Abstract
In the paper we consider a general inequality \(|p_{n-1}p_{n+1}-p_n{}^2|\le 4-|p_1|^2\) involving coefficients of functions with a positive real part. We prove this inequality for \(n=2\) and \(n=3\). Consequently, the relative inequalities involving coefficients of Schwarz functions are obtained. As an application, the two sharp estimates of the Hankel determinants \(H_{3,1}\) and \(H_{2,3}\) are proved for functions in \({\mathcal S}^*(1/2)\) and \({\mathcal {M}}\), respectively.
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1 Introduction
Denote by \({\mathcal {P}}\) the class of functions analytic in \({\mathbb {D}}=\{z\in {\mathbb {C}}: |z|<1\}\), given by
and having a positive real part.
It is clear that
is true for all \(h\in {\mathcal {P}}\) and positive integers \(n\ge 2\). It immediately follows from an inequality
and the inequality \(|p_{k}p_{n-k}-p_{n}|\le 2\) proved by Livingston ( [10]). Equality in (2) holds for example for \(h(z)=\frac{1+z^n}{1-z^n}\) which means that Inequlity (2) is sharp.
On the other hand, Inequality (2) is very often not strong enough to derive good results in many extremal problems concerning coefficients of analytic functions. For this reason we shall improve it by substituting the constant 4 by an expression depending on the first coefficient of h, i.e. \(|p_1|\).
This idea appears in many papers concerning functions with positive real part. As an example it is worth recalling the result of Robertson ( [12])
or results obtained by Brown ( [1]) and Lecko ( [7]).
The functional \(F(h)=|p_{n-1}p_{n+1}-p_n{}^2|\) defined for \(h\in {\mathcal {P}}\) is rotationally invariant. It means that \(F(h)=F(h_\varphi )\), where \(h_\varphi (z)=h(e^{i\varphi } z)\). For this reason, if necessary, we can assume that \(p_1\in [0,2]\).
To prove our results we need the following lemma (see, [6]).
Lemma 1
If \(p \in {\mathcal {P}}\) is of the form (1) with \(p_1\ge 0\), then
and
for some x, y, \(w\in \overline{{\mathbb {D}}}:=\{z\in {\mathbb {C}}:|z|\le 1 \}.\)
At the begining we shall estimate \(|p_{1}p_{3}-p_2{}^2|\) in terms of \(p_1\).
Theorem 2
If \(h\in {\mathcal {P}}\) is given by (1), then
Equality holds for rotations \(h_t(\varepsilon z)\) and \(g_t(\varepsilon z)\), \(|\varepsilon |=1\) of
and
Proof
From Lemma 1,
Assuming \(p=p_1\in [0,2]\) and writing \(r=|x|\in [0,1]\), we obtain
which results in (6). \(\square \)
It is an easy exercise to show that equality in (6) holds for \(h_t\) and \(g_t\) defined by (7) and (8), respectively. In general case, if
holds true, then equality in (9) will be obtained also by \(h_t\) and \(g_t\). Indeed, for \(h_t\), \(t\in [-1,1]\) given by (7) we have
where \(T_n(t)\) stands for Chebyshev polynomials of the first kind. It is known for these polynomials that the inequality
called Turan’s inequality, holds for all integers \(n\ge 2\). This results in the equality in (9).
Analogously, for \(g_t\), \(t\in [-1,1]\) given by (8) there is
and so
Unfortunately, we are not able to prove the conjecture (9), but we can do it in one more case, i.e. for \(n=3\).
2 Main Result
Theorem 3
If \(h\in {\mathcal {P}}\) is given by (1), then
Equality holds for the same functions as in Theorem 2.
Proof
From Lemma 1, after tidious but elementary calculation,
From rotational invariance of \(|p_{2}p_{4}-p_3{}^2|\) we can assume that \(p=p_1\in [0,2]\) and write \(p=2t\), \(t\in [0,1]\). Therefore,
where
After rearranging terms and applying the triangle inequality we have
Let \(x=re^{i\varphi }\), \(y=\varrho e^{i\theta }\); hence
Denote \(\alpha =2\varphi -\theta \). The last inequality can be rewritten as follows
Now, we shall maximize the expression
with respect to \(\alpha \). Consequently,
where
Since \(g'(\tau )=2r^2\varrho t \tau +\left( r^2-(1-r^2)\varrho ^2 t^2\right) \), we discuss three cases.
I. If \(r\le \frac{\varrho t}{1+\varrho t}\), then \(g'(1)\le 0\), so \(g'(\tau )\le 0\) for all \(\tau \in [-1,1]\). Hence, g is a decreasing function of \(\tau \), so the lowest value of g is obtained for \(\tau =1\). Consequently, the lowest values of V and W are obtained for \(\alpha =0\). Moreover, for \(\alpha =0\) there is
For this reason,
where
The function h has at most one critical point \((\varrho _0,t_0)\) inside \([0,1]\times [0,1]\) depending on r, where \(\varrho _0=t_0=\sqrt{\frac{1}{2(1-r)}}\) and \(h(\varrho _0,t_0)=\frac{1+2r-2r^2}{2(1-r)}\). For this critical point we have
with equality for \(r=1/2\).
Considering values of h on the boundary of \([0,1]\times [0,1]\) results in
and so
Since \(h(\varrho ,0) = h(0,\varrho )\) and \(h(\varrho ,1) = h(1,\varrho )\), thus \(H(\varrho ,0)\) and \(h(\varrho ,1)\) are also bounded by 1.
Summing up, if \(r\le \frac{\varrho t}{1+\varrho t}\), then \(|W|\le 1\). Observe that here and on it is not necessary to check whether the greatest value of particular functions discussed above are obtained for \(\varrho \) or t in the interval [0, 1].
II. If \(r\ge \frac{\varrho t}{1-\varrho t}\), then \(g'(-1)\ge 0\), so \(g'(\tau )\ge 0\) for all \(\tau \in [-1,1]\). Hence, g is a increasing function of \(\tau \), so the lowest value of g is obtained for \(\tau =-1\). Consequently, the lowest values of V and W are obtained for \(\alpha =\pi \). Moreover, for \(\alpha =\pi \) there is
For this reason,
where
The only critical point of the function h inside \([0,1]\times [0,1]\) is \((\varrho _0,t_0)\), where \(\varrho _0=t_0=\frac{\sqrt{2}}{2}\) and \(h(\varrho _0,t_0)=\tfrac{1}{2}+r\). Additionally,
with equality for \(r=1\).
Considering values of h on the boundary of \([0,1]\times [0,1]\) results in
and so
Since \(h(\varrho ,0) = h(0,\varrho )\) and \(h(\varrho ,1) = h(1,\varrho )\), thus \(H(\varrho ,0)\) and \(h(\varrho ,1)\) are also bounded by 1.
Summing up, if \(r\ge \frac{\varrho t}{1-\varrho t}\), then \(|W|\le 1\).
III. Let \(\frac{\varrho t}{1+\varrho t}<r<\frac{\varrho t}{1-\varrho t}\). In this case the lowest value of g is equal to
where \(\tau _0=\frac{(1-r^2)\varrho ^2 t^2-r^2}{2r^2\varrho t}\). Consequently,
and
Combining cases I - III, we have \(|W|\le 1\) for all \(r, \varrho , t\in [0,1]\). This and (13) lead to (12). The sharpness of this result follows from the argument presented before Theorem 3. \(\square \)
3 Relative Inequalities for Schwarz Functions
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 \(\{z\in {\mathbb {C}}: |z|<1\}\). The function \(\omega \in {\mathcal {B}}_0\) can be written as a power series
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 \(p\in {\mathcal {P}}\).
Let p(z) and \(\omega (z)\) be of the form (1) and (14), respectively. Comparing coefficients at powers of z in
we obtain
Coefficient problems for functions \(\omega \in {\mathcal {B}}_0\) were studied in numerous papers; for details, see ( [2, 3, 11, 13, 15]). Here, we need the inequality obtained in [15].
Theorem 4
If \(\omega \in {\mathcal {B}}_0\) is given by (14), then
Equality holds for rotations \(\varepsilon ^{-1} \omega (\varepsilon z)\), \(|\varepsilon |=1\) of
Directly from Theorem 3 and (15) we get the following theorem.
Theorem 5
If \(\omega \in {\mathcal {B}}_0\) is given by (14), then
Equality holds for rotations \(\varepsilon ^{-1} \omega (\varepsilon z)\), \(|\varepsilon |=1\) of
Finally, we can combine the inequalities from Theorem 5 and Theorem 4 to obtain more general inequality.
Theorem 6
If \(\omega \in {\mathcal {B}}_0\) is given by (14), then for all \(\alpha \in [0,1]\)
4 Applications
As an application of the theorems presented in the two previous sections we derive the sharp bounds of Hankel determinants for two classes of analytic functions, namely for the class \({\mathcal {S}}^*(1/2)\) of starlike functions of order 1/2 and the class \({\mathcal {M}}\) which consists of functions which are not necessarily univalent. Let us recall the definitions of these classes.
Denote by \({\mathcal {S}}^*(\alpha )\), \(\alpha \in [0,1)\) the class of functions analytic in \({\mathbb {D}}\) such that
and satisfying the condition
In a similar way the class \({\mathcal {M}}(\alpha )\), \(\alpha >0\) is defined. Namely, this family consists of functions analytic in \({\mathbb {D}}\) with a power series expanion (20) such that
Let \({\mathcal {M}}={\mathcal {M}}(1)\). Clearly, if \(f\in {\mathcal {M}}\), then
The Hankel determinants which we shall discuss are defined as follows:
and
The problem of deriving bounds of Hankel determinants has focused the attention of many mathematicians in the recent years. It was solved for the majority of classes of analytic functions in case of the second Hankel determinant given by \(H_{2,2} = a_2a_4-a_3{}^2\). Generally speaking, this problem for \(H_{3,1}\) is difficult, but today it is solved, among others, for main classes of function, for example for \({\mathcal {S}}^*\) or \({\mathcal {K}}\) (see, [4, 5]). Usually, the calculation is very hard. The studies on \(H_{2,3}\) are not so intensive as in case of \(H_{3,1}\), only a few sharp results are known (see, [14]).
Theorem 7
If \(f\in {\mathcal {S}}^*(1/2)\) is given by (20), then
Equality holds for
Proof
Each function \(f\in {\mathcal {S}}^*(1/2)\) can be written in terms of \(h\in {\mathcal {P}}\)
or equivalently, in terms of \(\omega \in {\mathcal {B}}_0\)
Hence, comparing coefficients in the equality \(zf'(z)(1-\omega (z))=f(z)\), where f and \(\omega \) are of the form (20) and (14), respectively, we obtain
Consequently,
and
where
If \(c\in [0,9/11)\), then
Denoting the latter by \(g_1(c)\) we can see that
and so in this case
If \(c\in [9/11,1]\), then
Denoting the latter by \(g_2(c)\) we obtain
and in this case
Combining both cases we have proven that \(G(c,d)\le 27\) and so \(|H_{2,3}| \le \frac{3}{16}\) with equality for \(c=0\) and \(d=1\). It means that the extremal function \(f\in {\mathcal {S}}^*(1/2)\) is obtained when in (24) we put \(\omega (z)=z^2\). \(\square \)
Theorem 8
If \(f\in {\mathcal {M}}\) is given by (20), then
Equality holds for
Proof
Each function \(f\in {\mathcal {M}}\) can be written in terms of \(h\in {\mathcal {P}}\)
or equivalently, in terms of \(\omega \in {\mathcal {B}}_0\)
Hence, comparing coefficients in the equality \(zf'(z)(1-\omega (z))=f(z)(1-2\omega (z))\), where f and \(\omega \) are of the form (20) and (14), respectively, we obtain
Consequently,
Note that \(H_{3,1}\) for \({\mathcal {M}}\) is exactly the same as \(H_{2,3}\) for \({\mathcal {S}}^*(1/2)\). From the proof of Theorem 7 it follows our result. \(\square \)
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This work was supported by FD-20/IM-5/140 from Lublin University of Technology.
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Zaprawa, P. On a Coefficient Inequality for Carathéodory Functions. Results Math 79, 30 (2024). https://doi.org/10.1007/s00025-023-02061-8
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DOI: https://doi.org/10.1007/s00025-023-02061-8