals are also given. All results are sharp.

1 Introduction

Let ${\mathcal {B}}_0$ be the class of Schwarz functions, i.e., analytic functions $\omega :{\mathbb {D}}\rightarrow {\mathbb {D}}$, $\omega (0)=0$, where ${\mathbb {D}}$ stands for the open unit disk $\{z\in {\mathbb {C}}: \vert z\vert <1\}$. The function $\omega \in {\mathcal {B}}_0$ can be written as a power series

$$\begin{aligned} \omega (z)=\sum _{n=1}^\infty c_nz^n\,\ z\in {\mathbb {D}}. \end{aligned}$$

(1)

Denote by ${\mathcal {P}}$ the class of functions analytic in ${\mathbb {D}}$, given by

$$\begin{aligned} p(z) = 1+\sum _{n=1}^\infty p_n z^n \end{aligned}$$

(2)

and having a positive real part.

It is clear that if

$$\begin{aligned} p(z) = \frac{1+\omega (z)}{1-\omega (z)}\, \end{aligned}$$

then

$$\begin{aligned} p\in {\mathcal {P}} \quad \text {if and only if}\quad \omega \in {\mathcal {B}}_0. \end{aligned}$$

This property makes it possible to discuss problems in ${\mathcal {B}}_0$ considering the class ${\mathcal {P}}$ and vice versa. Further, we apply this property to establish a relation between the initial coefficients of $\omega \in {\mathcal {B}}_0$ and $p\in {\mathcal {P}}$.

From the Schwarz–Pick lemma, it follows that for $\omega \in {\mathcal {B}}_0$ of the form (1),

$$\begin{aligned} \vert c_2\vert \le 1-\vert c_1\vert ^2. \end{aligned}$$

(3)

This inequality can be easily improved as follows. For any $\mu \in {\mathbb {C}}$,

$$\begin{aligned} \vert c_2-\mu c_1^2\vert \le \max \{1,\vert \mu \vert \}. \end{aligned}$$

(4)

Carlson in [2] obtained another generalization of the Schwarz–Pick lemma. In fact, he established some inequalities for the set ${\mathcal {B}}$ of functions bounded by 1 (the assumption $\omega (0)=0$ is not necessarily satisfied). Here, we adapt these inequalities for the class ${\mathcal {B}}_0$ (for details, see [8]).

Theorem 1

([2]) Let $\omega (z)=\sum _{n=1}^\infty c_{n}z^{n}$ be in ${\mathcal {B}}_0$. Then,

$$\begin{aligned} \vert c_{2n}\vert \le 1-\vert c_1\vert ^2-\vert c_2\vert ^2-\ldots -\vert c_n\vert ^2\quad , n=1,2,\ldots \end{aligned}$$

(5)

and

$$\begin{aligned} \vert c_{2n+1}\vert \le 1-\vert c_1\vert ^2-\vert c_2\vert ^2-\ldots -\vert c_{n}\vert ^2-\frac{\vert c_{n+1}\vert ^2}{1+\vert c_1\vert }\quad , n=1,2,\ldots . \end{aligned}$$

(6)

Equality in (5) holds for

$$\begin{aligned} f(z)=\frac{c_1z+c_2z^2+\ldots +c_nz^n+\varepsilon z^{2n}}{1+\left( \overline{c_n} z^{n} +\overline{c_{n-1}} z^{n+1}+\ldots +\overline{c_1} z^{2n-1}\right) \varepsilon }\,\ \vert \varepsilon \vert = 1 \end{aligned}$$

and in (6) for

$$\begin{aligned} f(z)=\frac{c_1z+c_2z^2+\ldots +c_{n}z^{n}+\tfrac{c_{n+1}}{1+\vert c_1\vert }z^{n+1}+\varepsilon z^{2n+1}}{1+\left( \tfrac{\overline{c_{n+1}}}{1+\vert c_1\vert } z^{n} +\overline{c_{n}} z^{n+1}+\ldots +\overline{c_1} z^{2n}\right) \varepsilon }\,\ \vert \varepsilon \vert = 1\, \end{aligned}$$

where $c_1 \overline{c_{n+1}}^2 \varepsilon$ is non-positive real.

It is worth recalling the inequality similar to these given in Theorem 1. Namely, for all $\omega \in {\mathcal {B}}_0$ and any positive integer N we have

$$\begin{aligned} \sum _{k=1}^N \vert c_k\vert ^2 \le 1. \end{aligned}$$

(7)

To derive our main results, we need the theorem proved by Schur.

Theorem 2

([9]) For a function $\omega$ analytic in $\Delta$ with the power series expansion (1), the following conditions are equivalent:

• 1. $\omega \in {\mathcal {B}}_0$

• 2. for all positive integers N and for all $\lambda _j\in {\mathbb {C}}$, $j=1,2,\ldots ,N$ we have

  $$\begin{aligned} \sum _{j=1}^N \left| \sum _{k=j}^N c_{k+1-j}\lambda _k \right| ^2 \le \sum _{j=1}^N \left| \lambda _k \right| ^2.\end{aligned}$$

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 $\omega \in {\mathcal {B}}_0$ by the corresponding coefficients of a function with a positive real part $p\in {\mathcal {P}}$.

Let p(z) and $\omega (z)$ be of the form (2) and (1), respectively. Comparing coefficients at powers of z in

$$\begin{aligned} \left[ 1-\omega (z)\right] p(z)=1+\omega (z)\, \end{aligned}$$

we obtain

$$\begin{aligned} \begin{aligned} p_1= & {} 2c_1 \\ p_2= & {} 2c_2+2c_1^2 \\ p_3= & {} 2c_3+4c_1c_2+2c_1^3 \\ p_4= & {} 2c_4+4c_1c_3+2c_2^2+6c_1^2c_2+2c_1^4. \end{aligned} \end{aligned}$$

(8)

Consequently, we need the following lemma (see, [4]).

Lemma 3

If $p \in {\mathcal {P}}$ is of the form (2) with $p_1\ge 0$, then

$$\begin{aligned} 2p_2= & {} p_1^2 + (4-p_1^2)x, \end{aligned}$$

(9)

$$\begin{aligned} 4p_3= & {} p_1^3 +(4-p_1^2)p_1 x (2-x) + 2(4-p_1^2)( 1 - \vert x\vert ^2) y \end{aligned}$$

(10)

and

$$\begin{aligned} 8p_4 =&p_1^4+(4-p_1^2) x \left[ p_1^2(x^2-3x+3)+4x \right] \nonumber \\&-4(4-p_1^2)(1-\vert x\vert ^2)\left[ p_1(x-1)y+{\overline{x}}y^2-\left( 1-\vert y\vert ^2\right) w\right] \end{aligned}$$

(11)

for some x, y, $w\in \overline{{\mathbb {D}}}:=\{z\in {\mathbb {C}}:\vert z\vert \le 1 \}.$

Throughout the paper, we assume that the first coefficient of $\omega \in {\mathcal {B}}_0$ is a non-negative real number. Consequently, we assume that $c_1=c\in [0,1]$. This assumption does not restrict the generality of our consideration because for any $\varphi \in {\mathbb {R}}$

$$\begin{aligned} \omega (z)\in {\mathcal {B}}_0 \quad \text {if and only if}\quad \omega (z e^{i\varphi })\in {\mathcal {B}}_0. \end{aligned}$$

A suitable choice of $\varphi$ makes $c_1$ being real and greater than or equal to 0.

2 Zalcman Functionals

For a given $\omega \in {\mathcal {B}}_0$ of the form (1), consider the functional $\Phi (\omega )=c_{n}-c_{k}c_{n-k}$. A related functional ${\widetilde{\Phi }}(f)=a_{n}-a_{k}a_{n-k+1}$ defined for an analytic function

$$\begin{aligned} f(z)=a_0+a_1z+a_2z^2+\ldots \end{aligned}$$

(12)

is called a general Zalcman functional. Its classical version ${\widetilde{\Phi }}_0(f)=a_{2n-1}-{a_{n}}^2$ appeared in the late 1960 s and was connected with the famous Zalcman conjecture for analytic univalent functions in ${\mathcal {S}}$. Zalcman conjectured (see, [1]) that $\vert a_{2n-1}-{a_{n}}^2\vert \le (n-1)^2$ for $f\in {\mathcal {S}}$ and $n\ge 2$. This conjecture was verified for ${\mathcal {S}}$ and $n=2,3,4,5,6$ as well as for many other subclasses of ${\mathcal {S}}$.

For a function $p\in {\mathcal {P}}$ of the form (2), an analogous functional is defined by ${\widehat{\Phi }}(p)=p_{n}-p_{k}p_{n-k}$. It was Livingston who proved in [5] that $\vert p_{n}-p_{k}p_{n-k}\vert \le 2$ for $p\in {\mathcal {P}}$ and $0\le k\le n$.

Special cases of $\Phi (\omega )$ for $\omega \in {\mathcal {B}}_0$ appeared in [3]. From this paper, Formulae (17) and (21) with $\lambda =0$ we know that $\left| c_3-c_1c_2 \right| \le 1$ and $\left| c_4-c_1c_3 \right| \le 1$. Moreover, $\left| c_5-c_1c_4 \right| \le 1$, as it was shown in [10] (Formula (1.8) with $\mu =0$). We shall show that an analogous inequality

$$\begin{aligned} \left| c_n-c_1c_{n-1} \right| \le 1 \end{aligned}$$

(13)

holds for all integers $n\ge 2$.

Theorem 4

If $\omega \in {\mathcal {B}}_0$ is given by (1), then the following sharp inequality holds for all $n\in {\mathbb {N}}$

$$\begin{aligned} \sum _{j=2}^n \left| c_{j}-c_{1}c_{j-1}\right| ^2 \le 1. \end{aligned}$$

(14)

Equality holds for each $\omega (z)=z^j$, $j\in {\mathbb {N}}$, $2\le j\le n$.

Consequently, we have

Corollary 5

If $\omega \in {\mathcal {B}}_0$ is given by (1), then (13) is true for all $n\in {\mathbb {N}}$, $n\ge 2$.

Proof of Theorem 4

Applying Theorem 2 for $N=n$ with $\lambda _1=\ldots =\lambda _{n-2}=0$ and $\lambda _{n-1}=-c_{1}$, $\lambda _{n}=1$, we immediately get

$$\begin{aligned} \sum _{j=2}^n \left| c_{j}-c_{1}c_{j-1}\right| ^2 + \vert c_1\vert ^2 \le 1 + \vert c_1\vert ^2 \end{aligned}$$

which is equivalent to (14). $\square$

If in Theorem 4 instead of $\lambda _{n-1}=-c_{1}$ we take $\lambda _{n-1}=-\mu c_{1}$, $\mu \in {\mathbb {R}}$, then we obtain

$$\begin{aligned} \sum _{j=2}^n \left| c_{j}-\mu c_{1}c_{j-1}\right| ^2 \le 1 - \vert c_1\vert ^2(1-\mu ^2). \end{aligned}$$

This results in

Theorem 6

If $\omega \in {\mathcal {B}}_0$ is given by (1), then the following sharp inequalities hold for all $n\in {\mathbb {N}}$ and $\mu \in {\mathbb {R}}$

$$\begin{aligned} \sum _{j=2}^n \left| c_{j} - \mu c_{1}c_{j-1}\right| ^2 \le \max \{1,\mu ^2\} \end{aligned}$$

(15)

and

$$\begin{aligned} \left| c_{n} - \mu c_{1}c_{n-1}\right| \le \max \{1,\vert \mu \vert \}. \end{aligned}$$

(16)

Equalities hold for each $\omega (z)=z^j$, $j\in {\mathbb {N}}$, $2\le j\le n$.

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 $\lambda _{n}=1$, $\lambda _{n-k}=-c_{k}$ and $\lambda _i=0$ for all $i\ne k$ where an integer k is chosen to satisfy $2\le k<n$, leads to

$$\begin{aligned} \sum _{j=k+1}^n \left| c_{j}-c_{k}c_{j-k}\right| ^2 + \sum _{j=1}^k \vert c_j\vert ^2 \le 1 + \vert c_k\vert ^2. \end{aligned}$$

This results in

Theorem 7

If $\omega \in {\mathcal {B}}_0$ is given by (1), then the following sharp inequality

$$\begin{aligned} \sum _{j=k+1}^n \left| c_{j}-c_{k}c_{j-k}\right| ^2 \le 1 - \sum _{j=1}^{k-1} \vert c_j\vert ^2 \end{aligned}$$

(17)

holds for all $n,k\in {\mathbb {N}}$ such that $2\le k<n$. Equality holds for each $\omega (z)=z^j$, $j\in {\mathbb {N}}\setminus \{k\}$, $j\le n$.

Consequently,

Corollary 8

If $\omega \in {\mathcal {B}}_0$ is given by (1), then

$$\begin{aligned} \left| c_{n}-c_{k}c_{n-k}\right| ^2 \le 1 - \sum _{j=1}^{k-1} \vert c_j\vert ^2. \end{aligned}$$

(18)

is true for all $n,k\in {\mathbb {N}}$ and $2\le k<n$.

Taking $k=n-1$ results in the following improvement in (13).

Corollary 9

If $\omega \in {\mathcal {B}}_0$ is given by (1), then

$$\begin{aligned} \left| c_{n}-c_{n-1}c_{1}\right| ^2 \le 1 - \sum _{j=1}^{n-2} \vert c_j\vert ^2. \end{aligned}$$

(19)

is true for all $n\in {\mathbb {N}}$, $n\ge 3$.

Remark 10

A slight modification of the proof of Theorem 7 yields

$$\begin{aligned} \left| c_{n}-\mu c_{k}c_{n-k}\right| ^2 \le 1 - \sum _{j=1}^{k-1} \vert c_j\vert ^2 \end{aligned}$$

(20)

and

$$\begin{aligned} \left| c_{n}-\mu c_{n-1}c_{1}\right| ^2 \le 1 - \sum _{j=1}^{n-2} \vert c_j\vert ^2\, \end{aligned}$$

(21)

which are true under the conditions of Corollaries 8 and 9 and for $\mu \in [-1,1]$.

The choice of $n=2\,m$ and $k=m$, $m\ge 2$ in Corollary 8 leads to

$$\begin{aligned} \left| c_{2m}-{c_{m}}^2\right| ^2 \le 1 - \sum _{j=1}^{m-1} \vert c_j\vert ^2. \end{aligned}$$

(22)

Interestingly, this inequality can be improved by applying Carlson’s theorem. We can write

$$\begin{aligned} \left| c_{2m}-{c_{m}}^2\right| \le \vert c_{2m}\vert +\vert c_m\vert ^2 \le \left( 1 - \sum _{j=1}^{m} \vert c_j\vert ^2\right) + \vert c_m\vert ^2. \end{aligned}$$

This means that

$$\begin{aligned} \left| c_{2m}-{c_{m}}^2\right| \le 1 - \sum _{j=1}^{m-1} \vert c_j\vert ^2 \end{aligned}$$

(23)

holds for all $\omega \in {\mathcal {B}}_0$ given by (1) and all integers $m\ge 2$.

From Corollary 9, we know that for $\omega \in {\mathcal {B}}_0$ given by (1) with $c=c_1\in [0,1]$ there is $\left| c_{3}-c_{1}c_{2}\right| \le \sqrt{1-c^2}$. This inequality can be slightly improved if we apply Carlson’s theorem once again.

Theorem 11

If $\omega \in {\mathcal {B}}_0$ is given by (1) and $c=c_1\in [0,1]$, then the following inequality holds

$$\begin{aligned} \left| c_3-c_1c_2 \right| \le {\left\{ \begin{array}{ll} \tfrac{1}{4} (1+c)(2-c)^2, &{}\quad c\in [0,\tfrac{2}{3}] \\ 2c(1-c^2), &{}\quad c\in [\tfrac{2}{3},1]. \end{array}\right. } \end{aligned}$$

(24)

Inequality (24) is sharp for $c=0$ and $c\in [\tfrac{2}{3},1]$. In the first case, the extremal function is $\omega (z)=z^3$. In the other, the extremal function is given by

$$\begin{aligned} \omega (z)=\frac{cz+z^2}{1+cz}=cz+(1-c^2)z^2-c(1-c^2)z^3+\ldots . \end{aligned}$$

(25)

Comparing two bounds for $\vert c_3-c_1c_2\vert$, i.e., the bound in (24) and $\sqrt{1-c^2}$ which follows from (19), we can see that the first bound is better and the equality in [0, 1] holds only for $c=0$, $c=\sqrt{2}/2$ and $c=1$.

Proof of Theorem 11

From the triangle inequality and Theorem 1,

$$\begin{aligned} \left| c_3-c_1c_2\right| \le 1-\vert c_1\vert ^2-\frac{\vert c_2\vert ^2}{1+\vert c_1\vert }+\vert c_1\vert \vert c_2\vert = h(\vert c_1\vert ,\vert c_2\vert )\, \end{aligned}$$

where the set of variability of $(\vert c_1\vert ,\vert c_2\vert )$ coincides with $\Omega =\{(x,y): x\in [0,1], 0\le y\le 1-x^2\}$.

For a fixed $x\in [0,1]$, the function $h(\cdot ,y)$ is increasing for $y<\tfrac{1}{2} x(1+x)$ and decreasing for $y>\tfrac{1}{2} x(1+x)$. Hence,

$$\begin{aligned}h(x,y) \le {\left\{ \begin{array}{ll} h(x,\tfrac{1}{2} x(1+x)), &{}\quad x\in [0,\tfrac{2}{3}] \\ h(x,1-x^2), &{}\quad x\in [\tfrac{2}{3},1]. \end{array}\right. } \end{aligned}$$

This results in (24). $\square$

In the same way, we can prove what follows.

Theorem 12

If $\omega \in {\mathcal {B}}_0$ is given by (1) and $c=c_1\in [0,1]$, $\mu \in {\mathbb {R}}$, then the following inequality holds

$$\begin{aligned} \left| c_3 - \mu c_1c_2 \right| \le {\left\{ \begin{array}{ll} \tfrac{1}{4} (1+c)[4(1-c)+\mu ^2 c^2], &{}\quad c\in [0,\tfrac{2}{2+\vert \mu \vert }] \\ (1+\vert \mu \vert )c(1-c^2), &{}\quad c\in [\tfrac{2}{2+\vert \mu \vert },1]. \end{array}\right. } \end{aligned}$$

(26)

In particular, if $\mu =2$, then

Theorem 13

If $\omega \in {\mathcal {B}}_0$ is given by (1) and $c=c_1\in [0,1]$, then the following inequality holds

$$\begin{aligned} \left| c_3 - 2c_1c_2 \right| \le {\left\{ \begin{array}{ll} 1+ c^3, &{}\quad c\in [0,\tfrac{1}{2}] \\ 3c(1-c^2), &{}\quad c\in [\tfrac{1}{2},1]. \end{array}\right. } \end{aligned}$$

(27)

3 Hankel Determinants

For a given analytic function f of the form (12), we define the second Hankel determinant as

$$\begin{aligned} H_2(n)=\left| \begin{array}{ll} a_2 &{} a_3 \\ a_3 &{} a_4 \end{array} \right| = a_2a_4-{a_3}^2. \end{aligned}$$

In recent years, the second Hankel determinant has been widely discussed for various subclasses of ${\mathcal {S}}$ 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 $a_2a_4-{a_3}^2$ for the whole class ${\mathcal {S}}$ is still not known. In this section, we derive the sharp bounds of $H_2(n)$ for $\omega \in {\mathcal {B}}_0$ and $n\in {\mathbb {N}}$.

Theorem 14

If $\omega \in {\mathcal {B}}_0$ is given by (1) and $c=c_1\in [0,1]$, then the following sharp inequality holds

$$\begin{aligned} \left| c_1c_3 - c_2^2 \right| \le 1-c^2 \end{aligned}$$

(28)

with equality for the function defined by (25)

Proof

From the triangle inequality and from (3),

$$\begin{aligned} \left| c_1c_3 - c_2^2\right|\le & {} \vert c_1\vert \left( 1-\vert c_1\vert ^2-\frac{\vert c_2\vert ^2}{1+\vert c_1\vert }\right) +\vert c_2\vert ^2 = \vert c_1\vert (1-\vert c_1\vert ^2)+\frac{\vert c_2\vert ^2}{1+\vert c_1\vert } \\\le & {} \vert c_1\vert (1-\vert c_1\vert ^2)+\frac{(1-\vert c_1\vert ^2)^2}{1+\vert c_1\vert } = 1-\vert c_1\vert ^2. \end{aligned}$$

$\square$

Remark 15

The same bound can be obtained replacing $\vert c_1c_3 - c_2^2\vert$ by $\vert c_1c_3 + c_2^2\vert$. In this case, the bound is not sharp for all $c\in [0,1]$, but for $c=0$ and $c=1$. The extremal functions are $\omega (z)=z^2$ and $\omega (z)=z$, respectively.

A slight modification of the above proof leads to a more general version of Theorem 14.

Theorem 16

If $\omega \in {\mathcal {B}}_0$ is given by (1) and $c=c_1\in [0,1]$, $\mu \in {\mathbb {R}}$, then

$$\begin{aligned} \left| c_1c_3 - \mu c_2^2\right| \le {\left\{ \begin{array}{ll} c(1-c^2), &{}\quad \vert \mu \vert \le \frac{c}{1+c} \\ (1-c^2)[c^2+\vert \mu \vert (1-c^2)], &{}\quad \vert \mu \vert \ge \frac{c}{1+c}. \end{array}\right. } \end{aligned}$$

(29)

The result is sharp.

In particular, if $\mu =1/2$, then

Corollary 17

If $\omega \in {\mathcal {B}}_0$ is given by (1) and $c=c_1\in [0,1]$, then

$$\begin{aligned} \left| 2c_1c_3 - c_2^2\right| \le 1-c^4. \end{aligned}$$

(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 $\left| c_{n-1}c_{n+1} - c_n^2\right|$. Applying (7), we are able to obtain the following general result.

Theorem 18

If $\omega \in {\mathcal {B}}_0$ is given by (1) and $c=c_1\in [0,1]$, then for all $n\in {\mathbb {N}}$, $n\ge 3$,

$$\begin{aligned} \left| c_{n-1}c_{n+1} - c_n^2\right| \le 1-c^2. \end{aligned}$$

(31)

Proof

Inequality (7) applied with $N=n+1$ implies

$$\begin{aligned} \vert c_n\vert ^2 \le 1-\vert c_1\vert ^2-\vert c_2\vert ^2-\ldots -\vert c_{n-1}\vert ^2-\vert c_{n+1}\vert ^2. \end{aligned}$$

Consequently,

$$\begin{aligned} \left| c_{n-1}c_{n+1} - c_n^2\right|\le & {} \vert c_{n-1}c_{n+1}\vert +\vert c_n^2\vert \\\le & {} 1-\vert c_1\vert ^2-\vert c_2\vert ^2-\ldots -\vert c_{n-2}\vert ^2\\{} & {} -\left( \vert c_{n+1}\vert -\vert c_{n-1}\vert \right) ^2 - \vert c_{n-1}c_{n+1}\vert . \end{aligned}$$

By omitting the last two components which are non-positive, we have

$$\begin{aligned} \left| c_{n-1}c_{n+1} - c_n^2\right| \le 1-\vert c_1\vert ^2-\vert c_2\vert ^2-\ldots -\vert c_{n-2}\vert ^2\, \end{aligned}$$

(32)

which results in (31) with the equality for $\omega (z)=z$ and $\omega (z)=z^n$. $\square$

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,

$$\begin{aligned} \left| c_{k}c_{m} - c_n^2\right| \le 1-\vert c_1\vert ^2-\vert c_2\vert ^2-\ldots -\vert c_{N-1}\vert ^2. \end{aligned}$$

(33)

The equality holds for $\omega (z)=z^j$, $j=1,2,\ldots , N$ or $j=n$.

To obtain the other generalization, we discuss two cases. Assume that $\mu \in [0,1]$. Hence,

$$\begin{aligned} \left| c_{n-1}c_{n+1} - \mu c_n^2\right|= & {} \left| (1-\mu )c_{n-1}c_{n+1} + \mu \left( c_{n-1}c_{n+1} - c_n^2\right) \right| \\\le & {} 1-\mu +\mu (1-c^2) = 1-\mu c^2. \end{aligned}$$

Let now $\mu \ge 1$. We have

$$\begin{aligned} \left| c_{n-1}c_{n+1} - \mu c_n^2\right|= & {} \left| \left( c_{n-1}c_{n+1} - c_n^2\right) + (1-\mu ) c_n^2\right| \\\le & {} 1-c^2 +(\mu -1) = \mu - c^2. \end{aligned}$$

Finally, observe that Theorem 18 is still valid if we replace $c_{n-1}c_{n+1} - c_n^2$ by $c_{n-1}c_{n+1} + c_n^2$. Combining information given above, we have

Theorem 19

If $\omega \in {\mathcal {B}}_0$ is given by (1) and $c=c_1\in [0,1]$, then for all $n\in {\mathbb {N}}$, $n\ge 3$, $\mu \in {\mathbb {R}}$,

$$\begin{aligned} \left| c_{n-1}c_{n+1} - \mu c_n^2\right| \le {\left\{ \begin{array}{ll} 1-\vert \mu \vert c^2 &{} \vert \mu \vert \le 1 \\ \vert \mu \vert -c^2 &{} \vert \mu \vert \ge 1. \end{array}\right. } \end{aligned}$$

(34)

Although Theorem 14 is sharp for all $c\in [0,1]$, the equality in Theorem 18 holds only for $c=0$ and $c=1$. We shall find the sharp estimate for all $c\in [0,1]$ also for case $n=3$.

Theorem 20

If $\omega \in {\mathcal {B}}_0$ is given by (1) and $c=c_1\in [0,1]$, then

$$\begin{aligned} \left| c_2c_4 - c_3^2\right| \le (1-c^2)^2. \end{aligned}$$

(35)

Equality holds for rotations $\varepsilon ^{-1} \omega (\varepsilon z)$ of

$$\begin{aligned} \omega (z)=\frac{z(c+z^2)}{1+cz^2}=cz+(1-c^2)z^3-c(1-c^2)z^5+\ldots \, \end{aligned}$$

(36)

where $\vert \varepsilon \vert =1$.

Proof

Let $\Psi \equiv c_2c_4 - c_3^2$. From the assumption $c_1\in [0,1]$, it follows that $p_1\in [0,2]$. From (8), we get

$$\begin{aligned} \Psi = \tfrac{1}{8}\left( 2p_2p_4+2p_1p_2p_3-p_2^3-p_1^2p_4-2p_3^2\right) . \end{aligned}$$

Writing $p=p_1$ and $t=4-p^2$ and applying Lemma 3, it follows that

$$\begin{aligned} p_4(2p_2-p_1^2)= & {} \tfrac{1}{8} tx \left[ p^4+tx \left[ p^2(x^2-3x+3)+4x \right] \right. \nonumber \\{} & {} \left. -4t(1-\vert x\vert ^2)\left[ p(x-1)y+{\overline{x}}y^2-\left( 1-\vert y\vert ^2\right) w\right] \right] \end{aligned}$$

(37)

and

$$\begin{aligned} 2p_3\left( p_1p_2-p_3\right)= & {} \tfrac{1}{8}\left[ p^3 +tpx(2-x) + 2t(1-\vert x\vert ^2)y\right] \cdot \nonumber \\{} & {} \left[ p^3+tpx^2-2t(1-\vert x\vert ^2)y\right] \end{aligned}$$

(38)

and

$$\begin{aligned} p_2^3 =\tfrac{1}{8}\left[ p^6 + 3tp^4x + 3t^2p^2x^2 + t^3x^3\right] . \end{aligned}$$

(39)

Hence, after tedious yet noncomplicated computations

$$\begin{aligned} \Psi = \tfrac{1}{16} t^2(1-\vert x\vert ^2)\left[ -y^2+x(1-\vert y\vert ^2)w\right] . \end{aligned}$$

Consequently,

$$\begin{aligned} \vert \Psi \vert \le \tfrac{1}{16} t^2(1-r^2)\left[ \varrho ^2+r(1-\varrho ^2)\right] \, \end{aligned}$$

where $r=\vert x\vert \in [0,1]$ and $\varrho =\vert y\vert \in [0,1]$. But

$$\begin{aligned} (1-r^2)\left[ \varrho ^2+r(1-\varrho ^2)\right] = (1-r^2)\left[ \varrho ^2(1-r)+r\right] \le 1-r^2 \le 1\, \end{aligned}$$

so

$$\begin{aligned} \vert \Psi \vert \le \tfrac{1}{16} t^2 = \tfrac{1}{16} (4-p^2)^2. \end{aligned}$$

Substituting $c=c_1=\tfrac{1}{2} p$ completes the proof. $\square$

For the rotation of a function given by (36), there is

$$\begin{aligned} c_2c_4 - c_3^2 = -\varepsilon ^2(1-c^2)^2. \end{aligned}$$

It is clear that for $n\ge 3$ and

$$\begin{aligned} \omega (z)=\frac{z(c+\varepsilon z^{n-1})}{1+c\varepsilon z^{n-1}}=cz+\varepsilon (1-c^2)z^n-\varepsilon ^2 c(1-c^2)z^{2n-1}+\ldots \, \end{aligned}$$

(40)

we have

$$\begin{aligned} c_{n-1}c_{n+1} - c_n^2 = -\varepsilon ^2(1-c^2)^2. \end{aligned}$$

This suggests that the exact bound of $\vert c_{n-1}c_{n+1} - c_n^2\vert$ for all $n\ge 3$ is equal to $(1-c^2)^2$.

Now, we shall estimate the sum

$$\begin{aligned} \sum _{n=1}^\infty \vert c_{n-1}c_{n+1} - c_n^2\vert . \end{aligned}$$

(41)

At the beginning, we prove the following lemma.

Lemma 21

If $\omega \in {\mathcal {B}}_0$ is given by (1), then for all $n\in {\mathbb {N}}$, $n\ge 4$,

$$\begin{aligned} \left| c_{n-2}c_{n} - c_{n-1}^2\right| ^2 + \left| c_{n-3}c_{n-1} - c_{n-2}^2\right| ^2 \le \vert c_{n-2}\vert ^2+\vert c_{n-1}\vert ^2. \end{aligned}$$

(42)

Proof

It is enough to apply Theorem 2 for $N=n$ with $\lambda _1=\ldots =\lambda _{n-2}=0$ and $\lambda _{n-1}=c_{n-1}$, $\lambda _{n}=-c_{n-2}$. Hence,

$$\begin{aligned}{} & {} \left| c_{n-1}^2-c_{n}c_{n-2} \right| ^2 + \left| c_{n-2}c_{n-1}-c_{n-1}c_{n-2} \right| ^2 + \left| c_{n-3}c_{n-1} - c_{n-2}^2\right| ^2 + \sum _{j=4}^n \vert A_j\vert ^2 \nonumber \\{} & {} \quad \le \vert c_{n-2}\vert ^2+\vert c_{n-1}\vert ^2 \end{aligned}$$

(43)

where the exact values of $A_j$ are not important. By omitting non-negative terms of the left-hand side of this inequality, we obtain (44). $\square$

Although Formula (42) holds true, if we separate it into two inequalities

$$\begin{aligned} \left| c_{n-2}c_{n} - c_{n-1}^2\right| \le \vert c_{n-1}\vert \quad \text {and}\quad \left| c_{n-3}c_{n-1} - c_{n-2}^2\right| \le \vert c_{n-2}\vert \, \end{aligned}$$

then the two new inequalities are, in general, false. For example, if $\omega \in {\mathcal {B}}_0$ is such that $c_{n-2}=c_{n}=1/2$ and all other $c_k$ vanish, then the first inequality is false. Similarly, if $\omega \in {\mathcal {B}}_0$ is such that $c_{n-3}=c_{n-1}=1/2$ and $c_k=0$ for $k\ne n-3$ and $k\ne n-1$, then the second inequality is false.

Taking the sum of inequalities (42) over all even integers $n\ge 4$, we obtain

Theorem 22

If $\omega \in {\mathcal {B}}_0$ is given by (1), then

$$\begin{aligned} \sum _{n=2}^\infty \left| c_{n-1}c_{n+1} - c_{n}^2\right| ^2 \le \sum _{n=2}^\infty \vert c_{n}\vert ^2. \end{aligned}$$

(44)

Consequently,

Corollary 23

If $\omega \in {\mathcal {B}}_0$ is given by (1), then

$$\begin{aligned} \sum _{n=2}^\infty \left| c_{n-1}c_{n+1} - c_{n}^2\right| ^2 \le 1-\vert c_{1}\vert ^2. \end{aligned}$$

(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 $n\ge 2$. The first one is the inequality

$$\begin{aligned} \sum _{j=1}^n \vert c_j\vert ^2 \le 1. \end{aligned}$$

From Theorem 4, we know that

$$\begin{aligned} \sum _{j=2}^n \left| c_{j}-c_{1}c_{j-1}\right| ^2 \le 1. \end{aligned}$$

Finally, Corollary 23 yields

$$\begin{aligned} \sum _{j=2}^n \left| c_{j-1}c_{j+1} - c_{j}^2\right| ^2 \le 1. \end{aligned}$$

Hence, for all $j\ge 2$,

$$\begin{aligned} \vert c_j\vert \le 1\quad ,\quad \left| c_{j}-c_{1}c_{j-1}\right| \le 1\quad ,\quad \left| c_{j-1}c_{j+1} - c_{j}^2\right| \le 1. \end{aligned}$$