Abstract
In the present paper, we proved the sharp inequality \(|H_{3,1}(f)|\le 1/9\) for analytic functions f with \(a_n:=f^{(n)}(0)/n!,\ n\in {\mathbb {N}},\ a_1:=1,\) such that
where
is the third Hankel determinant.
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1 Introduction
Let \({{\mathcal {H}}}\) be the class of analytic functions in \({\mathbb {D}}:= \left\{ z \in {\mathbb {C}} : |z|<1 \right\} \) and let \({\mathcal {A}}\) be its subclass of functions f normalized by \(f(0):=0\), \(f'(0):=1,\) i.e., of the form
Given \(\alpha \in [0,1),\) let \({\mathcal {S}}^*(\alpha )\) denote the subclass of \({\mathcal {A}}\) of functions f such that
called starlike of order \(\alpha .\) In particular, \({\mathcal {S}}^*(0)=:\mathcal S^*\) is the class of starlike functions, i.e., the family of all univalent functions in \({\mathcal {A}}\) which map \({\mathbb {D}}\) onto starlike domains (with respect to the origin). Starlike functions of order \(\alpha \) were introduced by Robertson [19] (see also [7, Vol. I, p. 138]). An important role is played by the class \({\mathcal {S}}^*(1/2).\) One of the significant results belongs to Marx [15] and Strohhäcker [23]. They proved that
(see also [16, Theorem 2.6a, p. 57]), where \({\mathcal {S}}^c\) means the class of convex functions, i.e., the family of all univalent functions in \({\mathcal {A}}\) which map \({\mathbb {D}}\) onto convex domains. By the well known result due to Study ([24], see also [6, p. 42]) a function f is in \({\mathcal {S}}^c\) if and only if
What is interesting, a function
is extremal for many computational problems in both these two classes, i.e., in \({\mathcal {S}}^c\) and \({\mathcal {S}}^*(1/2).\)
For \(q,n \in {\mathbb {N}},\) the Hankel determinant \(H_{q,n}(f)\) of function \(f \in {{\mathcal {A}}}\) of the form (1.1) is defined as
Given a subfamily \({\mathcal {F}}\) of \({\mathcal {A}},\)q and n, computing the upper bound of \(H_{q,n}\) is an interesting problem to study. Recently many authors examined the Hankel determinant \(H_{2,2}(f)=a_2a_4-a_3^2\) of order 2 (see e.g., [4, 5, 8, 9, 12, 17]). Note also that \(H_{2,1}(f)=a_3-a_2^2\) is the well known coefficient functional which for \({\mathcal {S}}\) was estimated in 1916 by Bieberbach (see e.g., [7, Vol. I, p. 35]). To find the upper bound of the Hankel determinant
of the third kind, is more difficult if we expect to get sharp estimate. Results in this direction however not sharp were obtained by various authors, e.g., [1, 2, 4, 5, 20,21,22, 25].
In this paper, we found the sharp bound of the Hankel determinant \(H_{3,1}\) over the class \({\mathcal {S}}^*(1/2),\) namely, we proved that \(|H_{3,1}(f)|\le 1/9\) for \(f\in {\mathcal {S}}^*(1/2)\) and that the inequality is sharp. Since the class \({\mathcal {S}}^*(1/2)\) 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 \({\mathcal {S}}^*(1/2)\) have a suitable representation expressed by coefficients of functions in \({\mathcal {P}}.\) Therefore to get the upper bound of \(H_{3,1},\) we based our computing on the well known formulas on coefficient \(c_2\) (e.g., [18, p. 166]), the formula \(c_3\) due to Libera and Zlotkiewicz [13, 14] and the formula for \(c_4\) recently found in [11].
At the end let us mention that in [10] the authors proved that \(|H_{3,1}(f)|\le 4/135=0.0296\ldots \) for \(f\in \mathcal S^c\) and that the result is sharp. Looking at the inclusion (1.3) we can state that the the corresponding bounds of \(H_{3,1}\) carry some information about the richness of classes. Classical estimates of coefficients does not necessarily include such a distinction, e.g., both in the class \({\mathcal {S}}^c\) and in the class \({\mathcal {S}}^*(1/2)\) modules of all coefficients are bounded by 1 (see [7, Theorem 7, p. 117; Theorem 2, p. 140]) with the extremal function given by (1.4).
2 Main Result
The basis for proof of the main result is the following lemma which contains the well known formula for \(c_2\) (e.g., [18, p. 166]), the formula for \(c_3\) due to Libera and Zlotkiewicz [13, 14] and the formula for \(c_4\) found in [11].
Lemma 2.1
If \(p \in {{\mathcal {P}}}\) is of the form (1.6) with \(c_1\ge 0,\) then
and
for some \(\zeta ,\eta ,\xi \in \overline{{\mathbb {D}}}:=\{z\in {\mathbb {C}}:|z|\le 1 \}.\)
We will now estimate the third order Hankel determinant \(H_{3,1}(f)\) for \(f \in {\mathcal {S}}^*(1/2)\).
Theorem 2.2
with the extremal function
Proof
Let \(f \in {\mathcal {S}}^*(1/2)\) be of the form (1.1). Then by (1.2) we have
for some function \(p \in {\mathcal {P}}\) of the form (1.6). Since the classes \({\mathcal {P}}\) and \({\mathcal {S}}^*(1/2)\) are invariant under the rotations, by Carathéodory Theorem we may assume that \(c:=c_1 \in [0,2]\) ([3], see also [7, Vol. I, p. 80, Theorem 3]). Putting the series (1.1) and (1.6) into (2.6) and equating coefficients we get
Hence and by (1.5) we have
To simplify computation, let \(t:=4-c^2.\) Thus formulas (2.1)-(2.3) we can rewrite as
Hence by straightforward algebraic computation we have
Setting the above expression to (2.7) we get
where for \(\zeta ,\,\eta ,\,\xi \in \overline{{\mathbb {D}}},\)
and
Let \(x:=|\zeta | \in [0,1]\) and \(y:=|\eta | \in [0,1].\) Since \(|\xi |\le 1,\) from (2.8) we obtain
where
with
and
Now, we will show that
Since \(f_2(c,x)>0\) and
for \(c\in (0,2)\) and \(x\in (0,1),\) so for \(c\in (0,2)\) and \(x\in (0,1),\)
For \(x=1/2\) the function \((0,2)\ni c\mapsto G(c,1/2)\) has no critical point, obviously. When \(x\not =1/2,\) then \(\partial G/\partial c=0\) iff
which holds only for \(x\in ((2+3\sqrt{2})/7,1).\) Thus
iff
which after simplifying reduces to
for \(x\in ((2+3\sqrt{2})/7,1).\) As we can check the above equation has no solution in \(((2+3\sqrt{2})/7,1)\) (real solutions are \(x_1\approx -0.1513,\ x_2\approx 1.0622,\ x_3\approx 2.4952\)). Thus the function G has no critical point in \((0,2)\times (0,1).\)
For \(c=0\) and \(c=2\) both functions
and
are decreasing, so
For \(x=0\) and \(x=1\) we have respectively,
and
Hence, by (2.12) and (2.11) it follows that the (2.9) holds. This together with (2.9) shows that \(|H_{3,1}(f)|\le 1/9.\)
For the function (2.5) which is in \({\mathcal {S}}^*(1/2),\) we have \(a_2=a_3=a_5=0\) and \(a_4=1/3.\) Thus \(H_{3,1}(f)=-1/9,\) which makes equality in (2.4). \(\square \)
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Acknowledgements
This work was partially supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science, ICT and Future Planning) (No. NRF-2017R1C1B5076778).
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Communicated by David Shoikhet.
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Lecko, A., Sim, Y.J. & Śmiarowska, B. The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions of Order 1 / 2. Complex Anal. Oper. Theory 13, 2231–2238 (2019). https://doi.org/10.1007/s11785-018-0819-0
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DOI: https://doi.org/10.1007/s11785-018-0819-0