Abstract
This article comprises the study of class \(\mathcal{S}_{E}^{\ast }\) that represents the class of normalized analytic functions f satisfying \({\varsigma \mathsf{f}}^{\prime }(z)/\mathsf{f}( {\varsigma })\prec \sec h ( \varsigma ) \). The geometry of functions of class \(\mathcal{S}_{E}^{\ast }\) is star-shaped, which is confined in the symmetric domain of a secant hyperbolic function. We find sharp coefficient results and sharp Hankel determinants of order two and three for functions in the class \(\mathcal{S}_{E}^{\ast }\). We also investigate the same sharp results for inverse coefficients.
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1 Introduction and preliminaries
Denote by \(\mathcal{A}\) a class of functions f that contains analytic functions in \(\mathbb{D}=\{\mathsf{\varsigma }\in \mathbb{C}:|\mathsf{\varsigma }|<1\}\) and having the Maclaurin series expansion of the form
A subclass of \(\mathcal{A}\) that contains univalent functions in \(\mathbb{D}\) is denoted by \(\mathcal{S}\). The classes of starlike and convex functions are, respectively, represented by \(\mathcal{S}^{\ast }\) and \(\mathcal{C}\) in \(\mathbb{D}\). These are analytically defined for the functions f by the relations \({Re} ( {\varsigma \mathsf{f}}^{\prime } ( {\varsigma } ) /\mathsf{f} ( {\varsigma } ) ) >0\) and \({Re} ( 1+ {\varsigma \mathsf{f}}^{\prime \prime } ( { \varsigma } ) /\mathsf{f}^{\prime } ( {\varsigma } ) ) >0\) in \(\mathbb{D}\), respectively. Let \(\mathcal{B}\) denote the family of functions w, which are analytic (holomorphic) in \(\mathbb{D}\) such that w maps zero onto zero and the disk to itself that is mathematically written as \(w(0)=0\) and \(\vert w \vert <1\) for \({\varsigma }\in \mathbb{D}\). The said function is called a Schwarz function. Considering functions f and g are both analytic (holomorphic) in \(\mathbb{D}\), then we write mathematically as \(f\prec g\) and read as f is subordinated to g if there exists a function w that is Schwarz in \(\mathbb{D}\) such that \(\mathsf{f}(\mathsf{\varsigma })=\mathsf{g} ( w(\varsigma) ) \) for \(\varsigma \in \mathbb{D}\). When g is univalent (one-to-one) and \(f ( 0 ) =\mathsf{g} ( 0 ) \), then \(\mathsf{f} ( \mathbb{D} ) \subset \mathsf{g} ( \mathbb{D} ) \).
Ma and Minda [12] introduced generalized subclasses of \(\mathcal{S}^{\ast }\) and \(\mathcal{C}\) denoted by \(\mathcal{S}^{\ast }(\Psi )\) and \(\mathcal{C}(\Psi )\), respectively, which are defined as follows:
and
The analytic and univalent function Ψ satisfies the conditions \(\Psi ( 0 ) =1\) and \({Re} \{ \Psi ^{\prime }(\varsigma ) \} >0\) in \(\mathbb{D}\) and \(\Psi (\mathbb{D})\) is convex.
Certain generating functions related with well-known numbers have attracted various authors. These functions are connected with starlike functions. For instance, Sokół [17, 18] used Fibonacci numbers to introduce a subclass that was strongly starlike. This concept was further utilized by Dziok et al [7, 8] to study various class of analytic functions. Deniz [6] studied a class of starlike functions related with Telephone numbers, also see [14]. The geometry of generating a function for Bell numbers was explored in [4, 9] and a subclass of starlike functions was introduced. Similar work for Bernoulli numbers was done by Raza et al. [16].
Recently, Bano et al. [2] introduced the class \(\mathcal{S}_{E}^{\ast }\) related with Euler numbers defined as
The constants \(\mathbf{E}_{m}\) are known as Eulers numbers. These numbers hold the relation \(\mathbf{E}_{2m+1}=0\), \(m=0,1,2,\ldots \) . The first few Euler numbers are given as \(\mathbf{E}_{0}=1\), \(\mathbf{E}_{2}=-1\), \(\mathbf{E}_{4}=5\), and \(\mathbf{E}_{6}=-61\). These numbers have very close association with Bernoulli, Genocchi, tangent, and Stirling numbers of two kinds. These are also related with the Euler polynomials and Riemann zeta function. Due to this relation, Euler numbers are very useful in combinatorics and number theory, see [1, 10, 13, 21] and references therein.
2 Coefficient bounds and Hankel determinants
The qth Hankel determinant for analytic functions was first given by Pommerenke [15] and is defined by
where \(m\geq 1\) and \(q\geq 1\). We see that
and
The Hankel determinant \(H_{2,1} ( \mathsf{f} ) \) is known as a Fekete–Szegö functional that is generalized as a \(\vert a_{3}-\mu a_{2}^{2} \vert \) for \(\mu \in \mathbb{C} \). For some recent work on it, we refer the readers to [3, 19, 20].
The class \(\mathcal{P}\) has great importance in function theory as most of the functions in various classes can be related with this class. The main tool in this study is to convert the coefficients for functions in class \(\mathcal{S}_{E}^{\ast }\) to the class \(\mathcal{P}\). We define it with some of its useful results for our investigation as follows.
Let \(\mathcal{P}\) represent the class of analytic functions p of the form
Lemma 2.1
[11] Let \(\mathsf{p}\in \mathcal{P}\) and be of the form (2.4). Then,
for some ρ and η such that \(\vert \rho \vert \leq 1\) and \(\vert \eta \vert \leq 1\).
Lemma 2.2
[12] Let \(\mathsf{p}\in \mathcal{P}\) be given by (2.4). Then,
for \(0<\upsilon \leq \frac{1}{2}\).
Lemma 2.3
[5]. Let \(\overline{\mathbb{D}}:=\{\rho \in \mathbb{C}:|\rho |\leq 1\}\), and for J, K, \(L\in \mathbb{R} \), let
If \(JL\geq 0\), then
If \(JL<0\), then
where
Theorem 2.4
Let \(\mathsf{f}\in \mathcal{S}_{E}^{\ast }\) and be of the form (1.1). Then,
These bounds are sharp.
Proof
Let \(\mathsf{f}\in \mathcal{S}_{E}^{\ast }\). Then, by using the definition of subordination, we can write
where \(w\in \mathcal{B}\). Let \(p\in \mathcal{P}\). Then, we can write
Now, using (2.6) and (2.7) and comparing the coefficients, we obtain
and
(i) The bound for \(a_{3}\) follows by applying well-known coefficient bounds for class \(\mathcal{P}\). This bound is sharp for the function
(ii) For the bound on \(a_{4}\), we write
Using Lemma 2.2, we obtain
Since the class \(\mathcal{S}_{E}^{\ast }\) is rotationally invariant, we take \(q:=q_{1}\), so that \(0\leq q\leq 2\). Hence,
Let
Then, it is easy to see that \(\Psi ( q ) \leq \frac{2}{27}\sqrt{3}\). For the sharpness, consider \(t_{0}=\frac{2}{3}\sqrt{3}\) and \(q_{0}(\mathsf{\varsigma })= \frac{1-\varsigma ^{2}}{1-t_{0}\varsigma + \varsigma ^{2}}\) such that
It is easy to see that \(w_{0}(0)=0\) and \(|w_{0}(\mathsf{\varsigma })|<1\). Hence, the function
is in class \(\mathcal{S}_{E}^{\ast }\). Here, \(a_{4}=\frac{2}{27}\sqrt{3}\).
(iii) For the bound on \(a_{5}\), using the inequalities from Lemma 2.1, we can write
where \(\rho , \eta \in \overline{\mathbb{D}}\), and
Letting \(t:=|\rho |\), and \(u:=|\eta |\), we have
where
with
Now, we have to find the maximum value in \(S:[0,2]\times [ 0,1]\times [ 0,1]\). We determine the maximum value of the function \(J(q,t,u)\) in the interior of S on the edges and vertices of S.
I. First, we consider the interior points of S. Differentiating (2.12) with respect to u and after some simplifications, we obtain
Since equation (2.13) is independent of u, we have no maximum value in \((0,2)\times (0,1)\times (0,1)\).
II. Next, we discuss the optimum value in the interior of six faces of S.
On the face \(q=0\), \(J(q,t,u)\) reduces to
The function \(\kappa _{1}\) has no maxima in \((0,1)\times (0,1)\) since
On \(q=2\), \(J(q,t,u)\) takes the form
On \(t=0\), \(J(q,t,u)\) can be written as
This implies function \(\kappa _{2}\) has maxima \((0,2)\times (0,1)\), since
On \(t=1\), \(J(q,t,u)\) reduces to
There is no critical point for the function \(\kappa _{3}\) in \((0,2)\times (0,1)\), since
On \(u=0\), \(J(q,t,u)\) reduces to
Therefore,
and
By implementing numerical methods, we see that the system \(\frac{\partial \kappa _{4}}{\partial q}=0\) and \(\frac{\partial \kappa _{4}}{\partial t}=0\) has no solution in \((0,2)\times (0,1)\).
On \(u=1\), the function \(J(q,t,u)\) takes the form
Similarly, on the face \(u=0\), the system of equations \(\frac{\partial \kappa _{5}}{\partial q}=0\) and \(\frac{\partial \kappa _{5}}{\partial t}=0\) has no solution in \((0,2)\times (0,1)\).
III. Now, we investigate the maximum of \(J(q,t,u)\) on the edges of S. From (2.16), we obtain \(J(q,0,0)=:s_{1}(q)=q^{4}/192\). Since \(s_{1}^{\prime }(q)>0\) for \([0,2]\). Therefore, \(s_{1}\) is increasing in \([0,2]\) and hence a maximum is achieved at \(q=2\). Therefore,
Also, from (2.16) at \(u=1\), we write \(J(q,0,1):=s_{2}(q)=(q^{4}-6q^{3}+24q)/192\), since \(s_{2}^{\prime }=0\) for \(q=1.388684545\) in \([0,2]\). Thus,
Putting \(q=0\) in (2.16), we have
Since (2.17) is independent of t, we obtain \(J(q,1,1)=J(q,1,0)=s_{3}(q):=\frac{-q^{4}+48}{384}\). Since \(s_{3}^{\prime }(q)<0\) for \([0,2]\), \(s_{3}(q)\) is decreasing in \([0,2]\) and hence a maximum is achieved at \(q=0\). Thus,
Putting \(q=0\) in (2.17), we obtain
Equation (2.15) is independent of variables u and t. Thus,
As equation (2.14) is independent of the variable u, we have \(J(0,t,0)=J(0,t,1)=s_{4}(t):=t^{2}/8\). Since \(s_{4}^{\prime }(t)>0\) for \([0,1] \), \(s_{4}(t)\) is increasing in \([0,1]\) and hence a maximum is achieved at \(t=1\). Thus,
The bound for \(\vert a_{5} \vert \) is sharp for the function
This completes the result. □
Theorem 2.5
Let \(\mathsf{f}\in \mathcal{S}_{E}^{\ast }\) be given by (1.1). Then,
The result is sharp.
Proof
Since
now using (2.8), we obtain
Now, using well-known coefficient bounds for class \(\mathcal{P}\), we obtain the required result. The function \(\mathsf{f}_{1}\) given in (2.10) that gives a sharp result. □
Theorem 2.6
Let \(\mathsf{f}\in \mathcal{S}_{E}^{\ast }\) be given by (1.1). Then,
This inequality is sharp.
Proof
Using (2.8)–(2.9) in (2.2), we obtain
where
As we see that the functional \(H_{3,1} ( \mathsf{f} ) \) and the class \(\mathcal{S}_{E}^{\ast }\) are rotationally invariant, we may take \(q:=q_{1}\), such that \(q\in [ 0,2 ] \). Then, by using Lemma 2.1 and after some computations we may write
where ρ and η satisfy the relation \(|\rho |\leq 1\), \(|\eta |\leq 1\).
First, we consider the case when \(q=2\). Then, \(\vert \psi \vert =192\), therefore from (2.19) we obtain \(\vert H_{3,1} ( \mathsf{f} ) \vert = \frac{1}{192}\). Next, we assume that \(q=0\), then \(\vert \psi \vert =0\). Now, suppose that \(q\in ( 0,2 ) \) and using the well-known triangle inequality, we obtain
where
with \(J=\frac {-q^{3}}{24(4-q^{2})}\), \(K=0\) and \(L=\frac {- ( 184-10q^{2} ) }{72q}\), then clearly
Also,
so that \(K>2(1- \vert L \vert )\), and applying Lemma 2.3, we can have
where
Clearly, \(g^{\prime } ( q ) =0\) has roots at \(q_{1}=0\) and \(q_{2}=\frac{2}{39}\sqrt{2184-39\sqrt{1342}}\). Also, \(g^{\prime \prime } ( q_{2} ) \approx -2328.5\), and so
which from (2.19) gives the required result.
For the sharpness, consider \(t_{0}=\frac{2}{39}\sqrt{2184-39\sqrt{1342}}\) and \(q_{0}(\mathsf{\varsigma })= \frac{1+t_{0}\varsigma + {\varsigma }^{2}}{1-\varsigma ^{2}}\) such that
It is easy to see that \(w_{0}(0)=0\) and \(|w_{0}(\mathsf{\varsigma })|<1\). Hence, the function
is in class \(\mathcal{S}_{E}^{\ast }\). Now, after simplifications, we see that
Hence, the required result is obtained. □
Theorem 2.7
Let \(\mathsf{f}\in \mathcal{S}_{E}^{\ast }\) and be of the form (1.1). Then,
The result is sharp.
Proof
Using (2.8)–(2.9) in (2.3), we obtain
where
As we see that the functional \(H_{2,3} ( \mathsf{f} ) \) and the class \(\mathcal{S}_{E}^{\ast }\) are rotationally invariant, we may take \(q:=q_{1}\), such that \(q\in [ 0,2 ] \). Then, by using Lemma 2.1 and after some computations we may write
where ρ and η satisfy the relation \(|\rho |\leq 1\), \(|\eta |\leq 1\).
First, we consider the case when \(q=0\). Then, \(\vert \psi \vert =0\), therefore from (2.21) we obtain \(\vert H_{2,3} ( \mathsf{f} ) \vert =0\). Next, we assume that \(q=2\), then \(\vert \psi \vert =384\) and \(\vert H_{2,3} ( \mathsf{f} ) \vert = \frac{1}{48}\). Now, suppose that \(q\in ( 0,2 ) \) and using the well-known triangle inequality, we obtain
where
with \(J=\frac {-q^{3}}{6(4-q^{2})}\), \(K=0\) and \(L=\frac {- ( 92-5q^{2} ) }{36q}\), then clearly
Now,
Thus, we conclude that \(\vert K \vert >2 ( 1- \vert L \vert ) \) and applying Lemma 2.3, we can have
where
Clearly, \(g^{\prime } ( q ) =2q ( 4-q^{2} ) (92-33q^{2})>0\) for \(q\in ( 0,2 ) \), therefore
which from (2.21) gives the result. It is sharp for \(\mathsf{f}_{1}\) given in (2.10). Hence, the required result is obtained. □
3 Inverse coefficients
For a function f that is univalent, its inverse function \(f^{-1}\) is defined on some disc \(\vert w \vert \leq 1/4\leq r_{0}(f)\), having a series expansion of the form
Next, note that since \(f ( f^{-1}(w) ) =w\), using (3.1) it follows that
Now, using (2.8) in the above relations, we can write
Theorem 3.1
Let \(\mathsf{f}\in \mathcal{S}_{E}^{\ast }\) and be of the form (1.1). Then,
Proof
(i) The bound for \(B_{3}\) follows by applying well-known coefficient bounds for class \(\mathcal{P}\). Sharpness can be found by \(\mathsf{f}_{1}\) given in (2.10). Here, \(a_{2}=0\), \(a_{3}=\frac{-1}{4}\). Now, \(B_{3}=2a_{2}^{2}-a_{3}\) implies \(B_{3}=\frac{1}{4}\).
(ii) For the bound on \(B_{4}\), we write
Using Lemma 2.2, we obtain
Since the class \(\mathcal{S}_{E}^{\ast }\) is rotationally invariant, we take \(q:=q_{1}\), so that \(0\leq q\leq 2\). Hence,
Let
Then, \(f ( q ) \leq \frac{2}{27}\sqrt{3}\). For the sharpness, consider the function \(\mathsf{f}_{0}\) given by (2.11). Here, \(a_{2}=0\), \(a_{3}=-\frac{1}{12}\), \(a_{4}=\frac{2}{27}\sqrt{3}\). Since \(B_{4}=5a_{2}a_{3}-5a_{2}^{3}-a_{4}\), \(B_{4}=-\frac{2}{27}\sqrt{3} \).
(iii) Using Lemma 2.1 in (3.7), we obtain
where
As we see that the functional \(B_{5}\) and the class \(\mathcal{S}_{E}^{\ast }\) are rotationally invariant, we may take \(q:=q_{1}\), such that \(q\in [ 0,2 ] \). Then, by using Lemma 2.1 and after some computations we may write
where ρ and η satisfy the relation \(|\rho |\leq 1\), \(|\eta |\leq 1\).
First, we consider the case when \(q=0\). Then, \(\vert \psi \vert =96 \vert \rho ^{2}\vert \), and from (3.8) we obtain \(\vert B_{5} \vert \leq 1/8\). Next, we assume that \(q=2 \), then \(\vert \psi \vert =80\) and \(\vert B_{5} \vert =\frac{5}{48}\). Now, suppose that \(q\in ( 0,2 ) \) and using the well-known triangle inequality, we obtain
where
with \(J=\frac {5q^{3}}{24(4-q^{2})}\), \(K=0\) and \(L=\frac { ( 4-3q^{2} ) }{4q}\), then clearly
Now,
Here, we have two cases: \(4q ( 1- \vert L \vert ) \leq 0\) for \(q\in ( 0,\frac{2}{3} ] \) and \(4q ( 1- \vert L \vert ) >0\) for \(( \frac{2}{3},\frac{2}{\sqrt{3}} ] \). Thus, we conclude that \(\vert K \vert \geq 2 ( 1- \vert L \vert ) \) for \(q\in ( 0,\frac{2}{3} ] \) and applying Lemma 2.3, we can have
where
Clearly, \(g_{1}^{\prime } ( q ) =4q(23q^{2}-48)<0\) for \(( 0,\frac{2}{3} ] \), therefore,
which from (3.8) gives the required result.
For the case \(( \frac{2}{3},\frac{2}{\sqrt{3}} ] \), we see that \(4q ( 1- \vert L \vert ) >0\) and \(\vert K \vert >2 ( 1- \vert L \vert ) \) and applying Lemma 2.3, we can have
where
We see that \(g_{2}^{\prime } ( q ) =20q^{3}-72q^{2}+96>0\), therefore,
Lastly, we consider the case when \(q\in ( \frac{2}{\sqrt{3}},2 ) \). Then, \(JL<0\). Now,
Also,
This shows that \(K^{2}<\min \{ 4(1+|L|)^{2},-4JL ( L^{-2}-1 ) \} \). Now, applying Lemma 2.3, we can have
where \(g_{2}\) is given by (3.9). The result is sharp for the function \(\mathsf{f}_{2}\) given in (2.18). We see that \(a_{2}=0\), \(a_{3}=0\), \(a_{4}=0\), \(a_{5}=-\frac{1}{8}\). Now, using (3.5), we have \(\vert B_{5} \vert =\frac{1}{8}\). Hence, the required result is obtained. □
Theorem 3.2
Let \(\mathsf{f}\in \mathcal{S}_{E}^{\ast }\) be given by (1.1). Then,
This result is sharp.
Proof
Since
Now, using (3.6), we obtain
Now, using well-known coefficient bounds for class \(\mathcal{P}\), we obtain the required result. This result is sharp for the function \(\mathsf{f}_{1}\) given in (2.10). □
Theorem 3.3
Let \(\mathsf{f}\in \mathcal{S}_{E}^{\ast }\) be given by (1.1). Then,
Proof
Using (3.6)–(3.7) in (2.2), we obtain
where
As we see that the functional \(H_{3,1} ( \mathsf{f}^{-1} ) \) and the class \(\mathcal{S}_{E}^{\ast }\) are rotationally invariant, we may take \(q:=q_{1}\), such that \(q\in [ 0,2 ] \). Then, by using Lemma 2.1 and after some computations we may write
where ρ and η satisfy the relation \(|\rho |\leq 1\), \(|\eta |\leq 1\).
First, we consider the case when \(q=0\). Then, \(\vert \psi \vert =0\). Next, we assume that \(q=2\), then \(\vert \psi \vert =96\), therefore, from (3.10) we obtain \(\vert H_{3,1} ( \mathsf{f}^{-1} ) \vert = \frac{1}{96}\). Now, suppose that \(q\in ( 0,2 ) \) and using the well-known triangle inequality, we obtain
where
with \(J=\frac {q^{3}}{12(4-q^{2})}\), \(K=0\) and \(L=\frac {- ( 92-5q^{2} ) }{36q}\), then clearly,
Now,
and
This shows that \(-4JL ( L^{-2}-1 ) \leq K^{2}\wedge |K|<2(1-|L|)\) and \(K^{2}<\min \{ 4(1+|L|)^{2},-4JL ( L^{-2}-1 ) \} \) do not hold. We also see that
and
We conclude that \(|L|( \vert K \vert +4|J|)\nleq \vert JK \vert \) and \(\vert JK \vert \nleq |L|( \vert K \vert -4|J|)\). Applying Lemma 2.3, we can have
where
Clearly, \(g_{3}^{\prime } ( q ) =0\) has roots at \(q_{1}=0\) and \(q_{2}=\frac{1}{3}\sqrt{42-3\sqrt{58}}\). Also, \(g_{3}^{\prime \prime } ( q_{2} ) \approx -518.6\), and so
which from (3.10) gives the required result. □
Theorem 3.4
Let \(\mathsf{f}\in \mathcal{S}_{E}^{\ast }\) and be of the form (1.1). Then,
This inequality is sharp.
Proof
Using (3.6)–(3.7) in (2.3), we obtain
where
As we see that the functional \(H_{2,3} ( \mathsf{f}^{-1} ) \) and the class \(\mathcal{S}_{E}^{\ast }\) is rotationally invariant, we may take \(q:=q_{1}\), such that \(q\in [ 0,2 ] \). Then, by using Lemma 2.1 and after some computations we may write
where ρ and η satisfy the relation \(|\rho |\leq 1\), \(|\eta |\leq 1\).
First, we consider the case when \(q=0\). Then, \(\vert \psi \vert =0\). Next, we assume that \(q=2\), then \(\vert \psi \vert =960\), therefore, from (3.11) we obtain \(\vert H_{2,3} ( \mathsf{f}^{-1} ) \vert = \frac{5}{192}\). Now, suppose that \(q\in ( 0,2 ) \) and using the well-known triangle inequality, we obtain
where
with \(J=\frac {15q^{3}}{72(4-q^{2})}\), \(K=0\) and \(L=\frac {- ( 92-5q^{2} ) }{36q}\), then clearly
Now,
and
Thus, we see that \(\vert K \vert \nleq 2 ( 1- \vert L \vert ) \). We also note that \(K^{2}\nless \min \{ 4(1+|L|)^{2},-4JL ( L^{-2}-1 ) \} \). Also,
and
Thus, \(|L|( \vert K \vert +4|J|)\nleq \vert JK \vert \) and \(\vert JK \vert \nleq |L|( \vert K \vert -4|J|)\). Now, applying Lemma 2.3, we can have
where
Clearly, \(g_{4}^{\prime } ( q ) =2q75q^{4}-448q^{2}+736)>0\) for \(q\in ( 0,2 ) \), therefore
which from (3.11) gives the result. The result is sharp for \(\mathsf{f}_{1}\) given in (2.10). Here, \(a_{2}=0\), \(\ a_{3}=-\frac{1}{4}\), \(a_{4}=0\), and \(a_{5}=\frac{1}{12}\). Now, using (3.2)–(3.5), we have \(B_{2}=0\), \(B_{3}=\frac{1}{4}\), \(B_{4}=0\), and \(B_{5}=\frac{5}{48}\). Lastly, by using (3.11), we have the required result. □
4 Conclusion
We have found sharp coefficient results and sharp Hankel determinants of orders two and three for starlike functions related with Euler numbers. We have obtained the fifth coefficient by using the results of Libera and Zlotkiewicz [11]. This approach can be utilized for the investigations of fifth coefficients bound for various classes of univalent functions. Furthermore, we have obtained the bounds on \(H_{1,3} ( \mathsf{f} ) \) and \(H_{2,3} ( \mathsf{f} ) \) by using the results of Choi et al [5]. This approach is useful for obtaining these kind of results for functions whose second coefficients are zero.
Availability of data and materials
No data or material were used in this study.
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The research of first author was supported by Higher Education Commission (HEC) of Pakistan under NRPU with Grant No. 20-16367/NRPU/R&D/HEC/2021-2020. The research of fourth author was supported by the researchers Supporting Project Number (RSP2024R401), King Saud University, Riyadh, Saudi Arabia.
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M. Raza, K. Bano and Q. **n contributed in conceptualization, methodology and investigation. S. N. Malik and F. Tchier did formal analysis, wrote the main manuscript and check for validation of results. All authors read and approved the final manuscript.
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Raza, M., Bano, K., **n, Q. et al. Sharp coefficient inequalities of starlike functions connected with secant hyperbolic function. J Inequal Appl 2024, 56 (2024). https://doi.org/10.1186/s13660-024-03134-0
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DOI: https://doi.org/10.1186/s13660-024-03134-0