Symmetry, Hopf bifurcation, and offset boosting in a novel chameleon system

Jie Liu; Bo Sang; Lihua Fan; Chun Wang; Xueqing Liu; Ning Wang; Irfan Ahmad; Jie Liu; Bo Sang; Lihua Fan; Chun Wang; Xueqing Liu; Ning Wang; Irfan Ahmad

doi:10.3934/math.2025225

AIMS Mathematics

2025, Volume 10, Issue 3: 4915-4937. doi: 10.3934/math.2025225

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Research article Special Issues

Symmetry, Hopf bifurcation, and offset boosting in a novel chameleon system

1.
School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, Shandong, China
2.
School of Microelectronics and Control Engineering, Changzhou University, Changzhou 213159, Jiangsu, China
3.
Faculty of Science and Engineering, Maynooth International Engineering College, Maynooth University, Maynooth, Co. Kildare W23, Ireland

Received: 23 December 2024 Revised: 02 February 2025 Accepted: 17 February 2025 Published: 06 March 2025
MSC : 37D45, 37G35

Chameleon systems are dynamical systems that exhibit either self-excited or hidden oscillations depending on the parameter values. This paper presents a comprehensive investigation of a quadratic chameleon system, including an analysis of its symmetry, dissipation, local stability, Hopf bifurcation, and various chaotic dynamics as the control parameters $(\mu, a, c)$ vary. Here, $\mu$ serves as the dissipation parameter in the $y$ direction. Bifurcation analysis for four scenarios with $\mu = 0$ was performed, revealing the emergence of various dynamical phenomena under different parameter settings. Offset boosting means introducing a constant into one of the state variables of the system for boosting the variable to a different level. Additionally, hidden chaotic bistability with offset boosting was exhibited by varying $\mu$ . The parameter $\mu$ serves as both the Hopf bifurcation parameter and the offset boosting parameter, while the other parameters $(a, c)$ also play critical roles as control parameters, resulting in period-doubling routes to self-excited or hidden chaotic attractors. These findings enrich our understanding of nonlinear dynamics in quadratic chameleon systems.

Keywords:

Citation: Jie Liu, Bo Sang, Lihua Fan, Chun Wang, Xueqing Liu, Ning Wang, Irfan Ahmad. Symmetry, Hopf bifurcation, and offset boosting in a novel chameleon system[J]. AIMS Mathematics, 2025, 10(3): 4915-4937. doi: 10.3934/math.2025225

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Abstract

1. Introduction

First, some fundamental ideas must be explained in order to fully comprehend the basic concepts utilized throughout the attainment of our major findings. For this, let $\mathcal{A}\$ denote the family of all holomorphic (regular) functions $f$ defined in the open unit disc $\mathbb{D} = \{z:z\in \mathbb{C}$ and $\left \vert z\right \vert < 1\}, \$ whose Taylor series representation is given as follows:

$\begin{equation} f\left( z\right) = z+\sum \limits_{j = 2}^{\infty }\mathcal{\xi }_{j}z^{j}, { \ \ \ \ \ \ \ \ \ }z\in \mathbb{D}\text{.} \end{equation}$

(1.1)

A subfamily containing all of the univalent functions of the family $\mathcal{A}$ in $\mathbb{D}$ is denoted by $\mathcal{S}$ . A useful technique for examining different inclusion and radii concerns for families of holomorphic functions is known as subordination. A function $f$ is subordinate to $g$ in $\mathbb{D}$ written as $f\prec g,$ if there exists a Schwarz function $\omega$ , which is regular in $\mathbb{D}$ and $\omega (0) = 0$ with $\left \vert \omega (z)\right \vert < 1$ , such that $f\left(z\right) = g(\omega \left(z\right)).\$ In addition, if the function $g$ is univalent in $\mathbb{D}$ then we have

$\begin{equation*} f\left( 0\right) = g\left( 0\right) \text{ and }f\left( \mathbb{D}\right) \subset g\left( \mathbb{D}\right) . \end{equation*}$

The known subclasses of S are represented by the letters $\mathcal{S}^{\ast }, \ \mathcal{C}, \ \mathcal{K}$ and $\mathcal{R}$ . These subclasses include starlike, convex, close to convex, and functions with bounded turnings. Two regular functions, $f$ and $\varsigma$ , are convolved in $\mathbb{D}$ , the series representation of $f$ is provided in (1.1) and $\varsigma = z+\sum \limits_{j = 2}^{\infty }b_{j}z^{j}$ is defined as follows:

$\begin{equation} \left( f\ast \varsigma \right) \left( z\right) = z+\sum \limits_{j = 2}^{\infty }\mathcal{\xi }_{j}b_{j}z^{j},\ \ \ \ \ \ \ \ \ \ \ \ \ z\in \mathbb{D}\text{ .} \end{equation}$

(1.2)

The integrated families of starlike and convex functions were developed in 1985 by Padmanabhan and Parvatham ^[1] who utilized the theory of convolution along with the function $\frac{z}{\left(1-z\right) ^{a}}$ , where $a\in \mathbb{R}$ . By taking a regular function $\phi (z)$ with $\phi (0) = 1,$ and $h\left(z\right) \in \mathcal{A},$ , Shanmugam ^[2] expanded on the concept presented in ^[1] and introduced the generic form of the function class $\mathcal{S} _{h}^{\ast }(\phi)$ as follows:

$\begin{equation} \mathcal{S}_{h}^{\ast }(\phi ) = \left \{ f\in \mathcal{A}:\frac{z(f\ast h)^{\prime }}{(f\ast h)}\prec \phi (z),{ \ \ \ \ }z\in \mathbb{D}\right \} \text{.} \end{equation}$

(1.3)

By taking $h\left(z\right) = \frac{z}{1-z}$ or $\frac{z}{\left(1-z\right) ^{2}}$ , we derive the famous classes $\mathcal{S}^{\ast }(\phi)$ and $\mathcal{C}(\phi)$ of Ma and Minda type starlike and convex functions defined in ^[3]. Further, by choosing $\phi (z) = \frac{1+z}{1-z}$ these classes can be reduced to $\mathcal{S}^{\ast }$ and $\mathcal{C}$ .

By limiting $\phi (z)$ in the generic form of $\mathcal{S}^{\ast }(\phi)$ and $\mathcal{C}(\phi)$ , numerous scholars have defined and investigated a variety of intriguing subclasses of analytic and univalent functions in the recent past. Here, we highlight few of them.

Let $\phi (z) = \frac{1+Fz}{1+Gz},$ $-1\leq G < F\leq 1$ . Then $\mathcal{S} ^{\ast }[F, G] = \mathcal{S}^{\ast }\left(\frac{1+Fz}{1+Gz}\right)$ is the class of Janowski starlike functions; see ^[4]. For $\phi (z) = \cos z,$ the class $\mathcal{S}_{\cos z}^{\ast }$ was studied by Bano and Raza ^[5], while for $\phi (z) = \cosh z,$ the function class $\mathcal{S}_{\cosh z}^{\ast }$ was introduced and studied by Alotaibi et al. ^[6]. For $\phi (z) = e^{z},$ the class $\mathcal{S}_{e}^{\ast }$ was defined and studied by Mendiratta et al. ^[7]. For $\phi (z) = 1+\sin z,$ the class $\mathcal{S} ^{\ast }\left(\phi \right)$ reduces to $\mathcal{S}_{\sin }^{\ast }$ , as presented and examined by Cho et al. ^[8]. For $\phi (z) = 1+z-\frac{ 1}{3}z^{3},$ we get the family $\mathcal{S}_{nep}^{\ast }$ that was examined by Wani and Swaminathan ^[9]. For $\phi (z) = 1+\sinh ^{-1}\left(z\right)$ , the family $\mathcal{S}^{\ast }\left(\phi \right)$ was established and studied by Kumar and Arora ^[10] for more details see ^[11]. For $\phi (z) = \frac{2}{1+e^{-z}}\mathbf{, }$ the class $\mathcal{S}^{\ast }\left(\phi \right)$ reduces to $\mathcal{S}_{sig}^{\ast }$ ; see ^[12] and ^[13,14]. For $\phi (z) = \sqrt{1+z},$ we obtain the family $\mathcal{S}^{\ast }\left(\sqrt{1+z}\right) = \mathcal{S}_{L}^{\ast }$ as studied by Sokol and Stankiewicz ^[15]. The class $\mathcal{S}_{\tanh z}^{\ast } = \mathcal{S}^{\ast }\left(\phi (z)\right),$ for $\phi (z) = 1+\tanh z,$ was established by Ullah et al. ^[16] see also ^[17].

For the given parameters $n,$ $r\in \mathbb{N}$ , the rth Hankel determinant $\mathcal{H}_{r, \text{ }n}$ was defined in ^[18] as follows:

$\begin{equation*} \mathcal{H}_{r,\text{ }n}(f) = \left \vert \begin{array}{cccccc} \mathcal{\xi }_{n}\text{ } & \mathcal{\xi }_{n+1} & . & . & . & \mathcal{\xi }_{n+r-1} \\ \mathcal{\xi }_{n+1} & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ \mathcal{\xi }_{n+r-1} & . & . & . & . & \mathcal{\xi }_{n+2\left( r-1\right) } \end{array} \right \vert \text{.} \end{equation*}$

For the given values of $n$ , $r$ and $\mathcal{\xi }_{1} = 1$ the second and third Hankel determinants are defined as follows:

$\begin{equation} \mathcal{H}_{2,1}(f) = \left \vert \begin{array}{cc} 1 & \mathcal{\xi }_{2} \\ \mathcal{\xi }_{2} & \mathcal{\xi }_{3} \end{array} \right \vert = \mathcal{\xi }_{3}-\mathcal{\xi }_{2}^{2}, \mathcal{H} _{2,2}(f) = \left \vert \begin{array}{cc} \mathcal{\xi }_{2} & \mathcal{\xi }_{3} \\ \mathcal{\xi }_{3} & \mathcal{\xi }_{4} \end{array} \right \vert = \mathcal{\xi }_{2}\mathcal{\xi }_{4}-\mathcal{\xi }_{3}^{2} \text{.} \end{equation}$

(1.4)

This technique has proven to be usful when examining power series with integral coefficients and singularities by taking the Hankel determinant into account; see ^[19]. Bounds of $\mathcal{H}_{r, \text{ }n}(f)$ for several kinds of univalent functions have been examined recently. For a detailed study on the Hankel determinant, we refer the reader to ^[20,21,22].

Scholars in the field of geometric function theory of complex analysis are still motivated by the study of coefficient problems, which include the Fekete–Szegö and Hankel determinant problems. To encourage and motivate interested readers, we have included numerous recent works (see, e.g., ^[20,21,22]) on a variety of the Fekete–Szegö and Hankel determinant problems, along with ongoing applications of the $q$ -calculus in the study of other analytic or meromorphic univalent and multivalent function classes. Motivated and inspired by the work mentioned above, in this article, we first define a new subclass of holomorphic convex functions that are related to the tangent functions. We then derive geometric properties like the necessary and sufficient conditions, radius of convexity, growth, and distortion estimates for our defined function class. Furthermore, the sharp coefficient bounds, sharp Fekete-Szegö inequality, sharp 2nd order Hankel determinant, and Krushkal inequalities are given. Moreover, we calculate the sharp coefficient bounds, sharp Fekete-Szegö inequality, and sharp second-order Hankel determinant for the functions whose coefficients are logarithmic.

We present the following subfamily of holomorphic functions.

Definition 1.1. Let $f\in \mathcal{A}, \$ be given in (1.1). Then $f\in \mathcal{C}_{\tan }$ if the following condition holds true:

$\begin{equation} f\mathcal{\in C}_{\tan }\iff f\in \mathcal{A}\text{ and }\frac{\left( zf^{\prime }(z)\right) ^{\prime }}{f^{\prime }(z)}\prec 1+\frac{\tan z}{2}, { \ \ \ \ \ }z\in \mathbb{D}\text{.} \end{equation}$

(1.5)

Geometrically, the family $\mathcal{C}_{\tan }$ comprises all of the functions $f$ that lie within the image domain of $1+\frac{\tan z}{2},$ for a specified radius.

2. Set of lemmas

We utilize the following lemmas in our major conclusion.

Let $\mathcal{P}$ stand for the family of all holomorphic functions $p$ that have a positive real portion and are represented by the following series:

$\begin{equation} p\left( \kappa \right) = 1+\sum\limits_{j = 1}^{\infty }c_{j}z^{j},\text{ }\kappa \in \Omega . \end{equation}$

(2.1)

Lemma 2.1. If $p\in {\mathcal{P}}$ , then the following estimations hold:

$\begin{eqnarray} \left \vert c_{j}\right \vert &\leq &2,\mathit{\text{}}j\geq 1, \end{eqnarray}$

(2.2)

$\begin{eqnarray} \left \vert c_{j+n}- {\mu} c_{j}c_{n}\right \vert & < &2,\ 0 < {\mu} \leq 1, \end{eqnarray}$

(2.3)

and for $\eta \in \mathbb{C}$ , we have

$\begin{equation} \left \vert c_{2}-\eta c_{1}^{2}\right \vert < 2\max \left \{ 1,\left \vert 2\eta -1\right \vert \right \} \text{.} \end{equation}$

(2.4)

Regarding the inequalities (2.2)–(2.4) are detailed in ^[23].

Lemma 2.2. ^[24] If $p\in {\mathcal{P}}$ and it has the form (2.1), then

$\begin{equation} |\alpha _{1}c_{1}^{3}-\alpha _{2}c_{1}c_{2}+\alpha _{3}c_{3}|\leq 2|\alpha _{1}|+2|\alpha _{2}-2\alpha _{1}|+2|\alpha _{1}-\alpha _{2}+\alpha _{3}|, \end{equation}$

(2.5)

where $\alpha _{1}, \alpha _{2}$ and $\alpha _{3}$ are real numbers.

Lemma 2.3. ^[25] Let $\chi _{1}, \sigma _{1}, \psi _{1}$ and $\varrho _{1}$ satisfy the inequalities for $\chi _{1}, \varrho _{1}\in \left(0, 1\right)$ and

$\begin{align*} & 8\varrho _{1}\left( 1-\varrho _{1}\right) \left[ \left( \chi _{1}\sigma _{1}-2\psi _{1}\right) ^{2}+\left( \chi _{1}\left( \varrho _{1}+\chi _{1}\right) -\sigma _{1}\right) ^{2}\right] \\ & +\chi _{1}\left( 1-\chi _{1}\right) \left( \sigma _{1}-2\varrho _{1}\chi _{1}\right) ^{2} \\ & \leq 4\chi _{1}^{2}\left( 1-\chi _{1}\right) ^{2}\varrho _{1}\left( 1-\varrho _{1}\right) . \end{align*}$

If $h\in \mathcal{P}$ and is of the form (2.1), then

$\begin{equation*} \left \vert \psi _{1}c_{1}^{4}+\varrho _{1}c_{2}^{2}+2\chi _{1}c_{1}c_{3}- \frac{3}{2}\sigma _{1}c_{1}^{2}c_{2}-c_{4}\right \vert \leq 2. \end{equation*}$

Lemma 2.4. Let $p\in {\mathcal{P}}$ and $x$ and $z$ belong to $\Lambda$ , then, we have

$\begin{eqnarray*} 2c_{2} & = &c_{1}^{2}+x\left( 4-c_{1}^{2}\right) , \\ 4c_{3} & = &2x\left( 4-c_{1}^{2}\right) c_{1}-x^{2}\left( 4-c_{1}^{2}\right) c_{1}+2z\left( 1-\left \vert x\right \vert ^{2}\right) \left( 4-c_{1}^{2}\right) +c_{1}^{3}, \end{eqnarray*}$

where $c_{2}$ and $c_{3}$ are discussed in ^[26] and ^[27] respectively.

The goal of the current study was to derive the necessary and sufficient conditions, radius of convexity, growth and distortion estimates, sharp coefficient bounds, sharp Fekete-Szegö inequality, Krushkal inequality, and logarithmic coefficient estimates for the subclass $\mathcal{C}_{\tan }$ of class $\mathcal{A}$ which is related to tangent functions.

3. Main results

Theorem 3.1. Let $f\in \mathcal{C}_{\tan }$ be as given in (1.1). Then

$\begin{equation} \frac{1}{z}\left[ f\left( z\right) \ast \left( \frac{z-Mz^{2}}{\left( 1-z\right) ^{3}}\right) \right] \neq 0, \end{equation}$

(3.1)

where

$\begin{equation} M = \frac{4+\tanh \left( e^{i\theta }\right) }{2}\mathit{\text{.}} \end{equation}$

(3.2)

Proof. Because $f\in \mathcal{C}_{\tan }$ is analytic in $\mathbb{D},$ $\frac{1}{z} f\left(z\right) \neq 0$ for all $z$ in $\mathbb{D}$ then, by using the definition of subordination and (1.5), we have

$\begin{equation} \frac{\left( zf^{\prime }(z)\right) ^{\prime }}{f^{\prime }(z)} = 1+\tanh \omega \left( z\right) \text{,} \end{equation}$

(3.3)

where $\omega \left(z\right)$ is the Schwarz function. Let $\omega \left(z\right) = e^{i\theta },$ $-\pi \leq \theta \leq \pi.$ Then (3.3) becomes

$\begin{equation*} \frac{zf^{\prime \prime }\left( z\right) }{f^{\prime }\left( z\right) }\neq \frac{\tan \left( e^{i\theta }\right) }{2}\text{,} \end{equation*}$

which implies that

$\begin{equation} z^{2}f^{\prime \prime }\left( z\right) -zf^{\prime }\left( z\right) \frac{ \tan \left( e^{i\theta }\right) }{2}\neq 0\text{.} \end{equation}$

(3.4)

It can be easily seen that

$\begin{equation} z^{2}f^{\prime \prime }\left( z\right) +zf^{\prime }\left( z\right) = f\left( z\right) \ast \frac{z\left( 1+z\right) }{\left( 1-z\right) ^{3}}\text{ and } zf^{\prime }\left( z\right) = f\left( z\right) \ast \frac{z}{\left( 1-z\right) ^{2}}\text{.} \end{equation}$

(3.5)

Using (3.5), and through some simple calculations (3.4) becomes

$\begin{equation} f\left( z\right) \ast \left( \frac{z-Mz^{2}}{\left( 1-z\right) ^{3}}\right) \neq 0\text{.} \end{equation}$

(3.6)

From (3.6), we will obtain (3.1) $,$ where $M$ is given in (3.2). □

Theorem 3.2. Let $f\in \mathcal{A}$ . Then $f\in \mathcal{C}_{\tan }$ if

$\begin{equation} \sum \limits_{n = 2}^{\infty }\left[ \frac{2n\left( 2+\tan \left( e^{i\theta }\right) \right) -4n^{2}}{\tan \left( e^{i\theta }\right) }\right] \mathcal{ \xi }_{n}z^{n-1}-1\neq 0. \end{equation}$

(3.7)

Proof. If $f\in \mathcal{C}_{\tan }$ then from Theorem 3.1, we have

$\begin{equation*} \frac{1}{z}\left[ f\left( z\right) \ast \left( \frac{z-Mz^{2}}{\left( 1-z\right) ^{3}}\right) \right] \neq 0, \end{equation*}$

where $M$ is given in (3.2). The above relation implies that

$\begin{equation*} \frac{1}{z}\left[ \left( f\left( z\right) \ast \frac{z}{\left( 1-z\right) ^{3}}\right) -\left( f\left( z\right) \ast \frac{Mz^{2}}{\left( 1-z\right) ^{3}}\right) \right] \neq 0\text{.} \end{equation*}$

Since $z^{2} = z\left(1+z\right) -z,$ so we have

$\begin{equation} \frac{1}{z}\left[ \left( f\left( z\right) \ast \frac{z}{\left( 1-z\right) ^{3}}\right) -M\left( f\left( z\right) \ast \frac{z\left( 1+z\right) }{ \left( 1-z\right) ^{3}}-f\left( z\right) \ast \frac{z}{\left( 1-z\right) ^{3} }\right) \right] \neq 0\text{.} \end{equation}$

(3.8)

Now applying (3.5) and some properties of convolution, (3.8), reduces to

$\begin{equation*} \frac{1}{z}\left[ \left( \frac{1}{2}z^{2}f^{\prime \prime }\left( z\right) +zf^{\prime }\left( z\right) \right) -M\left( z^{2}f^{\prime \prime }\left( z\right) \right) \right] \neq 0\text{.} \end{equation*}$

Using (1.1) and after some simplification, we obtain (3.7). □

Theorem 3.3. Let $f\in \mathcal{A}$ be as given in (1.1). Then $f\in \mathcal{C}_{\tan }$ if

$\begin{equation} \sum \limits_{n = 2}^{\infty }\left( \left \vert \frac{4n^{2}-2n\left( 2+\tan \left( e^{i\theta }\right) \right) }{\tan \left( e^{i\theta }\right) }\right \vert \right) \left \vert \mathcal{\xi }_{n}\right \vert < 1\mathit{\text{.}} \end{equation}$

(3.9)

Proof. To demonstrate the necessary outcome, we employ relation (3.7) as follows:

$\begin{eqnarray} &&\left \vert 1-\sum \limits_{n = 2}^{\infty }\frac{4n^{2}-2n\left( 2+\tan \left( e^{i\theta }\right) \right) }{\tan \left( e^{i\theta }\right) } \mathcal{\xi }_{n}z^{n-1}\right \vert \\ & > &1-\sum \limits_{n = 2}^{\infty }\left \vert \frac{4n^{2}-2n\left( 2+\tan \left( e^{i\theta }\right) \right) }{\tan \left( e^{i\theta }\right) }\right \vert \left \vert \mathcal{\xi }_{n}\right \vert \left \vert z\right \vert ^{n-1}. \end{eqnarray}$

(3.10)

From (3.9), we have

$\begin{equation} 1-\sum \limits_{n = 2}^{\infty }\left \vert \frac{4n^{2}-2n\left( 2+\tan \left( e^{i\theta }\right) \right) }{\tan \left( e^{i\theta }\right) }\right \vert \left \vert \mathcal{\xi }_{n}\right \vert > 0\text{.} \end{equation}$

(3.11)

From (3.10) and (3.11), we obtain the intended outcome by applying Theorem 3.2. □

Theorem 3.4. Let $f\in \mathcal{C}_{\tan }.$ Then $f$ is convex and of order $\alpha,$ $0\leq \alpha < 1$ and $\left \vert z\right \vert < r_{1}$ , where

$\begin{equation} r_{1} = \inf\limits_{n\geq 2}\left( \frac{\left \vert 4+n\left( n-3\right) \tan \left( e^{i\theta }\right) \right \vert }{\left \vert 2\tan \left( e^{i\theta }\right) \right \vert }\frac{\left( 1-\alpha \right) }{n\left( n-\alpha \right) }\right) ^{\frac{1}{n-1}}\mathit{\text{.}} \end{equation}$

(3.12)

Proof. It is sufficient to show that

$\begin{equation} \left \vert \frac{\left( zf^{\prime }(z)\right) ^{\prime }}{f^{\prime }(z)} -1\right \vert \leq 1-\alpha . \end{equation}$

(3.13)

From (1.1) $,$ we have

$\begin{equation} \left \vert \frac{zf^{\prime \prime }\left( z\right) }{f^{\prime }\left( z\right) }\right \vert \leq \frac{\sum \limits_{n = 2}^{\infty }n\left( n-1\right) \mathcal{\xi }_{n}\left \vert z\right \vert ^{n-1}}{1-\sum \limits_{n = 2}^{\infty }n\mathcal{\xi }_{n}\left \vert z\right \vert ^{n-1}} \text{.} \end{equation}$

(3.14)

(3.14) is bounded above by $1-\alpha,$ if

$\begin{equation} \sum \limits_{n = 2}^{\infty }\left[ \frac{n\left( n-1\right) +n\left( 1-\alpha \right) }{1-\alpha }\right] \left \vert \mathcal{\xi }_{n}\right \vert \left \vert z\right \vert ^{n-1}\leq 1\text{.} \end{equation}$

(3.15)

But by Theorem 3.1, the above inequality is true if

$\begin{equation} \sum \limits_{n = 2}^{\infty }\left \vert \frac{4n^{2}-2n\left( 2+\tan \left( e^{i\theta }\right) \right) }{\tan \left( e^{i\theta }\right) }\right \vert \left \vert \mathcal{\xi }_{n}\right \vert < 1. \end{equation}$

(3.16)

Then the inequality (3.15), becomes

$\begin{equation*} \left[ \frac{n\left( n-\alpha \right) }{1-\alpha }\right] \left \vert z\right \vert ^{n-1}\leq \left \vert \frac{4n^{2}-2n\left( 2+\tan \left( e^{i\theta }\right) \right) }{\tan \left( e^{i\theta }\right) }\right \vert \text{.} \end{equation*}$

Simple math yields

$\begin{equation*} r_{1} = \inf\limits_{n\geq 2}\left( \left \vert \frac{\left( 1-\alpha \right) \left( 4n-2\left( 2+\tan \left( e^{i\theta }\right) \right) \right) }{\left( n-\alpha \right) \tan \left( e^{i\theta }\right) }\right \vert \right) ^{ \frac{1}{n-1}}\text{.} \end{equation*}$

The desired outcome is demonstrated. □

4. Growth and distortion estimates

Theorem 4.1. Let $f\in \mathcal{C}_{\tan }$ and $\left \vert z\right \vert = r$ . Then

$\begin{equation} r-\left \vert \frac{\tan \left( e^{i\theta }\right) }{8-4\tan \left( e^{i\theta }\right) }\right \vert r^{2}\leq \left \vert f\left( z\right) \right \vert \leq r+\left \vert \frac{\tan \left( e^{i\theta }\right) }{ 8-4\tan \left( e^{i\theta }\right) }\right \vert r^{2}. \end{equation}$

(4.1)

Proof. Consider that

$\begin{eqnarray*} \left \vert f\left( z\right) \right \vert & = &\left \vert z+\sum\limits_{n = 2}^{\infty }\mathcal{\xi }_{n}z^{n}\right \vert \\ &\leq &r+\sum\limits_{n = 2}^{\infty }\left \vert \mathcal{\xi }_{n}\right \vert r^{n}. \end{eqnarray*}$

Since $r^{n}\leq r^{2\text{ }}$ for $n\geq 2$ and $r < 1,$ we have

$\begin{equation} \left \vert f\left( z\right) \right \vert \leq r+r^{2}\sum\limits_{n = 2}^{\infty }\left \vert \mathcal{\xi }_{n}\right \vert . \end{equation}$

(4.2)

Similarly

$\begin{equation} \left \vert f\left( z\right) \right \vert \geq r-r^{2}\sum\limits_{n = 2}^{\infty }\left \vert \mathcal{\xi }_{n}\right \vert . \end{equation}$

(4.3)

Now, applying (3.9) implies that

$\begin{equation*} \sum\limits_{n = 2}^{\infty }\left \vert \frac{4n^{2}-2n\left( 2+\tan \left( e^{i\theta }\right) \right) }{\tan \left( e^{i\theta }\right) }\right \vert \left \vert \mathcal{\xi }_{n}\right \vert < 1. \end{equation*}$

Since

$\begin{equation*} \left \vert \frac{16-4\left( 2+\tan \left( e^{i\theta }\right) \right) }{ \tan \left( e^{i\theta }\right) }\right \vert \sum\limits_{n = 2}^{\infty }\left \vert \mathcal{\xi }_{n}\right \vert \leq \sum\limits_{n = 2}^{\infty }\left \vert \frac{4n^{2}-2n\left( 2+\tan \left( e^{i\theta }\right) \right) }{\tan \left( e^{i\theta }\right) }\right \vert \left \vert \mathcal{\xi } _{n}\right \vert , \end{equation*}$

we get

$\begin{equation*} \left \vert \frac{8-4\tan \left( e^{i\theta }\right) }{\tan \left( e^{i\theta }\right) }\right \vert \sum\limits_{n = 2}^{\infty }\left \vert \mathcal{ \xi }_{n}\right \vert < 1, \end{equation*}$

One can easily write this as follows:

$\begin{equation*} \sum\limits_{n = 2}^{\infty }\left \vert \mathcal{\xi }_{n}\right \vert < \left \vert \frac{\tan \left( e^{i\theta }\right) }{16-4\left( 2+\tan \left( e^{i\theta }\right) \right) }\right \vert , \end{equation*}$

Placing this value in (4.2) and (4.3) the necessary inequality is obtained. □

Theorem 4.2. Let $f\in \mathcal{C}_{\tan }$ and $\left \vert z\right \vert = r$ . Then,

$\begin{equation*} 1-2\left \vert \frac{\tan \left( e^{i\theta }\right) }{8-4\tan \left( e^{i\theta }\right) }\right \vert r\leq \left \vert f^{\prime }\left( z\right) \right \vert \leq 1+2\left \vert \frac{\tan \left( e^{i\theta }\right) }{8-4\tan \left( e^{i\theta }\right) }\right \vert r. \end{equation*}$

Proof. Consider that

$\begin{eqnarray*} \left \vert f^{\prime }\left( z\right) \right \vert & = &\left \vert 1+\sum\limits_{n = 2}^{\infty }n\mathcal{\xi }_{n}z^{n}\right \vert \\ &\leq &1+\sum\limits_{n = 2}^{\infty }\left \vert \mathcal{\xi }_{n}\right \vert r^{n-1}. \end{eqnarray*}$

Since $r^{n-1}\leq r^{\text{ }}$ for $n\geq 2$ and $r < 1,$ we have

$\begin{equation} \left \vert f^{\prime }\left( z\right) \right \vert \leq 1+2r\sum\limits_{n = 2}^{\infty }\left \vert \mathcal{\xi }_{n}\right \vert . \end{equation}$

(4.4)

Similarly

$\begin{equation} \left \vert f^{\prime }\left( z\right) \right \vert \geq 1-2r\sum\limits_{n = 2}^{\infty }\left \vert \mathcal{\xi }_{n}\right \vert . \end{equation}$

(4.5)

Now, applying (3.9) implies that

Since

we get

one can easily write this as follows:

$\begin{equation*} \sum\limits_{n = 2}^{\infty }\left \vert \mathcal{\xi }_{n}\right \vert < \left \vert \frac{\tan \left( e^{i\theta }\right) }{8-4\tan \left( e^{i\theta }\right) } \right \vert. \end{equation*}$

Setting this value in (4.4) and (4.5), we accomplish what is needed. □

Theorem 4.3. For $f(z)\in \mathcal{C}_{\tan },$ the coefficient bounds are given by

$\begin{eqnarray} \left \vert \mathcal{\xi }_{2}\right \vert &\leq &\frac{1}{4}, \end{eqnarray}$

(4.6)

$\begin{eqnarray} \left \vert \mathcal{\xi }_{3}\right \vert &\leq &\frac{1}{12}, \end{eqnarray}$

(4.7)

$\begin{eqnarray} \left \vert \mathcal{\xi }_{4}\right \vert &\leq &\frac{1}{24}, \end{eqnarray}$

(4.8)

$\begin{eqnarray} \left \vert \mathcal{\xi }_{5}\right \vert &\leq &\frac{1}{24}\mathit{\text{.}} \end{eqnarray}$

(4.9)

and

$\begin{equation} \left \vert \mathcal{\xi }_{3}-\eta \mathcal{\xi }_{2}^{2}\right \vert \leq \frac{1}{12}\max \left \{ 1,\left \vert \frac{3\eta -2}{4}\right \vert \right \} . \end{equation}$

(4.10)

The above outcomes (4.6)–(4.9) are sharp for the functions given below:

$\begin{equation} f_{1}\left( z\right) = \int \limits_{0}^{z}\exp \int \limits_{0}^{z}\frac{ \frac{\tan x}{2}}{x}dx = z+\frac{1}{4}z^{2}+\frac{1}{24}z^{3}+\cdots \mathit{\text{,}} \end{equation}$

(4.11)

$\begin{eqnarray} f_{2}\left( z\right) & = &\int \limits_{0}^{z}\exp \int \limits_{0}^{z}\frac{ \frac{\tan x^{2}}{2}}{x}dx = z+\frac{z^{3}}{12}+\frac{z^{5}}{160}+\cdots , \end{eqnarray}$

(4.12)

$\begin{eqnarray} f_{3}\left( z\right) & = &\int \limits_{0}^{z}\exp \int \limits_{0}^{z}\frac{ \frac{\tan x^{3}}{2}}{x}dx = z+\frac{z^{4}}{24}+\frac{z^{7}}{504}+\cdots , \end{eqnarray}$

(4.13)

$\begin{eqnarray} f_{4}\left( z\right) & = &\int \limits_{0}^{z}\exp \int \limits_{0}^{z}\frac{ \frac{\tan x^{4}}{2}}{x}dx = z+\frac{z^{5}}{40}+\frac{z^{9}}{1152}+\cdots . \end{eqnarray}$

(4.14)

And the bound (4.10) is extreme for the function defined in (4.12).

Proof. Because $f(z)\in \mathcal{C}_{\tan },$ we have the definition

$\begin{equation*} \frac{\left( zf^{\prime }(z)\right) ^{\prime }}{f^{\prime }(z)}\prec \frac{ 2+\tan \left( z\right) }{2}\text{,} \end{equation*}$

which can be written as

$\begin{equation*} \frac{\left( zf^{\prime }(z)\right) ^{\prime }}{f^{\prime }(z)} = \frac{2+\tan \left( \omega \left( z\right) \right) }{2}, \end{equation*}$

where $\omega \left(z\right)$ is the holomorphic function with the following properties:

$\begin{equation*} \omega \left( 0\right) = 0\text{ and }\left \vert \omega \left( z\right) \right \vert < 1. \end{equation*}$

Now let

$\begin{equation} \tfrac{\left( zf^{\prime }(z)\right) ^{\prime }}{f^{\prime }(z)} = 1+2\mathcal{ \xi }_{2}z+\left( 6\mathcal{\xi }_{3}-4\mathcal{\xi }_{2}^{2}\right) z^{2}+\left( 12\mathcal{\xi }_{4}-18\mathcal{\xi }_{2}\mathcal{\xi }_{3}+8 \mathcal{\xi }_{2}^{3}\right) z^{3}+\cdots , \end{equation}$

(4.15)

and

$\begin{eqnarray} 1+\frac{\tan \left( \omega \left( z\right) \right) }{2} & = &1+\frac{1}{4} c_{1}z+\left( \frac{1}{4}c_{2}-\frac{1}{8}c_{1}^{2}\right) z^{2}+\left( \frac{1}{12}c_{1}^{3}-\frac{1}{4}c_{2}c_{1}+\frac{1}{4}c_{3}\right) z^{3} \\ &&+\left( -\frac{1}{16}c_{1}^{4}+\frac{1}{4}c_{1}^{2}c_{2}-\frac{ 1}{4}c_{3}c_{1}-\frac{1}{8}c_{2}^{2}+\frac{1}{4}c_{4}\right) z^{4}+\ldots \text{.} \end{eqnarray}$

(4.16)

Comparing (4.15) and (4.16), we have

$\begin{eqnarray} \mathcal{\xi }_{2} & = &\frac{1}{8}c_{1}\text{,} \end{eqnarray}$

(4.17)

$\begin{eqnarray} \mathcal{\xi }_{3} & = &\frac{1}{24}c_{2}-\frac{1}{96}c_{1}^{2}\text{,} \end{eqnarray}$

(4.18)

$\begin{eqnarray} \mathcal{\xi }_{4} & = &\frac{17}{4608}c_{1}^{3}-\frac{5}{384}c_{2}c_{1}+\frac{ 1}{48}c_{3}\text{.} \end{eqnarray}$

(4.19)

$\begin{eqnarray} \mathcal{\xi }_{5} & = &-\frac{1}{80} \left( \frac{157}{1152} c_{1}^{4}-\frac{29}{48}c_{1}^{2}c_{2}+\frac{2}{3}c_{3}c_{1}+\frac{3}{8} c_{2}^{2}-c_{4}\right). \end{eqnarray}$

(4.20)

Then by applying (2.2) to (4.17), we have

$\begin{equation*} \left \vert \mathcal{\xi }_{2}\right \vert \leq \frac{1}{4}\text{.} \end{equation*}$

And applying (2.3) with $n = k = 1$ to (4.18), we get

$\begin{equation*} \left \vert \mathcal{\xi }_{3}\right \vert \leq \frac{1}{12}. \end{equation*}$

For (4.19), applying Lemma 2.2 yields

$\begin{equation*} \left \vert \mathcal{\xi }_{4}\right \vert \leq \frac{1}{24}. \end{equation*}$

And for (4.20), we have

$\begin{eqnarray*} \left \vert \mathcal{\xi }_{5}\right \vert & = &\left \vert -\frac{1}{80} \right \vert \left \vert \frac{157}{1152}c_{1}^{4}-\frac{29}{48} c_{1}^{2}c_{2}+\frac{2}{3}c_{3}c_{1}+\frac{3}{8}c_{2}^{2}-c_{4}\right \vert \\ &\leq &\frac{1}{40}\text{ (by Lemma 2.3).} \end{eqnarray*}$

Now from (4.17) and (4.18), we have

$\begin{equation*} \left \vert \mathcal{\xi }_{3}-\eta \mathcal{\xi }_{2}^{2}\right \vert = \frac{1}{24}\left \vert c_{2}-\frac{3\eta -2}{4}c_{1}^{2}\right \vert . \end{equation*}$

And applying (2.4) to the above relation $,$ we achieve our goals. □

The following outcome occurs if we set $\eta = 1$ in the above result.

Remark 4.4. If we set $\eta = 1$ in (4.10), we get the following result

$\begin{equation*} \left \vert \mathcal{\xi }_{3}-\mathcal{\xi }_{2}^{2}\right \vert \leq \frac{ 1}{12}. \end{equation*}$

The outcome is precise for the function defined in (4.12), and it cannot be further enhanced.

Theorem 4.5. Let $f\left(z\right) \in \mathcal{C}_{\tan }.$ Then

$\begin{equation*} \left \vert \mathcal{\xi }_{2}\mathcal{\xi }_{3}-\mathcal{\xi }_{4}\right \vert \leq \frac{1}{24}. \end{equation*}$

The outcome is sharp for the function defined in (4.13).

Proof. From (4.17)–(4.19), we have

$\begin{equation*} \left \vert \mathcal{\xi }_{2}\mathcal{\xi }_{3}-\mathcal{\xi }_{4}\right \vert = \left \vert \frac{23}{4608}c_{1}^{3}+\frac{7}{384}c_{2}c_{1}-\frac{1}{ 48}c_{3}\right \vert. \end{equation*}$

Applying Lemma 2.2, we achieve the intended outcomes. □

Theorem 4.6. Let $f\left(z\right) \in \mathcal{C}_{\tan }.$ Then

$\begin{equation*} \left \vert \mathcal{\xi }_{2}\mathcal{\xi }_{4}-\mathcal{\xi } _{3}^{2}\right \vert \leq \frac{1}{144}. \end{equation*}$

The outcome is sharp for the function defined in (4.12).

Proof. From (4.17)–(4.19), we have

$\begin{equation*} \left \vert \mathcal{\xi }_{2}\mathcal{\xi }_{4}-\mathcal{\xi } _{3}^{2}\right \vert = \left \vert \frac{13}{36\,864}c_{1}^{4}-\frac{7}{9216} c_{1}^{2}c_{2}+\frac{1}{384}c_{3}c_{1}-\frac{1}{576} c_{2}^{2} \right \vert . \end{equation*}$

Now using Lemma 2.4, with $c_{1} = c$ and $\left \vert x\right \vert = y,$ we have

$\begin{eqnarray*} \left \vert \mathcal{\xi }_{2}\mathcal{\xi }_{4}-\mathcal{\xi } _{3}^{2}\right \vert &\leq &\frac{7}{36\,864}c^{4}+\frac{1}{1536}c^{2}\left( 4-c^{2}\right) y^{2}+\frac{1}{18\,432}c^{2}\left( 4-c^{2}\right) y \\ &&+\frac{1}{768} c\left( 1-y^{2}\right) \left( 4-c^{2}\right) + \frac{1}{2304}\left( 4-c^{2}\right) ^{2}y^{2} \\ & = &G\left( y,c\right) \text{ (say).} \end{eqnarray*}$

Further,

$\begin{equation*} \frac{\partial G\left( y,c\right) \text{ }}{\partial y} = \frac{1}{18\,432} \left( 4-c^{2}\right) \left( \left( 64+8c^{2}-48c\right) y+c^{2}\right) > 0 . \end{equation*}$

Clearly $\frac{\partial G\left(y, c\right) \text{ }}{\partial y} > 0$ in $y\in \left[ 0, 1\right]$ so the maximum is attained at $y = 1,$ i.e.,

$\begin{equation*} G\left( 1,c\right) = \frac{7}{36\,864}c^{4}+\frac{1}{1536}c^{2}\left( 4-c^{2}\right) +\frac{1}{18\,432}c^{2}\left( 4-c^{2}\right) + \frac{1}{2304}\left( 4-c^{2}\right) ^{2} = H\left( c\right) . \end{equation*}$

Further,

$\begin{equation*} H^{\prime }\left( c\right) = -\frac{1}{3072}c\left( c^{2}+4\right) , \end{equation*}$

since $H^{\prime }\left(c\right) = 0$ has three roots namely $c = 0,$ $-2i$ and $2i.$ The only root lying in the interval $\left[ 0, 2\right]$ is $0.$ Also, one may check easily that $H^{^{\prime \prime }}\left(c\right) \leq 0$ for $c = 0;$ thus, the maximum is attained at $c = 0,$ that is

$\begin{equation*} \left \vert \mathcal{\xi }_{2}\mathcal{\xi }_{4}-\mathcal{\xi } _{3}^{2}\right \vert \leq \frac{1}{144}. \end{equation*}$

□

5. Krushkal inequality

Here, we will provide direct evidence of the inequality

$\begin{equation*} \left \vert \mathcal{\xi }_{n}^{p}-\mathcal{\xi }_{2}^{p\left( n-1\right) }\right \vert \leq 2^{p\left( n-1\right) }-n^{p}, \end{equation*}$

over the class $\mathcal{C}_{\tan }$ for the choice of $n = 4,$ $p = 1$ , and for $n = 5,$ $p = 1$ . For a class of univalent functions as a whole, Krushkal introduced and demonstrated this inequality in ^[28]. For some recent investigations into the Krushkal inequality, we refer the readers to ^[14,29].

Theorem 5.1. For $f(z)\in \mathcal{C}_{\tan }$ , we have

$\begin{equation*} \left \vert \mathcal{\xi }_{4}-\mathcal{\xi }_{2}^{3}\right \vert \leq \frac{ 1}{24}. \end{equation*}$

The outcome is sharp for the function defined in (4.13).

Proof. From (4.17) and (4.19), we have

$\begin{equation*} \left \vert \mathcal{\xi }_{4}-\mathcal{\xi }_{2}^{3}\right \vert = \left \vert \frac{1}{576}c_{1}^{3}-\frac{5}{384}c_{2}c_{1}+\frac{1}{48}c_{3}\right \vert . \end{equation*}$

By applying Lemma 2.2, we get

$\begin{equation*} \left \vert \mathcal{\xi }_{4}-\mathcal{\xi }_{2}^{3}\right \vert \leq \frac{ 1}{24}. \end{equation*}$

□

Theorem 5.2. For $f(z)\in \mathcal{C}_{\tan },$ we have

$\begin{equation*} \left \vert \mathcal{\xi }_{5}-\mathcal{\xi }_{2}^{4}\right \vert \leq \frac{ 1}{40}. \end{equation*}$

The outcome is sharp for the function defined in (4.14).

Proof. From (4.17) and (4.20), we have

$\begin{eqnarray*} \left \vert \mathcal{\xi }_{5}-\mathcal{\xi }_{2}^{4}\right \vert & = &\left \vert -\frac{1}{80}\right \vert \left \vert \frac{359}{2304}c_{1}^{4}-\frac{ 29}{48}c_{1}^{2}c_{2}+\frac{2}{3}c_{3}c_{1}+\frac{3}{8}c_{2}^{2}-c_{4}\right \vert \\ &\leq &\frac{1}{40}\text{ (by Lemma 2.3).} \end{eqnarray*}$

□

6. Logarithmic coefficients for the family $\mathcal{C}_{\tan }$

The logarithmic coefficients of $f\in \mathcal{S}$ denoted by $\kappa _{n} = \kappa _{n}\left(f\right),$ are defined by the following series expansion:

$\begin{equation*} \log \frac{f\left( z\right) }{z} = 2\sum\limits_{n = 1}^{\infty }\kappa _{n}z^{n}. \end{equation*}$

For the function $f$ given by $\left(1.1\right),$ the logarithmic coefficients are as follows:

$\begin{eqnarray} \kappa _{1} & = &\frac{1}{2}\mathcal{\xi }_{2}, \end{eqnarray}$

(6.1)

$\begin{eqnarray} \kappa _{2} & = &\frac{1}{2}\left( \mathcal{\xi }_{3}-\frac{1}{2}\mathcal{\xi } _{2}^{2}\right) , \end{eqnarray}$

(6.2)

$\begin{eqnarray} \kappa _{3} & = &\frac{1}{2}\left( \mathcal{\xi }_{4}-\mathcal{\xi }_{2} \mathcal{\xi }_{3}+\frac{1}{3}\mathcal{\xi }_{2}^{3}\right) , \end{eqnarray}$

(6.3)

$\begin{eqnarray} \kappa _{4} & = &\frac{1}{2}\left( \mathcal{\xi }_{5}-\mathcal{\xi }_{2} \mathcal{\xi }_{4}-\mathcal{\xi }_{2}^{2}\mathcal{\xi }_{3}-\frac{1}{2} \mathcal{\xi }_{3}^{2}-\frac{1}{4}\mathcal{\xi }_{2}^{4}\right) . \end{eqnarray}$

(6.4)

Theorem 6.1. If $f$ has the form (1.1) and belongs to $\mathcal{C}_{\tan },$ then

$\begin{eqnarray*} \left \vert \kappa _{1}\right \vert &\leq &\frac{1}{8}, \\ \left \vert \kappa _{2}\right \vert &\leq &\frac{1}{24}, \\ \left \vert \kappa _{3}\right \vert &\leq &\frac{1}{48}, \\ \left \vert \kappa _{4}\right \vert &\leq &\frac{1}{80}. \end{eqnarray*}$

The bounds of Theorem 6.1 are precise and cannot be improved further.

Proof. Now from (6.1) to (6.4) and (4.17) to (4.20), we get

$\begin{eqnarray} \kappa _{1} & = &\frac{1}{16}c_{1}, \end{eqnarray}$

(6.5)

$\begin{eqnarray} \kappa _{2} & = &\frac{1}{48}c_{2}-\frac{7}{768}c_{1}^{2}, \end{eqnarray}$

(6.6)

$\begin{eqnarray} \kappa _{3} & = &\frac{13}{4608}c_{1}^{3}-\frac{7}{768}c_{2}c_{1}+\frac{1}{96} c_{3}, \end{eqnarray}$

(6.7)

$\begin{eqnarray} \kappa _{4} & = &-\frac{1561}{1474\,560}c_{1}^{4}+\frac{413}{92\,160} c_{1}^{2}c_{2}-\frac{7}{1280}c_{3}c_{1}- \frac{1}{360}c_{2}^{2}+ \frac{1}{160}c_{4}, \end{eqnarray}$

(6.8)

Applying (2.2) to (6.5), we get

$\begin{equation*} \left \vert \kappa _{1}\right \vert \leq \frac{1}{8}. \end{equation*}$

From (6.6), using (2.3), we get

$\begin{equation*} \left \vert \kappa _{2}\right \vert \leq \frac{1}{24}. \end{equation*}$

Applying Lemma 2.2 to (6.7), we get

$\begin{equation*} \left \vert \kappa _{3}\right \vert \leq \frac{1}{48}. \end{equation*}$

Also, applying Lemma 2.3 to (6.8), we get

$\begin{equation*} \left \vert \kappa _{4}\right \vert \leq \frac{1}{80}. \end{equation*}$

Proof for sharpness: Since

$\begin{eqnarray*} \log \frac{f_{1}\left( z\right) }{z} & = &2\sum\limits_{n = 2}^{\infty }\kappa \left( f_{1}\right) z^{n} = \frac{1}{4}z+\cdots , \\ \log \frac{f_{2}\left( z\right) }{z} & = &2\sum\limits_{n = 2}^{\infty }\kappa \left( f_{2}\right) z^{n} = \frac{1}{12}z^{2}+\cdots , \\ \log \frac{f_{3}\left( z\right) }{z} & = &2\sum\limits_{n = 2}^{\infty }\kappa \left( f_{2}\right) z^{n} = \frac{1}{24}z^{3}+\cdots , \\ \log \frac{f_{4}\left( z\right) }{z} & = &2\sum\limits_{n = 2}^{\infty }\kappa \left( f_{2}\right) z^{n} = \frac{1}{40}z^{4}+\cdots , \end{eqnarray*}$

it follows that these inequalities can be obtained for the functions denoted by $f_{n}\left(z\right)$ for $n = 1, 2, 3$ and 4 as defined in (4.11) to (4.14). □

Theorem 6.2. Let $f\in \mathcal{C}_{\tan }.$ Then for a complex number $\lambda,$ we have

$\begin{equation*} \left \vert \kappa _{2}-\lambda \kappa _{1}^{2}\right \vert \leq \frac{1}{24} \max \left \{ 1,\frac{\left \vert 3\lambda -1\right \vert }{8}\right \} . \end{equation*}$

The result is the best possible.

Proof. From (6.5) and (6.6), we have

$\begin{equation*} \left \vert \kappa _{2}-\lambda \kappa _{1}^{2}\right \vert = \frac{1}{48} \left \vert c_{2}-\frac{7+3\lambda }{16}c_{1}^{2}\right \vert . \end{equation*}$

Applying (2.4) to the preceding equation yields the desired outcome. □

Theorem 6.3. Let $f\in \mathcal{C}_{\tan }.$ Then

$\begin{equation*} \left \vert \kappa _{1}\kappa _{2}-\kappa _{3}\right \vert \leq \frac{1}{48}. \end{equation*}$

The outcome is extremal.

Proof. From (6.5)–(6.7), we have

$\begin{equation*} \left \vert \kappa _{1}\kappa _{2}-\kappa _{3}\right \vert = \left \vert \frac{125}{36\,864}c_{1}^{3}-\frac{1}{96}c_{2}c_{1}+\frac{1}{96}c_{3}\right \vert. \end{equation*}$

Applying Lemma 2.2, we achieve the intended outcomes. □

Theorem 6.4. Let $f\in \mathcal{C}_{\tan }.$ Then

$\begin{equation*} \left \vert \kappa _{1}\kappa _{3}-\kappa _{2}^{2}\right \vert \leq \frac{1}{ 576}. \end{equation*}$

The outcome is sharp.

Proof. From (6.5)–(6.7), we have

$\begin{equation*} \left \vert \kappa _{1}\kappa _{3}-\kappa _{2}^{2}\right \vert = \left \vert \frac{55}{589\,824}c_{1}^{4}-\frac{7}{36\,864}c_{1}^{2}c_{2}+\frac{1}{1536} c_{3}c_{1}- \frac{1}{2304}c_{2}^{2} \right \vert . \end{equation*}$

Now using Lemma 2.4, with $c_{1} = c,$ $\left \vert z\right \vert = 1$ and $\left \vert x\right \vert = y,$ we have

$\begin{eqnarray*} \left \vert \kappa _{1}\kappa _{3}-\kappa _{2}^{2}\right \vert &\leq &\frac{ 31}{589\,824}c^{4}+\frac{1}{6144}c^{2}\left( 4-c^{2}\right) y^{2}+\frac{1}{ 73\,728}c^{2}\left( 4-c^{2}\right) y \\ &&+\frac{1}{3072}c \left( 1-y^{2}\right) \left( 4-c^{2}\right) + \frac{1}{9216}\left( 4-c^{2}\right) ^{2}y^{2} \\ & = &G\left( y,c\right) \text{ (say).} \end{eqnarray*}$

Further,

$\begin{equation*} \frac{\partial G\left( y,c\right) \text{ }}{\partial y} = \frac{1}{73\,728} \left( 4-c^{2}\right) \left( 64y+8c^{2}y-48cy+c^{2}\right). \end{equation*}$

Clearly, $\frac{\partial G\left(y, c\right) \text{ }}{\partial y} > 0$ in $y\in \left[ 0, 1\right]$ so the maximum is attained at $y = 1,$ i.e.,

$\begin{equation*} G\left( 1,c\right) = \frac{31}{589\,824}c^{4}+\frac{1}{6144}c^{2}\left( 4-c^{2}\right) +\frac{1}{73\,728}c^{2}\left( 4-c^{2}\right) +\frac{1}{9216} \left( 4-c^{2}\right) ^{2} = H\left( c\right) . \end{equation*}$

Further,

$\begin{equation*} H^{\prime }\left( c\right) = -\frac{1}{49\,152}c\left( 3c^{2}+16\right) , \end{equation*}$

since $H^{\prime }\left(c\right) = 0$ has only one solution $c = 0,$ that lies in the interval $\left[ 0, 2\right].$ Also, one may check easily that $H^{^{\prime \prime }}\left(c\right) \leq 0$ for $c = 0;$ thus, the maximum can be attained at $c = 0,$ that is

$\begin{equation*} H\left( 0\right) \leq \frac{1}{576}. \end{equation*}$

□

7. Conclusions

In this study, we were motivated by the recent research and the sharp bounds of Hankel inequalities, and have have defined a new subclass of holomorphic convex functions that are related to the tangent functions. We then derived geometric properties like the necessary and sufficient conditions, radius of convexity, growth, and distortion estimates for our defined function class. Furthermore, the sharp coefficient bounds, sharp Fekete-Szegö inequality, sharp 2nd order Hankel determinant, and Krushkal inequalities have been given. Moreover, we have calculated the sharp coefficient bounds, sharp Fekete-Szegö inequality, and sharp second-order Hankel determinant for the functions whose coefficients are logarithmic. Hopefully, this work will open new directions for those working in geometric function theory and related areas. One can extend the work here by replacing the ordinal derivative with a certain q-derivative operator.

Use of AI tools declaration

The authors declare that they have not used artificial intelligence tools in the creation of this article.

Conflict of interest

All authors declare no conflicts of interest.

References

[1]	R. A. Meyers, Encyclopedia of physical science and technology, San Diego: Academic Press, 1992.
[2]	C. Li, W. Hu, J. C. Sprott, X. Wang, Multistability in symmetric chaotic systems, Eur. Phys. J. Spec. Top., 224 (2015), 1493–1506. https://doi.org/10.1140/epjst/e2015-02475-x doi: 10.1140/epjst/e2015-02475-x
[3]	T. A. Alexeeva, N. V. Kuznetsov, T. N. Mokaev, Study of irregular dynamics in an economic model: attractor localization and Lyapunov exponents, Chaos Soliton. Fract., 152 (2021), 111365. https://doi.org/10.1016/j.chaos.2021.111365 doi: 10.1016/j.chaos.2021.111365
[4]	S. Vaidyanathan, A. S. T. Kammogne, E. Tlelo-Cuautle, C. N. Talonang, B. Abd-El-Atty, A. A. Abd El-Latif, et al., A novel 3-D jerk system, its bifurcation analysis, electronic circuit design and a cryptographic application, Electronics, 12 (2023), 2818. https://doi.org/10.3390/electronics12132818 doi: 10.3390/electronics12132818
[5]	C. Nwachioma, J. H. P'erez-Cruz, Analysis of a new chaotic system, electronic realization and use in navigation of differential drive mobile robot, Chaos Soliton. Fract., 144 (2021), 110684. https://doi.org/10.1016/j.chaos.2021.110684 doi: 10.1016/j.chaos.2021.110684
[6]	N. V. Kuznetsov, G. A. Leonov, V. I. Vagaitsev, Analytical-numerical method for attractor localization of generalized Chua's system, IFAC Proc. Vol., 43 (2010), 29–33. https://doi.org/10.3182/20100826-3-TR-4016.00009 doi: 10.3182/20100826-3-TR-4016.00009
[7]	G. A. Leonov, N. V. Kuznetsov, V. I. Vagaitsev, Localization of hidden Chua's attractors, Phys. Lett. A, 375 (2011), 2230–2233. https://doi.org/10.1016/j.physleta.2011.04.037 doi: 10.1016/j.physleta.2011.04.037
[8]	G. A. Leonov, N. V. Kuznetsov, Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, Int. J. Bifurcat. Chaos, 23 (2013), 1330002. https://dx.doi.org/10.1142/S0218127413300024 doi: 10.1142/S0218127413300024
[9]	N. V. Stankevich, N. V. Kuznetsov, G. A. Leonov, L. O. Chua, Scenario of the birth of hidden attractors in the Chua circuit, Int. J. Bifurcat. Chaos, 27 (2017), 1730038. https://doi.org/10.1142/S0218127417300385 doi: 10.1142/S0218127417300385
[10]	N. Kuznetsov, T. Mokaev, V. Ponomarenko, E. Seleznev, N. Stankevich, L. Chua, Hidden attractors in Chua circuit: mathematical theory meets physical experiments, Nonlinear Dyn., 111 (2023), 5859–5887. https://doi.org/10.1007/s11071-022-08078-y doi: 10.1007/s11071-022-08078-y
[11]	Q. Wu, Q. Hong, X. Liu, X. Wang, Z. Zeng, A novel amplitude control method for constructing nested hidden multi-butterfly and multiscroll chaotic attractors, Chaos Soliton. Fract., 134 (2020), 109727. https://doi.org/10.1016/j.chaos.2020.109727 doi: 10.1016/j.chaos.2020.109727
[12]	H. Tian, Z. Wang, P. Zhang, M. Chen, Y. Wang, Dynamic analysis and robust control of a chaotic system with hidden attractor, Complexity, 2021 (2021), 8865522. https://doi.org/10.1155/2021/8865522 doi: 10.1155/2021/8865522
[13]	F. Bao, S. Yu, How to generate chaos from switching system: a saddle focus of index 1 and heteroclinic loop-based approach, Math. Probl. Eng., 2011 (2011), 756462. https://doi.org/10.1155/2011/756462 doi: 10.1155/2011/756462
[14]	T. Zhou, G. Chen, Classification of chaos in 3-D autonomous quadratic systems-Ⅰ: basic framework and methods, Int. J. Bifurcat. Chaos, 16 (2006), 2459–2479. https://doi.org/10.1142/S0218127406016203 doi: 10.1142/S0218127406016203
[15]	Z. Wei, J. C. Sprott, H. Chen, Elementary quadratic chaotic flows with a single non-hyperbolic equilibrium, Phys. Lett. A, 379 (2015), 2184–2187. https://doi.org/10.1016/j.physleta.2015.06.040 doi: 10.1016/j.physleta.2015.06.040
[16]	K. Rajagopal, V. T. Pham, F. R. Tahir, A. Akgul, H. R. Abdolmohammadi, S. Jafari, A chaotic jerk system with non-hyperbolic equilibrium: dynamics, effect of time delay and circuit realisation, Pramana, 90 (2018), 52. https://doi.org/10.1007/s12043-018-1545-x doi: 10.1007/s12043-018-1545-x
[17]	X. Cai, L. Liu, Y. Wang, C. Liu, A 3D chaotic system with piece-wise lines shape non-hyperbolic equilibria and its predefined-time control, Chaos Soliton. Fract., 146 (2021), 110904. https://doi.org/10.1016/j.chaos.2021.110904 doi: 10.1016/j.chaos.2021.110904
[18]	C. Li, W. Hai, Constructing multiwing attractors from a robust chaotic system with non-hyperbolic equilibrium points, Automatika, 59 (2018), 184–193. https://doi.org/10.1080/00051144.2018.1516273 doi: 10.1080/00051144.2018.1516273
[19]	C. Li, J. C. Sprott, Chaotic flows with a single nonquadratic term, Phys. Lett. A, 378 (2014), 178–183. https://doi.org/10.1016/j.physleta.2013.11.004 doi: 10.1016/j.physleta.2013.11.004
[20]	S. Jafari, J. C. Sprott, F. Nazarimehr, Recent new examples of hidden attractors, Eur. Phys. J. Spec. Top., 224 (2015), 1469–1476. https://doi.org/10.1140/epjst/e2015-02472-1 doi: 10.1140/epjst/e2015-02472-1
[21]	X. Wang, G. Chen, A chaotic system with only one stable equilibrium, Commun. Nonlinear Sci., 17 (2012), 1264–1272. https://doi.org/10.1016/j.cnsns.2011.07.017 doi: 10.1016/j.cnsns.2011.07.017
[22]	X. Wang, A. Akgul, S. Cicek, V. T. Pham, D. V. Hoang, A chaotic system with two stable equilibrium points: dynamics, circuit realization and communication application, Int. J. Bifurcat. Chaos, 27 (2017), 1750130. https://doi.org/10.1142/S0218127417501309 doi: 10.1142/S0218127417501309
[23]	P. C. Rech, Self-excited and hidden attractors in a multistable jerk system, Chaos Soliton. Fract., 164 (2022), 112614. https://doi.org/10.1016/j.chaos.2022.112614 doi: 10.1016/j.chaos.2022.112614
[24]	V. T. Pham, S. Jafari, C. Volos, T. Kapitaniak, A gallery of chaotic systems with an infinite number of equilibrium points, Chaos Soliton. Fract., 93 (2016), 58–63. https://doi.org/10.1016/j.chaos.2016.10.002 doi: 10.1016/j.chaos.2016.10.002
[25]	M. Molaie, S. Jafari, J. C. Sprott, S. M. Golpayegani, Simple chaotic flows with one stable equilibrium, Int. J. Bifurcat. Chaos, 23 (2013), 1350188. https://doi.org/10.1142/S0218127413501885 doi: 10.1142/S0218127413501885
[26]	S. Jafari, J. C. Sprott, Simple chaotic flows with a line equilibrium, Chaos Soliton. Fract., 57 (2013), 79–84. https://doi.org/10.1016/j.chaos.2013.08.018 doi: 10.1016/j.chaos.2013.08.018
[27]	S. Kumarasamy, M. Banerjee, V. Varshney, M. D. Shrimali, N. V. Kuznetsov, A. Prasad, Saddle-node bifurcation of periodic orbit route to hidden attractors, Phys. Rev. E, 107 (2023), L052201. https://doi.org/10.1103/PhysRevE.107.L052201 doi: 10.1103/PhysRevE.107.L052201
[28]	R. Balamurali, J. Kengne, R. G. Chengui, K. Rajagopal, Coupled van der Pol and Duffing oscillators: emergence of antimonotonicity and coexisting multiple self-excited and hidden oscillations, Eur. Phys. J. Plus, 137 (2022), 789. https://doi.org/10.1140/epjp/s13360-022-03000-2 doi: 10.1140/epjp/s13360-022-03000-2
[29]	B. Li, B. Sang, M. Liu, X. Hu, X. Zhang, N. Wang, Some jerk systems with hidden chaotic dynamics, Int. J. Bifurcat. Chaos, 33 (2023), 2350069. https://doi.org/10.1142/S0218127423500694 doi: 10.1142/S0218127423500694
[30]	C. Dong, M. Yang, L. Jia, Z. Li, Dynamics investigation and chaos-based application of a novel no-equilibrium system with coexisting hidden attractors, Phys. A, 633 (2024), 129391. https://doi.org/10.1016/j.physa.2023.129391 doi: 10.1016/j.physa.2023.129391
[31]	J. C. Sprott, Strange attractors with various equilibrium types, Eur. Phys. J. Spec. Top., 224 (2015), 1409–1419. https://doi.org/10.1140/epjst/e2015-02469-8 doi: 10.1140/epjst/e2015-02469-8
[32]	M. A. Jafari, E. Mliki, A. Akgul, V. T. Pham, S. T. Kingni, X. Wang, S. Jafari, Chameleon: the most hidden chaotic flow, Nonlinear Dyn., 88 (2017), 2303–2317. https://doi.org/10.1007/s11071-017-3378-4 doi: 10.1007/s11071-017-3378-4
[33]	K. Rajagopal, A. Karthikeyan, P. Duraisamy, Hyperchaotic chameleon: fractional order FPGA implementation, Complexity, 2017 (2017), 8979408. https://doi.org/10.1155/2017/8979408 doi: 10.1155/2017/8979408
[34]	H. Natiq, M. R. Said, M. R. Ariffin, S. He, L. Rondoni, S. Banerjee, Self-excited and hidden attractors in a novel chaotic system with complicated multistability, Eur. Phys. J. Plus, 133 (2018), 557. https://doi.org/10.1140/epjp/i2018-12360-y doi: 10.1140/epjp/i2018-12360-y
[35]	S. Cang, Y. Li, R. Zhang, Z. Wang, Hidden and self-excited coexisting attractors in a Lorenz-like system with two equilibrium points, Nonlinear Dyn., 95 (2019), 381–390. https://doi.org/10.1007/s11071-018-4570-x doi: 10.1007/s11071-018-4570-x
[36]	Q. Yang, Z. Wei, G. Chen, An unusual 3D autonomous quadratic chaotic system with two stable node-foci, Int. J. Bifurcat. Chaos, 20 (2010), 1061–1083. https://doi.org/10.1142/S0218127410026320 doi: 10.1142/S0218127410026320
[37]	V. F. Signing, G. G. Tegue, M. Kountchou, Z. T. Njitacke, N. Tsafack, J. D. Nkapkop, et al., A cryptosystem based on a chameleon chaotic system and dynamic DNA coding, Chaos Soliton. Fract., 155 (2022), 111777. https://doi.org/10.1016/j.chaos.2021.111777 doi: 10.1016/j.chaos.2021.111777
[38]	W. Fan, D. Xu, Z. Chen, N. Wang, Q. Xu, On two-parameter bifurcation and analog circuit implementation of a chameleon chaotic system, Phys. Scr., 99 (2023), 015218. https://doi.org/10.1088/1402-4896/ad1231 doi: 10.1088/1402-4896/ad1231
[39]	A. Tiwari, R. Nathasarma, B. K. Roy, A new time-reversible 3D chaotic system with coexisting dissipative and conservative behaviors and its active non-linear control, J. Franklin I., 361 (2024), 106637. https://doi.org/10.1016/j.jfranklin.2024.01.038 doi: 10.1016/j.jfranklin.2024.01.038
[40]	R. Zhou, Y. Gu, J. Cui, G. Ren, S. Yu, Nonlinear dynamic analysis of supercritical and subcritical Hopf bifurcations in gas foil bearing-rotor systems, Nonlinear Dyn., 103 (2021), 2241–2256. https://doi.org/10.1007/s11071-021-06234-4 doi: 10.1007/s11071-021-06234-4
[41]	N. V. Stankevich, N. V. Kuznetsov, G. A. Leonov, L. O. Chua, Scenario of the birth of hidden attractors in the Chua circuit, Int. J. Bifurcat. Chaos, 27 (2017), 1730038. https://doi.org/10.1142/S0218127417300385 doi: 10.1142/S0218127417300385
[42]	H. Zhao, Y. Lin, Y. Dai, Hopf bifurcation and hidden attractor of a modified Chua's equation, Nonlinear Dyn., 90 (2017), 2013–2021. https://doi.org/10.1007/s11071-017-3777-6 doi: 10.1007/s11071-017-3777-6
[43]	M. Liu, B. Sang, N. Wang, I. Ahmad, Chaotic dynamics by some quadratic jerk systems, Axioms, 10 (2021), 227. https://doi.org/10.3390/axioms10030227 doi: 10.3390/axioms10030227
[44]	Q. Yang, D. Zhu, L. Yang, A new 7D hyperchaotic system with five positive Lyapunov exponents coined, Int. J. Bifurcat. Chaos, 28 (2018), 1850057. https://doi.org/10.1142/S0218127418500578 doi: 10.1142/S0218127418500578
[45]	Z. Li, K. Chen, Neuromorphic behaviors in a neuron circuit based on current-controlled Chua Corsage Memristor, Chaos Soliton. Fract., 175 (2023), 114017. https://doi.org/10.1016/j.chaos.2023.114017 doi: 10.1016/j.chaos.2023.114017
[46]	Y. Liu, Y. Zhou, B. Guo, Hopf bifurcation, periodic solutions, and control of a new 4D hyperchaotic system, Mathematics, 11 (2023), 2699. https://doi.org/10.3390/math11122699 doi: 10.3390/math11122699
[47]	C. Li, J. C. Sprott, Variable-boostable chaotic flows, Optik, 127 (2016), 10389–10398. https://doi.org/10.1016/j.ijleo.2016.08.046 doi: 10.1016/j.ijleo.2016.08.046
[48]	C. Li, A. Akgul, L. Bi, Y. Xu, C. Zhang, A chaotic jerk oscillator with interlocked offset boosting, Eur. Phys. J. Plus, 139 (2024), 242. https://doi.org/10.1140/epjp/s13360-024-05040-2 doi: 10.1140/epjp/s13360-024-05040-2
[49]	C. Li, X. Wang, G. Chen, Diagnosing multistability by offset boosting, Nonlinear Dyn., 90 (2017), 1335–1341. https://doi.org/10.1007/s11071-017-3729-1 doi: 10.1007/s11071-017-3729-1
[50]	X. Gao, Hamilton energy of a complex chaotic system and offset boosting, Phys. Scr., 99 (2024), 015244. https://doi.org/10.1088/1402-4896/ad1739 doi: 10.1088/1402-4896/ad1739
[51]	X. Zhang, C. Li, T. Lei, H. Fu, Z. Liu, Offset boosting in a memristive hyperchaotic system, Phys. Scr., 99 (2024), 015247. https://doi.org/10.1088/1402-4896/ad156e doi: 10.1088/1402-4896/ad156e
[52]	H. Dang-Vu, C. Delcarte, Bifurcations et Chaos: une introduction à la dynamique contemporaine avec des programmes en Pascal, Fortran et Mathematica, Paris: Ellipses, 2000.
[53]	C. Grebogi, E. Ott, J. A. Yorke, Chaos, strange attractors, and fractal basin boundaries in nonlinear dynamics, Science, 238 (1987), 632–638. https://doi.org/10.1126/science.238.4827.632 doi: 10.1126/science.238.4827.632

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