Properties of a Class of Analytic Functions Influenced by Multiplicative Calculus

Abstract

Motivated by the notion of multiplicative calculus, more precisely multiplicative derivatives, we used the concept of subordination to create a new class of starlike functions. Because we attempted to operate within the existing framework of the design of analytic functions, a number of restrictions, which are in fact strong constraints, have been placed. We redefined our new class of functions using the three-parameter Mittag–Leffler function (Srivastava–Tomovski generalization of the Mittag–Leffler function), in order to increase the study’s adaptability. Coefficient estimates and their Fekete-Szegő inequalities are our main results. We have included a couple of examples to show the closure and inclusion properties of our defined class. Further, interesting bounds of logarithmic coefficients and their corresponding Fekete–Szegő functionals have also been obtained.

1. Introduction

Throughout this paper, we let $\mathbb{C}$ and $\mathbb{N}$ to denote the respective sets of complex numbers and natural numbers. Furthermore, ${\left(x\right)}_n$ will be used to denote the usual Pochhammer symbol and $\mathbb{U}=\left\{\xi \in \mathbb{C};\left|\xi \right|<1\right\}$, a unit disc. Denote by $\mathcal{A},$ the class of analytic function,

$$\mathcal{A}=\{\xi \in \mathbb{U};\phi \left(\xi \right)=\xi +a_2{\xi }^2+a_3{\xi }^3+\dots \}$$

where $a_n\in \mathbb{C}$, $\forall n\ge 2.$ We represent by $\mathcal{S}$, the class of functions in $\mathcal{A}$ which are univalent in $\mathbb{U}$.

Let $\mathcal{P}$ signify the category of functions that are analytic in $\mathbb{U}$ with $p\left(0\right)=1$ and $Re\left\{p\left(\xi \right)\right\}>0$ for all $\xi$ in $\mathbb{U}$.

An analytic function $\phi$ is subordinate to an analytic function $\psi ,$ written $\phi \left(\xi \right)\prec \psi \left(\xi \right)$(≺ denotes the subordination), provided that there is an analytic function $\varpi$ defined on $\mathbf{U}$ with $\varpi \left(0\right)=0$ and $\left|\varpi \right(\xi \left)\right|<1$ sustaining $\phi \left(\xi \right)=\psi \left(\varpi \right(\xi \left)\right).$ Starlike and convex functions, the well-known geometrically defined subclasses of $\mathcal{S}$, have the following respective analytic characterizations

$$Re\left(\frac{\xi {\phi }^{\prime }\left(\xi \right)}{\phi \left(\xi \right)}\right)>0\mathrm{and}Re\left(1+\frac{\xi {\phi }^{\prime \prime }\left(\xi \right)}{{\phi }^{\prime }\left(\xi \right)}\right)>0$$

(1)

and are denoted by ${\mathcal{S}}^{*}$ and $\mathcal{C}$, respectively. Different subclasses of ${\mathcal{S}}^{*}$ and $\mathcal{C}$ can be obtained by replacing the respective conditions in (1) with the following subordination condition

$$\frac{\xi {\phi }^{\prime }\left(\xi \right)}{\phi \left(\xi \right)}\prec \psi \left(\xi \right)$$

(2)

and

$$1+\frac{\xi {\phi }^{\prime \prime }\left(\xi \right)}{{\phi }^{\prime }\left(\xi \right)}\prec \psi \left(\xi \right)$$

(3)

where $\psi \left(\xi \right)\in \mathcal{P}$. By choosing $\psi$ to map the unit disc on to some specific regions like parabolas, cardioid, lemniscate of Bernoulli, and booth lemniscate in the right-half of the complex plane, various interesting subclasses of starlike and convex functions can be obtained. Here, we will list only a few of those studies, which are well known among the researchers in this field.

• Cho et al. [1] and Mendiratta et al. [2] studied various geometric properties of starlike functions by replacing $\psi \left(\xi \right)$ in (2) with $\psi \left(\xi \right)=1+sin\xi$ and $\psi \left(\xi \right)=e^{\xi }$, respectively.
• Fixing $\psi \left(\xi \right)=1+\frac{4}{3}\xi +\frac{2}{3}{\xi }^2$ in (2), Sharma et al. [3] studied a class of starlike function associated with petal-shaped domain, whereas Wani and Swaminathan [4] fixed $\psi \left(\xi \right)=1+\xi -\frac{1}{3}{\xi }^3,$ which maps $\mathbb{U}$ onto the interior of the 2-cusped kidney-shaped region and discussed application to the general coefficient problem for some subclasses of $\mathcal{S}.$
• Fixing $\psi \left(\xi \right)=\sqrt{1+\xi }$ in (2), Raina and Sokól [5] introduced a class of starlike functions which are bounded by the lemniscate of Bernoulli in the right-half plane.

For detailed study of various subclasses of $\mathcal{S}$ involving a conic region, refer to [6,7,8,9,10,11,12,13,14,15,16,17].

The Mittag–Leffler function is a special transcendental function that has gained attention due to its application in time-fractional differential equations and boundary value problems. We will not go into applications or their mathematical properties here. We referred the survey-cum-expository papers of Srivastava [18,19,20,21] for this study and found that they supplied enough material to analyze this duality theory. However, see Srivastava et al. [22,23,24,25,26] for thorough study on the Mittag–Leffler function. Srivastava et al. [24] introduced the following multi-index Mittag–Leffler functions as a kernel of specific fractional-calculus operators and as given below:

$$E_{{({\alpha }_j,{\beta }_j)}_m}^{\gamma ,k,\delta ,ϵ}\left(\xi \right)=\sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_{kn}{\left(\delta \right)}_{ϵn}}{{\prod }_{j=1}^m\mathsf{\Gamma }\left({\alpha }_{j}n+{\beta }_j\right)}\frac{{\xi }^n}{n!},$$

(4)

$$\left({\alpha }_j,{\beta }_j,\gamma ,k,\delta ,ϵ\in \mathbb{C};Re\left({\alpha }_j\right)>0,(j=1,\dots ,m);Re\left(\sum _{j=1}^m{\alpha }_j\right)>Re(k+ϵ)-1\right).$$

A special case of the multi-index Mittag–Leffler function defined by (4), when $m=2$ corresponding to the Srivastava–Tomovski generalization of the Mittag–Leffler function [25], is given by

$$E_{\alpha ,\beta }^{\gamma ,k}\left(\xi \right)=\sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_{nk}{\xi }^n}{\mathsf{\Gamma }\left(\alpha n+\beta \right)n!},\xi ,\alpha ,\beta ,\gamma ,k\in \mathbb{C},Re\left(\alpha \right)>0,Re\left(k\right)>0.$$

(5)

Note that by fixing the parameters the Mittag–Leffler function $E_{\alpha ,\beta }^{\gamma ,k}\left(\xi \right)$, this includes various well-known elementary functions and some special functions. For example,

$$E_{1,2}^{1,1}\left(\xi \right)=\frac{e^{\xi }-1}{\xi },E_{3,1}^{1,1}\left(\xi \right)=\frac{1}{2}\left[e^{{\xi }^{1/3}}+2e^{-\frac{1}{2}{\xi }^{1/3}}cos\left(\frac{\sqrt{3}}{2}{\xi }^{1/3}\right)\right].$$

$$E_{\frac{1}{2},1}^{1,1}\left(\xi \right)=e^{{\xi }^2}erfc(-\xi ),E_{\frac{1}{2},1}^{1,1}(\pm {\xi }^{1/2})=e^{\xi }\left[1+erf(\pm {\xi }^{1/2})\right]$$

where error function $erf\left(\xi \right)$ and the complementary error function $erfc\left(\xi \right)$ are defined by the formula

$$erfc\left(\xi \right)=1-erf\left(\xi \right)=1-\frac{2}{\sqrt{\pi }}{\int }_0^{\xi }{e}^{-t^2}dt.$$

The Mittag–Leffler function has been extensively used in areas such as fluid flow, electric networks, stochastic processes, and statistical distribution theory. Notably, it is used in almost all fractional dynamical systems. Using (5), Cang and Liu [27] defined an operator, which, explicitly for $p=1$, is given by

$$H_{\alpha ,\beta }^{\gamma ,k}\phi \left(\xi \right)=\xi +\sum _{n=2}^{\infty }\frac{\mathsf{\Gamma }(\gamma +nk)\mathsf{\Gamma }(\alpha +\beta )}{\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(\alpha n+\beta )n!}{a}_n{\xi }^n.$$

(6)

We will now provide a brief overview of multiplicative calculus, a type of non-Newtonian calculus. The importance of a calculus known as multiplicative calculus was highlighted by Bashirov, Kurpinar, and Özyapıin ([28] [pg. 37]) (also see [29,30,31]). Although it is not as versatile as classical calculus in terms of applications, it is nevertheless very interesting and a useful mathematical tool for economics and finance. For a positive real valued function $\phi :\mathbb{R}⟶\mathbb{R}$, the multiplicative derivative denoted by ${\phi }^{*}\left(x\right)$ is defined as

$${\phi }^{*}\left(x\right)=\underset{h\to 0}{lim}{\left(\frac{\phi (x+h)}{\phi \left(x\right)}\right)}^{\frac{1}{h}}=e^{\frac{{\phi }^{\prime }\left(x\right)}{\phi \left(x\right)}}=e^{{[ln\phi \left(x\right)]}^{\prime }}$$

where ${\phi }^{\prime }\left(x\right)$ is the classical derivative. In a similar way, the $n-$th ∗derivative of $\phi$ which is denoted by ${\phi }^{*\left(n\right)}$ for $n=0,1,\dots ,$ with ${\phi }^{*\left(0\right)}=\phi$ can be defined by ${\phi }^{*\left(n\right)}=e^{{[ln\phi \left(x\right)]}^{\left(n\right)}}$, provided that the $n-$th derivative of $\phi$ at x exists.

Assume that $\phi$ is a nowhere-vanishing differentiable complex-valued function on an open connected set $\mathbb{D}$ of the complex plane. Select a sufficiently small neighborhood $\Delta \subseteq \mathbb{D}$ of the point $\xi \in \mathbb{D}$ such that the branches of $log\phi$ on $\Delta$ exist in the form of the composition of the respective branches of $\mathbb{U}$ and the restriction of $\phi$ to $\mathbb{U}$, and the log-differentiation formula is valid for $log\phi$ on $\Delta$. The ∗-derivative of $\phi$ at $\xi \in \Delta$ by

$${\phi }^{*}\left(x\right)=e^{{\phi }^{\prime }\left(\xi \right)/\phi \left(\xi \right)}\mathrm{and}{\phi }^{*\left(n\right)}\left(x\right)=e^{{\left[{\phi }^{\prime }\left(\xi \right)/\phi \left(\xi \right)\right]}^{\left(n\right)}},n=1,2,\dots .$$

2. Definitions, Preliminaries and Results

Motivated by the definition of ∗-derivative, we let $\mathcal{R}\left(\psi \right)$ to denote the class of functions satisfying the conditions

$$\frac{\xi F^{*}\left(\xi \right)}{\phi \left(\xi \right)}\prec \psi \left(\xi \right),$$

(7)

where $F^{*}\left(\xi \right)=e^{\frac{{\xi }^2{\phi }^{\prime }\left(\xi \right)}{\phi \left(\xi \right)}}$, and $\psi \in \mathcal{P}$ has a series expansion of the form

$$\psi \left(\xi \right)=1+{\mathfrak{M}}_1\xi +{\mathfrak{M}}_2{\xi }^2+{\mathfrak{M}}_3{\xi }^3+\cdots ,({\mathfrak{M}}_1\ne 0;\xi \in \mathbb{U}).$$

(8)

Remark 1.

Here, we will discuss the purpose and limitations of the class $\mathcal{R}\left(\psi \right)$.

• The multiplicative derivative is defined for $\phi \left(\xi \right)$, which does not vanish in the chosen domain, but the general existing framework in $\mathcal{A}$ is that it vanishes at $\xi =0$. Hence, instead of using the multiplicative derivative directly, we have replaced ${\phi }^{\prime }\left(\xi \right)$ in the definition of a starlike function with $F^{*}\left(\xi \right)=e^{\frac{{\xi }^2{\phi }^{\prime }\left(\xi \right)}{\phi \left(\xi \right)}}$.
• Alternatively, we could have defined $\mathcal{R}\left(\psi \right)$ as

  $$\frac{z{e}^{{[ln\phi \left(\xi \right)]}^{\prime }}}{e\phi \left(\xi \right)}\prec \psi \left(\xi \right).$$

  However, we choose to keep the differential characterization as in (7), since it had some good geometrical implications.
• It is well-known that functions $\phi \in \mathcal{A}$ satisfying the condition $Re\left(\frac{\xi {\phi }^{\prime }\left(\xi \right)}{\phi \left(\xi \right)}\right)>0$ are univalent in $\mathbb{U}$. However, the functions in $\mathcal{R}\left(\psi \right)$ need not be univalent in $\mathbb{U}$.
• Since the multiplicative derivatives involves differential characterization in the exponent of an exponential, establishing the coefficient estimates (other than initial coefficients) of the class $\mathcal{R}\left(\psi \right)$ are computationally tedious. In fact, the sharp bounds of $|a_n|$ are unobtainable with the existing tools and techniques.

Example 1.

The class $\mathcal{R}\left(\psi \right)$ is non-empty. Let $\phi \left(\xi \right)=\xi -\frac{{\xi }^2}{5}$, $\left|\xi \right|<1$. Clearly, $\phi \left(\xi \right)$ is univalent in $\mathbb{U}$ and $\mathit{Area}\left(\phi \left(\mathbb{U}\right)\right)=\pi {\sum }_{n=1}^{\infty }n{\left|a_n\right|}^2=\pi \left(1·{\left|1\right|}^2+2·{\left|-1/5\right|}^2\right)=\frac{27\pi }{25}$. Now,

$$\frac{{\xi }^2{\phi }^{\prime }\left(\xi \right)}{\phi \left(\xi \right)}=\frac{\xi \left(5-2\xi \right)}{5-\xi }⟹G\left(\xi \right)=\frac{\xi e^{\frac{{\xi }^2{\phi }^{\prime }\left(\xi \right)}{\phi \left(\xi \right)}}}{\phi \left(\xi \right)}=\frac{1}{\left(1-\frac{\xi }{5}\right)}{e}^{\frac{\xi \left(5-2\xi \right)}{5-\xi }}.$$

Wecan see that the function $G\left(\xi \right)=\frac{1}{\left(1-\frac{\xi }{5}\right)}{e}^{\frac{\xi \left(5-2\xi \right)}{5-\xi }}$ maps the unit disc onto a close to cardioid shaped region in the right-half plane (see Figure 1). Figure 1 shows the respective 3D view (Figure 1a) and 2D view (Figure 1b) of the mapping of unit disc under the mapping $G\left(\xi \right)$. Hence, $\phi \left(\xi \right)=\xi -\frac{{\xi }^2}{5}\in \mathcal{R}\left(\psi \right)$.

In the following example, we will show that functions which are starlike need not be in $\mathcal{R}\left(\psi \right)$.

Example 2.

Let $h\left(\xi \right)=1+\frac{\xi }{k}\left(\frac{k+\xi }{k-\xi }\right)$, $k=1+\sqrt{2}$. The function is well known for mapping the unit disc onto a cardioid with cusp on the left-hand side. Further, the function is convex with respect to the point 1. Now,

$$\begin{array}{ccc}\hfill \frac{{\xi }^2{h}^{\prime }\left(\xi \right)}{h\left(\xi \right)}& =& \frac{k^2{\xi }^2+2k{\xi }^3-{\xi }^4}{\left(k-\xi \right)\left({\xi }^2+k^2\right)}\hfill \\ \hfill \mathrm{w}hichimplies\\ \hfill H\left(\xi \right)& =& \frac{z{e}^{\frac{{\xi }^2{\phi }^{\prime }\left(\xi \right)}{\phi \left(\xi \right)}}}{\phi \left(\xi \right)}=\frac{\xi }{1+\frac{\xi }{k}\left(\frac{k+\xi }{k-\xi }\right)}{e}^{\frac{k^2{\xi }^2+2k{\xi }^3-{\xi }^4}{\left(k-\xi \right)\left({\xi }^2+k^2\right)}}.\hfill \end{array}$$

We can see that the function $H\left(\xi \right)$maps the unit disc onto a region as shown in Figure 2a. Figure 2a illustrates that $h\left(\xi \right)$ does not belong to $\mathcal{R}\left(\psi \right)$, even though $h\left(\xi \right)\prec \psi \left(\xi \right)$, for ψ defined as in (8).

Remark 2.

Notice that $\mathcal{R}\left(\psi \right)$ is not a subclass nor a generalization of the class $\mathcal{S}$. For example, the function $\phi \left(\xi \right)=\xi -\frac{{\xi }^2}{2}$ is in $\mathcal{S}$ but does not belong to $\mathcal{R}\left(\psi \right)$ (see Figure 2b).

Motivated by the definition of the operator (6), Breaz et al. [32] defined the following operator ${\mathsf{\Omega }}_k^m(\alpha ,\beta ,\gamma )\phi \left(\xi \right):\mathcal{A}\to \mathcal{A}$ by

$${\mathsf{\Omega }}_{k,\lambda }^m(\alpha ,\beta ,\gamma )\phi \left(\xi \right)=\xi +\sum _{n=2}^{\infty }{\left[1-\lambda +\lambda n\right]}^m\frac{\mathsf{\Gamma }(\gamma +nk)\mathsf{\Gamma }(\alpha +\beta )}{\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(\alpha n+\beta )n!}{a}_n{\xi }^n.$$

(9)

$$\left(m\in \mathbb{N}\cup \left\{0\right\};0\le \lambda \le 1;\xi ,\alpha ,\beta ,\gamma ,k\in \mathbb{C},Re\left(\alpha \right)>0,Re\left(k\right)>0\right).$$

Remark 3.

Note that ${\mathsf{\Omega }}_{1,\lambda }^0(0,\beta ,1)\phi \left(\xi \right)=\phi \left(\xi \right)$ and ${\mathsf{\Omega }}_{1,1}^1(0,\beta ,1)\phi \left(\xi \right)=\xi {\phi }^{\prime }\left(\xi \right)$. The operator ${\mathsf{\Omega }}_{k,\lambda }^m(\alpha ,\beta ,\gamma )\phi \left(\xi \right)$ includes many operators previously studied by various authors as its special cases. We list some of the special cases:

• ${\mathsf{\Omega }}_{1,\lambda }^m(0,\beta ,1)\phi \left(\xi \right)=D_{\lambda }^m\phi \left(\xi \right)=\xi +{\sum }_{n=2}^{\infty }{\left[1-\lambda +\lambda n\right]}^m{a}_n{\xi }^n$. The operator $D_{\lambda }^m\phi \left(\xi \right)$ is the differential operator introduced and studied by Al-Oboudi [33].
• ${\mathsf{\Omega }}_{1,1}^m(0,\beta ,1)\phi \left(\xi \right)=D^m\phi \left(\xi \right)$, the operator $D^m\phi \left(\xi \right)$ is the well-known Sălăgean operator.
• Obviously, fixing $m=0$ and $\lambda =1$ in (9), then ${\mathsf{\Omega }}_{k,\lambda }^m(\alpha ,\beta ,\gamma )\phi \left(\xi \right)$ reduces to $H_{\alpha ,\beta }^{\gamma ,k}\phi \left(\xi \right)$ an operator defined and studied by Cang and Liu [27].

For a latest study of differential operators involving the Mittag–Leffler function, refer to Yassen and Attiya [34].

Motivated by the definition of $\mathcal{R}\left(\psi \right)$, we now introduce and study a new class of analytic functions which is defined as follows.

Definition 1.

Let ${\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$ denote the class of functions satisfying the conditions

$$\frac{\xi {\mathsf{\Omega }}_{k,\lambda }^m(\alpha ,\beta ,\gamma )F^{*}\left(\xi \right)}{\left[{\mathsf{\Omega }}_{k,\lambda }^m(\alpha ,\beta ,\gamma )\phi \left(\xi \right)\right]}\prec \psi \left(\xi \right),$$

(10)

where ${\mathsf{\Omega }}_{k,\lambda }^m(\alpha ,\beta ,\gamma )F^{*}\left(\xi \right)=e^{\frac{{\xi }^2{\mathsf{\Omega }}_{k,\lambda }^m(\alpha ,\beta ,\gamma ){\phi }^{\prime }\left(\xi \right)}{{\mathsf{\Omega }}_{k,\lambda }^m(\alpha ,\beta ,\gamma )\phi \left(\xi \right)}}$, $\psi \in \mathcal{P}$, and $\psi \left(\mathbb{U}\right)$ is defined as in (8).

Remark 4.

Here, we list some special cases of ${\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$:

• Letting $m=\alpha =0$, $k=\gamma =\lambda =1$ and $\psi \left(\xi \right)=\frac{1+\xi }{1-\xi }$ in Definition 1, then the class ${\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$ reduces to

  $$\mathcal{R}=\left\{\phi \in \mathcal{A};Re\left(\frac{\xi F^{*}\left(\xi \right)}{\phi \left(\xi \right)}\right)>0\right\},$$

  the class $\mathcal{R}$ is the multiplicative analogue of the well-known class of starlike functions.
• Letting $\alpha =0$ and $m=k=\gamma =\lambda =1$ in Definition 1, then the class ${\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )\equiv \mathcal{RC}$ given by

  $$\mathcal{RC}=\left\{\phi \in \mathcal{A};\frac{e^{\frac{\xi {\left[\xi {\phi }^{\prime }\left(\xi \right)\right]}^{\prime }}{{\phi }^{\prime }\left(\xi \right)}}}{{\phi }^{\prime }\left(\xi \right)}\prec \psi \left(\xi \right)\right\},$$

  the class $\mathcal{RC}$ is a new class motivated by the relationship between starlike and convex function.

The ${\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$ does not have any well-known classes that describe it as a special case. Nonetheless, we will attempt to determine its connection to certain analytic function subclasses that have already been studied by other authors.

This paper is structured as follows. In Section 3, we will obtain the coefficient bounds of $a_2$, $a_3$ and solve the Fekete-Szegő problem for the defined function class ${\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$. Applications of our main results pertaining to vertical domain are presented as corollaries. Section 4 is devoted to discuss the existence of an inverse function in the class ${\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$ and obtain its coefficient estimates. Finally, Section 5 has been devoted to present the bounds of logarithmic coefficients and their corresponding Fekete-Szegő functionals.

Now, we will state the following lemma, which we will use it to find the coefficient bounds.

Lemma 1

([35]). If $d\left(\xi \right)=1+{\displaystyle \sum _{k=1}^{\infty }}{d}_k{\xi }^k\in \mathcal{P}$, and ρ is complex number, then

$$\left|d_2-\rho d_1^2\right|\le 2max\left\{1;|2v-1|\right\},$$

and the result is sharp.

3. Initial Coefficients and Fekete-Szegő Inequality

We will begin the solution to the Fekete-Szegő problem for $\phi \in {\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$.

Theorem 1.

If $\phi \left(\xi \right)\in {\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$, then we have

$$\left|a_2\right|\le \frac{2\left|\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(2\alpha +\beta )\right|}{{\left[1+\lambda \right]}^m\left|\mathsf{\Gamma }(\gamma +2k)\mathsf{\Gamma }(\alpha +\beta )\right|}\left[1+\left|{\mathfrak{M}}_1\right|\right]$$

(11)

$$\left|a_3\right|\le \frac{6|{\mathfrak{M}}_1|\left|\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(3\alpha +\beta )\right|}{{\left[1+2\lambda \right]}^m\left|\mathsf{\Gamma }(\gamma +3k)\mathsf{\Gamma }(\alpha +\beta )\right|}\left[max\left\{1;\left|\frac{{\mathfrak{M}}_2}{{\mathfrak{M}}_1}-{\mathfrak{M}}_1\right|\right\}+\frac{3}{2\left|{\mathfrak{M}}_1\right|}+2\right]$$

(12)

and for all $\rho \in \mathbb{C}$

$$\begin{array}{c}\left|a_3-\rho a_2^2\right|\le \frac{6\left|{\mathfrak{M}}_1\right|\left|\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(3\alpha +\beta )\right|}{{\left[1+2\lambda \right]}^m\left|\mathsf{\Gamma }(\gamma +3k)\mathsf{\Gamma }(\alpha +\beta )\right|}\left[max\left\{1;\left|\frac{{\mathfrak{M}}_2}{{\mathfrak{M}}_1}-{\mathfrak{M}}_1\left(1-{\mathcal{K}}_1\right)\right|\right\}\right.\\ \left.+2\left|1-{\mathcal{K}}_1\right|+\frac{1}{2\left|{\mathfrak{M}}_1\right|}\left|3-2{\mathcal{K}}_1\right|\right],\end{array}$$

(13)

where ${\mathcal{K}}_1$ is given by

$$\begin{array}{c}{\mathcal{K}}_1=\frac{2\rho {\left[1+2\lambda \right]}^m\mathsf{\Gamma }(\gamma +3k)\mathsf{\Gamma }(\gamma +k){\left(\mathsf{\Gamma }(2\alpha +\beta )\right)}^2}{3{\left[1+\lambda \right]}^{2m}{\left(\mathsf{\Gamma }(\gamma +2k)\right)}^2\mathsf{\Gamma }(3\alpha +\beta )\mathsf{\Gamma }(\alpha +\beta )}.\end{array}$$

(14)

The inequality is sharp for each $\rho \in \mathbb{C}$.

Proof. 

As $\phi \in {\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$, by (10) we have

$$\frac{\xi {\mathsf{\Omega }}_{k,\lambda }^m(\alpha ,\beta ,\gamma )F^{*}\left(\xi \right)}{{\mathsf{\Omega }}_{k,\lambda }^m(\alpha ,\beta ,\gamma )\phi \left(\xi \right)}=\psi \left[w\left(\xi \right)\right].$$

(15)

Thus, let $\vartheta \in \mathcal{P}$ be of the form $\vartheta \left(\xi \right)=1+{\sum }_{k=1}^{\infty }{\vartheta }_n{\xi }^n$ and defined by

$$\vartheta \left(\xi \right)=\frac{1+w\left(\xi \right)}{1-w\left(\xi \right)},\xi \in \mathbb{U}.$$

On computation, the right-hand side of (15)

$$\begin{array}{c}\psi \left[w\left(\xi \right)\right]=1+\frac{{\vartheta }_1{\mathfrak{M}}_1}{2}\xi +\frac{{\mathfrak{M}}_1}{2}\left[{\vartheta }_2-\frac{{\vartheta }_1^2}{2}\left(1-\frac{{\mathfrak{M}}_2}{{\mathfrak{M}}_1}\right)\right]{\xi }^2+\cdots .\end{array}$$

(16)

The left-hand side of (15) will be of the form

$$\begin{array}{c}\frac{\xi {\mathsf{\Omega }}_{k,\lambda }^m(\alpha ,\beta ,\gamma )F^{*}\left(\xi \right)}{{\mathsf{\Omega }}_{k,\lambda }^m(\alpha ,\beta ,\gamma )\phi \left(\xi \right)}=1+\left[1-a_2{\left[1+\lambda \right]}^m\frac{\mathsf{\Gamma }(\gamma +2k)\mathsf{\Gamma }(\alpha +\beta )}{\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(2\alpha +\beta )2!}\right]\xi \\ +\left[\frac{1}{2}+{\left(\frac{{\left[1+\lambda \right]}^m\mathsf{\Gamma }(\gamma +2k)\mathsf{\Gamma }(\alpha +\beta )}{\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(2\alpha +\beta )2!}\right)}^2{a}_2^2-\frac{{\left[1+2\lambda \right]}^m\mathsf{\Gamma }(\gamma +3k)\mathsf{\Gamma }(\alpha +\beta )}{\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(3\alpha +\beta )3!}{a}_3\right]{\xi }^2+\cdots .\end{array}$$

(17)

From (17) and (16), we obtain

$$a_2=-\frac{2\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(2\alpha +\beta )}{{\left[1+\lambda \right]}^m\mathsf{\Gamma }(\gamma +2k)\mathsf{\Gamma }(\alpha +\beta )}\left[\frac{{\vartheta }_1{\mathfrak{M}}_1}{2}-1\right]$$

(18)

and

$$a_3=-\frac{3{\mathfrak{M}}_1\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(3\alpha +\beta )}{{\left[1+2\lambda \right]}^m\mathsf{\Gamma }(\gamma +3k)\mathsf{\Gamma }(\alpha +\beta )}\left[{\vartheta }_2-\frac{{\vartheta }_1^2}{2}\left(1-\frac{{\mathfrak{M}}_2}{{\mathfrak{M}}_1}+{\mathfrak{M}}_1\right)+2{\vartheta }_1-\frac{3}{{\mathfrak{M}}_1}\right].$$

(19)

Equation (11) can be obtained by applying the well-known result of $\left|{\vartheta }_1\right|\le 2$ in (18). Applying Lemma 1 together with inequality $\left|{\vartheta }_1\right|\le 2$ in (12), we obtain (12).

Now, to prove the Fekete-Szegő inequality for the class ${\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$, we consider

$$\begin{array}{c}\left|a_3-\rho a_2^2\right|=|\frac{3{\mathfrak{M}}_1\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(3\alpha +\beta )}{{\left[1+2\lambda \right]}^m\mathsf{\Gamma }(\gamma +3k)\mathsf{\Gamma }(\alpha +\beta )}\left[{\vartheta }_2-\frac{{\vartheta }_1^2}{2}\left(1-\frac{{\mathfrak{M}}_2}{{\mathfrak{M}}_1}+{\mathfrak{M}}_1\right)+2{\vartheta }_1-\frac{3}{{\mathfrak{M}}_1}\right]\\ +\frac{4\rho {\left(\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(2\alpha +\beta )\right)}^2}{{\left[1+\lambda \right]}^{2m}{\left(\mathsf{\Gamma }(\gamma +2k)\mathsf{\Gamma }(\alpha +\beta )\right)}^2}\left[\frac{{\vartheta }_1^2{\mathfrak{M}}_1^2}{4}-{\vartheta }_1{\mathfrak{M}}_1+1\right]|\end{array}$$

$$\begin{array}{c}=|\frac{3{\mathfrak{M}}_1\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(3\alpha +\beta )}{{\left[1+2\lambda \right]}^m\mathsf{\Gamma }(\gamma +3k)\mathsf{\Gamma }(\alpha +\beta )}\left[{\vartheta }_2-\frac{{\vartheta }_1^2}{2}\left(1-\frac{{\mathfrak{M}}_2}{{\mathfrak{M}}_1}+{\mathfrak{M}}_1\right.\right.\\ \left.-\frac{2{\mathfrak{M}}_1\rho {\left[1+2\lambda \right]}^m\mathsf{\Gamma }(\gamma +3k)\mathsf{\Gamma }(\gamma +k){\left(\mathsf{\Gamma }(2\alpha +\beta )\right)}^2}{3{\left[1+\lambda \right]}^{2m}{\left(\mathsf{\Gamma }(\gamma +2k)\right)}^2\mathsf{\Gamma }(3\alpha +\beta )\mathsf{\Gamma }(\alpha +\beta )}\right)\\ +2{\vartheta }_1\left(1-\frac{4\rho {\left[1+2\lambda \right]}^m\mathsf{\Gamma }(\gamma +3k)\mathsf{\Gamma }(\gamma +k){\left(\mathsf{\Gamma }(2\alpha +\beta )\right)}^2}{3{\left[1+\lambda \right]}^{2m}{\left(\mathsf{\Gamma }(\gamma +2k)\right)}^2\mathsf{\Gamma }(3\alpha +\beta )\mathsf{\Gamma }(\alpha +\beta )}\right)\\ \left.-\frac{1}{{\mathfrak{M}}_1}\left(3-\frac{4\rho {\left[1+2\lambda \right]}^m\mathsf{\Gamma }(\gamma +3k)\mathsf{\Gamma }(\gamma +k){\left(\mathsf{\Gamma }(2\alpha +\beta )\right)}^2}{3{\left[1+\lambda \right]}^{2m}{\left(\mathsf{\Gamma }(\gamma +2k)\right)}^2\mathsf{\Gamma }(3\alpha +\beta )\mathsf{\Gamma }(\alpha +\beta )}\right)\right]|\end{array}$$

Using the triangle inequality and Lemma 1 in the above equality, we can obtain (13). □

Let $m=\alpha =0$ and $k=\gamma =1$ in Theorem 1, we have the following.

Corollary 1.

Let $\phi \in \mathcal{R}\left(\psi \right)$ (see (7)). Then,

$$\left|a_2\right|\le 1+\left|{\mathfrak{M}}_1\right|$$

$$\left|a_3\right|\le \left|{\mathfrak{M}}_1\right|\left[max\left\{1;\left|\frac{{\mathfrak{M}}_2}{{\mathfrak{M}}_1}-{\mathfrak{M}}_1\right|\right\}+\frac{3}{2\left|{\mathfrak{M}}_1\right|}+2\right]$$

and for a complex number ρ,

$$\begin{array}{c}\left|a_3-\rho a_2^2\right|\le \left|{\mathfrak{M}}_1\right|\left[max\left\{1;\left|\frac{{\mathfrak{M}}_2}{{\mathfrak{M}}_1}-{\mathfrak{M}}_1\left(1-\rho \right)\right|\right\}+2\left|1-\rho \right|+\frac{1}{2\left|{\mathfrak{M}}_1\right|}\left|3-2\rho \right|\right].\end{array}$$

Letting $\psi \left(\xi \right)=(1+\xi )/(1-\xi )$ in Corollary 1, we get the following.

Corollary 2.

Let $\phi \in \mathcal{A}$ satisfy the condition

$$Re\left(\frac{z{e}^{\frac{{\xi }^2{\phi }^{\prime }\left(\xi \right)}{\phi \left(\xi \right)}}}{\phi \left(\xi \right)}\right)>0.$$

Then,

$$\left|a_2\right|\le 3,\left|a_3\right|\le \frac{15}{2}$$

and for a complex number ρ,

$$\begin{array}{c}\left|a_3-\rho a_2^2\right|\le \left|{\mathfrak{M}}_1\right|\left[max\left\{1;\left|2\rho -1\right|\right\}+2\left|1-\rho \right|+\frac{1}{4}\left|3-2\rho \right|\right].\end{array}$$

Remark 5.

It is well-known that for $\phi \left(\xi \right)=\xi +{\sum }_{n=2}^{\infty }{a}_n{\xi }^n\in {\mathcal{S}}^{*}$, then $\left|a_n\right|\le n$ for $n\ge 2$ and the Fekete-Szegő for function in ${\mathcal{S}}^{*}$ is known to be $\left|a_3-\rho a_2^2\right|\le max\left\{1,\left|3-4\rho \right|\right\}$, and ρ is a complex number. In comparison with Corollary 2, we can conclude that the $\mathcal{R}\left(1+\xi /1-\xi \right)$ is neither a subclass nor generalization of the class ${\mathcal{S}}^{*}$.

Letting $\psi \left(\xi \right)=1+\frac{\theta -\eta }{\pi }ilog\left(\frac{1-e^{2\pi i\left((1-\eta )/(\theta -\eta )\right)}\xi }{1-\xi }\right)$ ($\theta$ and $\eta$ are real numbers such that $0\le \eta <1<\theta$) in Corollary 1, we get the following.

Corollary 3.

Let $\phi \in \mathcal{A}$ satisfy the condition

$$\eta <Re\left(\frac{z{e}^{\frac{{\xi }^2{\phi }^{\prime }\left(\xi \right)}{\phi \left(\xi \right)}}}{\phi \left(\xi \right)}\right)<\theta .$$

Then,

$$\left|a_2\right|\le 1+\frac{2(\theta -\eta )}{\pi }sin\left[\frac{\pi (1-\eta )}{(\theta -\eta )}\right]$$

$$\begin{array}{c}\left|a_3\right|\le \frac{2(\theta -\eta )}{\pi }sin\left[\frac{\pi (1-\eta )}{(\theta -\eta )}\right]\left[max\left\{1;cos\left[\frac{\pi (1-\eta )}{(\theta -\eta )}\right]\left(1-2\frac{(\theta -\eta )}{\pi }tan\left[\frac{\pi (1-\eta )}{(\theta -\eta )}\right]\right)\right\}\right.\\ \left.+\frac{3\pi }{4(\theta -\eta )}csc\left[\frac{\pi (1-\eta )}{(\theta -\eta )}\right]+2\right]\end{array}$$

and for a complex number ρ,

$$\begin{array}{c}\left|a_3-\rho a_2^2\right|\le \frac{2(\theta -\eta )}{\pi }sin\left[\frac{\pi (1-\eta )}{(\theta -\eta )}\right]\left[max\left\{1;cos\left[\frac{\pi (1-\eta )}{(\theta -\eta )}\right]\right.\right.\\ \left.\left(1-2\frac{(1-\rho )(\theta -\eta )}{\pi }tan\left[\frac{\pi (1-\eta )}{(\theta -\eta )}\right]\right)\right\}+2\left|1-\rho \right|\\ \left.+\frac{\left|3-2\rho \right|\pi }{4(\theta -\eta )}csc\left[\frac{\pi (1-\eta )}{(\theta -\eta )}\right]\right].\end{array}$$

Proof. 

We note that the function $\psi \left(z\right)=1+\frac{\theta -\eta }{\pi }ilog\left(\frac{1-e^{2\pi i\left((1-\eta )/(\theta -\eta )\right)}z}{1-z}\right)$ maps the open unit disk $\mathbb{U}$ onto a convex domain and is of the form

$$\mathcal{T}\left(z\right)=1+\sum _{k=1}^{\infty }{\mathfrak{M}}_k{z}^k,z\in \mathbb{U},$$

where ${\mathfrak{M}}_k={\displaystyle \frac{\theta -\eta }{k\pi }}i\left(1-e^{2k\pi i\left((1-\eta )/(\theta -\eta )\right)}\right)$, $k\ge 1$. We can obtain

$$\begin{array}{cc}\hfill \left|{\mathfrak{M}}_1\right|& =\left|{\displaystyle \frac{\theta -\eta }{\pi }}i\left(1-cos\left(\frac{2\pi (1-\eta )}{(\theta -\eta )}\right)-isin\left(\frac{2\pi (1-\eta )}{(\theta -\eta )}\right)\right)\right|\hfill \\ & ={\displaystyle \frac{\theta -\eta }{\pi }}\sqrt{2-2cos\left(\frac{2\pi (1-\eta )}{(\theta -\eta )}\right)}=\frac{2(\theta -\eta )}{\pi }sin\left(\frac{\pi (1-\eta )}{(\theta -\eta )}\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left|\frac{{\mathfrak{M}}_2}{{\mathfrak{M}}_1^2}-1\right|& =\left|{\displaystyle \frac{\pi }{2i\left(\theta -\eta \right)}}\left(\frac{\left(1-e^{4k\pi i\left((1-\eta )/(\theta -\eta )\right)}\right)}{{\left(1-e^{2k\pi i\left((1-\eta )/(\theta -\eta )\right)}\right)}^2}\right)-1\right|\hfill \\ & ={\displaystyle \frac{\pi }{2\left(\theta -\eta \right)}}\left|\frac{1}{i}\left(\frac{\left(1+e^{2k\pi i\left((1-\eta )/(\theta -\eta )\right)}\right)}{1-e^{2k\pi i\left((1-\eta )/(\theta -\eta )\right)}}\right)-\frac{2\left(\theta -\eta \right)}{\pi }\right|\hfill \\ & ={\displaystyle \frac{\pi }{2\left(\theta -\eta \right)}}\left|\frac{1}{-i}\left(\frac{e^{k\pi i\left((1-\eta )/(\theta -\eta )\right)}+e^{-k\pi i\left((1-\eta )/(\theta -\eta )\right)}}{e^{k\pi i\left((1-\eta )/(\theta -\eta )\right)}-e^{-k\pi i\left((1-\eta )/(\theta -\eta )\right)}}\right)-\frac{2\left(\theta -\eta \right)}{\pi }\right|\hfill \\ & ={\displaystyle \frac{\pi }{2\left(\theta -\eta \right)}}cot\left(\frac{\pi (1-\eta )}{(\theta -\eta )}\right)\left[1-\frac{2\left(\theta -\eta \right)}{\pi }tan\left(\frac{\pi (1-\eta )}{(\theta -\eta )}\right)\right]\hfill \end{array}$$

(20)

Substituting the values of $|{\mathfrak{M}}_1|$ and (20) in Theorem 1, we obtain assertion of our corollary. □

4. Coefficient Estimates of ${\mathbf{\phi }}^{-\mathbf{1}}\left(\mathbf{\xi }\right)$

We let $\mathcal{S}$ to denote the class of functions univalent in $\mathbb{U}$. It is well known from Koebe $\frac{1}{4}$-quarter theorem that every function $\phi \left(\xi \right)=\xi +{\sum }_{n=2}^{\infty }{a}_n{\xi }^n$ in $\mathcal{S}$ has an inverse ${\phi }^{-1}$, defined by ${\phi }^{-1}\left(\phi \left(\xi \right)\right)=\xi ,\xi \in \mathbb{U}$ and $\phi \left({\phi }^{-1}\left(w\right)\right)=w,\left(\right|w|<r;r\ge 1/4)$, where

$$\chi \left(w\right)={\phi }^{-1}\left(w\right)=w-a_2{w}^2+(2a_2^2-a_3)w^2-\left(5a_2^2-5a_2{a}_3+a_4\right)w^4+\cdots .$$

(21)

The functions in ${\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$ need not be univalent, but since ${\phi }^{\prime }\left(0\right)=1\ne 0$ for all $\phi \in {\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$ and $\phi \left(0\right)=0$, there exists an inverse function in some small disk with a center at $w=0$. Next, the result is valid only for the functions in ${\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$ which are univalent.

Theorem 2.

Let $\phi \in {\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$ and let ${\phi }^{-1}$ be the inverse of φ defined by

$${\phi }^{-1}\left(w\right)=w+\sum _{k=2}^{\infty }{b}_k{w}^k,\left(\right|w|<r;r\ge 1/4),$$

then

$$\left|b_2\right|\le \frac{2\left|\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(2\alpha +\beta )\right|}{{\left[1+\lambda \right]}^m\left|\mathsf{\Gamma }(\gamma +2k)\mathsf{\Gamma }(\alpha +\beta )\right|}\left[1+\left|{\mathfrak{M}}_1\right|\right]$$

and

$$\begin{array}{c}\left|b_3\right|\le \frac{6\left|{\mathfrak{M}}_1\right|\left|\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(3\alpha +\beta )\right|}{{\left[1+2\lambda \right]}^m\left|\mathsf{\Gamma }(\gamma +3k)\mathsf{\Gamma }(\alpha +\beta )\right|}\left[max\left\{1;\left|\frac{{\mathfrak{M}}_2}{{\mathfrak{M}}_1}-{\mathfrak{M}}_1\left(1-\mathcal{M}\right)\right|\right\}\right.\\ \left.+2\left|1-\mathcal{M}\right|+\frac{1}{2\left|{\mathfrak{M}}_1\right|}\left|3-2\mathcal{M}\right|\right],\end{array}$$

where $\mathcal{M}={\left[{\mathcal{K}}_1\right]}_{\mathrm{with}\rho =2}$, ${\mathcal{K}}_1$ is defined as in (14). Furthermore, for all $\tau \in \mathbb{C}$

$$\begin{array}{c}\left|b_3-\tau b_2^2\right|\le \frac{6\left|{\mathfrak{M}}_1\right|\left|\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(3\alpha +\beta )\right|}{{\left[1+2\lambda \right]}^m\left|\mathsf{\Gamma }(\gamma +3k)\mathsf{\Gamma }(\alpha +\beta )\right|}\left[max\left\{1;\left|\frac{{\mathfrak{M}}_2}{{\mathfrak{M}}_1}-{\mathfrak{M}}_1\left(1-\mathsf{\Pi }\right)\right|\right\}\right.\\ \left.+2\left|1-\mathsf{\Pi }\right|+\frac{1}{2\left|{\mathfrak{M}}_1\right|}\left|3-2\mathsf{\Pi }\right|\right],\end{array}$$

where $\mathsf{\Pi }={\left[{\mathcal{K}}_1\right]}_{\mathrm{with}\rho =\tau -2}$. The inequality is sharp for each $\rho \in \mathbb{C}$.

Proof. 

From $\phi \left(\xi \right)=\xi +{\sum }_{n=2}^{\infty }{a}_n{\xi }^n$ and (21), we have

$$b_2=-a_2\mathrm{and}{b}_3=2a_2^2-a_3.$$

The estimate for $|b_2|=|a_2|$ follows immediately from (18). Letting $\rho =2$ in (13), we obtain the estimate $|b_3|$. To find the Fekete-Szegő inequality for the inverse function, consider

$$\left|b_3-\tau b_2^2\right|=\left|2a_2^2-a_3-\tau a_2^2\right|=\left|a_3-(\tau -2)a_2^2\right|.$$

Changing $\rho =(\tau -2)$ in (13), we obtain the desired result. □

5. Logarithmic Coefficients for Functions Belonging to ${\mathcal{R}}_{\mathbf{k},\mathbf{\lambda }}^{\mathbf{m}}(\mathbf{\alpha },\mathbf{\beta },\mathbf{\gamma };\mathbf{\psi })$

Inspired by recent works like [36,37,38], in this section we determine the coefficient bounds and Fekete-Szegő problem associated with the logarithmic function.

If the function $\phi$ is analytic in $\mathbb{U}$, such that ${\displaystyle \frac{\phi \left(\xi \right)}{\xi }}\ne 0$ for all $\xi \in \mathbb{U}$, then the well-known logarithmic coefficients $d_n:=d_n\left(\phi \right)$, $n\in \mathbb{N}$, of $\phi$ are given by

$$log\frac{\phi \left(\xi \right)}{\xi }=2\sum _{n=1}^{\infty }{d}_n{\xi }^n,z\in \mathbb{U}.$$

(22)

For a function $\phi \left(\xi \right)\in {\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$, the left-hand side of the subordination of Definition 1 should be an analytic function in $\mathbb{U}$, hence ${\displaystyle \frac{\phi \left(\xi \right)}{\xi }}\ne 0$ for all $\xi \in \mathbb{U}$. Therefore, for all functions $\phi \left(\xi \right)\in {\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$, the relation (22) is well defined.

Theorem 3.

If $\phi \left(\xi \right)\in {\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$, with the logarithmic coefficients given by (22), then

$$\begin{array}{cc}\hfill |d_1|\le & \frac{\left|\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(2\alpha +\beta )\right|}{{\left[1+\lambda \right]}^m\left|\mathsf{\Gamma }(\gamma +2k)\mathsf{\Gamma }(\alpha +\beta )\right|}\left[1+\left|{\mathfrak{M}}_1\right|\right],\hfill \\ \hfill |d_2|\le & \frac{6\left|{\mathfrak{M}}_1\right|\left|\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(3\alpha +\beta )\right|}{{\left[1+2\lambda \right]}^m\left|\mathsf{\Gamma }(\gamma +3k)\mathsf{\Gamma }(\alpha +\beta )\right|}\left[max\left\{1;\left|\frac{{\mathfrak{M}}_2}{{\mathfrak{M}}_1}-{\mathfrak{M}}_1\left(1-{\mathcal{K}}_2\right)\right|\right\}\right.\hfill \\ \hfill & \left.+2\left|1-{\mathcal{K}}_2\right|+\frac{1}{2\left|{\mathfrak{M}}_1\right|}\left|3-2{\mathcal{K}}_2\right|\right]\hfill \end{array}$$

with

$${\mathcal{K}}_2=\frac{{\left[1+2\lambda \right]}^m\mathsf{\Gamma }(\gamma +3k)\mathsf{\Gamma }(\gamma +k){\left(\mathsf{\Gamma }(2\alpha +\beta )\right)}^2}{3{\left[1+\lambda \right]}^{2m}{\left(\mathsf{\Gamma }(\gamma +2k)\right)}^2\mathsf{\Gamma }(3\alpha +\beta )\mathsf{\Gamma }(\alpha +\beta )}.$$

(23)

For $\mu \in \mathbb{C}$, we have

$$\begin{array}{c}|d_2-\mu d_1^2|\le \frac{3\left|{\mathfrak{M}}_1\right|\left|\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(3\alpha +\beta )\right|}{{\left[1+2\lambda \right]}^m\left|\mathsf{\Gamma }(\gamma +3k)\mathsf{\Gamma }(\alpha +\beta )\right|}\left[max\left\{1;\left|\frac{{\mathfrak{M}}_2}{{\mathfrak{M}}_1}-{\mathfrak{M}}_1\left(1-\mathcal{W}\right)\right|\right\}\right.\\ \left.+2\left|1-\mathcal{W}\right|+\frac{1}{2\left|{\mathfrak{M}}_1\right|}\left|3-2\mathcal{W}\right|\right],\end{array}$$

(24)

where $\mathcal{W}={\left[{\mathcal{K}}_1\right]}_{\mathrm{with}\rho =\frac{1+\mu }{2}}$, ${\mathcal{K}}_1$ is defined as in (14).

Proof. 

From $\phi \left(\xi \right)=\xi +{\sum }_{n=2}^{\infty }{a}_n{\xi }^n$ and equating the first two coefficients of relation (22), we get

$$d_1=\frac{a_2}{2},d_2=\frac{1}{2}\left(a_3-\frac{a_2^2}{2}\right).$$

Using (18) and (19), we obtain

$$\begin{array}{cc}\hfill d_1=& \frac{\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(2\alpha +\beta )}{{\left[1+\lambda \right]}^m\mathsf{\Gamma }(\gamma +2k)\mathsf{\Gamma }(\alpha +\beta )}\left[\frac{{\vartheta }_1{\mathfrak{M}}_1}{2}-1\right],\hfill \\ \hfill d_2=& \frac{1}{2}\left(a_3-\frac{a_2^2}{2}\right)\hfill \\ \hfill =& -\frac{3{\mathfrak{M}}_1\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(3\alpha +\beta )}{2{\left[1+2\lambda \right]}^m\mathsf{\Gamma }(\gamma +3k)\mathsf{\Gamma }(\alpha +\beta )}\left[{\vartheta }_2-\frac{{\vartheta }_1^2}{2}\left(1-\frac{{\mathfrak{M}}_2}{{\mathfrak{M}}_1}+{\mathfrak{M}}_1\right)+2{\vartheta }_1-\frac{3}{{\mathfrak{M}}_1}\right]\hfill \\ \hfill -& {\left[\frac{\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(2\alpha +\beta )}{{\left[1+\lambda \right]}^m\mathsf{\Gamma }(\gamma +2k)\mathsf{\Gamma }(\alpha +\beta )}\left(\frac{{\vartheta }_1{\mathfrak{M}}_1}{2}-1\right)\right]}^2.\hfill \end{array}$$

Using (11), it follows that

$$|d_1|\le \frac{\left|\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(2\alpha +\beta )\right|}{{\left[1+\lambda \right]}^m\left|\mathsf{\Gamma }(\gamma +2k)\mathsf{\Gamma }(\alpha +\beta )\right|}\left[1+\left|{\mathfrak{M}}_1\right|\right],$$

and fixing $\rho =\frac{1}{2}$ in (13), we get

$$\begin{array}{c}|d_2|\le \left|a_3-\frac{1}{2}{a}_2^2\right|=\frac{6\left|{\mathfrak{M}}_1\right|\left|\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(3\alpha +\beta )\right|}{{\left[1+2\lambda \right]}^m\left|\mathsf{\Gamma }(\gamma +3k)\mathsf{\Gamma }(\alpha +\beta )\right|}\left[max\left\{1;\left|\frac{{\mathfrak{M}}_2}{{\mathfrak{M}}_1}-{\mathfrak{M}}_1\left(1-{\mathcal{K}}_2\right)\right|\right\}\right.\\ \left.+2\left|1-{\mathcal{K}}_2\right|+\frac{1}{2\left|{\mathfrak{M}}_1\right|}\left|3-2{\mathcal{K}}_2\right|\right],\end{array}$$

where ${\mathcal{K}}_2$ is given by (23). To find estimate (24), consider

$$\left|d_2-\mu d_1^2\right|=\frac{1}{2}\left[a_3-\frac{(1+\mu )}{2}{a}_2^2\right].$$

Changing $\rho =\frac{1+\mu }{2}$ in (13) and simplifying, we obtain the desired result. □

6. Conclusions

Under varying selections of the function $\psi$ and parameters in the Definition 1, the function class ${\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$ reduces to classes with good geometrical implications but not to well-known classes like spiral-like, starlike, and convex. So our main results have lots of applications; here, we restricted ourselves to pointing out only a few of them.

Given the fact that $F^{**}\left(\xi \right)=e^{{[lnf\left(\xi \right)]}^{\prime \prime }}$, computing the estimates involving long differential characterization, which is in the exponent of an exponential, is cumbersome. Hence, extending this study to the class of convex functions would be very complicated. Further, from the coefficient estimates (see Theorem 1), it is very clear that functions in ${\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$ are not univalent. Now, the question arises: what should be the radius of the disc so that functions in ${\mathcal{R}}_{k,\lambda }^m(\alpha ,\beta ,\gamma ;\psi )$ are univalent?