Abstract
Sharp upper and lower bounds of the Hermitian Toeplitz determinants of the second and third orders are found for various subclasses of close-to-convex functions.
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1 Introduction and Definitions
Let \({\mathbb {D}}:=\{ z \in {\mathbb {C}} : |z|<1 \}\) and \(\overline{\mathbb {D}}:=\{z\in {\mathbb {C}}: |z|\le 1\},\) and for \(r>0,\) let \({\mathbb {T}}_r:=\{ z \in {\mathbb {C}} : |z|=r\}\) and \({\mathbb {T}}:={\mathbb {T}}_1.\) Denote by \({\mathcal {H}}\) be the class of all analytic functions f in \(\mathbb {D}\) and by \({\mathcal {A}}\) the subclass of \({\mathcal {H}}\) with f normalized such that \(f(0)=0\) and \(f'(0)=1,\) so that f(z) is of the form
Let \({\mathcal {S}}\) be the subclass of \({\mathcal {A}}\) consisting of univalent functions.
For \(q,n\in {\mathbb {N}},\) consider the matrix \(T_{q,n}(f)\) with \(f\in {\mathcal {A}}\) given by (1) defined by
where \({\overline{a}}_k:=\overline{a_k}.\) In the case when \(a_n\) is a real number, \(T_{q,n}(f)\) is called an Hermitian Toeplitz matrix.
In recent years, a great many papers have been devoted to the estimation of determinants whose entries are coefficients of functions in \({\mathcal {A}}\) or its subclasses. Hankel matrices, i.e., square matrices which have constant entries along the reverse diagonal and the generalized Zalcman functional \(J_{m,n}(f):=a_{m+n-1}-a_ma_n,\ m,n\in \mathbb {N},\) are of particular interest (see, e.g., [5, 6, 8, 13, 15, 16, 18,19,20, 25]). Also of interest are the determinants of symmetric Toeplitz matrices, the study of which was initiated in [1].
In [9, 11, 14], research was investigated into the study of Hermitian Toeplitz determinants whose entries are the coefficients of functions in subclasses of \({\mathcal {A}}.\)
In this paper, we continue this research by computing the sharp upper and lower bounds of the second- and third-order Hermitian Toeplitz determinants over some subclasses of close-to-convex functions, but first noting that the following general result was proved in [11].
Theorem 1
([11]) Let \({\mathcal {F}}\) be a subclass of \({\mathcal {A}}\) such that \(\{f\in {\mathcal {F}}: a_2=0\}\not =\emptyset \) and \(A_2({\mathcal {F}}):=\max \{|a_2|: f\in {\mathcal {F}}\}\) exists. Then
Both inequalities are sharp.
We next define the classes of close-to-convex functions considered in this paper. First denote by \({\mathcal {S}}^*\) the subclass of \({\mathcal {S}}\) consisting of the starlike functions, i.e., \(f\in {\mathcal {S}}^*\) if and only if \(f\in {\mathcal {A}}\) and
A function \(f\in {\mathcal {A}}\) is called close-to-convex if there exist \(g\in {\mathcal {S}}^*\) and \(\delta \in \mathbb {R}\) such that
The class \({\mathcal {C}}\) of all close-to-convex functions (which is necessarily a subclass \({\mathcal {S}}\)), was introduced by Kaplan [12] (see also [10, Vol. II, p. 3]), where the following geometrical interpretation was given: \(f\in {\mathcal {A}}\) is close-to-convex if and only if there are no sections of the curve \(f(\mathbb {T}_r),\) for every \(r\in (0,1),\) in which tangent vector turns backward through an angle not less than \(\pi \) (cf. [10, Vol. II, p. 4]). Lewandowski [22, 23] proved that the class of close-to-convex functions is identical with the class of linearly accessible functions introduced by Biernacki [3].
Given \(g\in {\mathcal {S}}^*\) and \(\delta \in {\mathbb {R}},\) let \({\mathcal {C}}_\delta (g)\) be the subclass of \({\mathcal {C}}\) of all f satisfying (2). The four classes \({\mathcal {C}}_0(g_i),\ i=1,\ldots ,4,\) where
and
are particularly interesting and have been studied by various authors (e.g., [2, 7, 17]). In [14], the sharp bounds of the second- and third-order Hermitian Toeplitz determinants were found for the classes \({\mathcal {C}}_0(g_1)\) and \({\mathcal {C}}_0(g_2).\) In this paper, we will do the same for the other two classes, i.e., for \({\mathcal {C}}_0(g_3)=:{\mathcal {F}}_1\) and \({\mathcal {C}}_0(g_4)=:{\mathcal {F}}_2\) which f in view of (2) satisfy the conditions
and
respectively. We note here that in [21] the classes \({\mathcal {C}}(\delta ,\xi _1,\xi _2),\) where \(\delta \in (-\pi /2,\pi /2),\) \(\xi _1,\xi _2\in {\overline{\mathbb {D}}},\) of univalent functions were introduced by generalizing Robertson’s condition for convexity in the direction of the imaginary axis [26]. In particular, the class \({\mathcal {C}}(0,(-1-\sqrt{3}\mathrm {i})/2,(-1+\sqrt{3}\mathrm {i})/2)\) is identical to the class \({\mathcal {F}}_1\) and the class \({\mathcal {C}}(0,0,1)\) is identical to the class \({\mathcal {F}}_2.\) A geometrical property of functions in classes \({\mathcal {C}}(\delta ,\xi _1,\xi _2)\) relating to the hyperbolic or parabolic family of arcs related to \(\xi _1\) and \(\xi _2\) was presented in [21].
Let \({\mathcal {P}}\) be the class of all \(p\in {\mathcal {H}}\) of the form
having a positive real part in \(\mathbb {D}.\)
2 Lemmas
In the proof of our main result, we will use the following lemma, see ([4, 24, p. 166]).
Lemma 1
If \(p \in {{\mathcal {P}}}\) is of the form (5), then
Moreover,
and
for some \(\zeta _i \in \overline{{\mathbb {D}}}\), \(i \in \{ 1,2 \}\).
For \(\zeta _1 \in {\mathbb {T}}\), there is a unique function \(p \in {{\mathcal {P}}}\) with \(c_1\) as in (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 (7) and (8), namely
3 The Class \(\mathcal {F}_1\)
Let \(f \in {\mathcal {F}}_1\) be the form (1). Then by (3) there exists \(p\in {{\mathcal {P}}}\) of the form (5) such that
Substituting (1) and (5) into (10) and equating the coefficients, we obtain
Hence, using (6) it follows that \(A_2({\mathcal {F}}_1)=3/2\) with \(f_1\in \mathcal {F}_1\) satisfying
i.e.,
Also \(f_2\in \mathcal {F}_1\) such that
i.e.,
serves as the extreme function, since \(a_2=0\). Thus, Theorem 1 gives
Theorem 2
If \(f\in \mathcal {F}_1\), then
Both inequalities are sharp.
We now find the upper and lower bounds of \(\det T_{3,1}(f)\) in the class \(\mathcal {F}_1\), first noting that
Theorem 3
If \(f \in \mathcal {F}_1,\) then
Both inequalities are sharp.
Proof
We first find the upper bound.
By (11) and (6), we see that \(|a_2| \le 3/2\) and \(|a_3| \le 4/3.\) Since \({{\,\mathrm{Re}\,}}(a_2^2 {\overline{a}}_3) \le |a_2|^2|a_3|,\) it follows from (13) that
where
Observe now that the point (1, 1) is the unique solution in \((0,3/2)\times (0,4/3)\) of the system of equations
However,
so (1, 1) is a saddle point of F.
We now consider F on the boundary of \([0,3/2]\times [0,4/3].\)
-
(1)
On the side \(x=0,\)
$$\begin{aligned} F(0,y)=1-y^2 \le 1,\quad 0\le y\le \frac{4}{3}. \end{aligned}$$ -
(2)
On the side \(x=3/2,\)
$$\begin{aligned} F\left( \frac{3}{2},y\right) =-\frac{7}{2}+\frac{9}{2}y-y^2 \le F\left( \frac{3}{2},\frac{4}{3}\right) =\frac{13}{18},\quad 0\le y\le \frac{4}{3}. \end{aligned}$$ -
(3)
On the side \(y=0,\)
$$\begin{aligned} F(x,0)=1-2x^2 \le 1,\quad 0\le x\le \frac{3}{2}. \end{aligned}$$ -
(4)
On the side \(y=4/3\)
$$\begin{aligned} F\left( x,\frac{4}{3}\right) =-\frac{7}{9}+\frac{2}{3}x^2 \le \frac{13}{18},\quad 0\le x\le \frac{3}{2}. \end{aligned}$$
Therefore, the inequality \(F(x,y) \le 1\) holds for all \((x,y)\in [0,3/2]\times [0,4/3],\) which in view of (15) gives the upper bound.
For the lower bound, we substitute (7) and (8) into (11) to obtain
with \(\zeta _i \in \overline{{\mathbb {D}}},\) \(i=1,2.\) Therefore, from (13) we have
where
and
We now consider various cases.
A. Suppose that \(\zeta _1\zeta _2\not =0.\) Then \(\zeta _1 =r \mathrm {e}^{\mathrm {i}\theta }\) and \(\zeta _2 = s \mathrm {e}^{\mathrm {i}\psi }\) with \(r,s\in (0,1]\) and \(\theta ,\psi \in [0,2\pi ).\) Then
where
and
where \(\alpha \in \mathbb {R}\) satisfies
with
Since \(\sin (\psi +\alpha ) \ge -1\) and \(s\le 1\), we have
Therefore, from (18)–(20), we obtain
Noting that \(|\zeta _2|\le 1\) from (17) we have
Thus, from (16), (23) and (24) it follows that
where
with
and
for \(t\in [0,1]\) and \(x\in [-1,1].\)
Let \(\varOmega :=[0,1]\times [-1,1].\) Now we will show that
A1. We next deal with the critical points of G in the interior of \(\varOmega \), i.e., in \((0,1)\times (-1,1).\) Note that \(g_2(t,x)\ge 0.\) Moreover, \(g_2(t,x)=0\) holds only for \(t=\sqrt{3}/2\) and \(x=\sqrt{3}/3.\)
A1.1. When \(t=\sqrt{3}/2\) and \(x=\sqrt{3}/3,\) from (17), (19)and (20) we have
which gives
A1.2. We now consider case \(g_2(t,x)>0.\) Differentiating G with respect to x yields
A1.2.a. Assume first that \(\partial g_1/\partial x=0.\) Then from (27), it follows that \(\partial g_2/\partial x=0,\) which is possible only for \(t=\sqrt{6}/2>1\) and \(x=\sqrt{6}/4.\)
A1.2.b. Assume next that \(\partial g_1/\partial x\ne 0.\) Then we can write (27) as
or equivalently, by substituting (26), as
Now note that by (28) the inequality
holds for
Differentiating G with respect to t, we have
where for \((t,x)\in (0,1)\times (-1,1),\)
Therefore, each critical point of G satisfies
or
I. Assume that (32) holds, then \( x=x(t)= (-3+4t^2)/(4t). \) Thus, by (29) we see that
occurs only when \(t=1/2.\) Thus, \(x=x(1/2 )=-1 .\) However, it can be seen that the right side of (30) equals to \(-2\) for \(t=1/2\) and \(x=-1,\) which means that then the inequality (30) is not true. Therefore, G does not have critical point in the interior of \(\varOmega \) in the case of (32).
II. Suppose now that (33) is satisfied. Since Eq. (33) is a quadratic in x with \(\Delta :=225-408t^2+208t^4>0,\ t\in (0,1),\) it has two roots, namely
a. Let \(x=x_1.\) Then \(\varPhi (t,x_1(t))=0\) is equivalent to the equation
Squaring both sides of (35) leads to
where for \(t\in (0,1),\)
and
We see that there is a unique root \(t=\sqrt{3}/2\) of the Eq. (36), which also satisfies (35). Then by (34), \(x_1(\sqrt{3}/2)=\sqrt{3}/3.\) However, this case was discussed in A1.1.
b. Let \(x=x_2.\) Then \(\varPhi (t,x_2(t))=0\) is equivalent to the equation
Squaring both sides of (37) yields again the Eq. (36) having a unique root \(t=\sqrt{3}/2,\) which does not satisfy (37)
A2. It therefore remains to consider G on the boundary of \(\varOmega \).
(1) On the side \(t=0\),
(2) On the side \(t=1\),
(3) On the side \(x=-1\),
Since \(\varrho _1'(t) = 0\) occurs only when \(t=1/2\) and \(\varrho _1''(1/2)=168>0\), we have
(4) On the side \(x=1\),
Since \(\varrho _2'(t) = 0\) occurs only when \(t=(19-\sqrt{57})/16 \in [0,1] \) and \(\varrho _2''((19-\sqrt{57})/16)=(57-27\sqrt{57})/2<0\), we have
B. Suppose that \(\zeta _1=0.\) Then
and therefore,
C. Suppose that \(\zeta _2=0\) and \(\zeta _1=r \mathrm {e}^{\mathrm {i}\theta }\not =0,\) where \(r\in (0,1]\) and \(\theta \in [0,2\pi ).\) Then
and
Thus,
where
for \(t\in (0,1]\) and \(x\in [-1,1].\) Set
Note that \(-1<x_w\) occurs for \(t>(\sqrt{69}-3)/8= 0.663328\cdots ,\) and \(x_w<1\) holds for \(t\in (0,1).\) Hence, for \(t\in \left( (\sqrt{69}-3)/8,1\right) \) we have
Since \(\phi _1\) is increasing in \(\left( (\sqrt{69}-3)/8,1\right) ,\) so
for \(t\in \left( (\sqrt{69}-3)/8,1\right) . \) Further, \(x_w<-1\) occurs for \(t\in \left( 0,(\sqrt{69}-3)/8\right) .\) Hence,
Since \(\phi _2'(t) = 0\) have a unique root \(t_0=(-2+\sqrt{19})/4=0.589724\cdots \) and \(\phi _2''(t_0)=152>0\), then
Summarizing, form Parts A–C it follows the lower bound in (14).
We discuss now sharpness of (14). The function \(f_2\) defined by (12), for which \(a_2=a_3=0,\) is extremal for the upper bound in (14). It is observed from (16), (23), (24) and (25) that equality for the lower bound in (14) holds when the following conditions are satisfied:
where \(\alpha \) is determined by the condition (21) with \(\kappa _1\) and \(\kappa _2\) given in (22). Thus, \(\theta =\pi ,\) \(\alpha =\pi /2\) and \(\psi =\pi .\) Consequently, \(\zeta _1=-1/2\) and \(\zeta _2=-1,\) which in view of (9) holds for the function
in the class \(\mathcal {P}.\) Therefore, the extremal function f in the class \({\mathcal {F}}_1\) for the lower bound in (14) satisfies (10) with p given as above, having the coefficients \(a_2=-1\) and \(a_3=0.\) \(\square \)
4 The Class \(\mathcal {F}_2\)
Let \(f \in \mathcal {F}_2\) be the form (1). Then by (4) there exists \(p\in {{\mathcal {P}}}\) of the form (5) such that
Putting the series (1) and (5) into (40) by equating the coefficients, we get
Hence and by (6), it follows that \(A_2(\mathcal {F}_2)=3/2\) with the extremal function \(f_1\in \mathcal {F}_2\) such that
Observe also that \(a_2=0\) for the function \(f_2\in \mathcal {F}_2\) such that
Therefore, by Theorem 1 we have
Theorem 4
If \(f\in \mathcal {F}_2\), then
Both inequalities are sharp.
Now we estimate \(\det T_{3,1}(f)\) for functions in the class \(\mathcal {F}_2.\)
Theorem 5
If \(f \in \mathcal {F}_2,\) then
The inequality is sharp.
Proof
By (41) and (6), we see that \(|a_2| \le 3/2\) and \(|a_3| \le 5/3.\) As in the proof of Theorem 3, the inequality (15) holds with the function
Repeating argumentation in the proof of Theorem 3, we see that the function F does not have any relative maxima in \((0,3/2)\times (0,5/3).\)
We consider F on the boundary of \([0,3/2]\times [0,5/3].\)
-
(1)
On the side \(x=0\),
$$\begin{aligned} F(0,y)=1-y^2 \le 1,\quad 0\le y\le \frac{5}{3}. \end{aligned}$$ -
(2)
On the side \(x=3/2\),
$$\begin{aligned} F\left( \frac{3}{2},y\right) = -\frac{7}{2}+\frac{9}{2}y-y^2 \le F\left( \frac{3}{2},\frac{5}{3}\right) = \frac{11}{9},\quad 0\le y\le \frac{5}{3}. \end{aligned}$$ -
(3)
On the side \(y=0\),
$$\begin{aligned} F(x,0)=1-2x^2 \le 1,\quad 0\le x\le \frac{3}{2}. \end{aligned}$$ -
(4)
On the side \(y=5/3\),
$$\begin{aligned} F\left( x,\frac{5}{3}\right) = -\frac{16}{9} + \frac{4}{3}x^2 \le F\left( \frac{3}{2},\frac{5}{3}\right) = \frac{11}{9},\quad 0\le x\le \frac{3}{2}. \end{aligned}$$
Therefore, the inequality \(F(x,y) \le 11/9\) holds for all \((x,y)\in [0,3/2]\times [0,5/3]\) which in view of (15) shows (43).
For the function \(f_1\) given by (42), \(a_2=3/2\) and \(a_3=5/3\) which makes equality in (43). \(\square \)
Theorem 6
If \(f \in \mathcal {F}_2,\) then
The inequality is sharp.
Proof
Substituting (7) and (8) into (41) yields
for some \(\zeta _i \in \overline{{\mathbb {D}}}\) (\(i=1,2\)). Therefore, from (13) we get
where
and
A. Suppose that \(\zeta _1\zeta _2\not =0.\) Thus, \(\zeta _1 =r \mathrm {e}^{\mathrm {i}\theta }\) and \(\zeta _2 = s \mathrm {e}^{\mathrm {i}\psi }\) with \(r,s\in (0,1]\) and \(\theta ,\psi \in [0,2\pi ).\) Then
where
and
where \(\alpha \) is the quantity satisfying (21) with
Since \(\sin (\psi +\alpha ) \ge -1\) and \(s\le 1\), we have
Therefore, from (47), (48) and (49), we get
Taking into account that \(|\zeta _2|\le 1\) from (46) we have
Thus, from (45), (51) and (52) it follows that
where
with
and
for \(t\in [0,1]\) and \(x\in [-1,1].\)
Let \(\varOmega :=[0,1]\times [-1,1]\) and
Now we will show that
A1. For this, we first we find the critical points of G in the interior of \(\varOmega \), i.e., in \((0,1)\times (-1,1).\) Since \(g_2(t,x)=0\) holds only for \(t=(\sqrt{2}-1)/2\) and \(x=1,\) it follows that \(g_2(t,x)> 0\) for \((t,x)\in (0,1)\times (-1,1).\)
Differentiating G with respect to x yields
A1.1 Assume first that \(\partial g_1/\partial x=0.\) Then by (55), it follows that \(\partial g_2/\partial x=0,\) which is possible only for \(t=\sqrt{2}/2\) and \(x=\sqrt{2}/4.\) From (53), we have
A1.2 Assume now that \(\partial g_1/\partial x\ne 0.\) Then we can write the Eq. (55) as
or equivalently, by substituting (54), as
Furthermore, note that by (56) the inequality
is true for
or
Differentiating G with respect to t yields
where
Therefore, each critical point of G satisfies
or
I. Assume that (60) holds. Then \( x=x(t)= (3-4t^2)/(4t). \) Thus, by (57) we see that
occurs only when \(t={\hat{t}}_i,\ i=1,2,\) where
Thus,
and
However, it can be seen that
which means that the inequality (58) is not satisfied for \(t={\hat{t}}_1\) and \(x={\hat{x}}_1\). Note that the case \(t={\hat{t}}_2\) and \(x={\hat{x}}_2\) reduces to A1.1.
Therefore, G does not have critical point in the interior of \(\varOmega .\)
II. Suppose now that (61) is satisfied. Equation (61) as a quadratic equation of x with \(\Delta :=441-216t^2-176t^4>0,\ t\in (0,1)\) has two roots, namely
a. Let \(x=x_1.\) Then \(\varPhi (t,x_1(t))=0\) is equivalent to the equation
Squaring the both sides of (63) leads to
where for \(t\in (0,1),\)
and
Thus, there are two roots \(t_1\) and \(t_2\) in (0, 1) of the Eq. (64), namely
For \(t=t_1\), by (62), \({\tilde{x}}_1:=x_1(t_1)=\sqrt{2}/4\) and this case was discussed in A1.1.
For \(t=t_2\), by (62),
It can be verified that \({\tilde{\Phi }}(t_2,{\tilde{x}}_2) =0\) and the inequality (58) holds for \(t=t_2\) and \(x={\tilde{x}}_2\). Therefore, G has a critical point at \((t_2,{\tilde{x}}_2)\).
b. Let \(x=x_2.\) Then \(\varPhi (t,x_2(t))=0\) is equivalent to the equation
Squaring both sides of (67) yields again the Eq. (64) having roots \(t_1\) and \(t_2\) given by (65), which do not satisfy (67).
Therefore, by A, B1 and B2, the function G has a unique critical point at \((t_2,{\tilde{x}}_2)\). Denote
Since \(\lambda _1 >0\) and
the function G has a local minimum at \((t_2,{\tilde{x}}_2).\)
A2. It remains to consider G in the boundary of \(\Omega \).
(1) On the side \(t=0\),
(2) On the side \(t=1\),
(3) On the side \(x=-1\),
(4) On the side \(x=1\),
When \(t \in \left[ 0,(-1+\sqrt{2})/2\right] =:I_1\), we have \(-1+4t+4t^2 \le 0\). Therefore,
When \(t\in \left[ (-1+\sqrt{2})/2,1\right] =:I_2\), we have \(-1+4t+4t^2 \ge 0\) and \(\varrho (t) = (-2+t+2t^2) (-1+8t+4t^2) \). Since \(\varrho '(t) = 0\) occurs only when \(t=1/2 \in I_2\) and \(\varrho ''(1/2)=80>0\),we have
B. Suppose that \(\zeta _1=0.\) Then
and therefore,
C. Suppose that \(\zeta _2=0\) and \(\zeta _1\not =0.\) Then
Thus, taking \(\zeta _1=r \mathrm {e}^{\mathrm {i}\theta },\) where \(r\in (0,1]\) and \(\theta \in [0,2\pi )\) we have
Then
where
for \(t\in (0,1]\) and \(x\in [-1,1].\) Set
Note that \(-1<x_w\) occurs for \(t\in (0,1)\) and \(x_w<1\) holds for \(t>(\sqrt{77}-5)/8=0.471870\cdots .\) Hence, for \(t\in \left( (\sqrt{77}-5)/8,1\right) \) we have
Since \(\phi _1\) is decreasing in \(\left( (\sqrt{77}-5)/8,1\right) ,\)
On the other hand, \(x_w>1\) occurs for \(t\in \left( 0,(\sqrt{77}-5)/8\right) .\) Hence,
Since \(\phi _2\) is decreasing in \((0,(\sqrt{77}-5)/8)\),
Summarizing, form Parts A–C it follows that the inequality (44) holds.
It remains to show that the inequality (44) is sharp. It is observed from (45), (51), (52) and (53) that \(9\det T_{3,1}(f) = \Theta \) holds when the following conditions are satisfied:
where \(t_2\) and \({\tilde{x}}_2\) are given by (65) and (66), and where \(\alpha \) is determined by the condition (21) with \(\kappa _1\) and \(\kappa _2\) given by (50).
Now set \(\theta =\mathrm{Arccos}({\tilde{x}}_2)\) so that it satisfies the second condition in (68). Then \(\kappa _1 = 3.309903\cdots >0\) and \(\kappa _2 = -0.241293\cdots <0\). Thus, (21) is satisfied if we take
Thus, if we put
then \(\psi \) satisfies the fourth condition in (68). Now consider a function \({\tilde{p}}\) which has the form (9) with \(\zeta _1 = t_2\mathrm {e}^{\mathrm {i}\theta }\) and \(\zeta _2 = \mathrm {e}^{\mathrm {i}\psi }\). Since \(\zeta _1\in {\mathbb {D}}\) and \(\zeta _2 \in {\mathbb {T}}\), from Lemma 1 it follows that \({\tilde{p}}\in \mathcal {P}\), and so the extremal function f in the class \({\mathcal {F}}_2\) for which equality in (44) holds satisfies (40) with \(p:={\tilde{p}}.\) \(\square \)
References
Ali, Md Firoz, Thomas, D.K., Vasudevarao, A.: Toeplitz determinants whose elements are the coefficients of analytic and univalent functions. Bull. Austr. Math. Soc. 97(2), 253–264 (2018)
Ali, M.F., Vasudevarao, A.: On logarithmic coefficients of some close-to-convex functions. Proc. Am. Math. Soc. 146, 1131–1142 (2018)
Biernacki, M.: Sur la représentation conforme des domaines linéairement accessibles. Prace Mat.-Fiz. 44, 293–314 (1936)
Carathéodory, C.: Über den Variabilitatsbereich der Koeffizienten von Potenzreihen, die gegebene werte nicht annehmen. Math. Ann. 64, 95–115 (1907)
Cho, N.E., Kowalczyk, B., Kwon, O.S., Lecko, A., Sim, Y.J.: Some coefficient inequalities related to the Hankel determinant for strongly starlike functions of order alpha. J. Math. Inequal. 11(2), 429–439 (2017)
Cho, N.E., Kowalczyk, B., Kwon, O.S., Lecko, A., Sim, Y.J.: The bounds of some determinants for starlike functions of order alpha. Bull. Malays. Math. Sci. Soc. 41(1), 523–535 (2018)
Cho, N.E., Kowalczyk, B., Kwon, O.S., Lecko, A., Sim, Y.J.: On the third logarithmic coefficient in some subclasses of close-to-convex functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. 114(52), 1–14 (2020)
Cho, N.E., Kowalczyk, B., Lecko, A.: Sharp bounds of some coefficient functionals over the class of functions convex in the direction of the imaginary axis. Bull. Aust. Math. Soc. 100, 86–96 (2019)
Cudna, K., Kwon, O.S., Lecko, A., Sim, Y.J., Śmiarowska, B.: The second and third-order Hermitian Toeplitz determinants for starlike and convex functions of order \(\alpha \). Bol. Soc. Mat. Mex. 26, 361–375 (2020)
Goodman, A.W.: Univalent Functions. Mariner, Tampa (1983)
Jastrzȩbski, P., Kowalczyk, B., Kwon, O.S., Lecko, A., Sim, Y.J.: Hermitian Toeplitz determinants of the second and third-order for classes of close-to-star functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. 114(166), 1–14 (2020)
Kaplan, W.: Close to convex schlicht functions. Mich. Math. J. 1, 169–185 (1952)
Kowalczyk, B., Kwon, O.S., Lecko, A., Sim, Y.J.: The bounds of some determinants for functions of bounded turning of order alpha. Bulletin de la Société Des Sciences et des Lettres de Łódź: Recherches sur les déformations LXVII(1), 107–118 (2017)
Kowalczyk, B., Kwon, O.S., Lecko, A., Sim, Y.J., Śmiarowska, B.: The Third-Order Hermitian Toeplitz Determinant for Classes of Functions Convex in One Direction. Bull. Malays. Math. Sci. Soc. 43, 3143–3158 (2020)
Kowalczyk, B., Lecko, A., Sim, Y.J.: The sharp bound of the Hankel determinant of the third kind for convex functions. Bull. Aust. Math. Soc. 97, 435–445 (2018)
Kowalczyk, B., Lecko, A., Lecko, M., Sim, Y.J.: The sharp bound of the third Hankel determinant for some classes of analytic functions. Bull. Korean Math. Soc. 55(6), 1859–1868 (2018)
Kumar, U.P., Vasudevarao, A.: Logarithmic coefficients for certain subclasses of close-to-convex functions. Monatsh. Math. 187, 543–563 (2018)
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)
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)
Lee, S.K., Ravichandran, V., Supramanian, S.: Bound for the second Hankel determinant of certain univalent functions. J. Inequal. Appl. 2013(281), 1–17 (2013)
Lecko, A.: A generalization of analytic condition for convexity in one direction. Demonstr. Math. XXX(1), 155–170 (2002)
Lewandowski, Z.: Sur l’identiteé de certaines classes de fonctions univalentes, I. Ann. Univ. Mariae Curie-Skłodowska Sect. A 12, 131–146 (1958)
Lewandowski, Z.: Sur l’identiteé de certaines classes de fonctions univalentes, II. Ann. Univ. Mariae Curie-Skłodowska Sect. A 14, 19–46 (1960)
Pommerenke, C.: Univalent Functions. Vandenhoeck & Ruprecht, Göttingen (1975)
Ravichandran, V., Verma, S.: Generalized Zalcman conjecture for some classes of analytic functions. J. Math. Anal. Appl. 450(1), 592–605 (2017)
Robertson, M.S.: Analytic functions star-like in one direction. Am. J. Math. 58, 465–472 (1936)
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Lecko, A., Śmiarowska, B. Sharp Bounds of the Hermitian Toeplitz Determinants for Some Classes of Close-to-Convex Functions. Bull. Malays. Math. Sci. Soc. 44, 3391–3412 (2021). https://doi.org/10.1007/s40840-021-01122-x
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DOI: https://doi.org/10.1007/s40840-021-01122-x