Abstract
Let f be analytic in \(D=\{z: |z|< 1\}\) with \(f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}\). Suppose that \(S^*\) is the class of starlike functions, and K is the class of close-to-convex functions. The paper instigates a study of finding estimates for Toeplitz determinants whose elements are the coefficients \(a_{n}\) for f in \(S^*\) and K.
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1 Introduction
In the theory of univalent functions, a great deal of attention (see e.g., [2, 3, 5, 6]) has been given to estimate the size of determinants of Hankel matrices, whose entries are the coefficients of analytic functions f defined in the unit disc \(D=\{z:|z|<1\}\) with Taylor series
Hankel matrices (and determinants) play an important role in several branches of mathematics and have many applications [7]. Closely related to Hankel determinants are the Toepliz determinants. A Toeplitz matrix can be thought of as an ‘upside-down’ Hankel matrix, in that Hankel matrices have constant entries along the reverse diagonal, whereas Toeplitz matrices have constant entries along the diagonal. A good summary of the applications of Toeplitz matrices to a wide range of areas of pure and applied mathematics can also be found in [7].
In this paper we instigate research into the determinants of symmetric Toeplitz determinants, whose entries are the coefficients \(a_{n}\) of starlike and close-to-convex functions.
We recall the definition of the Hankel determinant \(H_{q}(n) \) for f with the form as in (1) as follows:
and define the symmetric Toeplitz determinant \(T_{q}(n) \) as follows:
So for example
For \(f\in S\), the problem of finding the best possible bounds for \(||{a_{n+1}}|-|a_{n}||\) has a long history [1]. It is well-known [1] that \(||{a_{n+1}}|-|a_{n}||\le C\); however, finding exact values of the constant C for S and its subclasses has proved difficult. It is clear from the definition that finding estimates for \(T_{n}(q)\) is related to finding bounds for \(A(n):=|{a_{n+1}}-a_{n}|\). However, the function \(k(z)=z/(1+z)^2\) shows that the best possible upper bound obtainable for A(n) is \(2n+1\), and so obtaining bounds for A(n) is different to finding bounds for \(||{a_{n+1}}|-|a_{n}||\).
In this paper we give some sharp estimates for \(T_{n}(q)\) for low values of n and q when f is starlike and close-to-convex.
2 Definitions and Preliminaries
We first recall the definitions of starlike and close-to-convex functions.
Let f be analytic in D and be given by (1). Then a function f is starlike if, and only if,
We denote the class of starlike functions by \(S^*\).
An analytic function f is close-to-convex in D if, and only if, there exists \(g\in S^*\) such that
We denote the class of close-to-convex functions by K.
For \(f\in S^*\), we can write \(zf'(z)=f(z)h(z)\), where \(h\in P\), the class of function satisfying \(Re\ h(z)>0\) for \(z\in D\) and
For \(f\in K\), we can write \(zf'(z)=g(z)p(z)\), where \(p\in P\) and
We shall use the following result [4], which has been used widely.
Lemma 1
If \(h\in P\) with coefficients \(c_{n}\) as above, then for some complex valued x with \(|x|\le 1\) and some complex valued \(\zeta \) with \(|\zeta |\le 1\),
Similarly for \(p\in P\) with coefficients \(p_{n}\) as above, there exist some complex valued y with \(|y|\le 1\) and some complex valued \(\eta \) with \(|\eta |\le 1\), such that
We first prove the following, noting that a weaker result is proved for close-to-convex functions in Theorem 5.
3 Results
Theorem 1
For \(f\in S^*\) given by (1),
The inequality is sharp.
Proof
First note that equating coefficients in the equation \(zf'(z)=f(z)h(z)\), we have
and so
We now use Lemma 1 to express \(c_{2}\) in terms of \(c_{1}\) to obtain
where for simplicity we have written \(X=4-c_{1}^2\).
Without loss of generality we assume that \(c_{1}=c\), where \(0\le c \le 2\). Using the triangle inequality, we obtain (with now \(X=4-c^2\))
Clearly \(\phi '(|x|)>0\) on [0, 1] and so \(\phi (|x|)\le \phi (1)\).
Hence
Treating the cases when the absolute term is either positive or negative, it is a trivial exercise to show that this expression has maximum value 5 on [0, 2], when \(c=2\).
Clearly the inequality is sharp when \(f(z)=z/(1-z)^2\). \(\square \)
Theorem 2
For \(f\in S^*\) given by (1),
Proof
Using (2) and Lemma 1 to express \(c_{2}\) and \(c_{3}\) in terms of \(c_{1}\), we obtain, with \(X=4-c_{1}^2\) and \(Z=(1-|x|^2)\zeta \),
As in the proof of Theorem 1, without loss of generality we can write \(c_{1}=c\), where \(0\le c\le 2\), by using the triangle inequality,
where now \(X=4-c^2\) and \(Z=1-|x|^2\).
Substituting for X and Z in \(\phi (c,|x|)\), and differentiating with respect to |x|, we find that
Simplifying the above expression we note that \(\dfrac{20 c}{9} + \dfrac{3 c^2}{2} - \dfrac{10 c^3}{9} + \dfrac{31 c^4}{24} + \dfrac{5 c^5}{36} - \dfrac{ 5 c^6}{12} \ge 0\) for \(c\in [0,2]\). Considering the discriminant of the resulting quadratic expression in |x|, then shows that \(\phi '(c,|x|)\ge 0\) for \(|x|\in [0,1]\) and fixed \(c\in [0,2]\). It thus follows that \(\phi (c,|x|)\) increases with |x|, and so \(\phi (c,|x|)\le \phi (c,1)\). Hence
It is now an elementary exercise to show that this expression has maximum value 7, which completes the proof of the theorem.
The inequality is again sharp when \(f(z)=z/(1-z)^2\). \(\square \)
Theorem 3
For \(f\in S^*\) given by (1),
The inequality is sharp.
Proof
Write
Using the same techniques as above, it is an easy exercise to show that \(|a_{2} - a_{4}|\le 2\). Thus we need to show that \(|a_{2}^2 - 2 a_{3}^2 + a_{2}a_{4}|\le 6\).
From (2), we obtain
As before, we use Lemma 1 to express \(c_{2}\) and \(c_{3}\) in terms of \(c_{1}\) to obtain, with \(X=4-c_{1}^2\) and \(Z=(1-|x|^2)\zeta \),
Using the triangle inequality and assuming that \(c_{1}=c\) where \(0\le c\le 2 \), we obtain
Thus we need to find the maximum value of \(\mu (c,|x|)\) on \([0,2]\times [0,1]\). First assume that there is a maximum at an interior point \((c_{0}, |x_{0}|)\) of \([0,2]\times [0,1]\). Then differentiating \(\mu (c,|x|)\) with respect to |x| and equalling it to 0 would imply that \(c_{0}=2\), which is a contradiction. Thus to find the maximum of \(\mu (c,|x|)\), we need only consider the end points of \([0,2]\times [0,1]\).
When \(c=0\), \(\mu (0,|x|)=2|x|^2\le 2\).
When \(c=2\), \(\mu (2,|x|)=6\).
When \(|x|=0\), \(\mu (c,0)=\left| c^2 - \dfrac{5}{8}c^4\right| +\dfrac{1}{6}c \left( 4 - c^2\right) \), which has maximum value 6 on [0, 2].
Finally when \(|x|=1\), \(\mu (c,1)=\left| c^2 - \dfrac{5}{8}c^4\right| +\dfrac{5}{12}c^2 (4 - c^2) + \dfrac{1}{8}(4 - c^2)^2\), which also has maximum value 6 on [0, 2], which completes the proof of the theorem.
The inequality is again sharp when \(f(z)=z/(1-z)^2\). \(\square \)
Theorem 4
For \(f\in S^*\) given by (1),
The inequality is sharp.
Proof
Expanding the determinant by using (2) and Lemma 1, we obtain
As before, without loss in generality we can assume that \(c_{1}=c\), where \(0\le c\le 2\). Then, by using the triangle inequality and the fact that \(|x|\le 1\) we obtain
It is now a simple exercise in elementary calculus to show that this expression has a maximum value of 8 when \(c=2\), which completes the proof.
The inequality is again sharp when \(f(z)=z/(1-z)^2\). \(\square \)
Theorem 5
Let \(f\in K\) and be given by (1) with the associated starlike function g be defined by
Then
provided \(b_{2}\) is real.
The inequality is sharp.
Proof
Write \(zf'(z)=g(z)h(z)\), and \(zg'(z)=g(z)p(z)\), with
and
Then equating the coefficients in \(zf'(z)=g(z)h(z)\) where coefficients’ relations from \(zg'(z)=g(z)p(z)\) is also used, we obtain
so that
We now use Lemma 1 to express \(c_{2}\) and \(p_{2}\) in terms of \(c_{1}\) and \(p_{1}\) and writing \(X=4-c_{1}^2\) and \(Y=4-p_{1}^2\) for simplicity to obtain
Again without loss in generality we can assume that \(c_{1}=c\), where \(0\le c \le 2\). Also since we are assuming \(b_{2}=p_{1}\) to be real, we can write \(p_{1}=q\), with \(0\le |q| \le 2\), and write \(|q|=p\). We note at this point a further normalisation of \(p_{1}\) to be real would remove the requirement that \(p_{1}=b_{2}\) is real, but such a normalisation does not appear to be justified.
It follows from Lemma 1 that with now \(X=4-c^2\) and \(Y=4-p^2\)
We now assume \(|x| \le 1\) and \(|y|\le 1\) and simplify to obtain
Suppose that the expression between the modulus signs is positive, then
Two variable calculus now shows that \(\psi _{1}(c,p)\) has a maximum value of 5 at [0, 2].
If the expression between the modulus signs is negative, we obtain
and two variable calculus shows that \(\psi _{2}(c,p)\) has a maximum value less than 3.
Thus the proof of Theorem 5 is complete.
The inequality is again sharp when \(f(z)=z/(1-z)^2\). \(\square \)
Using the same technique, it is possible to prove the following. We omit the proof.
Theorem 6
Let \(f\in K\) and be given by (1) with associated starlike function g defined by
Then
provided \(b_{2}\) is real.
The inequality is sharp.
Remark
It is most likely that the restriction \(b_{2}\) real can be removed in Theorems 5 and 6. However, as was pointed out, only a normalisation of either \(c_{1}\) or \(p_{1}\) can be justified, and so the method used requires that \(b_{2}=p_{1}\) is real.
Change history
13 March 2018
The authors have retracted this article because the article contains major flaws in the proof of the main results. The results of the paper are invalid, since the assumption that the functionals considered are rotationally invariant is not valid. All authors have agreed to this retraction.
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The authors wish to thank the referee for his/her comments and suggestions in improving the paper.
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Communicated by Rosihan M. Ali.
The authors have retracted this article because the article contains major flaws in the proof of the main results. The results of the paper are invalid, since the assumption that the functionals considered are rotationally invariant is not valid. All authors have agreed to this retraction
A correction to this article is available online at https://doi.org/10.1007/s40840-018-0620-2.
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Thomas, D.K., Abdul Halim, S. RETRACTED ARTICLE: Toeplitz Matrices Whose Elements are the Coefficients of Starlike and Close-to-Convex Functions. Bull. Malays. Math. Sci. Soc. 40, 1781–1790 (2017). https://doi.org/10.1007/s40840-016-0385-4
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DOI: https://doi.org/10.1007/s40840-016-0385-4