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.
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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
Denote by \({\mathcal {P}}\) the class of functions analytic in \({\mathbb {D}}\), given by
and having a positive real part.
It is clear that if
then
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),
This inequality can be easily improved as follows. For any \(\mu \in {\mathbb {C}}\),
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,
and
Equality in (5) holds for
and in (6) for
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
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
we obtain
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
and
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 \(\varphi \in {\mathbb {R}}\)
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
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
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 \(n\in {\mathbb {N}}\)
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
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
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 \(\mu \in {\mathbb {R}}\)
and
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
This results in
Theorem 7
If \(\omega \in {\mathcal {B}}_0\) is given by (1), then the following sharp inequality
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
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
is true for all \(n\in {\mathbb {N}}\), \(n\ge 3\).
Remark 10
A slight modification of the proof of Theorem 7 yields
and
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
Interestingly, this inequality can be improved by applying Carlson’s theorem. We can write
This means that
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
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
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,
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,
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
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
3 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 \({\mathcal {S}}\) 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
with equality for the function defined by (25)
Proof
From the triangle inequality and from (3),
\(\square \)
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
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
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 \(n\in {\mathbb {N}}\), \(n\ge 3\),
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 \(\omega (z)=z\) 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,
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,
Let now \(\mu \ge 1\). We have
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 \(n\in {\mathbb {N}}\), \(n\ge 3\), \(\mu \in {\mathbb {R}}\),
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
Equality holds for rotations \(\varepsilon ^{-1} \omega (\varepsilon z)\) of
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
Writing \(p=p_1\) and \(t=4-p^2\) and applying Lemma 3, it follows that
and
and
Hence, after tedious yet noncomplicated computations
Consequently,
where \(r=\vert x\vert \in [0,1]\) and \(\varrho =\vert y\vert \in [0,1]\). But
so
Substituting \(c=c_1=\tfrac{1}{2} p\) completes the proof. \(\square \)
For the rotation of a function given by (36), there is
It is clear that for \(n\ge 3\) and
we have
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
At the beginning, we prove the following lemma.
Lemma 21
If \(\omega \in {\mathcal {B}}_0\) is given by (1), then for all \(n\in {\mathbb {N}}\), \(n\ge 4\),
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,
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
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
Consequently,
Corollary 23
If \(\omega \in {\mathcal {B}}_0\) is given by (1), then
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
From Theorem 4, we know that
Finally, Corollary 23 yields
Hence, for all \(j\ge 2\),
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Zaprawa, P. Inequalities for the Coefficients of Schwarz Functions. Bull. Malays. Math. Sci. Soc. 46, 144 (2023). https://doi.org/10.1007/s40840-023-01538-7
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DOI: https://doi.org/10.1007/s40840-023-01538-7