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A study of sharp coefficient bounds for a new subfamily of starlike functions
Journal of Inequalities and Applications volume 2021, Article number: 194 (2021)
Abstract
In this article, by employing the hyperbolic tangent function tanhz, a subfamily \(\mathcal{S}_{\tanh }^{\ast }\) of starlike functions in the open unit disk \(\mathbb{D}\subset \mathbb{C}\):
is introduced and investigated. The main contribution of this article includes derivations of sharp inequalities involving the Taylor–Maclaurin coefficients for functions belonging to the class \(\mathcal{S}_{\tanh }^{\ast } \) of starlike functions in \(\mathbb{D}\). In particular, the bounds of the first three Taylor–Maclaurin coefficients, the estimates of the Fekete–Szegö type functionals, and the estimates of the second- and third-order Hankel determinants are the main problems that are proposed to be studied here.
1 Introduction, definitions, and preliminaries
Let us represent the family of analytic (or regular or holomorphic) functions in \(\mathbb{D}\) by the notation \(\mathcal{H} (\mathbb{D} )\) and suppose that \(\mathcal{A}\) is the subclass of \(\mathcal{H} (\mathbb{D} )\) defined as follows:
Further, all normalized univalent functions in \(\mathbb{D}\) are contained in the set \(\mathcal{S}\subset \mathcal{A}\). For two given functions \(g_{1},g_{2}\in \mathcal{H} (\mathbb{D} )\), we say that \(g_{1}\) is subordinate to \(g_{2}\), written symbolically as \(g_{1}\prec g_{2}\), if there exists a Schwarz function w, which is analytic in \(\mathbb{D}\) with
such that
Moreover, if the function \(g_{2}\) is univalent in \(\mathbb{D}\), then the following equivalence holds true:
Though the subject of function theory was founded in 1851, the coefficient conjecture presented by Bieberbach [13] in 1916 led to the field’s emergence as a promising area of new research. This conjecture was proved by de Branges [18] in 1985. Between 1916 and 1985, many of the finest scholars of the day sought to prove or disprove this Bieberbach conjecture. As a consequence, they discovered numerous sub-families of the class \(\mathcal{S}\) of normalized univalent functions connected to distinct image domains. The families of starlike and convex functions, respectively, denoted by \(\mathcal{S}^{\ast }\) and \(\mathcal{K}\), are the most fundamental and significant subclasses of the set \(\mathcal{S}\). In 1992, Ma and Minda [36] considered the general form of the family as follows:
where ϕ is a holomorphic function with \(\phi ^{\prime } (0 ) >0\) and has a positive real part in \(\mathbb{D}\). Also, the function ϕ maps \(\mathbb{D}\) onto a star-shaped region with respect to \(\phi (0 )=1\) and is symmetric about the real axis. They addressed some specific results such as distortion, growth, and covering theorems. In recent years, several sub-families of the normalized analytic function class \(\mathcal{A}\) were studied as a special case of the class \(\mathcal{S}^{\ast } (\phi )\). For example, we have:
-
(i)
If we choose
$$\begin{aligned} \phi (z)=\frac{1+Lz}{1+Mz}\quad (-1\leqq M< L\leqq 1), \end{aligned}$$then we achieve the class given by
$$\begin{aligned} \mathcal{S}^{\ast }[L,M]\equiv \mathcal{S}^{\ast } \biggl( \frac{1+Lz}{1+Mz} \biggr), \end{aligned}$$which is described as the functions of the Janowski starlike class investigated in [22]. Furthermore, the class \(\mathcal{S}^{\ast } (\xi )\) given by
$$\begin{aligned} \mathcal{S}^{\ast } (\xi ):=\mathcal{S}^{\ast }[1-2\xi,-1] \end{aligned}$$is the familiar starlike function family of order ξ with \(0\leqq \xi <1\).
-
(ii)
The following family:
$$\begin{aligned} \mathcal{S}_{\mathcal{L}}^{\ast }:=\mathcal{S}^{\ast }\bigl(\phi (z)\bigr)\quad \bigl(\phi (z)=\sqrt{1+z} \bigr) \end{aligned}$$was studied in [49] by Sokól and Stankiewicz. The function \(\phi (z)=\sqrt{1+z}\) maps the region \(\mathbb{D}\) onto the image domain which is bounded by \(|w^{2}-1|<1\).
-
(iii)
The class given by
$$\begin{aligned} \mathcal{S}_{car}^{\ast }:=\mathcal{S}^{\ast }\bigl(\phi (z)\bigr)\quad \biggl( \phi (z) =1+\frac{4}{3} z+\frac{2}{3} z^{2} \biggr) \end{aligned}$$was examined by Sharma et al. [46]. It consists of functions \(f\in \mathcal{A}\) in such a manner that
$$\begin{aligned} \frac{zf^{\prime }(z)}{f(z)} \end{aligned}$$is located in the region bounded by the cardioid given by
$$\begin{aligned} \bigl(9x^{2}+9y^{2}-18x+5\bigr)^{2}-16 \bigl(9x^{2}+9y^{2}-6x+1\bigr)=0. \end{aligned}$$ -
(iv)
By selecting \(\phi (z)=1+\sin z\), the class \(\mathcal{S}^{\ast } (\phi (z) )\) leads to the family \(\mathcal{S}_{\sin }^{\ast }\), which was investigated by Cho et al. [17]. On the other hand, the function class given by
$$\begin{aligned} \mathcal{S} _{e}^{\ast }\equiv \mathcal{S}^{\ast } \bigl(e^{z} \bigr) \end{aligned}$$was studied in [38] and, subsequently, in [48]. This function class was recently generalized by Srivastava et al. [56] in which the authors determined an upper bound of the Hankel determinant of the third order.
-
(v)
The following families:
$$\begin{aligned} \mathcal{S}_{\cos }^{\ast }:=\mathcal{S}^{\ast }(\cos z) \end{aligned}$$and
$$\begin{aligned} \mathcal{S}_{\cosh }^{\ast }:=\mathcal{S} ^{\ast }(\cosh z) \end{aligned}$$were considered, respectively, by Raza and Bano [9] and Alotaibi et al. [2]. In both of these papers, the authors studied some interesting properties of the families which they studied.
-
(vi)
By choosing \(\phi (z)=1+\sin z\), we obtain the following class:
$$\begin{aligned} \mathcal{S}_{\sin }^{\ast }:=\mathcal{S}^{\ast } \bigl(\phi (z) \bigr), \end{aligned}$$which was investigated in [17]. The authors in [17] addressed the radii problems for the defined class \(\mathcal{S}_{\sin }^{\ast }\).
-
(vii)
By considering the function \(\phi (z)=1+\sinh ^{-1}z\), we get the recently-examined family given by
$$\begin{aligned} \mathcal{S}_{\rho }^{\ast }:=\mathcal{S}^{\ast } \bigl(1+ \sinh ^{-1}z \bigr), \end{aligned}$$which was introduced by Kumar and Arora [29]. They discussed relationships of this class with the already known classes. In 2021, Barukab et al. [12] derived sharp bounds for the Hankel determinant of the third order for the following function class:
$$\begin{aligned} \mathcal{R}_{s}:= \bigl\{ f: f\in \mathcal{A} \text{ and } f^{ \prime } (z ) \prec 1+\sinh ^{-1}z\ (z\in \mathbb{D}) \bigr\} . \end{aligned}$$
In the present paper, we consider the following hyperbolic function:
Also, one can easily find that \({\Re } (\varphi _{1}(z) )>0\).
Definition 1
([59])
By using the above-defined hyperbolic function \(\varphi _{1}(z)\), we define the following family of functions:
In other words, a function f is in the class \(\mathcal{S}_{\tanh }^{\ast }\) if and only if there exists a holomorphic function q, fulfilling \(q ( z ) \prec q_{0} ( z ):=1+\tanh z\), such that
By taking
in (3), we get the function that plays the role of the extremal function in many problems of the class \(\mathcal{S}_{\tanh }^{\ast }\), given by
Definition 2
The Hankel determinant
for a function \(f\in \mathcal{S}\) of the series form (1) was given by Pommerenke [40, 41] as follows:
In particular, the following determinants are known as the first-, the second-, and the third-order Hankel determinants, respectively:
and
In the literature, there are just a few references to the Hankel determinant for functions belonging to the general family \(\mathcal{S}\). For the function \(f\in \mathcal{S}\), the best established sharp inequality is given by
where λ is an absolute constant. This result is due to Hayman [21]. Further, for the same class \(\mathcal{S}\), it was derived in [39] as follows:
and
The challenge of finding the sharp bounds of Hankel determinants for a particular family of functions drew the attention of numerous researchers. For example, the sharp bounds of \(\vert \mathcal{HD}_{2,2} ( f ) \vert \) for the sub-families \(\mathcal{K}\), \(\mathcal{S}^{\ast }\), and \(\mathcal{R}\) (the family of bounded turning functions) of the class \(\mathcal{S}\) were calculated by Janteng et al. [23, 24]. These estimates are given by
For the families
of starlike functions of order β and
of strongly starlike functions of order β, the authors in [15, 16] showed that \(\vert \mathcal{HD}_{2,2} ( f ) \vert \) is bounded by \((1-\beta ) ^{2}\) and \(\beta ^{2}\), respectively. The exact bound for the family \(\mathcal{S}^{\ast } ( \phi ) \) of the Ma–Minda type starlike functions was derived in [33] (see also [19]). For other works involving \(\vert \mathcal{HD}_{2,2} ( f ) \vert \), see (for example) [4, 10, 14, 25, 35].
It is quite clear from the formulas given in (5), (6), and (7) that the calculation of the bound for \(\vert \mathcal{HD}_{3,1} ( f ) \vert \) is far more challenging in comparison with the finding of the bound for \(\vert \mathcal{HD}_{2,2} ( f ) \vert \). In the year 2010, Babalola [8] investigated the bounds for the third-order Hankel determinant for the families of \(\mathcal{K}\), \(\mathcal{S}^{\ast }\), and \(\mathcal{R}\). Subsequently, by using the same or analogous approach, several authors in [3, 11, 28, 43, 45] derived bounds for the third-order Hankel determinant \(\vert \mathcal{HD}_{3,1} ( f ) \vert \) for various sub-families of analytic and univalent functions. On the other hand, in the year 2017, Zaprawa [61] improved the findings of Babalola [8] by applying a new methodology to show that
Zaprawa [61] remarked that such limits were indeed not the best ones. Later in the year 2018, Kwon et al. [31] strengthened Zaprawa’s result for \(f\in \mathcal{S}^{\ast }\) and showed that \(\vert \mathcal{HD}_{3,1} ( f ) \vert \leqq \frac{8}{9}\), and this bound was further improved by Zaprawa et al. [62] by showing in 2021 that
In recent years, the following sharp bounds for the third-order Hankel determinant \(\vert \mathcal{HD}_{3,1} ( f ) \vert \) were given by Kowalczyk et al. [27] and Lecko et al. [32]:
where \(\mathcal{S}^{\ast } ( \frac{1}{2} ) \) represents the family of starlike functions of order \(\frac{1}{2}\) in \(\mathbb{D}\). The interested readers may also refer to the research provided by Mahmood et al. [37] in which they calculated bounds for the third-order Hankel determinant for the basic (or \(\mathfrak{q}\)-) starlike functions in \(\mathbb{D}\).
For more contributions in this direction, the interested reader should see, for example, [20, 44, 47, 52–55]. In particular, Arif et al. [6], Srivastava et al. [55], Arif et al. [5], and Wang et al. [60] successfully investigated bounds for the fourth-order Hankel determinant for different subclasses of analytic functions.
In the present article, our aim is to calculate the sharp bounds of the coefficient inequalities, Fekete–Szegö type functional, and the Hankel determinants of order two and order three for the subclass \(\mathcal{S}_{\tanh }^{\ast }\) of starlike functions.
2 A set of lemmas
Definition 3
A function p is said to be in the class \(\mathcal{P}\) if and only if it has the following series expansion:
and satisfies the inequality given by
Lemma 1
Let the function \(p\in \mathcal{P}\) have the series form (8). Then, for \(x,\delta, \rho \in \overline{\mathbb{D}}=\mathbb{D}\cup \{1\}\),
and
Remark 2
In Lemma 1and elsewhere in this paper, for the formula for \(c_{2}\), see [42]. The formula for \(c_{3}\) is due to Libera and Złotkiewicz [34]. The formula for \(c_{4}\) was proved in [30].
Lemma 3
If the function \(p\in \mathcal{P}\) has the series form (8), then
and
If \(B\in [0,1 ]\) with \(B (2B-1 ) \leqq D\leqq B\), then
Remark 4
Inequalities (12), (13), and (14) in Lemma 3are taken from [26, 42] and [6, 7, 47], respectively.
3 Coefficient inequalities for the function class \(\mathcal{S}_{\tanh }^{\ast }\)
The first two findings, Theorem 5 and Theorem 6, are special cases of the results established in the paper [1], and that is why we omitted both the proofs.
Theorem 5
Let the function f of the form (1) be in the class \(\mathcal{S}_{\tanh }^{\ast }\). Then
Each of these bounds is sharp.
Theorem 6
Let the function f of the form (1) be in the class \(\mathcal{S}_{\tanh }^{\ast }\). Then
This inequality is sharp.
Theorem 7
Let the function f of the form (1) be in the class \(\mathcal{S}_{\tanh }^{\ast }\). Then
This result is sharp.
Proof
Let \(f\in \mathcal{S}_{\tanh }^{\ast }\). Then equation (2) can be written in the form of a hyperbolic function w as follows:
Let \(p\in \mathcal{P}\). Then, in terms of the Schwarz function w, we have
or, equivalently,
where
By using (1), we obtain
After some calculation and by using the series expansion given by (16), we get
Now, if we compare (17) and (18), we get
and
By using (19), (20), and (21), we obtain
which, in view of (9) and (10), together with \(c_{1}=c\in [ 0,1 ] \), yields
Now, upon applying \(\vert \delta \vert \leqq 1\) and \(\vert x \vert =b\leqq 1\), and using the triangle inequality, we get
It is a simple exercise to differentiate \(F (c,b )\) with respect to b and show that \(F^{\prime } (c,b ) \leqq 0\) on the rectangle \([0,2 ] \times [0,1 ]\). So, by putting \(b=0\), we obtain
We thus find that
Finally, upon taking \(G^{\prime }(c)=0\), we obtain \(c=0,1\). Thus, clearly, \(G(c)\) has its maximum value at \(c=0\), so that
in which the equality holds true for the extremal function given by
This evidently completes our demonstration of Theorem 7. □
Theorem 8
Let the function f of the form (1) be in the class \(\mathcal{S}_{\tanh }^{\ast }\). Then
This inequality is sharp.
Proof
We can write \(\mathcal{HD}_{2,2} (f )\) as follows:
From (19), (20), and (21), we have
Now, by using (9) and (10) in order to express \(c_{2}\) and \(c_{3}\) in terms of \(c_{1}\) and also \(c_{1}=c \ (0\leqq c\leqq 2)\), we obtain
By using \(\vert \delta \vert \leqq 1\) and \(\vert x \vert =b \leqq 1\) and applying the triangle inequality, if we take \(c\in [0,2 ]\), we obtain
Upon differentiating with respect to b, we have
It is a simple exercise to show that \(\Xi ^{\prime } (c,b ) \geqq 0\) on \([0,1 ]\), so that
Putting \(b=1\), we get
As \(G^{\prime } (c ) \leqq 0\), so \(G (c )\) is a decreasing function of c, so that it gives the maximum value at \(c=0\):
Finally, the above bound for \(\mathcal{HD}_{2,2} (f )\) is sharp and is achieved by the following extremal function:
We have thus completed the proof of Theorem 8. □
4 The third Hankel determinant
In this section, we determine the bounds of \(\vert \mathcal{HD}_{3,1} ( f ) \vert \) for the function \(f\in \mathcal{S}_{\tanh }^{\ast }\).
Theorem 9
Let the function f of the form (1) be in the class \(\mathcal{S}_{\tanh }^{\ast }\). Then
This result is sharp.
Proof
The third-order Hankel determinant can be written as follows:
By using (19), (20), (21), and (22), together with \(c_{1}=c\in [0,2 ]\), we have
For simplifying the computation, we let \(t=4-c^{2}\) in (9), (10), and (11). Then, by using the simplified form of these formulas, we have
and
Upon substituting these expressions into (25) and simplifying, we get
Now, since \(t= ( 4-c^{2} )\), we have
where
and
Thus, upon setting \(\vert \delta \vert =y\) and \(\vert x \vert =x\), and by taking \(\vert \rho \vert \leqq 1\), we obtain
where
with
and
Let the closed cuboid be of the following form:
We need to find the points of maxima inside this closed cuboid Δ, inside the six faces, and on the twelve edges in order to maximize the function \(H(c,x,y)\) given by (27). For this objective in view, we consider the following three cases.
I. Let \(c,x,y\in ( 0,2 ) \times (0,1 ) \times ( 0,1 )\). In order to find the points of maxima inside Δ, we take partial derivative of (27) with respect to y, so that we achieve
which can be seen to vanish when
If \(y_{0}\) is a critical point inside Δ, then \(y_{0}\in ( 0,1 )\), which is possible only if
and
We now have to get the solutions which satisfy both of inequalities (29) and (30) for the existence of the critical points. Let us set
Since \(h^{\prime } (x ) <0\) for \((0,1 )\), the function \(h (x ) \) is decreasing in \((0,1 ) \). Hence \(c^{2}>\frac{7}{4}\), and a simple exercise shows that (29) does not hold true in this case for all values of \(x\in (0,1 )\) and there is no critical point of \(H(c,x,y)\) in \((0,2 ) \times (0,1 ) \times (0,1 )\).
II. In order to find the points of maxima inside the six faces of the cuboid Δ, we deal with each face individually. On \(c=0\), \(H (c,x,y )\) reduces to
Clearly, \(q_{1}\) has no optimal points in \(( 0,1 ) \times ( 0,1 ) \) since
On \(c=2\), \(H ( c,x,y ) \) reduces to
On \(x=0\), \(H ( c,x,y ) \) reduces to
where \(y\in ( 0,1 ) \) and \(c\in ( 0,2 ) \). We now solve
in order to find the points of maxima. On solving
we obtain
for the given range of \(y,y_{1}\) that should belong to \(( 0,1 ) \). This is possible only if
A calculation shows that
implies that
By substituting from equation (35) into equation (36) and simplifying, we have
A further calculation gives the solution of (37) in \(( 0,2 ) \), that is, \(c\approx 1.16653673056906\). Thus \(q_{2}\) has no optimal point in \(( 0,2 ) \times ( 0,1 ) \).
On \(x=1\), \(H (c,x,y )\) reduces to
Solving
we obtain the critical points given by
Since \(c_{0}\) is the minimum point of \(q_{3}\), \(q_{3}\) attains its maximum value at \(c_{1}\), that is, at \(c=1717.98045\).
On \(y=0\), \(H (c,x,y ) \) reduces to
A computation reveals that the following system of equations has no solution:
in \((0,2 ) \times (0,1 ) \).
On \(y=1\), \(H (c,x,y )\) reduces to
A computation reveals that the following system of equations has no solution:
in \(( 0,2 ) \times ( 0,1 )\).
III. In this case, we find the maxima of \(H (c,x,y )\) on the edges of Δ. By putting \(y=0\) in (34), we have
Clearly, \(m_{1}^{\prime } ( c )=0\) for \(c=\eta _{0}=0\) and \(c=\eta _{1}=1.75122868295016\) in \([0,2 ]\), where \(\eta _{0}\) is the minimum point and the maximum point of \(m_{1} (c )\) is attained at \(\eta _{1}\). This implies that
Solving equation (34) at \(y=1\), we get
Since \(m_{2}^{\prime } (c ) <0\) for \(c\in [0,2 ]\), \(m_{2} (c ) \) is decreasing in \([0,2 ]\) and hence the maximum is obtained at \(c=0\). Thus
By putting \(c=0\) in (34), we get
A simple calculation gives
Equation (38) is independent of y, so we have
Now \(m_{3}^{\prime } (c ) =0\) for \(c=\eta _{0}=0\) and \(c=\eta _{1}=1.07838082301303\) in \([0,2 ]\), where \(\eta _{0}\) is the minimum point and the maximum point of \(m_{3} (c )\) is attained at \(\eta _{1}\). We conclude that
By putting \(c=0\) in (38), we obtain
As (33) is independent of c, x, and y, we find that
By putting \(y=0\) in (31), we have
Now \(m_{4}^{\prime } ( x ) =0\) for \(x=x_{0}=0.8164965809\) in \([ 0,1 ] \). Therefore, the function \(m_{4} ( x ) \) is increasing for \(x\leqq x_{0}\) and decreasing for \(x_{0}\leqq x\). Hence \(m_{4} ( x ) \) has its maximum at \(x=x_{0}\). We conclude that
By putting \(y=1\) in (31), we get
Since \(m_{5}^{\prime } ( x ) <0\) for \([ 0,1 ] \), therefore the function \(m_{4} ( x ) \) is decreasing in \([ 0,1 ] \) and hence attains its maximum value at \(x=0\), so that
Thus, from the above cases, we conclude that
From equation (26), we can write
If \(f\in \mathcal{S}_{\tanh }^{\ast }\), then the equality is achieved by the function given by
Theorem 9 has thus been proved as asserted. □
5 Concluding remarks and observations
In the present article, we have introduced and studied a new subfamily of starlike functions in the open unit disk \(\mathbb{D}\), which involves the hyperbolic function tanhz. For functions belonging to such a class of starlike functions, we have considered some interesting problems such as the bounds of the first three Taylor–Maclaurin coefficients, the estimates of the Fekete–Szegö type functional, and the estimates of the second- and third-order Hankel determinants. All of the bounds which we have investigated in this article have been shown to be sharp.
A potential direction for further research based upon our present investigation would involve the use of the familiar quantum or basic (or \(\mathfrak{q}\)-) calculus as (for example) in the related recent works [37, 44, 50, 53, 54, 56], [57], and [58]. However, as clearly pointed out in the survey-cum-expository review articles by Srivastava (see, for details, [50, p. 340]; see also [51, pp. 1511–1512]), any attempt to translate these suggested \(\mathfrak{q}\)-results in terms of the so-called trivial and inconsequential \((\mathfrak{p},\mathfrak{q})\)-calculus would obviously lead to a shallow research, because the additional forced-in parameter \(\mathfrak{p}\) is obviously redundant or superfluous.
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References
Ali, R.M., Ravichandran, V., Seenivasagan, N.: Coefficient bounds for p-valent functions. Appl. Math. Comput. 187, 35–46 (2007)
Alotaibi, A., Arif, M., Alghamdi, M.A., Hussain, S.: Starlikness associated with cosine hyperbolic function. Mathematics 8, Article ID 1118 (2020)
Altınkaya, Ş., Yalçın, S.: Third Hankel determinant for Bazilevič functions. Adv. Math.: Sci. J. 5, 91–96 (2016)
Altınkaya, Ş., Yalçın, S.: Upper bound of second Hankel determinant for bi-Bazilevic functions. Mediterr. J. Math. 13, 4081–4090 (2016)
Arif, M., Rani, L., Raza, M., Zaprawa, P.: Fourth Hankel determinant for the set of star-like functions. Math. Probl. Eng. 2021, Article ID 6674010 (2021). https://doi.org/10.1155/2021/6674010
Arif, M., Raza, M., Tang, H., Hussain, S., Khan, H.: Hankel determinant of order three for familiar subsets of analytic functions related with sine function. Open Math. 17, 1615–1630 (2019)
Arif, M., Umar, S., Raza, M., Bulboaca, T., Farooq, M.U., Khan, H.: On fourth Hankel determinant for functions associated with Bernoulli’s lemniscate. Hacet. J. Math. Stat. 49, 1777–1780 (2020)
Babalola, K.O.: On \(H_{3}(1)\) Hankel determinant for some classes of univalent functions. Inequal. Theory Appl. 6, 1–7 (2010)
Bano, K., Raza, M.: Starlike functions associated with cosine function. Bull. Iran. Math. Soc. (2020). https://doi.org/10.1007/s41980-020-00456-9
Bansal, D.: Upper bound of second Hankel determinant for a new class of analytic functions. Appl. Math. Lett. 26, 103–107 (2013)
Bansal, D., Maharana, S., Prajapat, J.K.: Third order Hankel determinant for certain univalent functions. J. Korean Math. Soc. 52, 1139–1148 (2015)
Barukab, O., Arif, M., Abbas, M., Khan, S.K.: Sharp bounds of the coefficient results for the family of bounded turning functions associated with petal shaped domain. J. Funct. Spaces 2021, Article ID 5535629 (2021)
Bieberbach, L.: Über dié koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln. Sitz.ber. Preuss. Akad. Wiss. 138, 940–955 (1916)
Çaglar, M., Deniz, E., Srivastava, H.M.: Second Hankel determinant for certain subclasses of bi-univalent functions. Turk. J. Math. 41, 694–706 (2017)
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, 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, 523–535 (2018)
Cho, N.E., Kumar, V., Kumar, S.S., Ravichandran, V.: Radius problems for starlike functions associated with the sine function. Bull. Iran. Math. Soc. 45, 213–232 (2019)
De Branges, L.: A proof of the Bieberbach conjecture. Acta Math. 154, 137–152 (1985)
Ebadian, A., Bulboacă, T., Cho, N.E., Adegani, E.A.: Coefficient bounds and differential subordinations for analytic functions associated with starlike functions. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 114, Article ID 128 (2020)
Güney, H.Ö., Murugusundaramoorthy, G., Srivastava, H.M.: The second Hankel determinant for a certain class of bi-close-to-convex functions. Results Math. 74, Article ID 93 (2019)
Hayman, W.K.: On second Hankel determinant of mean univalent functions. Proc. Lond. Math. Soc. (3) 18, 77–94 (1968)
Janowski, W.: Extremal problems for a family of functions with positive real part and for some related families. Ann. Pol. Math. 23, 159–177 (1970)
Janteng, A., Halim, S.A., Darus, M.: Coefficient inequality for a function whose derivative has a positive real part. J. Inequal. Pure Appl. Math. 7(2), Article ID 50 (2006)
Janteng, A., Halim, S.A., Darus, M.: Hankel determinant for starlike and convex functions. Int. J. Math. Anal. 1, 619–625 (2007)
Kanas, S., Adegani, E.A., Zireh, A.: An unified approach to second Hankel determinant of bi-subordinate functions. Mediterr. J. Math. 14, Article ID 233 (2017)
Keough, F., Merkes, E.: A coefficient inequality for certain subclasses of analytic functions. Proc. Am. Math. Soc. 20, 8–12 (1969)
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)
Krishna, D.V., Venkateswarlu, B., RamReddy, T.: Third Hankel determinant for bounded turning functions of order alpha. J. Niger. Math. Soc. 34, 121–127 (2015)
Kumar, S.S., Arora, K.: Starlike functions associated with a petal shaped domain (2020). arXiv preprint 2010.10072
Kwon, O.S., Lecko, A., Sim, Y.J.: On the fourth coefficient of functions in the Carathéodory class. Comput. Methods Funct. Theory 18, 307–314 (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 \(\frac{1}{2}\). Complex Anal. Oper. Theory 13, 2231–2238 (2019)
Lee, S.K., Ravichandran, V., Supramaniam, S.: Bounds for the second Hankel determinant of certain univalent functions. J. Inequal. Appl. 2013, Article ID 281 (2013)
Libera, R.J., Złotkiewicz, E.J.: Early coefficients of the inverse of a regular convex function. Proc. Am. Math. Soc. 85, 225–230 (1982)
Liu, M.-S., Xu, J.-F., Yang, M.: Upper bound of second Hankel determinant for certain subclasses of analytic functions. Abstr. Appl. Anal. 2014, Article ID 603180 (2014)
Ma, W.C., Minda, D.: A unified treatment of some special classes of univalent functions. In: Li, Z., Ren, F., Yang, L., Zhang, S. (eds.) Proceedings of the Conference on Complex Analysis, Tianjin, People’s Republic of China, June 19–22, 1992. Conference Proceedings and Lecture Notes in Analysis, vol. I, pp. 157–169. International Press, Cambridge (1994)
Mahmood, S., Srivastava, H.M., Khan, N., Ahmad, Q.Z., Khan, B., Ali, I.: Upper bound of the third Hankel determinant for a subclass of q-starlike functions. Symmetry 11, Article ID 347 (2019)
Mendiratta, R., Nagpal, S., Ravichandran, V.: On a subclass of strongly starlike functions associated with exponential function. Bull. Malays. Math. Sci. Soc. 38, 365–386 (2015)
Obradović, M., Tuneski, N.: Hankel determinants of second and third order for the class S of univalent functions. Math. Slovaca 71, 649–654 (2021)
Pommerenke, C.: On the coefficients and Hankel determinants of univalent functions. J. Lond. Math. Soc. 41, 111–122 (1966)
Pommerenke, C.: On the Hankel determinants of univalent functions. Mathematika 14, 108–112 (1967)
Pommerenke, C.: Univalent Function. Vanderhoeck & Ruprecht, Göttingen (1975)
Raza, M., Malik, S.N.: Upper bound of third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli. J. Inequal. Appl. 2013, Article ID 412 (2013)
Shafiq, M., Srivastava, H.M., Khan, N., Ahmad, Q.Z., Darus, M., Kiran, S.: An upper bound of the third Hankel determinant for a subclass of q-starlike functions associated with k-Fibonacci numbers. Symmetry 12, Article ID 1043 (2020)
Shanmugam, G., Stephen, B.A., Babalola, K.O.: Third Hankel determinant for α-starlike functions. Gulf J. Math. 2, 107113 (2014)
Sharma, K., Jain, N.K., Ravichandran, V.: Starlike functions associated with a cardioid. Afr. Math. 27, 923–939 (2016)
Shi, L., Ali, I., Arif, M., Cho, N.E., Hussain, S., Khan, H.: A study of third Hankel determinant problem for certain subfamilies of analytic functions involving cardioid domain. Mathematics 7, Article ID 418 (2019)
Shi, L., Srivastava, H.M., Arif, M., Hussain, S., Khan, H.: An investigation of the third Hankel determinant problem for certain subfamilies of univalent functions involving the exponential function. Symmetry 11, Article ID 598 (2019)
Sokół, J., Stankiewicz, J.: Radius of convexity of some subclasses of strongly starlike functions. Zesz. Nauk. Politech. Rzesz., Mat. Fiz. 19, 101–105 (1996)
Srivastava, H.M.: Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis. Iran. J. Sci. Technol. Trans. A, Sci. 44, 327–344 (2020)
Srivastava, H.M.: Some parametric and argument variations of the operators of fractional calculus and related special functions and integral transformations. J. Nonlinear Convex Anal. 22, 1501–1520 (2021)
Srivastava, H.M., Ahmad, Q.Z., Darus, M., Khan, N., Khan, B., Zaman, N., Shah, H.H.: Upper bound of the third Hankel determinant for a subclass of close-to-convex functions associated with the lemniscate of Bernoulli. Mathematics 7, Article ID 848 (2019)
Srivastava, H.M., Ahmad, Q.Z., Khan, N., Khan, B.: Hankel and Toeplitz determinants for a subclass of q-starlike functions associated with a general conic domain. Mathematics 7, Article ID 181 (2019)
Srivastava, H.M., Altınkaya, S., Yalcın, S.: Hankel determinant for a subclass of bi-univalent functions defined by using a symmetric q-derivative operator. Filomat 32, 503–516 (2018)
Srivastava, H.M., Kaur, G., Singh, G.: Estimates of the fourth Hankel determinant for a class of analytic functions with bounded turnings involving cardioid domains. J. Nonlinear Convex Anal. 22, 511–526 (2021)
Srivastava, H.M., Khan, B., Khan, N., Tahir, M., Ahmad, S., Khan, N.: Upper bound of the third Hankel determinant for a subclass of q-starlike functions associated with the q-exponential function. Bull. Sci. Math. 167, Article ID 102942 (2021)
Srivastava, H.M., Khan, N., Darus, M., Khan, S., Ahmad, Q.Z., Hussain, S.: Fekete–Szegö type problems and their applications for a subclass of q-starlike functions with respect to symmetrical points. Mathematics 8, Article ID 842 (2020)
Srivastava, H.M., Raza, N., AbuJarad, E.S.A., Srivastava, G., AbuJarad, M.H.: Fekete–Szegö inequality for classes of \((p,q)\)-starlike and \((p,q)\)-convex functions. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113, 3563–3584 (2019)
Ullah, K., Zainab, S., Arif, M., Darus, M., Shutaywi, M.: Radius problems for starlike functions associated with the tan hyperbolic function. J. Funct. Spaces 2021, Article ID 9967640 (2021). https://doi.org/10.1155/2021/9967640
Wang, Z.-G., Raza, M., Arif, M., Ahmad, K.: On the third and fourth Hankel determinants for a subclass of analytic functions. Bull. Malays. Math. Sci. Soc. (2021). https://doi.org/10.1007/s40840-021-01195-8
Zaprawa, P.: Third Hankel determinants for subclasses of univalent functions. Mediterr. J. Math. 14, Article ID 19 (2017). https://doi.org/10.1007/s00009-016-0829-y
Zaprawa, P., Obradović, M., Tuneski, N.: Third Hankel determinant for univalent starlike functions. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 115, Article ID 49 (2021)
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Ullah, K., Srivastava, H.M., Rafiq, A. et al. A study of sharp coefficient bounds for a new subfamily of starlike functions. J Inequal Appl 2021, 194 (2021). https://doi.org/10.1186/s13660-021-02729-1
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DOI: https://doi.org/10.1186/s13660-021-02729-1
Keywords
- Analytic (or regular or holomorphic) functions
- Univalent functions
- Starlike functions
- Principle of subordination
- Schwarz function
- Hyperbolic and trigonometric functions
- Coefficient bounds
- Fekete–Szegö functional
- The quantum or basic (or \(\mathfrak{q}\)-) calculus and its trivial \((\mathfrak{p},\mathfrak{q})\)-variation