Boundary distribution estimation for precise object detection

Peng Zhi; Haoran Zhou; Hang Huang; Rui Zhao; Rui Zhou; Qingguo Zhou; Peng Zhi; Haoran Zhou; Hang Huang; Rui Zhao; Rui Zhou; Qingguo Zhou

doi:10.3934/era.2023257

Electronic Research Archive

2023, Volume 31, Issue 8: 5025-5038. doi: 10.3934/era.2023257

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Research article Special Issues

Boundary distribution estimation for precise object detection

School of Information Science and Engineering, Lanzhou University, Lanzhou, China

Received: 30 May 2023 Revised: 20 June 2023 Accepted: 24 June 2023 Published: 14 July 2023

In the field of state-of-the-art object detection, the task of object localization is typically accomplished through a dedicated subnet that emphasizes bounding box regression. This subnet traditionally predicts the object's position by regressing the box's center position and scaling factors. Despite the widespread adoption of this approach, we have observed that the localization results often suffer from defects, leading to unsatisfactory detector performance. In this paper, we address the shortcomings of previous methods through theoretical analysis and experimental verification and present an innovative solution for precise object detection. Instead of solely focusing on the object's center and size, our approach enhances the accuracy of bounding box localization by refining the box edges based on the estimated distribution at the object's boundary. Experimental results demonstrate the potential and generalizability of our proposed method.

Keywords:

Citation: Peng Zhi, Haoran Zhou, Hang Huang, Rui Zhao, Rui Zhou, Qingguo Zhou. Boundary distribution estimation for precise object detection[J]. Electronic Research Archive, 2023, 31(8): 5025-5038. doi: 10.3934/era.2023257

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Abstract

1. Introduction

Let $\mathcal{A}_{p}$ denote the class of analytic and $p$ -valent functions $f(z)$ with the next form

$\begin{equation} f(z) = z^{p}+\sum\limits^{\infty}_{n = 1}a_{n+p}z^{n+p}, \; \; \; \; (p\in\mathbb{N} = \{1, 2, \cdots\}) \end{equation}$

(1.1)

in the open unit disk $\Delta = \{z\in\mathbb{C}: \mid z\mid < 1\}$ .

For $f\in\mathcal{A}_{p}$ , its $q$ -derivative or the $q$ -difference $\mathcal{D}_{q}f(z)$ is given by

$\begin{equation*} \mathcal{D}_{q}f(z) = [p]_{q}z^{p-1}+\sum\limits^{\infty}_{n = 1}[n+p]_{q}a_{n+p}z^{n+p-1}, \; \; \; \; (0 \lt q \lt 1), \end{equation*}$

where the $q$ -derivative operator $\mathcal{D}_{q}f(z)$ (refer to ^[13] and ^[14]) of the function $f$ is defined by

$\mathcal{D}_{q}f(z): = \left\{\begin{array}{ll} \frac{f(z)-f(qz)}{(1-q)z}, \; \; &(z\neq0; 0 \lt q \lt 1), \\ f'(0), \; \; &(z = 0) \end{array}\right.$

provided that $f'(0)$ exists, and the $q$ -number $[n]_{q}$ is just $[\chi]_{q}$ when $\chi = n\in\mathbb{N}$ , here

$[\chi]_{q} = \left\{\begin{array}{ll} \frac{1-q^{\chi}}{1-q}\; \; \; &\mbox{for}\; \; \chi\in\mathbb{C}, \\ \sum^{\chi-1}_{k = 0}q^{k}\; \; \; &\mbox{for}\; \; \chi = n\in\mathbb{N}. \end{array}\right.$

Note that $\mathcal{D}_{q}f(z)\longrightarrow f'(z)$ when $q\longrightarrow1_{-}$ , where $f'$ is the ordinary derivative of the function $f$ .

Consider the generalized Bernardi integral operator $\mathcal{J}^{\eta}_{p, q}:\mathcal{A}_{p}\longrightarrow\mathcal{A}_{p}$ with the next form

$\begin{equation} \mathcal{J}^{\eta}_{p, q}f(z) = \frac{[p+\eta]_{q}}{z^{\eta}}\int^{z}_{0}t^{\eta-1}f(t)d_{q}t, \; \; (z\in\Delta, \Re\eta \gt -1\; \text{and}\; f\in\mathcal{A}_{p}). \end{equation}$

(1.2)

Then, for $f\in\mathcal{A}_{p}$ , we obtain that

$\begin{equation} \mathcal{J}^{\eta}_{p, q}f(z) = z^{p}+\sum\limits^{\infty}_{n = 1}L^{\eta}_{p, q}(n)a_{n+p}z^{n+p}, \; \; (z\in\Delta), \end{equation}$

(1.3)

where

$\begin{equation} L^{\eta}_{p, q}(n) = \frac{[p+\eta]_{q}}{[n+\eta]_{q}}: = L_{n}. \end{equation}$

(1.4)

Here we remark that if $p = 1$ , it is exactly $q$ -Bernardi integral operator $\mathcal{J}^{\eta}_{q}$ ^[21]. Further, if $p = 1$ and $q\rightarrow1_{-}$ , obviously it is the classical Bernardi integral operator $\mathcal{J}_{\eta}$ ^[5]. In fact, Alexander ^[1] and Libera ^[18] integral operators are special versions of $\mathcal{J}_{\eta}$ for $\eta = 0$ and $\eta = 1$ , respectively.

For two analytic functions $f$ and $g$ , if there exists an analytic function $h$ satisfying $h(0) = 0$ and $\mid h(z)\mid < 1$ for $z\in\Delta$ so that $f(z) = g(h(z))$ , then $f$ is subordinate to $g$ , i.e., $f\prec g$ .

Let $\Lambda$ be the class of all analytic function $\phi$ via the form

$\begin{equation} \phi(z) = 1+\sum\limits^{\infty}_{n = 1}A_{n}z^{n}, \; \; (A_{1} \gt 0, z\in\Delta). \end{equation}$

(1.5)

It is well known that the $q$ -calculus ^[13,14], even the $(p, q)$ -calculus ^[6], is a generalization of the ordinary calculus without the limit symbol, and its related theory has been applied into mathematical, physical and engineering fields (see ^[11,15,25]). Since Ismail et al.^[12] firstly utilized the $q$ -derivative operator to investigate the $q$ -calculus of the class of starlike functions in disk, there had a great deal of work in this respect; for example, refer to Rehman et al. ^[24] for partial sums of generalized $q$ -Mittag-Leffler functions, Srivastava et al. ^[31] for Fekete-Szegö inequality for classes of $(p, q)$ -starlike and $(p, q)$ -convex functions and ^[27] for close-to-convexity of a certain family of $q$ -Mittag-Leffler functions, Seoudy and Aouf ^[26] for the coefficient estimates of $q$ -starlike and $q$ -convex functions and Uçar ^[33] for the coefficient inequality for $q$ -starlike functions. Besides, by involving some special functions and operators or increasing the complexity of function classes, many new subclasses of analytic functions associated with $q$ -calculus or $(p, q)$ -calculus were considered. Here we may refer to ^[9,23], Ahmad et al. ^[2] for convolution properties for a family of analytic functions involving $q$ -analogue of Ruscheweyh differential operator, Dweby and Darus ^[8] for subclass of harmonic univalent functions associated with $q$ -analogue of Dziok-Srivastava operator, Mahmmod and Sokól ^[20] for new subclass of analytic functions in conical domain associated with Ruscheweyh q-differential operator, Srivastava et al. ^[28] for coefficient inequalities for $q$ -starlike functions associated with the Janowski functions, ^[31] for Fekete-Szegö inequality for classes of $(p, q)$ -starlike and $(p, q)$ -convex functions using the $q$ -Bernardi integral operator and ^[32] for some results on the $q$ -analogues of the incomplete Fibonacci and Lucas polynomials. For the multivalent functions, Purohit ^[22] ever studied a new class of multivalently analytic functions associated with fractional $q$ -calculus operators, while Shi et al. ^[29] investigated the multivalent $q$ -starlike functions connected with circular domain. Moreover, Arif et al. ^[4] considered a $q$ -analogue of the Ruscheweyh type operator, and Srivastava et al. ^[30] dealt with basic and fractional q-calculus and associated Fekete-Szegö problems for $p$ -valently $q$ -starlike functions and $p$ -valently $q$ -convex functions of complex order using certain integral operators, and Khan et al. ^[17] for a new integral operator in $q$ -analog for multivalent functions. Stimulated by the previous results, in the paper we intend to introduce and investigate several new subclasses of $q$ -starlike and $q$ -convex type analytic and multivalent functions involving a generalized Bernardi integral operator, and establish the corresponding Fekete-Szegö type functional inequalities for these function classes. Besides, the corresponding bound estimates of the coefficients $a_{p+1}$ and $a_{p+2}$ are provided.

From now on we introduce some general subclasses of analytic and multivalent functions associated with the $q$ -derivative operator and the generalized Bernardi integral operator.

Definition 1.1. Let $f(z)\in\mathcal{A}_{p}$ and $\mu, \lambda\geq0$ . If the following subordination

$\begin{equation} (1-\lambda)\left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu}+\frac{\lambda \mathcal{D}_{q}\left(\mathcal{J}^{\eta}_{p, q}f\right)(z)}{[p]_{q}z^{p-1}}\left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu-1}\prec\phi(z) \end{equation}$

(1.6)

is satisfied for $z\in\Delta$ , then we call that $f(z)$ belongs to the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ .

Definition 1.2. Let $f(z)\in\mathcal{A}_{p}$ and $0\leq\lambda\leq1$ . If the following subordination

$\begin{equation} (1-\lambda)\frac{z\mathcal{D}_{q}(\mathcal{J}^{\eta}_{p, q}f)(z)}{[p]_{q}\mathcal{J}^{\eta}_{p, q}f(z)}+\frac{\lambda}{[p]_{q}} \left(1+\frac{qz\mathcal{D}_{q}[\mathcal{D}_{q}(\mathcal{J}^{\eta}_{p, q}f)](z)}{\mathcal{D}_{q}(\mathcal{J}^{\eta}_{p, q}f)(z)}\right)\prec\phi(z) \end{equation}$

(1.7)

is satisfied for $z\in\Delta$ , then we call that $f(z)$ belongs to the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ .

Definition 1.3. Let $f(z)\in\mathcal{A}_{p}$ and $\mu\geq0$ . If the following subordination

$\begin{equation} \left(\frac{z\mathcal{D}_{q}\left(\mathcal{J}^{\eta}_{p, q}f\right)(z)}{[p]_{q}\mathcal{J}^{\eta}_{p, q}f(z)}\right)\left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu}\prec\phi(z) \end{equation}$

(1.8)

is satisfied for $z\in\Delta$ , then we call that $f(z)$ belongs to the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ .

Remark 1.4. If we put

$\begin{equation*} \phi(z) = \left(\frac{1+z}{1-z}\right)^{\alpha}\; for\; 0 \lt \alpha\leq1 \end{equation*}$

$\begin{equation*} \phi(z) = \frac{1+(1-2\beta)z}{1-z}\; \; for\; 0\leq\beta \lt 1 \end{equation*}$

in Definition (1.1–1.3), then the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ (res. $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ and $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ ) reduces to $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \alpha)$ (res. $\mathcal{LM}^{\eta}_{p, q}(\lambda; \alpha)$ and $\mathcal{NS}^{\eta}_{p, q}(\mu; \alpha)$ ) or $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \beta)$ (res. $\mathcal{LM}^{\eta}_{p, q}(\lambda; \beta)$ and $\mathcal{NS}^{\eta}_{p, q}(\mu; \beta)$ ). Without the generalized Bernardi integral operator, the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ (res. $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ and $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ ) is the classical function class $\mathcal{LN}(\mu, \lambda; \phi)$ (res. $\mathcal{LM}(\lambda; \phi)$ and $\mathcal{NS}(\mu; \phi)$ ) when $p = 1$ and $q\rightarrow1_{-}$ .

Let $\Omega$ be the class of functions $\omega(z)$ denoted by

$\begin{equation} \omega(z) = \sum\limits^{\infty}_{n = 1}E_{n}z^{n}, \; \; (z\in\Delta) \end{equation}$

(1.9)

via the inequality $\vert\omega(z)\vert < 1(z\in\Delta)$ . Now we recall some necessary Lemmas below.

Lemma 1.5 (^[16]). Let the function $\omega\in\Omega$ . Then

$\begin{equation*} \vert E_{2}-\tau E^{2}_{1}\vert\leq\max\{1, \vert\tau\vert\}, \; \; \; \; (\tau\in\mathbb{C}). \end{equation*}$

Specially, the sharp result holds for the next function

$\begin{equation*} \omega(z) = z\; \; \; \; \mathit{\text{or}}\; \; \; \; \omega(z) = z^{2}, \; \; \; \; (z\in\Delta). \end{equation*}$

Lemma 1.6 (^[7,10]). Let $\mathcal{P}$ be the class of all analytic functions $h(z)$ of the following form

$\begin{equation*} h(z) = 1+\sum\limits^{\infty}_{n = 1}c_{n}z^{n}, \; \; (z\in\Delta) \end{equation*}$

satisfying $\Re h(z) > 0$ and $h(0) = 1$ . Then there exist the sharp coefficient estimates $\mid c_{n}\mid\leq2(n\in\mathbb{N})$ . In Particular, the equality holds for all $n$ for the next function

$\begin{equation*} h(z) = \frac{1+z}{1-z} = 1+\sum\limits^{\infty}_{n = 1}2z^{n}. \end{equation*}$

Lemma 1.7 (^[3,19]). Let the function $\omega\in\Omega$ . Then

$\mid E_{2}-\kappa E^{2}_{1}\mid\leq\left\{\begin{array}{ll} -\kappa\; \; &\mathit{\mbox{if}}\; \; \kappa\leq-1, \\ 1\; \; &\mathit{\mbox{if}}\; \; -1\leq\kappa\leq1, \\ \kappa\; \; &\mathit{\mbox{if}}\; \; \kappa\geq1. \end{array}\right.$

For $\kappa < -1$ or $\kappa > 1$ , the inequality holds literally if and only if $\omega(z) = z$ or one of its rotations. If $< \kappa < 1$ , the inequality holds literally if and only if $\omega(z) = z^{2}$ or one of its rotations. In Particular, if $\kappa = -1$ , then the sharp result holds for the next function

$\begin{equation*} \omega(z) = \frac{z(z+\xi)}{1+\xi z}, \; \; \; \; (0\leq\xi\leq1) \end{equation*}$

or one of its rotations. If $\kappa = 1$ , then the sharp result holds for the next function

$\begin{equation*} \omega(z) = -\frac{z(z+\xi)}{1+\xi z}, \; \; \; \; (0\leq\xi\leq1) \end{equation*}$

or one of its rotations. If $-1 < \kappa < 1$ , then the upper bound is sharp as the followings

$\begin{equation*} \vert E_{2}-\kappa E^{2}_{1}\vert+(\kappa+1)\vert E_{1}\vert^{2}\leq1, \; \; \; \; (-1 \lt \kappa\leq0) \end{equation*}$

and

$\begin{equation*} \vert E_{2}-\kappa E^{2}_{1}\vert+(1-\kappa)\vert E_{1}\vert^{2}\leq1, \; \; \; \; (0 \lt \kappa \lt 1). \end{equation*}$

2. Functional estimates for the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$

By (1.9) we give that

$\begin{eqnarray} \phi(\omega(z))& = &1+A_{1}E_{1}z+(A_{1}E_{2}+A_{2}E^{2}_{1})z^{2}\\ &+&(A_{1}E_{3}+2A_{2}E_{1}E_{2}+A_{3}E^{3}_{1})z^{3}+\ldots. \end{eqnarray}$

(2.1)

In the section, with Lemma 1.5 we study Fekete-Szegö functional problem for the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ and provide the following theorem.

Theorem 2.1. Let $\delta\in\mathbb{C}$ . If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ , then

$\begin{equation*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert\leq\frac{A_{1}[p]_{q}}{\vert L_{2}\vert[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}\max\left\{1;\left\vert\frac{A_{1}[p]_{q}\Theta}{2L^{2}_{1}[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}}-\frac{A_{2}}{A_{1}}\right\vert\right\}, \end{equation*}$

where

$\begin{eqnarray*} \Theta& = &2\delta L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]\notag\\ &+&L^{2}_{1}[(\mu-1)[\mu-\lambda(\lambda\mu-\mu+1)][p]_{q} +2\lambda(2\mu-\lambda\mu-1)[p+1]_{q}]. \end{eqnarray*}$

Moreover, the sharp result holds for the next function

$\begin{equation*} \omega(z) = z\; \; \; \; \mathit{\text{or}}\; \; \; \; \omega(z) = z^{2}, \; \; \; \; (z\in\Delta). \end{equation*}$

Proof. Assume that $f(z)\in\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ . Then, from Definition 1.1, there exists an analytic function $\omega(z)\in\Omega$ such that

(2.2)

Part case

$\begin{eqnarray*} &&\left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu-1}\\ & = &1+(\mu-1)L_{1}a_{p+1}z+\left[(\mu-1)L_{2}a_{p+2}+\frac{(\mu-1)(\mu-2)}{2}L^{2}_{1}a^{2}_{p+1}\right]z^{2}\notag\\ &+&\left[(\mu-1) L_{3}a_{p+3}+\frac{(\mu-1)(\mu-2)}{2}L_{1}L_{2}a_{p+1}a_{p+2}+\frac{(\mu-1)(\mu-2)(\mu-3)}{6}L^{3}_{1}a^{3}_{p+1}\right]z^{3}+\ldots, \end{eqnarray*}$

$\begin{eqnarray*} &&\frac{\lambda \mathcal{D}_{q}\left(\mathcal{J}^{\eta}_{p, q}f\right)(z)}{[p]_{q}z^{p-1}}\\ & = &\lambda+\frac{\lambda L_{1}a_{p+1}[p+1]_{q}}{[p]_{q}}z+\frac{\lambda L_{2}a_{p+2}[p+2]_{q}}{[p]_{q}}z^{2}\notag\\ &+&\frac{\lambda L_{3}a_{p+3}[p+3]_{q}}{[p]_{q}}z^{3}+\ldots, \end{eqnarray*}$

$\begin{eqnarray*} &&\frac{\lambda \mathcal{D}_{q}\left(\mathcal{J}^{\eta}_{p, q}f\right)(z)}{[p]_{q}z^{p-1}} \left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu-1}\\ & = &\lambda+\lambda\left[\mu+\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)\right]L_{1}a_{p+1}z\notag\\ &+&\lambda\left\{\left[\mu+\left(\frac{[p+2]_{q}}{[p]_{q}}-1\right)\right]L_{2}a_{p+2}+(\mu-1)\left[\frac{\mu}{2}+\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)\right]L^{2}_{1}a^{2}_{p+1}\right\}z^{2}+\ldots, \end{eqnarray*}$

$\begin{eqnarray*} &&(1-\lambda)\left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu}\\ & = &(1-\lambda)+(1-\lambda)\mu L_{1}a_{p+1}z+(1-\lambda)\left[\mu L_{2}a_{p+2}+\frac{\mu(\mu-1)}{2}L^{2}_{1}a^{2}_{p+1}\right]z^{2}\notag\\ &+&(1-\lambda)\left[\mu L_{3}a_{p+3}+\frac{\mu(\mu-1)}{2}L_{1}L_{2}a_{p+1}a_{p+2}+\frac{\mu(\mu-1)(\mu-2)}{6}L^{3}_{1}a^{3}_{p+1}\right]z^{3}+\ldots. \end{eqnarray*}$

Since

$\begin{eqnarray*} &&(1-\lambda)\left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu}+\frac{\lambda \mathcal{D}_{q}\left(\mathcal{J}^{\eta}_{p, q}f\right)(z)}{[p]_{q}z^{p-1}} \left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu-1}\\ & = &1+\left[\mu+\lambda\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)\right]L_{1}a_{p+1}z+\{\left[\mu+\lambda\left(\frac{[p+2]_{q}}{[p]_{q}}-1\right)\right]L_{2}a_{p+2} \notag\\ &+&\left[\frac{(\mu-1)}{2}[\mu-2\lambda(\lambda\mu-\mu+1)]+\lambda(2\mu-\lambda\mu-1)\frac{[p+1]_{q}}{[p]_{q}}\right]L^{2}_{1}a^{2}_{p+1}\}z^{2}+\ldots, \end{eqnarray*}$

by (2.1) and (2.2) we see that

$\begin{equation*} A_{1}E_{1} = \left[\mu+\lambda\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)\right]L_{1}a_{p+1}, \end{equation*}$

$\begin{eqnarray*} A_{1}E_{2}+A_{2}E^{2}_{1}& = &\left[\mu+\lambda\left(\frac{[p+2]_{q}}{[p]_{q}}-1\right)\right]L_{2}a_{p+2}\notag\\ &+&\left[\frac{\mu-1}{2}[\mu-2\lambda(\lambda\mu-\mu+1)]+\lambda(2\mu-\lambda\mu-1)\frac{[p+1]_{q}}{[p]_{q}}\right]L^{2}_{1}a^{2}_{p+1}. \end{eqnarray*}$

Thereby

$\begin{equation} a_{p+1} = \frac{A_{1}E_{1}[p]_{q}}{L_{1}[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]} \end{equation}$

(2.3)

and

$\begin{eqnarray} a_{p+2}& = &\frac{(A_{1}E_{2}+A_{2}E^{2}_{1})[p]_{q}}{L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}\\ &-&\frac{A^{2}_{1}E^{2}_{1}[p]^{2}_{q}\{(\mu-1)[\mu-2\lambda(\lambda\mu-\mu+1)][p]_{q}+2\lambda(2\mu-\lambda\mu-1)[p+1]_{q}\}}{2L_{2}[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}. \end{eqnarray}$

(2.4)

Further, with (2.3) and (2.4) we obtain that

$\begin{equation*} a_{p+2}-\delta a^{2}_{p+1} = \frac{A_{1}[p]_{q}}{L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}[ E_{2}-\hbar E^{2}_{1}], \end{equation*}$

where

$\begin{eqnarray*} \hbar& = &\{2\delta L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]+L^{2}_{1}[(\mu-1)[\mu-\lambda(\lambda\mu-\mu+1)][p]_{q}\notag\\ &+&2\lambda(2\mu-\lambda\mu-1)[p+1]_{q}]\}\frac{A_{1}[p]_{q}}{2L^{2}_{1}[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}}-\frac{A_{2}}{A_{1}}. \end{eqnarray*}$

Therefore, according to Lemma 1.5 we finish the proof of Theorem 2.1.

Corollary 2.2. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ , then

$\begin{eqnarray*} \vert a_{p+2}\vert&\leq&\frac{A_{1}[p]_{q}}{\vert L_{2}\vert[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}\\ &\times&\max\left\{1;\left\vert\frac{(\mu-1)[\mu-\lambda(\lambda\mu-\mu+1)][p]_{q}+2\lambda(2\mu-\lambda\mu-1)[p+1]_{q}}{2[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}}-\frac{A_{2}}{A_{1}}\right\vert\right\}. \end{eqnarray*}$

Moreover, the sharp result holds for the next function

$\begin{equation*} \omega(z) = z\; \; \; \; \mathit{\text{or}}\; \; \; \; \omega(z) = z^{2}, \; \; \; \; (z\in\Delta). \end{equation*}$

When $\phi\in\mathcal{P}$ , combining (2.3) and (2.4) with Lemma 1.6 we instantly establish the next corollary for the coefficient bounds of $a_{p+1}$ and $a_{p+2}$ .

Corollary 2.3. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ , then

$\begin{equation*} \vert a_{p+1}\vert\leq\frac{2\vert E_{1}\vert[p]_{q}}{\vert L_{1}\vert[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]} \end{equation*}$

and

$\begin{eqnarray*} \vert a_{p+2}\vert&\leq &\frac{2(\vert E_{2}\vert+\vert E_{1}\vert^{2})[p]_{q}}{\vert L_{2}\vert[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}\notag\\ &+&\frac{2E^{2}_{1}[p]^{2}_{q}\vert(\mu-1)[\mu-\lambda(\lambda\mu-\mu+1)][p]_{q}+2\lambda(2\mu-\lambda\mu-1)[p+1]_{q}\vert}{\vert L_{2}\vert[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}. \end{eqnarray*}$

If we choose real $\delta$ and $\eta$ , then by Lemma 1.7 we derive the next result for Fekete-Szegö problem.

Theorem 2.4. Let $\delta, \eta\in\mathbb{R}$ and $\phi\in\Lambda$ satisfying

$\begin{equation*} \phi(z) = 1+\sum\limits^{\infty}_{n = 1}A_{n}z^{n}, \; \; (A_{1}, A_{2} \gt 0, z\in\Delta). \end{equation*}$

If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ , then

$\vert a_{p+2}-\delta a^{2}_{p+1}\vert\leq\left\{\begin{array}{ll} \frac{[p]_{q}}{ L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}\left\{A_{2}-\frac{A^{2}_{1}[p]_{q}\Theta}{2L^{2}_{1}\vert\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}}\right\}, \; (\delta\leq\Upsilon_{1});\\ \frac{A_{1}[p]_{q}}{ L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}, \; (\Upsilon_{1}\leq\delta\leq\Upsilon_{2});\\ \frac{[p]_{q}}{ L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}\left\{-A_{2}+\frac{A^{2}_{1}[p]_{q}\Theta}{2L^{2}_{1}\vert\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}}\right\}, \; (\delta\geq\Upsilon_{2}), \end{array}\right.$

where

$\begin{eqnarray*} \Upsilon_{1}& = &\frac{(A_{2}-A_{1})L^{2}_{1}[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}} {A^{2}_{1}L_{2}[p]_{q}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}\notag\\ &-&\frac{L^{2}_{1}[(\mu-1)[\mu-\lambda(\lambda\mu-\mu+1)][p]_{q}+2\lambda(2\mu-\lambda\mu-1)[p+1]_{q}]} {2L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]} \end{eqnarray*}$

and

$\begin{eqnarray*} \Upsilon_{2}& = &\frac{(A_{2}+A_{1})L^{2}_{1}[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}} {A^{2}_{1}L_{2}[p]_{q}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}\notag\\ &-&\frac{L^{2}_{1}[(\mu-1)[\mu-\lambda(\lambda\mu-\mu+1)][p]_{q}+2\lambda(2\mu-\lambda\mu-1)[p+1]_{q}]} {2L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}. \end{eqnarray*}$

Moreover, we take

$\begin{eqnarray*} \Upsilon_{3}& = &\frac{A_{2}L^{2}_{1}[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}} {A^{2}_{1}L_{2}[p]_{q}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}\notag\\ &-&\frac{L^{2}_{1}[(\mu-1)[\mu-\lambda(\lambda\mu-\mu+1)][p]_{q}+2\lambda(2\mu-\lambda\mu-1)[p+1]_{q}]} {2L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}. \end{eqnarray*}$

Then, each of the following results is true:

(A) For $\delta\in[\Upsilon_{1}, \Upsilon_{3}]$ ,

$\begin{eqnarray*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert&+&\left\{2(A_{1}-A_{2})L^{2}_{1}[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}+A^{2}_{1}[p]_{q}\Theta\right\}\\ &\times&\frac{\vert a_{p+1}\vert^{2}}{2 A^{2}_{1}L_{2}[p]_{q}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]} \leq\frac{A_{1}[p]_{q}}{ L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}; \end{eqnarray*}$

(B) For $\delta\in[\Upsilon_{3}, \Upsilon_{2}]$ ,

$\begin{eqnarray*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert&+&\left\{2(A_{1}+A_{2})L^{2}_{1}[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}-A^{2}_{1}[p]_{q}\Theta\right\}\\ &\times&\frac{\vert a_{p+1}\vert^{2}}{2 A^{2}_{1}L_{2}[p]_{q}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]} \leq\frac{A_{1}[p]_{q}}{ L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}, \end{eqnarray*}$

where

Remark 2.5. Fixing the parameter $p = 1$ in Theorems 2.1 and 2.4, we can state the new results for the univalent function classes $\mathcal{LN}^{\eta}_{1, q}(\mu, \lambda; \phi) = \mathcal{LN}^{\eta}_{q}(\mu, \lambda; \phi)$ . As Remark 1.4, we may consider $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \alpha)$ or $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \beta)$ to establish latest results. On the other hand, for the different parameters $\mu$ and $\lambda$ , we can deduce new results for $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ .

3. Functional estimates for the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$

In the section we mainly consider Fekete-Szegö functional problem for the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ and establish the theorem as follows.

Theorem 3.1. Let $\delta\in\mathbb{C}$ . If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ , then

$\begin{eqnarray*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert&\leq&\frac{A_{1}[p]_{q}^{2}}{\vert L_{2}\vert([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}\notag\\ &\times&\max\left\{1;\left\vert\frac{ A_{1}[p]_{q}\Phi}{L^{2}_{1}([p+1]_{q}-[p]_{q})^{2}[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]^{2}}-\frac{A_{2}}{A_{1}}\right\vert\right\}, \end{eqnarray*}$

where

$\begin{equation*} \Phi = \delta L_{2}[p]_{q}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]+L^{2}_{1}([p+1]_{q}-[p]_{q})[\lambda([p+1]^{2}_{q}-[p]^{2}_{q})+[p]^{2}_{q}]. \end{equation*}$

Moreover, the sharp result holds for the next function

$\begin{equation*} \omega(z) = z\; \; \; \; \mathit{\text{or}}\; \; \; \; \omega(z) = z^{2}, \; \; \; \; (z\in\Delta). \end{equation*}$

Proof. If $f\in\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ , from Definition 1.2 there exists an analytic function $\omega(z)\in\Omega$ such that

(3.1)

Part case

$\begin{eqnarray*} &&\frac{z\mathcal{D}_{q}\left(\mathcal{J}^{\eta}_{p, q}f\right)(z)}{[p]_{q}\mathcal{J}^{\eta}_{p, q}f(z)}\\ & = &1+\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)L_{1}a_{p+1}z+\left[ \left(\frac{[p+2]_{q}}{[p]_{q}}-1\right)L_{2}a_{p+2}-\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)L^{2}_{1}a^{2}_{p+1}\right]z^{2}+\ldots, \end{eqnarray*}$

$\begin{eqnarray*} &&\frac{1}{[p]_{q}} \left(1+\frac{qz\mathcal{D}_{q}[\mathcal{D}_{q}(\mathcal{J}^{\eta}_{p, q}f)](z)}{\mathcal{D}_{q}(\mathcal{J}^{\eta}_{p, q}f)(z)}\right)\\ & = &1+\frac{L_{1}[p+1]_{q}}{[p]_{q}}\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)a_{p+1}z\notag\\ &+&\left[\frac{L_{2}[p+2]_{q}}{[p]_{q}}\left(\frac{[p+2]_{q}}{[p]_{q}}-1\right)a_{p+2}-\frac{L^{2}_{1}[p+1]^{2}_{q}}{[p]^{2}_{q}}\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)a^{2}_{p+1}\right]z^{2}+\ldots. \end{eqnarray*}$

Since

$\begin{eqnarray*} &&(1-\lambda)\frac{z\mathcal{D}_{q}(\mathcal{J}^{\eta}_{p, q}f)(z)}{[p]_{q}\mathcal{J}^{\eta}_{p, q}f(z)}+\frac{\lambda}{[p]_{q}} \left(1+\frac{qz\mathcal{D}_{q}[\mathcal{D}_{q}(\mathcal{J}^{\eta}_{p, q}f)](z)}{\mathcal{D}_{q}(\mathcal{J}^{\eta}_{p, q}f)(z)}\right)\\ & = &1+\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)\left[\lambda\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)+1\right]L_{1}a_{p+1}z\notag\\ &+&\left\{\left(\frac{[p+2]_{q}}{[p]_{q}}-1\right)\left[\lambda\left(\frac{[p+2]_{q}}{[p]_{q}}-1\right)+1\right]L_{2}a_{p+2}-\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)\left[\lambda\left(\frac{[p+1]^{2}_{q}}{[p]^{2}_{q}}-1\right)+1\right]L^{2}_{1} a^{2}_{p+1}\right\}z^{2}+\ldots, \end{eqnarray*}$

by (2.1) and (3.1) we note that

$\begin{equation*} A_{1}E_{1} = \left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)\left[\lambda\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)+1\right]L_{1}a_{p+1} \end{equation*}$

and

$\begin{eqnarray*} A_{1}E_{2}+A_{2}E^{2}_{1}& = &\left(\frac{[p+2]_{q}}{[p]_{q}}-1\right)\left[\lambda\left(\frac{[p+2]_{q}}{[p]_{q}}-1\right)+1\right]L_{2}a_{p+2}\\ &-&\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)\left[\lambda\left(\frac{[p+1]^{2}_{q}}{[p]^{2}_{q}}-1\right)+1\right]L^{2}_{1} a^{2}_{p+1}. \end{eqnarray*}$

Then, it leads to

$\begin{equation} a_{p+1} = \frac{A_{1}E_{1}[p]^{2}_{q}}{L_{1}([p+1]_{q}-[p]_{q})[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]} \end{equation}$

(3.2)

and

$\begin{eqnarray} a_{p+2}& = &\frac{[p]_{q}^{2}}{ L_{2}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}\\ &\times&\left[(A_{1}E_{2}+A_{2}E^{2}_{1})+\frac{A^{2}_{1}E^{2}_{1}[p]_{q}\{\lambda([p+1]^{2}_{q}-[p]^{2}_{q})+[p]^{2}_{q}\}}{ ([p+1]_{q}-[p]_{q})[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]^{2}}\right]. \end{eqnarray}$

(3.3)

Furthermore, in accordance with (3.2) and (3.3) we gain that

$\begin{equation*} a_{p+2}-\delta a^{2}_{p+1} = \frac{A_{1}[p]_{q}^{2}}{ L_{2}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}[ E_{2}-\varrho E^{2}_{1}], \end{equation*}$

where

$\begin{equation*} \varrho = \frac{A_{1}[p]_{q}\Phi}{L^{2}_{1}([p+1]_{q}-[p]_{q})^{2}[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]^{2}}-\frac{A_{2}}{A_{1}}. \end{equation*}$

Thus, from Lemma 1.5 we give the Fekete-Szegö functional inequality in Theorem 3.1.

Corollary 3.2. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ , then

$\begin{eqnarray*} \vert a_{p+2}\vert&\leq&\frac{A_{1}[p]_{q}^{2}}{\vert L_{2}\vert([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}\\ &\times&\max\left\{1;\left\vert\frac{A_{1}[p]_{q}[\lambda([p+1]^{2}_{q}-[p]^{2}_{q})+[p]^{2}_{q}]}{([p+1]_{q}-[p]_{q})[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]^{2}}-\frac{A_{2}}{A_{1}}\right\vert\right\}. \end{eqnarray*}$

Moreover, the sharp result holds for the next function

$\begin{equation*} \omega(z) = z\; \; \; \; \mathit{\text{or}}\; \; \; \; \omega(z) = z^{2}, \; \; \; \; (z\in\Delta). \end{equation*}$

If $\phi\in\mathcal{P}$ , by (3.2) and (3.3) we take Lemma 1.6 to prove the next corollary for the coefficient bounds of $a_{p+1}$ and $a_{p+2}$ .

Corollary 3.3. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ , then

$\begin{equation*} \vert a_{p+1}\vert\leq\frac{2\vert E_{1}\vert[p]^{2}_{q}}{\vert L_{1}\vert([p+1]_{q}-[p]_{q})[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]} \end{equation*}$

and

$\begin{eqnarray*} \vert a_{p+2}\vert&\leq&\frac{2[p]_{q}^{2}}{ \vert L_{2}\vert([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}\notag\\ &\times&\left[(\vert E_{2}\vert+\vert E_{1}\vert^{2})+\frac{2\vert E_{1}\vert^{2}[p]_{q}\{\lambda([p+1]^{2}_{q}-[p]^{2}_{q})+[p]^{2}_{q}\}}{ ([p+1]_{q}-[p]_{q})[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]^{2}}\right]. \end{eqnarray*}$

On the other hand, if we take real $\delta$ and $\eta$ , then by Lemma 1.7 we give the next result for Fekete-Szegö problem.

Theorem 3.4. Let $\delta, \eta\in\mathbb{R}$ and $\phi\in\Lambda$ satisfying

$\begin{equation*} \phi(z) = 1+\sum\limits^{\infty}_{n = 1}A_{n}z^{n}, \; \; (A_{1}, A_{2} \gt 0, z\in\Delta). \end{equation*}$

If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ , then

$\begin{equation*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert\leq \end{equation*}$

$\left\{\begin{array}{ll} \frac{[p]_{q}^{2}}{\vert L_{2}\vert([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}\left\{A_{2}-\frac{ A^{2}_{1}[p]_{q}\Phi}{L^{2}_{1}([p+1]_{q}-[p]_{q})^{2}[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]^{2}}\right\}, \; (\delta\leq\Gamma_{1});\\ \frac{A_{1}[p]_{q}^{2}}{\vert L_{2}\vert([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}, \; (\Gamma_{1}\leq\delta\leq\Gamma_{2});\\ \frac{[p]_{q}^{2}}{\vert L_{2}\vert([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}\left\{-A_{2}+\frac{ A^{2}_{1}[p]_{q}\Phi}{L^{2}_{1}([p+1]_{q}-[p]_{q})^{2}[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]^{2}}\right\}, \; (\delta\geq\Gamma_{2}), \end{array}\right.$

where

$\begin{eqnarray*} \Gamma_{1}& = &\{(A_{2}-A_{1})([p+1]_{q}-[p]_{q})\{\lambda([p+1]_{q}-[p]_{q})+[p]_{q}\}^{2}-A^{2}_{1}[p]_{q}[\lambda([p+1]^{2}_{q}-[p]^{2}_{q})+[p]^{2}_{q})]\} \notag\\ &\times&\frac{L^{2}_{1}([p+1]_{q}-[p]_{q})} {L_{2}A^{2}_{1}[p]^{2}_{q}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]} \end{eqnarray*}$

and

$\begin{eqnarray*} \Gamma_{2}& = &\{(A_{2}+A_{1})([p+1]_{q}-[p]_{q})\{\lambda([p+1]_{q}-[p]_{q})+[p]_{q}\}^{2}-A^{2}_{1}[p]_{q}[\lambda([p+1]^{2}_{q}-[p]^{2}_{q})+[p]^{2}_{q})]\} \notag\\ &\times&\frac{L^{2}_{1}([p+1]_{q}-[p]_{q})} {L_{2}A^{2}_{1}[p]^{2}_{q}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}. \end{eqnarray*}$

Moreover, we choose

$\begin{eqnarray*} \Gamma_{3}& = &\{A_{2}([p+1]_{q}-[p]_{q})\{\lambda([p+1]_{q}-[p]_{q})+[p]_{q}\}^{2}-A^{2}_{1}[p]_{q}[\lambda([p+1]^{2}_{q}-[p]^{2}_{q})+[p]^{2}_{q})]\} \notag\\ &\times&\frac{L^{2}_{1}([p+1]_{q}-[p]_{q})} {L_{2}A^{2}_{1}[p]^{2}_{q}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}. \end{eqnarray*}$

Then, each of the following results is true:

(A) For $\delta\in[\Gamma_{1}, \Gamma_{3}]$ ,

$\begin{eqnarray*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert&+&\frac{(A_{1}-A_{2})L^{2}_{1}([p+1]_{q}-[p]_{q})^{2}[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]^{2}+A^{2}_{1}[p]_{q}\Phi}{ A^{2}_{1}L_{2}[p]^{2}_{q}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}\vert a_{p+1}\vert^{2}\notag\\ &\leq&\frac{A_{1}[p]^{2}_{q}}{ L_{2}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}; \end{eqnarray*}$

(B) For $\delta\in[\Gamma_{3}, \Gamma_{2}]$ ,

$\begin{eqnarray*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert&+&\frac{(A_{1}+A_{2})L^{2}_{1}([p+1]_{q}-[p]_{q})^{2}[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]^{2}-A^{2}_{1}[p]_{q}\Phi}{ A^{2}_{1}L_{2}[p]^{2}_{q}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}\vert a_{p+1}\vert^{2}\notag\\ &\leq&\frac{A_{1}[p]^{2}_{q}}{ L_{2}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}, \end{eqnarray*}$

where

$\begin{equation*} \Phi = \delta L_{2}[p]_{q}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]+L^{2}_{1}[p+1]_{q}([p+1]_{q}-[p]_{q})[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]. \end{equation*}$

Remark 3.5. Similarly, by taking the parameter $p = 1$ in Theorems 3.1 and 3.4, we can obtain the new results for the univalent function classes $\mathcal{LM}^{\eta}_{1, q}(\lambda; \phi) = \mathcal{LM}^{\eta}_{q}(\lambda; \phi)$ . As Remark 1.4, we may consider $\mathcal{LM}^{\eta}_{p, q}(\lambda; \alpha)$ or $\mathcal{LM}^{\eta}_{p, q}(\lambda; \beta)$ to establish latest results. Clearly, for special parameter $\lambda$ , we can still imply new results for $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ .

4. Functional estimates for the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$

In the section we investigate Fekete-Szegö functional problem for the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ and obtain the corresponding theorem below.

Theorem 4.1. Let $\delta\in\mathbb{C}$ . If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ , then

$\begin{eqnarray*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert&\leq&\frac{A_{1}[p]_{q}}{\vert L_{2}\vert(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}\notag\\ &\times&\max\left\{1;\left\vert\frac{A_{1}[p]_{q}\Psi}{2L^{2}_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}}-\frac{A_{2}}{A_{1}}\right\vert\right\}, \end{eqnarray*}$

where

$\begin{equation*} \Psi = 2\delta L_{2}[p]_{q}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})+(\mu-1)[\mu[p]_{q}+2([p+1]_{q}-[p]_{q})]L^{2}_{1}. \end{equation*}$

Moreover, the sharp result holds for the next function

$\begin{equation*} \omega(z) = z\; \; \; \; \mathit{\text{or}}\; \; \; \; \omega(z) = z^{2}, \; \; \; \; (z\in\Delta). \end{equation*}$

Proof. Since $f\in\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ , from Definition 1.3 there exists an analytic function $\omega(z)\in\Omega$ such that

(4.1)

Part case

$\begin{eqnarray*} &&\left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu}\\ & = &1+\mu L_{1}a_{p+1}z+\left[\mu L_{2}a_{p+2}+\frac{\mu(\mu-1)}{2}L^{2}_{1}a^{2}_{p+1}\right]z^{2}\notag\\ &+&\left[\mu L_{3}a_{p+3}+\frac{\mu(\mu-1)}{2}L_{1}L_{2}a_{p+1}a_{p+2}+\frac{\mu(\mu-1)(\mu-2)}{6}L^{3}_{1}a^{3}_{p+1}\right]z^{3}+\ldots. \end{eqnarray*}$

Since

$\begin{equation*} \left(\frac{z\mathcal{D}_{q}\left(\mathcal{J}^{\eta}_{p, q}f\right)(z)}{[p]_{q}\mathcal{J}^{\eta}_{p, q}f(z)}\right)\left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu} = 1+\left(\frac{[p+1]_{q}}{[p]_{q}}-1+\mu\right)L_{1}a_{p+1}z \end{equation*}$

$\begin{equation*} +\left\{\left(\frac{[p+2]_{q}}{[p]_{q}}-1+\mu\right)L_{2}a_{p+2}+(\mu-1)\left[\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)+\frac{\mu}{2} \right]L^{2}_{1}a^{2}_{p+1}\right\}z^{2}+\ldots, \end{equation*}$

from (2.1) and (4.1) we know that

$\begin{equation*} A_{1}E_{1} = \left(\frac{[p+1]_{q}}{[p]_{q}}-1+\mu\right)L_{1}a_{p+1} \end{equation*}$

and

$\begin{eqnarray*} A_{1}E_{2}+A_{2}E^{2}_{1}& = &\left(\frac{[p+2]_{q}}{[p]_{q}}-1+\mu\right)L_{2}a_{p+2}\\ &+&(\mu-1)\left[\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)+\frac{\mu}{2} \right]L^{2}_{1}a^{2}_{p+1}. \end{eqnarray*}$

Thus it deduces that

$\begin{equation} a_{p+1} = \frac{A_{1}E_{1}[p]_{q}}{L_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})} \end{equation}$

(4.2)

and

$\begin{eqnarray} a_{p+2}& = &\frac{(A_{1}E_{2}+A_{2}E^{2}_{1})[p]_{q}}{L_{2}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}\\ &-&\frac{(\mu-1)A^{2}_{1}E^{2}_{1}[p]^{2}_{q}[\mu[p]_{q}+2([p+1]_{q}-[p]_{q})]}{2L_{2}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}}. \end{eqnarray}$

(4.3)

Moreover, in the light of (4.2) and (4.3) we know that

$\begin{equation*} a_{p+2}-\delta a^{2}_{p+1} = \frac{A_{1}[p]_{q}}{L_{2}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}[ E_{2}-\aleph E^{2}_{1}], \end{equation*}$

where

$\begin{equation*} \aleph = \frac{2\delta L_{2}[p]_{q}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})+(\mu-1)[\mu[p]_{q}+2([p+1]_{q}-[p]_{q})]L^{2}_{1}}{2L^{2}_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}}A_{1}[p]_{q}-\frac{A_{2}}{A_{1}}. \end{equation*}$

Hence, in view of Lemma 1.5 we get the Fekete-Szegö functional inequality in Theorem 4.1.

Corollary 4.2. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ , then

$\begin{eqnarray*} \vert a_{p+2}\vert&\leq&\frac{A_{1}[p]_{q}}{\vert L_{2}\vert(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}\\ &\times&\max\left\{1;\left\vert\frac{A_{1}(\mu-1)[p]_{q}\{\mu[p]_{q}+2([p+1]_{q}-[p]_{q})\}}{2(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}}-\frac{A_{2}}{A_{1}}\right\vert\right\}. \end{eqnarray*}$

Moreover, the sharp result holds for the next function

$\begin{equation*} \omega(z) = z\; \; \; \; \mathit{\text{or}}\; \; \; \; \omega(z) = z^{2}, \; \; \; \; (z\in\Delta). \end{equation*}$

Once $\phi\in\mathcal{P}$ , together with (4.2) and (4.3) we apply Lemma 1.6 to prove the next corollary for the coefficient bounds of $a_{p+1}$ and $a_{p+2}$ .

Corollary 4.3. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ , then

$\begin{equation*} \vert a_{p+1}\vert\leq\frac{2E_{1}[p]_{q}}{\vert L_{1}\vert(\mu[p]_{q}+[p+1]_{q}-[p]_{q})} \end{equation*}$

and

$\begin{eqnarray*} \vert a_{p+2}\vert&\leq&\frac{2(\vert E_{2}\vert+\vert E_{1}\vert^{2})[p]_{q}}{\vert L_{2}\vert(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}\notag\\ &+&\frac{2\vert\mu-1\vert\vert E_{1}\vert^{2}[p]^{2}_{q}[\mu[p]_{q}+2([p+1]_{q}-[p]_{q})]}{\vert L_{2}\vert(\mu[p]_{q}+[p+2]_{q}-[p]_{q})(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}}. \end{eqnarray*}$

Clearly, if we let $\delta$ and $\eta$ be real, then from Lemma 1.7 we also show the following result for Fekete-Szegö problem.

Theorem 4.4. Let $\delta, \eta\in\mathbb{R}$ and $\phi\in\Lambda$ satisfying

$\begin{equation*} \phi(z) = 1+\sum\limits^{\infty}_{n = 1}A_{n}z^{n}, \; \; (A_{1}, A_{2} \gt 0, z\in\Delta). \end{equation*}$

If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ , then

$\vert a_{p+2}-\delta a^{2}_{p+1}\vert\leq\left\{\begin{array}{ll} \frac{[p]_{q}}{ L_{2}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}\left\{A_{2}-\frac{A^{2}_{1}[p]_{q}\Psi}{2L^{2}_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}}\right\}, \; (\delta\leq\Pi_{1});\\ \frac{A_{1}[p]_{q}}{ L_{2}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})};\\ \frac{[p]_{q}}{ L_{2}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}\left\{-A_{2}+\frac{A^{2}_{1}[p]_{q}\Psi}{2L^{2}_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}}\right\}, \; (\delta\geq\Pi_{2}), \end{array}\right.$

where

$\begin{equation*} \Pi_{1} = \frac{2(A_{2}-A_{1})L^{2}_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}-A^{2}_{1}(\mu-1)L^{2}_{1}[p]_{q}[\mu[p]_{q}+2([p+1]_{q}-[p]_{q})]} {2L_{2}A^{2}_{1}[p]^{2}_{q}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})} \end{equation*}$

and

$\begin{equation*} \Pi_{2} = \frac{2(A_{2}+A_{1})L^{2}_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}-A^{2}_{1}(\mu-1)L^{2}_{1}[p]_{q}[\mu[p]_{q}+2([p+1]_{q}-[p]_{q})]} {2L_{2}A^{2}_{1}[p]^{2}_{q}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}. \end{equation*}$

Moreover, we put

$\begin{equation*} \Pi_{1} = \frac{2A_{2}L^{2}_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}-A^{2}_{1}(\mu-1)L^{2}_{1}[p]_{q}[\mu[p]_{q}+2([p+1]_{q}-[p]_{q})]} {2L_{2}A^{2}_{1}[p]^{2}_{q}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}. \end{equation*}$

Then, each of the following results is true:

(A) For $\delta\in[\Pi_{1}, \Pi_{3}]$ ,

$\begin{eqnarray*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert&+&\frac{2(A_{1}-A_{2})L^{2}_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}+A^{2}_{1}[p]_{q}\Psi}{ 2L_{2}A^{2}_{1}[p]_{q}[\mu[p]_{q}+([p+2]_{q}-[p]_{q})]}\vert a_{p+1}\vert^{2}\notag\\ &\leq&\frac{A_{1}[p]_{q}}{ L_{2}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}; \end{eqnarray*}$

(B) For $\delta\in[\Pi_{3}, \Pi_{2}]$ ,

$\begin{eqnarray*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert&+&\frac{2(A_{1}+A_{2})L^{2}_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}-A^{2}_{1}[p]_{q}\Psi}{ 2L_{2}A^{2}_{1}[p]_{q}[\mu[p]_{q}+([p+2]_{q}-[p]_{q})]}\vert a_{p+1}\vert^{2}\notag\\ &\leq&\frac{A_{1}[p]_{q}}{ L_{2}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}, \end{eqnarray*}$

where

$\begin{equation*} \Psi = 2\delta L_{2}[p]_{q}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})+(\mu-1)[\mu[p]_{q}+2([p+1]_{q}-[p]_{q})]L^{2}_{1}. \end{equation*}$

Remark 4.5. Similarly, by choose the parameter $p = 1$ in Theorems 4.1 and 4.4, we can provide the new results for the univalent function classes $\mathcal{NS}^{\eta}_{1, q}(\mu; \phi) = \mathcal{NS}^{\eta}_{q}(\mu; \phi)$ . As Remark 1.4, we may consider $\mathcal{NS}^{\eta}_{p, q}(\mu; \alpha)$ or $\mathcal{NS}^{\eta}_{p, q}(\mu; \beta)$ to establish latest results. Besides, for the fixed parameter $\mu$ , we can still infer new results for $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ .

5. Conclusion

By involving a generalized Bernardi integral operator, several new subclasses of $q$ -starlike and $q$ -convex type analytic and multivalent functions are introduced to generalize the classical starlike and convex functions. Meanwhile, for these classes we may know integral operator and $q$ -derivative as well as multivalency how to change the coefficients of functions. In our main results, we establish the Fekete-Szegö type functional inequalities for these function classes. Further, the corresponding bound estimates of the coefficients $a_{p+1}$ and $a_{p+2}$ are interpreted. In fact, if we use the other integral operators or take $(p, q)$ -operator when certain function is univalent but not multivalent, we may get many similar results as in this article.

Acknowledgments

We thank the referees for their careful readings and using comments so that this manuscript is greatly improved. This work is supported by Institution of Higher Education Scientific Research Project in Ningxia of the People's Republic of China under Grant NGY2017011, Natural Science Foundation of Ningxia of the People's Republic of China under Grant 2020AAC03066, the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region under Grant NJYT-18-A14, the Natural Science Foundation of Inner Mongolia of the People's Republic of China under Grant 2018MS01026, the Natural Science Foundation of the People's Republic of China under Grant 11561001, 42064004 and 11762016.

Conflict of interest

The authors declare no conflict of interest.

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Boundary distribution estimation for precise object detection

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Abstract

1. Introduction

2. Functional estimates for the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$

3. Functional estimates for the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$

4. Functional estimates for the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$

5. Conclusion

Acknowledgments

Conflict of interest

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Abstract

1. Introduction

2. Functional estimates for the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$

3. Functional estimates for the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$

4. Functional estimates for the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$

5. Conclusion

Acknowledgments

Conflict of interest

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Boundary distribution estimation for precise object detection

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Abstract

1. Introduction

2. Functional estimates for the class LNηp,q(μ,λ;ϕ) \mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)

3. Functional estimates for the class LMηp,q(λ;ϕ) \mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)

4. Functional estimates for the class NSηp,q(μ;ϕ) \mathcal{NS}^{\eta}_{p, q}(\mu; \phi)

5. Conclusion

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Abstract

1. Introduction

2. Functional estimates for the class LNηp,q(μ,λ;ϕ) \mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)

3. Functional estimates for the class LMηp,q(λ;ϕ) \mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)

4. Functional estimates for the class NSηp,q(μ;ϕ) \mathcal{NS}^{\eta}_{p, q}(\mu; \phi)

5. Conclusion

Acknowledgments

Conflict of interest

References

2. Functional estimates for the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$

3. Functional estimates for the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$

4. Functional estimates for the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$

2. Functional estimates for the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$

3. Functional estimates for the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$

4. Functional estimates for the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$