A comprehensive transfer news headline generation method based on semantic prototype transduction

Ting-Huai Ma; Xin Yu; Huan Rong; Ting-Huai Ma; Xin Yu; Huan Rong

doi:10.3934/mbe.2023055

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 1: 1195-1228. doi: 10.3934/mbe.2023055

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Research article Special Issues

A comprehensive transfer news headline generation method based on semantic prototype transduction

1.
School of Computer Science, Nanjing University of Information Science & Technology, Nanjing 210044, China
2.
School of Artificial Intelligence (School of Future Technology), Nanjing University of Information Science & Technology, Nanjing 210044, China

Academic Editor: Vladimir Mityushev

Received: 09 August 2022 Revised: 11 October 2022 Accepted: 14 October 2022 Published: 26 October 2022

Most current deep learning-based news headline generation models only target domain-specific news data. When a new news domain appears, it is usually costly to obtain a large amount of data with reference truth on the new domain for model training, so text generation models trained by traditional supervised approaches often do not generalize well on the new domain—inspired by the idea of transfer learning, this paper designs a cross-domain transfer text generation method based on domain data distribution alignment, intermediate domain redistribution, and zero-shot learning semantic prototype transduction, focusing on the data problem with no reference truth in the target domain. Eventually, the model can be guided by the most relevant source domain data to generate headlines from the target domain news text through the semantic correlation between source and target domain data during the training process of generating headlines for the target domain news, even without any reference truth of the news headlines in the target domain, which improves the usability of the text generation model in real scenarios. The experimental results show that the proposed transfer text generation method has a good domain transfer effect and outperforms other existing transfer text generation methods in various text generation evaluation indexes, proving the proposed method's effectiveness in this paper.

Keywords:

Citation: Ting-Huai Ma, Xin Yu, Huan Rong. A comprehensive transfer news headline generation method based on semantic prototype transduction[J]. Mathematical Biosciences and Engineering, 2023, 20(1): 1195-1228. doi: 10.3934/mbe.2023055

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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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