Abstract
For two subclasses of close-to-star functions we estimate early logarithmic coefficients, coefficients of inverse functions, Hankel determinant \(H_{2,2}\) and Zalcman functional \(J_{2,3}\). All results are sharp.
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1 Introduction
Given \(r>0,\) let \({\mathbb {D}}_r:=\{ z \in {\mathbb {C}}{:}\,|z|<r \},\) and let \({\mathbb {D}}:={\mathbb {D}}_1.\) Let \(\overline{{\mathbb {D}}}:=\{z\in \mathbb C{:}\,|z|\le 1\}\) and \(\mathbb T:=\partial {\mathbb {D}}.\) Let \({{\mathcal {H}}}\) be the class of all analytic functions in \({\mathbb {D}}\) and \({{\mathcal {A}}}\) be its subclass of f normalized by \(f(0):=0\) and \(f'(0):=1,\) i.e., of the form
Let \({\mathcal {S}}\) be the subclass of \({\mathcal {A}}\) of all univalent functions and \({\mathcal {S}}^*\) be the subclass of \({\mathcal {S}}\) of all starlike functions, namely, \(f\in {\mathcal {S}}^*\) if \(f\in {\mathcal {A}}\) and
A function \(f\in {\mathcal {A}}\) is called close-to-star if there exist \(g\in {\mathcal {S}}^*\) and \(\beta \in \mathbb R\) such that
Denote by \({\mathcal {C}}{\mathcal {S}}{\mathcal {T}}\) the class of all close-to-star functions introduced by Reade [30]. Note that \(f\in {\mathcal {C}}{\mathcal {S}}{\mathcal {T}}\) if and only if a function
is close-to-convex [15, 12, Vol. II, p. 3]. The class of close-to-star functions and its subclasses were intensively studied by various authors (e.g., MacGregor [25], Sakaguchi [32], Causey and Merkes [4]; for further references, see [12, Vol. II, pp. 97–104]). Given \(g\in {\mathcal {S}}^*\) and \(\beta \in \mathbb R,\) let \({\mathcal {C}}{\mathcal {S}}{\mathcal {T}}_\beta (g)\) be the subclass of \({\mathcal {C}}{\mathcal {S}}{\mathcal {T}}\) of all f satisfying (1.2). The classes \({\mathcal {C}}{\mathcal {S}}{\mathcal {T}}_0(g_i),\ i=1,2,3,\) where
are particularly interesting and were separately studied by authors. In this paper we deal with the classes \({\mathcal {C}}{\mathcal {S}}{\mathcal {T}}_0(g_1)=:{\mathcal {S}\mathcal {T}}(\mathrm {i})\) and \({\mathcal {C}}{\mathcal {S}}{\mathcal {T}}_0(g_2)=:{\mathcal {S}\mathcal {T}}(1)\) which elements f in view of (1.2) satisfy the condition
and
respectively. Let us add the inequality (1.4) defines the subclass of the class of functions starlike in the direction of the real axis introduced by Robertson [31]. Moreover, each function F given by (1.3) over the class \({\mathcal {S}\mathcal {T}}(\mathrm {i})\) maps univalently \({\mathbb {D}}\) onto a domain \(F({\mathbb {D}})\) convex in the direction of the imaginary axis. The concept of convexity in one direction belongs to Roberston [31] (see e.g., [12, p. 199]). Each function F given by (1.3) over the class \({\mathcal {S}\mathcal {T}}(1)\) maps univalently \({\mathbb {D}}\) onto a domain \(F({\mathbb {D}})\) called convex in the positive the direction of the real axis, i.e., \(\{w+it{:}\,t\ge 0\}\subset f({\mathbb {D}})\) for every \(w\in f({\mathbb {D}})\) [2, 8, 9, 11, 20, 21]. Let us remark that the condition (1.4) was generalized by replacing the expression \(1-z^2\) by the expression \(1-\alpha ^2z^2\) with \(\alpha \in [0,1]\) in [13].
In this paper we find the sharp estimates of early logarithmic coefficients (Sect. 2), of the Hankel determinant \(H_{2,2}\) and of Zalcman functional \(J_{2,3}\) (Sect. 3) and of the early inverse coefficients (Sect. 4) of functions in the classes \({\mathcal {S}\mathcal {T}}(\mathrm {i})\) and \({\mathcal {S}\mathcal {T}}(1).\) Since both classes \({\mathcal {S}\mathcal {T}}(\mathrm {i})\) and \({\mathcal {S}\mathcal {T}}(1)\) have a representation 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 \({\mathcal {S}\mathcal {T}}(\mathrm {i})\) and \({\mathcal {S}\mathcal {T}}(1)\) have a suitable representation expressed by the coefficients of functions in \({\mathcal {P}}.\) Therefore to get the upper bounds of considered functionals our computing is based on parametric formulas for the second and third coefficients in \({\mathcal {P}}.\) However both classes are rotation non-invariant. Thus to solve discussed problems we will apply a general formula for \(c_3\) recently found in [7]. The formula (1.7) was proved by Carathéodory [3] (see e.g., [10, p. 41]). The formula (1.8) can be found in [28, p. 166]. The formula (1.9) was shown in a recent paper [7], where the extremal functions (1.11) and (1.12) were computed also. For \(c_1\ge 0\) the formula (1.9) is due to by Libera and Zlotkiewicz [22, 23].
Lemma 1.1
If \(p \in {{\mathcal {P}}}\) is of the form (1.6), then
and
for some \(\zeta _i\in \overline{{\mathbb {D}}},\) \(i\in \{1,2,3\}.\)
For \(\zeta _1\in \mathbb T,\) there is a unique function \(p\in {\mathcal {P}}\) with \(c_1\) as in (1.7), namely,
For \(\zeta _1\in {\mathbb {D}}\) and \(\zeta _2\in \mathbb T,\) there is a unique function \(p\in {\mathcal {P}}\) with \(c_1\) and \(c_2\) as in (1.7)–(1.8), namely,
For \(\zeta _1,\zeta _2\in {\mathbb {D}}\) and \(\zeta _3\in \mathbb T,\) there is a unique function \(p\in {\mathcal {P}}\) with \(c_1,\) \(c_2\) and \(c_3\) as in (1.7)–(1.9), namely,
2 Logarithmic Coefficients
Given \(f\in {\mathcal {S}}\) let
The numbers \(\gamma _n\) are called logarithmic coefficients of f. Differentiating (2.1) and using (1.1) we get
As it well known, the logarithmic coefficients play a crucial role in Milin conjecture ([26], see also [10, p. 155]). It is surprising that for the class \({\mathcal {S}}\) the sharp estimates of single logarithmic coefficients \({\mathcal {S}}\) are known only for \(\gamma _1\) and \(\gamma _2,\) namely,
and are unknown for \(n\ge 3.\) Logarithmic coefficients is one of the topic recently being of interest by various authors (e.g., [1, 18, 33]).
Logarithmic coefficients can be considered for functions f from the class \({\mathcal {A}}\) however under the assumption that the branch of logarithm \({\mathbb {D}}\ni z\mapsto \log f(z)/z\) exists. From (1.4) and (1.5) it follows that \(g(z):=f(z)/z\not =0\) in \({\mathbb {D}}{\setminus }\{0\}\) for \(f\in {\mathcal {S}\mathcal {T}}(\mathrm {i})\) and \(f\in {\mathcal {S}\mathcal {T}}(1)\). However \(g({\mathbb {D}})\) needs not be necessarily a simply connected domain. Therefore, let \({\mathcal {S}\mathcal {T}}_0(\mathrm {i})\) and \({\mathcal {S}\mathcal {T}}_0(1)\) be the subclasses of \({\mathcal {S}\mathcal {T}}(\mathrm {i})\) and \({\mathcal {S}\mathcal {T}}(1)\) respectively, of all functions f for which the branch \({\mathbb {D}}\ni z\mapsto \log f(z)/z\) with \(\log 1:=0\) exists.
Theorem 2.1
If \(f \in {\mathcal {S}\mathcal {T}}_0(\mathrm {i})\) is of the form (1.1), then
All inequalities are sharp.
Proof
By (1.4) there exists \(p \in {{\mathcal {P}}}\) of the form (1.6) such that
Substituting the series (1.1) and (1.6) into (2.3) by equating the coefficients we get
The inequality \(|\gamma _1|\le 1\) follows directly from (2.2), (2.4) and (1.7) with sharpness for the function f given by (2.3), where p is as in (1.10).
Substituting (1.7) and (1.8) into (2.4) from (2.2) it follows that
with sharpness for the function f given by (2.3), where p is as in (1.11) with \(\zeta _1=0\) and any \(\zeta _2\in \mathbb T.\)
Hence and by (1.7)–(1.9) we get
where \(\zeta _i \in \overline{{\mathbb {D}}},\) \(i=1,2,3\). Thus by setting \(x:=|\zeta _1|\in [0,1]\) and \(y:=|\zeta _2|\in [0,1]\) we obtain
Thus \(|\gamma _3|\le 1\) with sharpness for the function f given by (2.3), where p is as in (1.12) with \(\zeta _1=\zeta _2=0\) and any \(\zeta _3\in \mathbb T.\) \(\square \)
Theorem 2.2
If \(f \in {\mathcal {S}\mathcal {T}}_0(1)\) is of the form (1.1), then
All inequalities are sharp.
Proof
By (1.5) there exists \(p \in {{\mathcal {P}}}\) of the form (1.6) such that
Substituting the series (1.1) and (1.6) into (2.5) by equating the coefficients we get
The inequality \(|\gamma _1|\le 2\) follows directly from (2.2), (2.6) and (1.7) with sharpness for the function f given by (2.5), where p is as in (1.10).
Substituting (1.7) and (1.8) into (2.6) from (2.2) it follows that
with sharpness for the function f given by (2.3), where p is as in (1.11) with \(\zeta _1=0\) and any \(\zeta _2\in \mathbb T.\)
Hence and by (1.7)–(1.9) we get
where \(\zeta _i \in \overline{{\mathbb {D}}},\) \(i=1,2,3\). Thus by setting \(x:=|\zeta _1|\in [0,1]\) and \(y:=|\zeta _2|\in [0,1]\) we obtain
We have \(F(1/3,y)=52/27.\) Moreover for \(x\in (1/3,1]\) and \(x\in [0,1/3)\) we get
and
respectively. Thus by (2.7), \(|\gamma _3| \le (1+\sqrt{2})/3\) with sharpness for the function f given by (2.3), where p is as in (1.12) with \(\zeta _1=1/\sqrt{2},\) \(\zeta _2=\mathrm {i}\) and any \(\zeta _3\in \mathbb T.\) \(\square \)
3 Zalcman Functional and Hankel Determinant
Now we compute the sharp upper bound of the Zalcman functional \(J_{2,3}(f):=a_2 a_3 - a_4\) being a special case of the generalized Zalcman functional \(J_{n,m}(f):=a_n a_m - a_{n+m-1},\ n,m \in {\mathbb {N}}{\setminus }\{1\},\) which was investigated by Ma [24] for \(f \in {{\mathcal {S}}}\) (see also [29] for relevant results on this functional). We will find also the sharp bound of the second Hankel determinant \(H_{2,2}(f)=a_2a_4-a_3^2.\) Both functionals \(J_{2,3}\) and \(H_{2,2}\) have been studied recently by various authors (see e.g., [5, 6, 14, 16, 17, 19, 27]).
Theorem 3.1
If \(f \in {\mathcal {S}\mathcal {T}}(\mathrm {i})\) is of the form (1.1), then
The inequality is sharp with the extremal function
Proof
From (2.4) by using (1.7)–(1.9) it follows that
with sharpness for the function (3.1).
To find sharp estimate for \(H_{2,2}\) over \({\mathcal {S}\mathcal {T}}(\mathrm {i})\) we use the following lemma.
\(\square \)
Proposition 3.2
Proof
Since the inequality (3.3) clearly holds for \(z=0,\) assume that \(z=r \mathrm {e}^{\mathrm {i}\theta }\) with \(0 < r\le 1\) and \(0\le \theta <2\pi \). A simple computation gives
where \(\varphi {:}\,[-1,1]\rightarrow {\mathbb {R}}\) is a function defined by
Note that \(\varphi '(x)=0\) occurs only when \(x=(1-4r^2)/(4r)=:x_0\).
When \(r \le (-1+\sqrt{2})/2,\) we have \(x_0>1\) or \(1-4r-4r^2>0\). Therefore
Hence we get
Thus from (3.4) and (3.5) it follows that the inequality (3.3) holds for \(|z|\le (-1+\sqrt{2})/2\).
When \((-1+\sqrt{2})/2 \le r \le 1\), we have \(x_0 \in [-1,1].\) Then
Combining (3.4) and (3.6) we see that the inequality (3.3) holds for \((-1+\sqrt{2})/2 \le |z| \le 1\). \(\square \)
Theorem 3.3
If \(f \in {\mathcal {S}\mathcal {T}}(\mathrm {i})\) is of the form (1.1), then
The inequality is sharp with the extremal function
Proof
From (2.4) by using (1.7)–(1.9)we have
where \(\zeta _i \in \overline{{\mathbb {D}}},\ i=1,2,3\). Let \(x:=|\zeta _1|\in [0,1]\) and \(y=|\zeta _2|\in [0,1].\)
Assume first that \(x\in [0,x_0],\) where \(x_0:=(-1+\sqrt{2})/2.\) Then by (3.9) and Proposition 3.2 for \(y\in [0,1]\) we get
Clearly, for each \(x\in [0,x_0],\) the function \([0,1]\ni y\mapsto F(\cdot ,y)\) is increasing and therefore for \(y\in [0,1],\)
Assume now that \(x\in [x_0,1].\) Then by (3.9) and Proposition 3.2 for \(y\in [0,1]\) we get
Note first that
Clearly, for each \(x\in [x_0,1],\) the function \([0,1]\ni y\mapsto G(\cdot ,y)\) is increasing and therefore for \(y\in [0,1],\)
Hence, from (3.10) and (3.11) it follows that the inequality (3.7) is true. Equality in (3.7) holds for the function f given by (2.3), where p is given by (1.12) with \(\zeta _1:=1/6\) and \(\zeta _2=\zeta _3:=1\), i.e., for the function (3.8). \(\square \)
Theorem 3.4
If \(f \in {\mathcal {S}\mathcal {T}}(1)\) is of the form (1.1), then
The inequality is sharp with the extremal function
Proof
From (2.6), by using (1.7) and the inequality \(|c_1c_2-c_3|\le 2\) which was proved in (3.2), we obtain
with sharpness for the function (3.12). \(\square \)
Theorem 3.5
If \(f \in {\mathcal {S}\mathcal {T}}(1)\) is of the form (1.1), then
The inequality is sharp with the extremal function (3.12).
Proof
From (2.6) by using (1.7)–(1.9)we have
where \(\zeta _i\in \overline{{\mathbb {D}}},\ i\in \{1,2,3\}\). Set \(x:=|\zeta _1| \in [0,1]\) and \(y=:|\zeta _2| \in [0,1]\). By (3.14) we have
Note first that
Let now \(x\in [0,1).\) Then for \(y\in [0,1]\) we have
iff \(y=1/2(1-x^2)=:y_0.\) Since \(y_0\ge 1\) for each \(x\in [1/\sqrt{2},1),\) so then the function \([0,1]\ni y\mapsto F(\cdot ,y)\) is increasing and therefore
For \(x\in [0,1/\sqrt{2})\) we have
Hence by (3.15) and (3.16) it follows that the inequality (3.13) is true. Equality in (3.13) holds for the function f defined by (3.12). \(\square \)
4 Inverse Coefficients
Since \({\mathcal {S}\mathcal {T}}(\mathrm {i})\) is a compact class and \(f'(0)=1\) for every \(f\in {\mathcal {S}\mathcal {T}}(\mathrm {i}),\) there exists \(r_0\in (0,1)\) such that every \(f\in {\mathcal {S}\mathcal {T}}(\mathrm {i})\) is invertible in the disk \({\mathbb {D}}_{r_0}.\) Thus there exists \(\delta >0\) such that the inverse function \({\hat{f}}\) of \(f_{|{\mathbb {D}}_{r_0}}\) has a series expansion in the disk \({\mathbb {D}}_{\delta }\) of the form
Thus for \(f\in {\mathcal {S}\mathcal {T}}(\mathrm {i})\) of the form (1.1) the following relations hold (see e.g., [12, Vol. I, p. 57])
Similar situation holds for the class \({\mathcal {S}\mathcal {T}}(1).\)
Theorem 4.1
If \({\hat{f}}\) is the inverse function of \(f \in {\mathcal {S}\mathcal {T}}(\mathrm {i})\) of the form (4.1), then
-
(i)
\(|\beta _2| \le 2;\)
-
(ii)
\(|\beta _3| \le 7;\)
-
(iii)
\(|\beta _4| \le 30\).
All inequalities are sharp with the extremal function
Proof
Substituting (2.4) into (4.2) we get
and
By (4.4) and (1.7) the inequality (i) follows immediately. From (4.4) with (1.7) and (1.8) we have
Now we prove (iii). By (4.5) and (1.7)–(1.9) we have
where \(\zeta _i \in \overline{{\mathbb {D}}},\ i=1,2,3,\) \(x:=|\zeta _1|\in [0,1]\) and \(y:=|\zeta _2|\in [0,1]\).
Note first that
Let now \(x\in [0,1).\) Then for \(y\in [0,1]\) we have
iff \(y=4x/(1-x)=:y_0.\) Since \(y_0\ge 1\) for each \(x\in [1/5,1),\) so then the function \([0,1]\ni y\mapsto F(\cdot ,y)\) is increasing and therefore
For \(x\in [0,1/5)\) we have
Hence by (4.6)–(4.8) it follows that the inequality in (iii) is true.
All inequalities are sharp with the extremal function (4.3). \(\square \)
Theorem 4.2
If \({\hat{f}}\) is the inverse function of \(f \in {\mathcal {S}\mathcal {T}}(1)\) of the form (4.1), then
-
(i)
\( |\beta _2| \le 4;\)
-
(ii)
\( |\beta _3| \le 23;\)
-
(iii)
\( |\beta _4| \le 156\).
All inequalities are sharp with the extremal function
Proof
Substituting (2.6) into (4.2) we get
and
By (4.10) and (1.7) the inequality (i) follows immediately. From (4.10) with (1.7) and (1.8) we have
Now we prove (iii). By (4.11) and (1.7)–(1.9) we have
where \(\zeta _i \in \overline{{\mathbb {D}}},\ i=1,2,3,\) \(x:=|\zeta _1|\in [0,1]\) and \(y:=|\zeta _2|\in [0,1]\).
Note first that
Let now \(x\in [0,1).\) Then for \(y\in [0,1]\) we have
iff \(y=4(1+x)/(1-x)=:y_0.\) Since \(y_0\ge 1\) for each \(x\in (0,1),\) so the function \([0,1]\ni y\mapsto F(\cdot ,y)\) is increasing and therefore
Hence and from (4.12) it follows that the inequality in (iii) is true.
All inequalities are sharp with the extremal function (4.9). \(\square \)
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Acknowledgements
This work was partially supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea Government (MSIP; Ministry of Science, ICT and Future Planning) (No. NRF-2017R1C1B5076778).
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Lecko, A., Sim, Y.J. Coefficient Problems in the Subclasses of Close-to-Star Functions. Results Math 74, 104 (2019). https://doi.org/10.1007/s00025-019-1030-y
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DOI: https://doi.org/10.1007/s00025-019-1030-y
Keywords
- Univalent functions
- close-to-star functions
- functions starlike in the direction of the real axis
- functions convex in the direction of the imaginary axis
- Hankel determinant
- Zalcman functional
- logarithmic coefficients
- inverse functions
- Carathéodory class