Stability and bifurcation analyses of p53 gene regulatory network with time delay

Jianmin Hou; Quansheng Liu; Hongwei Yang; Lixin Wang; Yuanhong Bi; Jianmin Hou; Quansheng Liu; Hongwei Yang; Lixin Wang; Yuanhong Bi

doi:10.3934/era.2022045

Electronic Research Archive

2022, Volume 30, Issue 3: 850-873. doi: 10.3934/era.2022045

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Research article

Stability and bifurcation analyses of p53 gene regulatory network with time delay

1.
School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, China
2.
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China
3.
School of Ecology and Environment, Inner Mongolia University, Hohhot, 010021, China
4.
School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot, 010070, China

Received: 29 September 2021 Revised: 27 November 2021 Accepted: 28 November 2021 Published: 01 March 2022

In this paper, based on a p53 gene regulatory network regulated by Programmed Cell Death 5(PDCD5), a time delay in transcription and translation of Mdm2 gene expression is introduced into the network, the effects of the time delay on oscillation dynamics of p53 are investigated through stability and bifurcation analyses. The local stability of the positive equilibrium in the network is proved through analyzing the characteristic values of the corresponding linearized systems, which give the conditions on undergoing Hopf bifurcation without and with time delay, respectively. The theoretical results are verified through numerical simulations of time series, characteristic values and potential landscapes. Furthermore, combined effect of time delay and several typical parameters in the network on oscillation dynamics of p53 are explored through two-parameter bifurcation diagrams. The results show p53 reaches a lower stable steady state under smaller PDCD5 level, the production rates of p53 and Mdm2 while reaches a higher stable steady state under these larger ones. But the case is the opposite for the degradation rate of p53. Specially, p53 oscillates at a smaller Mdm2 degradation rate, but a larger one makes p53 reach a low stable steady state. Besides, moderate time delay can make the steady state switch from stable to unstable and induce p53 oscillation for moderate value of these parameters. Theses results reveal that time delay has a significant impact on p53 oscillation and may provide a useful insight into developing anti-cancer therapy.

Keywords:

Citation: Jianmin Hou, Quansheng Liu, Hongwei Yang, Lixin Wang, Yuanhong Bi. Stability and bifurcation analyses of p53 gene regulatory network with time delay[J]. Electronic Research Archive, 2022, 30(3): 850-873. doi: 10.3934/era.2022045

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Abstract

1. Introduction and preliminaries

Let $\mathcal{A}$ indicate an analytic functions family, which is normalized under the condition $f$ $(0) =$ $f^{\prime}(0)-1 = 0$ in $\mathbb{U} = \{z:z\in \mathbb{C}$ and $\left \vert z \ \right \vert < 1\}$ and given by the following Taylor-Maclaurin series:

$\begin{equation} f \ (z) = z+\sum \limits_{n = 2}^{\infty}a_{n}z^{n}\ .\ \ \ \ \ \end{equation}$

(1.1)

Further, by $\mathcal{S}$ we shall denote the class of all functions in $\mathcal{A}$ which are univalent in $\mathbb{U}$ .

With a view to recalling the principle of subordination between analytic functions, let the functions $f$ and $g$ be analytic in $\mathbb{U}$ . Then we say that the function $f$ is subordinate to $g$ if there exists a Schwarz function $w\left(z\right)$ , analytic in $\mathbb{U}$ with

$\begin{equation*} \omega \left( 0\right) = 0, \ \left\vert \omega (z)\right\vert < 1, \ \left( z\in \mathbb{U}\right) \end{equation*}$

such that

$\begin{equation*} f \ \left( z\right) = g \ (\omega \left( z\right) ). \end{equation*}$

We denote this subordination by

$\begin{equation*} f\prec g\text{ or}\ f \ (z)\prec g \ (z). \end{equation*}$

In particular, if the function $g$ is univalent in $\mathbb{U}$ , the above subordination is equivalent to

$\begin{equation*} f \ (0) = g \ (0), \ f \ (\mathbb{U})\subset g \ ( \mathbb{U}). \end{equation*}$

The Koebe-One Quarter Theorem ^[11] asserts that image of $\mathbb{U}$ under every univalent function $f\in \mathcal{A}$ contains a disc of radius $\frac{1}{4}.\$ thus every univalent function $f$ has an inverse $\ f^{-1}\$ satisfying $\ f^{-1}(f(z)) = z$ and $f$ $(\ f^{-1}$ $(w)) = w$ $(\left\vert w\right\vert < r$ $_{0}(f$ $), r$ $_{0}(f$ $)$ $> \frac{ 1}{4}$ $)$ , where

$\begin{equation} \ f^{-1}(w) = w-a_{2}w^{2}+(2a_{2}^{2}-a_{3})w^{3}-(5a_{2}^{3}-5a_{2}a_{3}+a_{4})w^{4}+\cdots . \end{equation}$

(1.2)

A function $f\in \mathcal{A}$ is said to be bi-univalent functions in $\mathbb{U}$ if both $f$ and $\ f^{-1}$ are univalent in $\mathbb{U}$ . A function $f\in \mathcal{S}$ is said to be bi-univalent in $\mathbb{U}$ if there exists a function $g\in \mathcal{S}$ such that $g(z)$ is an univalent extension of $f^{-1}$ to $\mathbb{U}$ . Let $\Lambda$ denote the class of bi-univalent functions in $\mathbb{U}$ . The functions $\frac{z}{ 1-z}$ , $-\log (1-z)$ , $\frac{1}{2}\log \left(\frac{1+z}{1-z}\right)$ are in the class $\Lambda$ (see details in ^[20]). However, the familiar Koebe function is not bi-univalent. Lewin ^[17] investigated the class of bi-univalent functions $\Lambda$ and obtained a bound $|a_{2}|\leq 1.51$ . Motivated by the work of Lewin ^[17], Brannan and Clunie ^[9] conjectured that $|a_{2}|\leq \sqrt{2}.$ The coefficient estimate problem for $|a_{n}|\, \, \, \, (n\in \mathbb{N}, \, \, \, \, n\geq 3)$ is still open (^[20]). Brannan and Taha ^[10] also worked on certain subclasses of the bi-univalent function class $\Lambda$ and obtained estimates for their initial coefficients. Various classes of bi-univalent functions were introduced and studied in recent times, the study of bi-univalent functions gained momentum mainly due to the work of Srivastava et al. ^[20]. Motivated by this, many researchers ^[1], ^[4,5,6,7,8], ^[13,14,15], ^[20], ^[21], and ^[27,28,29], also the references cited there in) recently investigated several interesting subclasses of the class $\Lambda$ and found non-sharp estimates on the first two Taylor-Maclaurin coefficients. Recently, many researchers have been exploring bi-univalent functions, few to mention Fibonacci polynomials, Lucas polynomials, Chebyshev polynomials, Pell polynomials, Lucas–Lehmer polynomials, orthogonal polynomials and the other special polynomials and their generalizations are of great importance in a variety of branches such as physics, engineering, architecture, nature, art, number theory, combinatorics and numerical analysis. These polynomials have been studied in several papers from a theoretical point of view (see, for example, ^{[23,24,25,26,27,28,29,30]} also see references therein).

We recall the following results relevant for our study as stated in ^[3].

Let $p(x)$ and $q(x)$ be polynomials with real coefficients. The $(p, q)-$ Lucas polynomials $\mathcal{L}_{p, q, n}(x)$ are defined by the recurrence relation

$\begin{equation*} \mathcal{L}_{p, q, n}(x) = p(x)\mathcal{L}_{p, q, n-1}(x)+q(x)\mathcal{L} _{p, q, n-2}(x)\quad (n\geq 2), \end{equation*}$

from which the first few Lucas polynomials can be found as

$\begin{eqnarray} \mathcal{L}_{p, q, 0}(x) & = &2, \\ \mathcal{L}_{p, q, 1}(x) & = &p(x), \\ \mathcal{L}_{p, q, 2}(x) & = &p^{2}(x)+2q(x), \\ \mathcal{L}_{p, q, 3}(x) & = &p^{3}(x)+3p(x)q(x), ...\text{.} \end{eqnarray}$

(1.3)

For the special cases of $p(x)$ and $q(x),$ we can get the polynomials given $\mathcal{L}_{x, 1, n}(x)\equiv \mathcal{L}_{n}(x)$ Lucas polynomials, $\mathcal{L}_{2x, 1, n}(x)\equiv \mathcal{D}_{n}(x)$ Pell–Lucas polynomials, $\mathcal{L}_{1, 2x, n}(x)\equiv j_{n}(x)$ Jacobsthal–Lucas polynomials, $\mathcal{L}_{3x, -2, n}(x)\equiv F_{n}(x)$ Fermat–Lucas polynomials, $\mathcal{L}_{2x, -1, n}(x)\equiv T_{n}(x)$ Chebyshev polynomials first kind.

Lemma 1.1. ^[16] Let $G\{ \mathcal{L}(x)\}(z)$ be the generating function of the $(p, q)-$ Lucas polynomial sequence $\mathcal{L}_{p, q, n}(x).$ Then,

$\begin{equation*} G\{ \mathcal{L}(x)\}(z) = \sum\limits_{n = 0}^{\infty }\mathcal{L}_{p, q, n}(x)z^{n} = \frac{2-p(x)z}{1-p(x)z-q(x)z^{2}} \end{equation*}$

and

$\begin{equation*} \mathcal{G}_{\{ \mathcal{L}(x)\}}(z) = G\{ \mathcal{L}(x)\}(z)-1 = 1+ \sum\limits_{n = 1}^{\infty }\mathcal{L}_{p, q, n}(x)z^{n} = \frac{1+q(x)z^{2}}{ 1-p(x)z-q(x)z^{2}}. \end{equation*}$

Definition 1.2. ^[22] For $\vartheta \geq 0,$ $\delta \in \mathbb{R},$ $\vartheta +i\delta \neq 0$ and $f\in \mathcal{A}$ , let $\mathcal{B} (\vartheta, \delta)$ denote the class of Bazilevič function if and only if

$\begin{equation*} Re\left[ {\left( {\frac{{z{f^\prime }(z)}}{{f(z)}}} \right){{\left( {\frac{{f(z)}}{z}} \right)}^{\vartheta + i\delta }}} \right] > 0\;. \end{equation*}$

Several authors have researched different subfamilies of the well-known Bazilevič functions of type $\vartheta$ from various viewpoints (see ^[3] and ^[19]). For Bazilevič functions of order $\vartheta +i\delta$ , there is no much work associated with Lucas polynomials in the literature. Initiating an exploration of properties of Lucas polynomials associated with Bazilevič functions of order $\vartheta +i\delta$ is the main goal of this paper. To do so, we take into account the following definitions. In this paper motivated by the very recent work of Altinkaya and Yalcin ^[3] (also see ^[18]) we define a new class $\mathcal{B}(\vartheta, \delta)$ , bi-Bazilevič function of $\Lambda$ based on $(p, q)-$ Lucas polynomials as below:

Definition 1.3. For $f\in \Lambda$ , $\vartheta \geq0,$ $\delta \in \mathbb{R},$ $\vartheta+i\delta \neq0$ and let $\mathcal{B}(\vartheta, \delta)$ denote the class of Bi-Bazilevič functions of order $t\ \$ and type $\vartheta+i\delta$ if only if

$\begin{equation} \left[ \left( \frac{zf^{\prime}(z)}{f(z)}\right) \left( \frac{f(z)}{z} \right) ^{\vartheta+i\delta}\right] \prec \mathcal{G}_{\{ \mathcal{L} (x)\}}(z)\quad (z\in \mathbb{U}) \end{equation}$

(1.4)

and

$\begin{equation} \left[ \left( \frac{zg^{\prime}(w)}{g(w)}\right) \left( \frac{g(w)}{w}% \right) ^{\vartheta+i\delta}\right] \prec \mathcal{G}_{\{ \mathcal{L}% (x)\}}(w)\quad (w\in \mathbb{U}), \label{ieq4} \end{equation}$

(1.5)

where $\mathcal{G}_{\mathcal{L}_{p, q, n}} (z) \in \Phi$ and the function $g$ is described as $g(w) = f^{-1}(w).$

Remark 1.4. We note that for $\delta = 0$ the class $R(\vartheta, 0) = R(\vartheta)$ is defined by Altinkaya and Yalcin ^[2].

The class $\mathcal{B}(0, 0) = \mathcal{S}_\Lambda^*$ is defined as follows:

Definition 1.5. A function $f\in \Lambda$ is said to be in the class $\mathcal{S }_\Lambda^*$ , if the following subordinations hold

$\begin{equation*} \frac{zf^{\prime}(z)}{f(z)}\prec \mathcal{G}_{\{ \mathcal{L}(x)\}}(z)(z\in \mathbb{U}) \end{equation*}$

and

$\begin{equation*} \frac{wg^{\prime }(w)}{g(w)}\prec \mathcal{G}_{\{ \mathcal{L}(x)\}}(w)(w\in \mathbb{U}) \end{equation*}$

where $g(w) = f^{-1}(w).$

We begin this section by finding the estimates of the coefficients $|a_{2}|$ and $|a_{3}|$ for functions in the class $\mathcal{B}(\vartheta, \delta)$ .

2. Coefficient bounds for the function class $\mathcal{B}(\vartheta, \delta)$

Theorem 2.1. Let the function $f(z)$ given by 1.1 be in the class $\mathcal{B}(\vartheta, \delta)$ . Then

$\begin{equation*} \left \vert a_{2}\right \vert \leq \frac{p(x)\sqrt {2p(x)}}{ \sqrt{\vert \{ \left( \left( \vartheta +i\delta \right) ^{2}+3\left( \vartheta +i\delta \right) +2\right)-2\left( \vartheta +i\delta +1\right) ^{2}\}p^2(x)-4q(x)\left( \vartheta +i\delta +1\right) ^{2} \vert }}. \end{equation*}$

and

$\begin{equation*} \left \vert a_{3}\right \vert \leq \frac{p^{2}(x)}{(\vartheta +1)^2+ \delta^2 } +\frac{p(x)}{\sqrt {(\vartheta +2)^2+ \delta^2} } . \end{equation*}$

Proof. Let $f\in \mathcal{B}(\vartheta, \delta, x)$ there exist two analytic functions $u, v:\mathbb{U}\rightarrow \mathbb{U}$ with $u(0) = 0 = v(0)$ , such that $|u(z)| < 1$ , $|v(w)| < 1,$ we can write from (1.4) and (1.5), we have

$\begin{equation} \left[ \left( \frac{zf^{\prime }(z)}{f(z)}\right) \left( \frac{f(z)}{z} \right) ^{\vartheta +i\delta }\right] = \mathcal{G}_{\{ \mathcal{L} (x)\}}(z)\quad (z\in \mathbb{U}) \quad \end{equation}$

(2.1)

and

$\begin{equation} \left[ \left( \frac{zg^{\prime}(w)}{g(w)}\right) \left( \frac{g(w)}{w} \right) ^{\vartheta+i\delta}\right] = \mathcal{G}_{\{ \mathcal{L} (x)\}}(w)\quad (w\in \mathbb{U}), \end{equation}$

(2.2)

It is fairly well known that if

$\begin{equation*} |u(z)| = |u_{1}z+u_{2}z^{2}+\cdots | < 1 \end{equation*}$

and

$\begin{equation*} |v(w)| = |v_{1}w+v_{2}w^{2}+\cdots | < 1. \end{equation*}$

then

$\begin{equation*} |u_{k}|\leq 1\qquad and\qquad |v_{k}|\leq 1\quad (k\in \mathbb{N}) \end{equation*}$

It follows that, so we have

$\begin{eqnarray} \mathcal{G}_{\{ \mathcal{L}(x)\}}(u(z)) & = &1+\mathcal{L}_{p, q, 1}(x)u(z)+ \mathcal{L}_{p, q, 2}(x)u^{2}(z)+\ldots \\ & = &1+\mathcal{L}_{p, q, 1}(x)u_{1}z+[\mathcal{L}_{p, q, 1}(x)u_{2}+\mathcal{L} _{p, q, 2}(x)u_{1}^{2}]z^{2}+\ldots \end{eqnarray}$

(2.3)

and

$\begin{eqnarray} \mathcal{G}_{\{ \mathcal{L}(x)\}}(v(w)) & = &1+\mathcal{L}_{p, q, 1}(x)v(w)+ \mathcal{L}_{p, q, 2}(x)v^{2}(w)+\ldots \\ & = &1+\mathcal{L}_{p, q, 1}(x)v_{1}w+[\mathcal{L}_{p, q, 1}(x)v_{2}+\mathcal{L} _{p, q, 2}(x)v_{1}^{2}]w^{2}+\ldots \end{eqnarray}$

(2.4)

From the equalities (2.1) and (2.2), we obtain that

$\begin{equation} \left[ \left( \frac{zf^{\prime }(z)}{f(z)}\right) \left( \frac{f(z)}{z} \right) ^{\vartheta +i\delta }\right] = 1+\mathcal{L}_{p, q, 1}(x)u_{1}z+[ \mathcal{L}_{p, q, 1}(x)u_{2}+\mathcal{L}_{p, q, 2}(x)u_{1}^{2}]z^{2}+\ldots, \end{equation}$

(2.5)

and

$\begin{equation} \left[ \left( \frac{zg^{\prime}(w)}{g(w)}\right) \left( \frac{g(w)}{w} \right) ^{\vartheta+i\delta}\right] = 1+\mathcal{L}_{p, q, 1}(x)v_{1}w+[ \mathcal{L}_{p, q, 1}(x)v_{2}+\mathcal{L}_{p, q, 2}(x)v_{1}^{2}]w^{2}+\ldots, \end{equation}$

(2.6)

It follows from (2.5) and (2.6) that

$\begin{equation} \left( \vartheta +i\delta +1\right) a_{2} = \mathcal{L}_{p, q, 1}(x)u_1, , \end{equation}$

(2.7)

$\begin{equation} \frac{\left( \vartheta +i\delta -1\right) \left( \vartheta +i\delta +2\right) }{2}a_{2}^{2}-\left( \vartheta +i\delta +2\right) a_3 = \mathcal{L} _{p, q, 1}(x)u_2+\mathcal{L}_{p, q, 2}(x)u_1^2, \end{equation}$

(2.8)

and

$\begin{equation} -\left( \vartheta+i\delta+1\right) a_{2} = \mathcal{L}_{p, q, 1}(x)v_1, \end{equation}$

(2.9)

$\begin{equation} \frac{\left( \vartheta+i\delta+2\right) \left( \vartheta+i\delta+3\right) }{2 }a_{2}^{2}+\left( \vartheta+i\delta+2\right) a_3 = \mathcal{L}_{p, q, 1}(x)v_2+ \mathcal{L}_{p, q, 2}(x)v_1^2, \end{equation}$

(2.10)

From (2.7) and (2.9)

$\begin{equation} u_{1} = -v_{1} \end{equation}$

(2.11)

and

$\begin{equation} 2\left( \vartheta+i\delta+1\right) ^{2}a_{2}^{2} = \mathcal{L} _{p, q, 1}^{2}(x)(u_{1}^{2}+v_{1}^{2})., \end{equation}$

(2.12)

by adding (2.8) to (2.10), we get

$\begin{equation} \left( \left( \vartheta +i\delta \right) ^{2}+3\left( \vartheta +i\delta \right) +2\right) a_{2}^{2} = \mathcal{L}_{p, q, 1}(x)(u_{2}+v_{2})+\mathcal{L} _{p, q, 2}(x)(u_{1}^{2}+v_{1}^{2}), \end{equation}$

(2.13)

by using (2.12) in equality (2.13), we have

$\begin{eqnarray} \left[ \left( \left( \vartheta +i\delta \right) ^{2}+3\left( \vartheta +i\delta \right) +2\right) -\frac{2\mathcal{L}_{p, q, 2}(x)\left( \vartheta +i\delta +1\right) ^{2}}{\mathcal{L}^{2}_{p, q, 1}(x)}\right] a_{2}^{2}& = & \mathcal{L}_{p, q, 1}(x)(u_{2}+v_{2}), \end{eqnarray}$

$\begin{eqnarray} a_{2}^{2}& = &\frac{\mathcal{L}^3_{p, q, 1}(x)(u_{2}+v_{2})} {\left[ \left( \left( \vartheta +i\delta \right) ^{2}+3\left( \vartheta +i\delta \right) +2\right)\mathcal{L}^{2}_{p, q, 1}(x)-2\mathcal{L}_{p, q, 2}(x)\left( \vartheta +i\delta +1\right) ^{2}\right]}. \end{eqnarray}$

(2.14)

Thus, from (1.3) and (2.14) we get

$\begin{equation*} \left \vert a_{2}\right \vert \leq \frac{p(x)\sqrt {2p(x)}}{\sqrt{\vert \{ \left( \left( \vartheta +i\delta \right) ^{2}+3\left( \vartheta +i\delta \right) +2\right)-2\left( \vartheta +i\delta +1\right) ^{2}\}p^2(x)-4q(x)\left( \vartheta +i\delta +1\right) ^{2} \vert }}. \end{equation*}$

Next, in order to find the bound on $|a_{3}|$ , by subtracting (2.10) from (2.8), we obtain

$\begin{eqnarray} 2\left( \vartheta +i\delta +2\right) a_{3}-2\left( \vartheta +i\delta +2\right) a_{2}^{2}& = &\mathcal{L}_{p, q, 1}(x)(u_{2}-v_{2})+\mathcal{L} _{p, q, 2}(x)(u_{1}^{2}-v_{1}^{2}) \\ 2\left( \vartheta +i\delta +2\right) a_{3}& = &\mathcal{L} _{p, q, 1}(x)(u_{2}-v_{2})+2\left( \vartheta +i\delta +2\right) a_{2}^{2} \\ a_{3}& = &\frac{\mathcal{L}_{p, q, 1}(x)(u_{2}-v_{2})}{2\left( \vartheta +i\delta +2\right)}+a_{2}^{2} \end{eqnarray}$

(2.15)

Then, in view of (2.11) and (2.12), we have from (2.15)

$\begin{equation*} a_{3} = \frac{\mathcal{L}_{p, q, 1}^{2}(x)}{2\left( \vartheta +i\delta +2\right) ^{2}}(u_{1}^{2}+v_{1}^{2})+\frac{\mathcal{L}_{p, q, 1}(x)}{2\left( \vartheta +i\delta +2\right) }(u_{2}-v_{2}). \end{equation*}$

$\begin{eqnarray*} |a_{3}|&\leq&\frac{p^{2}(x)}{\left|\vartheta +i\delta +1\right| ^{2}}+\frac{ p(x)}{|\vartheta +i\delta +2| } \notag \\ & = &\frac{p^{2}(x)}{(\vartheta +1)^2+ \delta^2 } +\frac{p(x)}{\sqrt { (\vartheta +2)^2+ \delta^2} } \end{eqnarray*}$

This completes the proof.

Taking $\delta = 0,$ in Theorem 2.1, we get the following corollary.

Corollary 2.2. Let the function $f(z)$ given by (1.1) be in the class $\mathcal{B} (\vartheta)$ . Then

$\begin{equation*} \left \vert a_{2}\right \vert \leq \frac{p(x)\sqrt {2p(x)}}{\sqrt{\vert \{ \left( \vartheta ^{2}+3\vartheta +2\right)-2\left( \vartheta+1\right) ^{2}\}p^2(x)-4q(x)\left( \vartheta +1\right) ^{2} \vert }} \end{equation*}$

and

$\begin{equation*} \left \vert a_{3}\right \vert \leq \frac{p^{2}(x)}{(\vartheta +2)^2} +\frac{ p(x)}{\vartheta +2} \end{equation*}$

Also, taking $\vartheta = 0$ and $\delta = 0,$ in Theorem 2.1, we get the results given in ^[18].

3. Fekete-Szegö inequality for the class $\mathcal{B}(\vartheta, \delta)$

Fekete-Szegö inequality is one of the famous problems related to coefficients of univalent analytic functions. It was first given by ^[12], the classical Fekete-Szegö inequality for the coefficients of $f\in \mathcal{S}$ is

$\begin{equation*} \left \vert a_{3}-\mu a_{2}^{2}\right \vert \leq1+2\exp \left( -2\mu /(1-\mu)\right) \text{ for }\mu \in \left[ 0, 1\right) . \end{equation*}$

As ${\mu} \rightarrow1^{-}$ , we have the elementary inequality $\left \vert a_{3}-a_{2}^{2}\right \vert \leq1$ . Moreover, the coefficient functional

$\begin{equation*} \varsigma_{\mu}(f) = a_{3}-\mu a_{2}^{2} \end{equation*}$

on the normalized analytic functions $f$ in the unit disk $\mathbb{U}$ plays an important role in function theory. The problem of maximizing the absolute value of the functional $\varsigma {\mu} (f)$ is called the Fekete-Szegö problem.

In this section, we are ready to find the sharp bounds of Fekete-Szegö functional $\varsigma _{\mu }(f)$ defined for $f\in \mathcal{B}(\vartheta, \delta)$ given by (1.1).

Theorem 3.1. Let f given by (1.1) be in the class $\mathcal{B} (\vartheta, \delta)$ and ${\mu} \in \mathbb{R}$ . Then

$\begin{equation*} \left \vert a_{3}- {\mu} a_{2}^{2}\right \vert \leq \left \{ \begin{array}{c} \frac{p(x)}{\sqrt{(\vartheta+2)^{2}+\delta ^{2}}}, \ \ \ \ \ \ \ \ 0\leq \left \vert h(\mu )\right \vert \leq \frac{1}{2\sqrt{ (\vartheta+2)^{2}+\delta ^{2}}} \\ 2p(x)\left \vert h(\mu )\right \vert , \ \ \ \ \ \ \ \ \ \ \ \ \ \left \vert h(\mu )\right \vert \geq \frac{1}{2\sqrt{(\vartheta+2)^{2}+ \delta ^{2}}} \end{array} \right. \end{equation*}$

where

$\begin{equation*} h(\mu) = \frac{\mathcal{L}^2_{p, q, 1}(x)(1-\mu)}{\left( \left( \vartheta +i\delta \right) ^{2}+3\left( \vartheta +i\delta \right) +2\right)\mathcal{L} ^{2}_{p, q, 1}(x)-2\mathcal{L}_{p, q, 2}(x)\left( \vartheta +i\delta +1\right) ^{2}}. \end{equation*}$

Proof. From (2.14) and (2.15), we conclude that

$\begin{eqnarray*} a_{3}- {\mu} a_{2}^{2}& = &(1-\mu )\frac{\mathcal{L}^3_{p, q, 1}(x)(u_{2}+v_{2})} {\left[ \left( \left( \vartheta +i\delta \right) ^{2}+3\left( \vartheta +i\delta \right) +2\right)\mathcal{L}^{2}_{p, q, 1}(x)-2\mathcal{L}_{p, q, 2}(x)\left( \vartheta +i\delta +1\right) ^{2}\right]} \\ &+&\frac{\mathcal{L}_{p, q, 1}(x)}{2\left( \vartheta +i\delta +2\right) } (u_{2}-v_{2}) \end{eqnarray*}$

$\begin{eqnarray*} = \mathcal{L}_{p, q, 1}(x)\left[ \left( h(\mu )+\frac{1}{2(\vartheta +i\delta +2)}\right) u_{2}+\left( h(\mu )-\frac{1}{2(\vartheta +i\delta +2)}\right) v_{2}\right] \end{eqnarray*}$

where

Then, in view of (1.3), we obtain

We end this section with some corollaries.

Taking ${\mu} = 1$ in Theorem 3.1, we get the following corollary.

Corollary 3.2. If $f\in \mathcal{B}(\vartheta, \delta)$ , then

$\begin{equation*} \left \vert a_{3}-a_{2}^{2}\right \vert \leq \frac{p(x)}{\sqrt{ (\vartheta+2)^{2}+\delta ^{2}}}. \end{equation*}$

Taking $\delta = 0$ in Theorem 3.1, we get the following corollary.

Corollary 3.3. Let $f$ given by (1.1) be in the class $\mathcal{B}(\vartheta, 0)$ . Then

$\begin{equation*} \left \vert a_{3}- {\mu} a_{2}^{2}\right \vert \leq \left \{ \begin{array}{c} \frac{p(x)}{\vartheta+2}, \ \ \ \ \ \ \ \ 0\leq \left \vert h(\mu )\right \vert \leq \frac{1}{2(\vartheta+2)} \\ 2p(x)\left \vert h(\mu )\right \vert , \ \ \ \ \ \ \ \ \ \ \ \ \ \left \vert h(\mu )\right \vert \geq \frac{1}{2(\vartheta+2)} \end{array} \right. \end{equation*}$

Also, taking $\vartheta = 0,$ $\delta = 0$ and ${\mu} = 1$ in Theorem 3.1, we get the following corollary.

Corollary 3.4. Let $f$ given by (1.1) be in the class $\mathcal{B}$ . Then

$\begin{equation*} \left| {{a_3} - a_2^2} \right| \le \frac{{p(x)}}{2}. \end{equation*}$

Conflict of interest

All authors declare no conflicts of interest in this paper.

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