Abstract
In the present paper, the estimate of the third Hankel determinant
for the class of starlike functions, i.e., for the class of analytic functions f standardly normalized such that \({{\mathrm{Re}}}(zf'(z)/f(z)) > 0,\ z\in {{\mathbb {D}}}:=\{z \in {\mathbb {C}} : |z|<1\},\) is improved.
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1 Introduction
Let \({{\mathcal {H}}}\) be a class of analytic functions in \({\mathbb {D}}:= \left\{ z \in {\mathbb {C}} : |z|<1 \right\} \) and let \({{\mathcal {A}}}\) be its subclass normalized by \(f(0):=0\), \(f'(0):=1,\) i.e., of the form
Let \({{\mathcal {S}}}^{*}\) denote the class of starlike functions, namely, the subclass of \({{\mathcal {A}}}\) consisting of functions f such that
Given \(q,n \in {\mathbb {N}},\) the Hankel determinants \(H_{q,n}(f)\) of Taylor’s coefficients of functions \(f \in {{\mathcal {A}}}\) of the form (1.1) are defined as
Particularly, the third Hankel determinant \(H_{3,1}(f)\) is given by
To find the growth of the Hankel determinant \(H_{q,n}(f)\) dependent on q and n for the whole class \({\mathcal {S}}\subset {\mathcal {A}}\) of univalent functions as well as for its subclasses is an interesting problem to study. For the class \({\mathcal {S}}\) some important result was shown by Pommerenke [13]. For fixed q and n the growth problem can be reduced to an estimate of the Hankel determinant for the selected subclasses of \({\mathcal {A}}.\) Recently many authors examined the Hankel determinant \(H_{2,2}(f)=a_2a_4-a_3^2\) of order 2 (see, e.g., [3, 4, 6, 8, 12]). Note also that \(H_{2,1}(f)=a_3-a_2^2.\) Thus the Hankel determinant \(H_{2,1}(f)\) reduces to the well-known coefficient functional which for \({\mathcal {S}}\) was estimated in 1916 by Bieberbach (see, e.g., [5, Vol. I, p. 35]).
The problem to find the upper bound of the Hankel determinant \(H_{3,1}(f)\) of order 3 is more sophisticated if we expect to get sharp result. From (1.3) by using the triangle inequality we get at once the following inequality
This simple observation allowed to estimate of \(|H_{3,1}(f)|\) for compact subclasses \({\mathcal {F}}\) of \({\mathcal {A}}\) by various authors (see, e.g., [2, 15,16,17,18]). However, these results are far from sharpness. If case when a given subclass \({\mathcal {F}}\) of \({\mathcal {A}}\) has a representation with using the Carathéodory class \({\mathcal {P}}\), i.e., the class of functions \(p \in {{\mathcal {H}}}\) of the form
having a positive real part in \({\mathbb {D}},\) the coefficients of functions in \({\mathcal {F}}\) have a suitable representation expressed by the coefficients of functions in \({\mathcal {P}}.\) Therefore to get the upper bound of each term in (1.4) cited authors based their computing on the well-known formulas on coefficient \(c_2\) (e.g., [14, p. 166]) and on the formula \(c_3\) due to Libera and Zlotkiewicz [9].
In order to improve the bound of \(|H_{3,1}(f)|\) we have to use directly formula (1.3), where we need to apply a formula for \(c_4,\) similar to the formulas (2.1) and (2.2). In a recent paper [7] the authors found such a formula for \(c_4.\) According to the authors’ knowledge, formulas for the coefficients \(c_n\) for \(n \ge 5\) analogous to the formulas (2.1) and (2.2) are not known.
Basing on the formulas for \(c_2,\ c_3\) and \(c_4,\) we improve the known estimate of the Hankel determinant \(H_{3,1}(f)\) in the class \({{\mathcal {S}}}^{*}\) of starlike functions. We show that \(|H_{3,1}(f)|\le 8/9.\) Estimating each term of the right hand of (1.4) Babalola [1] showed that \(|H_{3,1}(f)|\le 16.\) In [19] Zaprawa by a suitable grouping and using Lemma 1 due to Livingston [11] proved that \(|H_{3,1}(f)|\le 1.\)
2 Main Result
The basis for proof of the main result is the following lemma. It contains the well-known formula for \(c_2\) (e.g., [14, p. 166]), the formula for \(c_3\) due to Libera and Zlotkiewicz [9, 10] and the formula for \(c_4\) found by the authors [7].
Lemma 2.1
If \(p \in {{\mathcal {P}}}\) is of the form (1.5) with \(c_1\ge 0,\) then
and
for some \(\zeta ,\eta ,\xi \in \overline{{\mathbb {D}}}:=\{z\in {\mathbb {C}}:|z|\le 1 \}.\)
Now, we will estimate the third-order Hankel determinant \(H_{3,1}(f)\) for \(f \in {{\mathcal {S}}}^{*}\). To this end, the following propositions are required.
Proposition 2.2
Let \(\Theta : [0,3] \times [0,1] \rightarrow {\mathbb {R}}\) be a function defined by
where for \(x\in [0,1],\)
and
Then \(\Theta (t,x) > 0\) for \(0 \le t \le 3\) and \(0 \le x \le 1\).
Proof
At first, note that the polynomial \(\theta _{3}\) has a unique zero \(x=:x_{1}\approx 0.9314\) in (0, 1). Since \(x_1\in (0.92,0.95)\) and for \(x\in (0.92,0.95),\)
it follows that
For \(x \not = x_1\), \((\partial /\partial t) \Theta (t,x)=0\) occurs at
We have
where
The polynomial \(\theta _{4}\) has exactly two zeros in (0, 1), namely, \(x=:x_2\approx 0.533701\) and \(x=:x_3\approx 0.811327.\) We will now show that
Since \(x_1>0.9,\) so \(\theta _3(x)>0,\) for \(x\in [0.5,0.9]\) and the inequality (2.5) is equivalent to
The above one can be equivalently written as
As the polynomial on the left hand of the above inequality has a unique zero \(x\approx 0.40928\) in [0, 1], the above inequality is true, so is the inequality (2.5). Thus the function \(\Theta \) has no critical point in \((0,3) \times (0,1).\) Hence it is sufficient to show that \(\Theta > 0\) on the boundary of \([0,3] \times [0,1]\). We can easily check that the following inequalities hold:
and
Thus the proof of the proposition is completed. \(\square \)
Proposition 2.3
Let \(\Psi : [1,4] \times [0,1] \rightarrow {\mathbb {R}}\) be a function defined by
where for \(x\in [0,1],\)
and
Then \(\Psi (t,x) > 0\) for \(1 \le t \le 4\) and \(0 \le x \le 1\).
Proof
At first, note that the function \(\psi _{3}\) has a unique zero \(x=:x_{1}\approx 0.51839\) in (0, 1). Since \(x_1\in (0.5,0.6)\) and for \(x\in (0.5,0.6),\)
it follows that
For \(x \not = x_1\), \((\partial /\partial t) \Psi (t,x)=0\) occurs at
We have
where
The polynomial \(\psi _{4}\) has a unique zero \(x=:x_{2}\approx 0.388025\) in (0, 1). Since \(x_1>0.5,\) so \(\psi _3(x)>0,\) for \(x\in (0,0.5).\) Additionally, since \(\psi _2\) has no zero in (0, 1), the inequality (2.7) is true on [0, 1]. Thus \(t_0(x)<0\) for \(x\in (0,0.5)\) and in consequence, the function \(\Psi \) has no critical point in \((1,4) \times (0,1).\) Hence it is sufficient to show that \(\Psi >0\) on the boundary of \([1,4] \times [0,1]\). We can easily check that the following inequalities hold:
and
Thus the proof of the proposition is completed. \(\square \)
Proposition 2.4
Let \(\Phi : [3,4] \times [0,1] \rightarrow {\mathbb {R}}\) be a function defined by
where for \(x\in [0,1],\)
and
Then \(\Phi (t,x) > 0\) for \(3 \le t \le 4\) and \(0 \le x \le 1\).
Proof
Since \(\phi _{3}=-\psi _3,\) by the part of proof of Proposition 2.3, we at once have
For \(x \not = x_1\), \((\partial /\partial t)\Phi (t,x)=0\) occurs at
We have
where
The polynomial \(\phi _4\) has exactly two zeros in (0, 1), namely \( x=:x_2\approx 0.414034\) and \(x=:x_3\approx 0.663886.\) We have \(t_{0}(x_2) \approx 3.59845\) and \(t_{0}(x_3) = -2.95522.\) Therefore the function \(\Phi \) has a unique critical point \((t_0(x_2),x_2)\) in \((3,4) \times (0,1)\). For \((t,x)\in [3.58,3.61]\times [0.39,0.43]\) by simple computing, we show that \(\Phi (t,x)>0.\) Thus, particularly \(\Phi (t_0(x_2),x_2)>0.\) Therefore it is sufficient to show that \(\Phi >0\) on the boundary of \([3,4] \times [0,1].\) We can easily check that the following inequalities hold:
and
Thus the proof of the proposition is completed. \(\square \)
Finally, we estimate now the third-order Hankel determinant \(H_{3,1}(f)\) for \(f \in {{\mathcal {S}}}^{*}\).
Theorem 2.5
If \(f \in {{\mathcal {S}}}^{*}\) is the form (1.1), then
Proof
Let \(f \in {{\mathcal {S}}}^{*}\) be of the form (1.1). Then by (1.2) we have
for some function \(p \in {{\mathcal {P}}}\) of the form (1.5). Since the class \({{\mathcal {P}}}\) is invariant under the rotations, we may assume that \(c:=c_1 \in [0,2]\) (e.g., [5, Vol. I, p. 80, Theorem 3]). Putting the series (1.1) and (1.5) into (2.10) and by equating the coefficients we get
and
Hence
Now using the equalities (2.1)–(2.3), by straightforward algebraic computation we have
where for \(\zeta ,\,\eta ,\,\xi \in \overline{{\mathbb {D}}},\)
and
Setting \(x:=|\zeta | \in [0,1],\)\(y:=|\eta | \in [0,1]\) and taking into account that \(|\xi |\le 1,\) from (2.11) we get
where
with
and
Now, we will show that
for \(c\in [0,2],\ x\in [0,1]\) and \(y\in [0,1].\)
I. Assume first that \(c\in [1,2].\) Then by (2.13) we have
(a) Consider the case \(f_{3}(c,x)\ge f_4(c,x)\) in \([1,2]\times [0,1].\) Let
By (2.15) we get
Set \(t:=4-c^2.\) Clearly, \(t\in [0,3].\) Define
A simple computing yields
Hence and by Proposition 2.2 we have
where the function \(\Theta \) is defined by (2.4). Thus the function \([0,3]\ni t\mapsto {\tilde{F}}_{1}(t,\cdot )\) is increasing, and therefore we have
Indeed, the last inequality is true since, as easy to verify the inequality
holds. Thus the inequality (2.16) confirms the inequality (2.14).
(b) Consider the case \(f_{3}(c,x)< f_4(c,x)\) in \([1,2]\times [0,1].\) Let
Since \(f_{2}(c,x)\ge 0\) in \([1,2]\times [0,1],\) so
If \(\sigma \ge 1,\) i.e., if \(cf_{2}(c,x)+2(f_{3}(c,x) - f_{4}(c,x))\ge 0,\) then
and repeating the argumentation of Part (a) we get the inequality (2.14).
If \(\sigma <1,\) i.e., if \(cf_{2}(c,x)+2(f_{3}(c,x) - f_{4}(c,x)) < 0,\) then
Set \(t:=c^2.\) Clearly, \(t\in [1,4].\) Define
A simple computing yields
Hence and by Proposition 2.3 we have
where the function \(\Psi \) is defined by (2.6). Thus the function \([1,4]\ni t\mapsto {\tilde{F}}_{2}(t,\cdot )\) is decreasing, and therefore we have
Indeed, the last inequality is true since, as easy to verify the inequality
holds. Thus the inequality (2.17) confirms the inequality (2.14).
II. Assume that \(c\in (0,1).\) Then by (2.13) we have
(a) Consider the case \(f_{3}(c,x)\ge f_4(c,x)\) in \((0,1)\times [0,1].\) Let
By (2.18) we get
Set \(t:=4-c^2.\) Clearly, \(t\in (3,4).\) Define
A simple computing yields
Hence and by Proposition 2.4 we have
where the function \(\Phi \) is defined by (2.8). Thus the function \((3,4)\ni t\mapsto {\tilde{F}}_{1}(t,\cdot )\) is increasing, and therefore we have
Indeed, the last inequality is true since so is the following one
Thus the inequality (2.19) confirms the inequality (2.14).
(b) Consider the case \(f_{3}(c,x)< f_4(c,x)\) in \((0,1)\times [0,1]\) which is equivalent to
for \(c\in (0,1)\) and \(x\in [0,1].\) Note that
Thus the inequality (2.20) can be written as
However,
Indeed, the above inequality is equivalent to
which by simplifying is equivalent to the true inequality
Thus by (2.21) and (2.22) it follows that \(c\ge 1\) which contradicts the assumption.
III. At the end assume that \(c=0.\) Then by (2.13) we have
Summarizing, from all considering cases it follows that the inequality (2.14) holds which together with (2.12) shows (2.9). \(\square \)
Remark 2.6
Although the constant 8 / 9 improves essentially the estimates found in [1] and [19], it is not the best possible. To find the sharp estimate of the Hankel determinant \(H_{3,1}(f)\) for starlike functions is still an open problem.
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We would like to express gratitude to the referees for their constructive comments and suggestions that helped to improve the clarity of this manuscript.
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Communicated by Saminathan Ponnusamy.
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Kwon, O.S., Lecko, A. & Sim, Y.J. The Bound of the Hankel Determinant of the Third Kind for Starlike Functions. Bull. Malays. Math. Sci. Soc. 42, 767–780 (2019). https://doi.org/10.1007/s40840-018-0683-0
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DOI: https://doi.org/10.1007/s40840-018-0683-0