Biotechnological conversion of methane to methanol: evaluation of progress and potential

Charlotte E. Bjorck; Paul D. Dobson; Jagroop Pandhal; Charlotte E. Bjorck; Paul D. Dobson; Jagroop Pandhal

doi:10.3934/bioeng.2018.1.1

AIMS Bioengineering

2018, Volume 5, Issue 1: 1-38. doi: 10.3934/bioeng.2018.1.1

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Review

Biotechnological conversion of methane to methanol: evaluation of progress and potential

1.
Department of Chemical and Biological Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK
2.
The National Centre for Text Mining, Manchester Institute of Biotechnology, Princess Street, Manchester M1 7DN, UK

Received: 14 November 2017 Accepted: 14 January 2018 Published: 19 January 2018

Sources of methane are numerous, and vary greatly in their use and sustainable credentials. A Jekyll and Hyde character, it is a valuable energy source present as geological deposits of natural gas, however it is also potent greenhouse gas, released during many waste management processes. Gas-to-liquid technologies are being investigated as a means to exploit and monetise non-traditional and unutilised methane sources. The product identified as having the greatest potential is methanol due to it being a robust, commercially mature conversion process from methane and its beneficial fuel characteristics. Commercial methane to methanol conversion requires high temperatures and pressures, in an energy intensive and costly process. In contrast methanotrophic bacteria perform the desired transformation under ambient conditions, using methane monooxygenase (MMO) enzymes. Despite the great potential of these bacteria a number of biotechnical difficulties are hindering progress towards an industrially suitable process. We have identified five major challenges that exist as barriers to a viable conversion process that, to our knowledge, have not previously been examined as distinct process challenges. Although biotechnological applications of methanotrophic bacteria have been reviewed in part, no review has comprehensively covered progress and challenges for a methane to methanol process from an industrial perspective. All published examples to date of methanotroph catalysed conversion of methane to methanol are collated, and standardised to allow direct comparison. The focus will be on conversion of methane to methanol by whole-cell, wild type, methanotroph cultures, and the potential for their application in an industrially relevant process. A recent shift in the research community focus from a mainly biological angle to an overall engineering approach, offers potential to exploit methanotrophs in an industrially relevant biotechnological gas-to-liquid process. Current innovations and future opportunities are discussed.

Keywords:

Citation: Charlotte E. Bjorck, Paul D. Dobson, Jagroop Pandhal. Biotechnological conversion of methane to methanol: evaluation of progress and potential[J]. AIMS Bioengineering, 2018, 5(1): 1-38. doi: 10.3934/bioeng.2018.1.1

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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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AIMS Bioengineering

Biotechnological conversion of methane to methanol: evaluation of progress and potential

Related Papers:

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

References

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Catalog

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

References

AIMS Bioengineering

Biotechnological conversion of methane to methanol: evaluation of progress and potential

Related Papers:

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

References

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Catalog

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

References