Applications of Caputo-Type Fractional Derivatives for Subclasses of Bi-Univalent Functions with Bounded Boundary Rotation

Abstract

In this article, for the first time by using Caputo-type fractional derivatives, we introduce three new subclasses of bi-univalent functions associated with bounded boundary rotation in an open unit disk to obtain non-sharp estimates of the first two Taylor–Maclaurin coefficients, $|a_2|$ and $|a_3|$. Furthermore, the famous Fekete–Szegö inequality is obtained for the newly defined subclasses of bi-univalent functions. Several consequences of our results are pointed out which are new and not yet discussed in association with bounded boundary rotation. Some improved results when compared with those already available in the literature are also stated as corollaries.

1. Introduction

Indicate $\mathcal{A}$ as the class of all functions $h:\mathbb{E}\to \mathbb{C}$ defined by

$$h\left(\xi \right)=\xi +\sum _{n=2}^{\infty }{d}_n{\xi }^n$$

(1)

which are analytic in open unit disk $\mathbb{E}:=\left\{\xi \in \mathbb{C}:\left|\xi \right|<1\right\}$ with the normalization $h^{\prime }\left(0\right)-1=h\left(0\right).$ Let $\mathcal{S}$ be the subclass of $\mathcal{A}$, which are univalent in $\mathbb{E}.$ The well-known subclasses of class $\mathcal{S}$ are, namely, the starlike and convex classes, and the class of functions whose derivatives have a positive real part. The analytic descriptions of these classes of functions of order $\alpha (0\le \alpha <1)$ are given by the following (see [1]):

$${\mathcal{S}}^{*}\left(\alpha \right):=\left\{h\in \mathcal{S}:\Re \left({\displaystyle \frac{\xi h^{\prime }\left(\xi \right)}{h\left(\xi \right)}}\right)>\alpha ,0\le \alpha <1\right\},$$

$$\mathcal{C}\left(\alpha \right):=\left\{h\in \mathcal{S}:\Re \left({\displaystyle \frac{{\left(\xi h^{\prime }\left(\xi \right)\right)}^{\prime }}{h^{\prime }\left(\xi \right)}}\right)>\alpha ,0\le \alpha <1\right\}$$

and

$$\mathcal{R}\left(\alpha \right):=\left\{h\in \mathcal{S}:\Re \left(h^{\prime }\left(\xi \right)\right)>\alpha ,0\le \alpha <1\right\}.$$

Indicate ${\mathcal{V}}_{\vartheta }$ as the class of functions h given in (1) which map open unit disk $\mathbb{E}$ conformally onto an image domain $h\left(\mathbb{E}\right)$ of a boundary rotation of, at most, $\vartheta \pi .$ The functions belonging to the class ${\mathcal{V}}_{\vartheta }$are known as functions of bounded boundary rotation. Pinchuk [2] introduced the class ${\mathcal{V}}_{\vartheta }.$ Any function $h\in {\mathcal{V}}_{\vartheta }$ is expressed as

$${\int }_0^{2\pi }\left|\Re \left({\displaystyle \frac{{\left(r{e}^{i\lambda }{h}^{\prime }\left(r{e}^{i\lambda }\right)\right)}^{\prime }}{h^{\prime }\left(r{e}^{i\lambda }\right)}}\right)\right|d\lambda \le \vartheta \pi .$$

Assume $R_{\vartheta }$ as the class of functions h given in (1) which map open unit $\mathbb{E}$ conformally onto an image domain $h\left(\mathbb{E}\right)$ of a boundary radius rotation of, at most, $\vartheta \pi .$ The functions belonging to the class $R_{\vartheta }$ are known as functions of bounded radius rotation. If a function $h\in R_{\vartheta },$ then it can be expressed as

$${\int }_0^{2\pi }\left|\Re \left({\displaystyle \frac{r{e}^{i\lambda }{h}^{\prime }\left(r{e}^{i\lambda }\right)}{h\left(r{e}^{i\lambda }\right)}}\right)\right|d\lambda \le \vartheta \pi .$$

Let ${\mathcal{P}}_{\vartheta }$ be the class of functions W with $W\left(0\right)=1$ in $\mathbb{E}$ with an integral representation

$$W\left(\xi \right)={\int }_0^{2\pi }{\displaystyle \frac{1+\xi e^{-i\lambda }}{1-\xi e^{-i\lambda }}}d\theta \left(\lambda \right),$$

where $\theta \left(\lambda \right)$ is a function of bounded variation satisfying

$${\int }_0^{2\pi }d\theta \left(\lambda \right)=2and{\int }_0^{2\pi }\left|d\theta \left(\lambda \right)\right|\le \vartheta .$$

Noonan [3] gave the concept of the order of a function for both ${\mathcal{V}}_{\vartheta }$ and $R_{\vartheta }$ in 1971, and Padmanabhan and Parvatham [4] introduced the concept of the order of a function for ${\mathcal{P}}_{\vartheta }$ in 1975. Let ${\mathcal{P}}_{\vartheta }\left(\beta \right)$ be the class of function t in $\mathbb{E}$ normalized by the conditions $t\left(0\right)=1$ and

$${\int }_0^{2\pi }\left|{\displaystyle \frac{\Re \left(t\right(\xi \left)\right)-\beta }{1-\beta }}\right|d\xi \le \vartheta \pi$$

where $\vartheta \ge 2$ and $0\le \beta <1.$

Lemma 1

([5]). If a function $t\in {\mathcal{P}}_{\vartheta }\left(\beta \right)$ is given in the form

$$t\left(\zeta \right)=1+t_1\zeta +t_2{\zeta }^2+t_3{\zeta }^3+\cdots ,\zeta \in \mathbb{E},$$

then, for each $n\ge 1,$

$$|t_n|\le \vartheta (1-\beta )$$

where $\vartheta \ge 2$ and $0\le \beta <1.$ This result is sharp.

It is well known that [6] every function $h\in \mathcal{S}$ has an inverse $h^{-1},$ defined by

$$\xi =h^{-1}\left(h\left(\xi \right)\right),\forall \xi \in \mathbb{E}$$

and

$$\omega =h\left(h^{-1}\left(\omega \right)\right),\forall \left|\omega \right|<r_0\left(h\right)and{r}_0\left(h\right)\ge {\displaystyle \frac{1}{4}}.$$

Hence, the inverse function $h^{-1}$ is given by

$$\gamma \left(\omega \right)=h^{-1}\left(\omega \right)=\omega -h_2{\omega }^2+(2h_2^2-h_3){\omega }^2-(5h_2^3-5h_2{h}_3+h_4){\omega }^4+\cdots .$$

(2)

If both h and $h^{-1}$ belong to the class $\mathcal{S},$ then h is said to be bi-univalent. Let us indicate $\Sigma$ as the class of bi-univalent functions in $\mathbb{E}.$ We notice that the class $\Sigma$ is not empty. For instance, the functions

$$f_1\left(\xi \right)={\displaystyle \frac{\xi }{1-\xi }},f_2\left(\xi \right)={\displaystyle \frac{1}{2}}log{\displaystyle \frac{1+\xi }{1-\xi }}\mathrm{and}{f}_3\left(\xi \right)=-log(1-\xi )$$

with their corresponding inverses

$$f_1^{-1}\left(\omega \right)={\displaystyle \frac{\omega }{1+\omega }},f_2^{-1}\left(\omega \right)={\displaystyle \frac{e^{2\omega }-1}{e^{2\omega }+1}}\mathrm{and}{f}_3^{-1}\left(\omega \right)={\displaystyle \frac{e^{\omega }-1}{e^{\omega }}},$$

are elements of $\Sigma .$ However, the Koebe function is not a member of $\Sigma$. Lewin [7] introduced the class $\Sigma$, and it was proved that $|d_2|<1.51.$ The coefficient problem for each of the following Taylor–Maclaurin coefficients

$$|d_n|,n\in \mathbb{N}\setminus \{1,2\},$$

is still an open problem. Subsequently, Brannan and Clunie [8] conjectured that $\left|d_2\right|\le \sqrt{2}$, and Netanyahu [9] showed that, for $h\in \Sigma ,$$max|d_2|={\displaystyle \frac{4}{3}}.$ Motivated by the pioneer work of Stivastava et al. [10], many authors introduced and investigated various subclasses of the class $\Sigma$ and obtained estimates for their coefficients $|d_2|$ and $|d_3|$ for the functions in these subclasses. Brannan and Taha [11] introduced the subclasses of bi-univalent functions ${\mathcal{S}}_{\Sigma }^{*}\left(\beta \right)$ and ${\mathcal{K}}_{\Sigma }\left(\beta \right)$, called bi-starlike functions of order $\beta$, and ${\mathcal{K}}_{\Sigma }\left(\beta \right)$ bi-convex functions of order $\beta$, respectively.

In geometric function theory and its related field, the study of operators plays an important role. Several authors [12,13,14,15,16] introduced and investigated various subclasses of the class $\Sigma$ by using different operators.

Definition 1.

Assume that, in a simply connected region of the $\zeta -$ plane, including the origin, the function h is analytic. For order σ, the fractional integral of h is defined as

$${\mathcal{D}}_{\zeta }^{-\sigma }h\left(\zeta \right)={\displaystyle \frac{1}{\Gamma \left(\sigma \right)}}\underset{0}{\overset{\zeta }{\int }}{\displaystyle \frac{h\left(t\right)}{{(\zeta -t)}^{1-\sigma }}}dt,\sigma >0,$$

(3)

and the fractional derivatives of h order σ are

$${\mathcal{D}}_{\zeta }^{\sigma }h\left(\zeta \right)={\displaystyle \frac{1}{\Gamma (1-\sigma )}}{\displaystyle \frac{d}{d\zeta }}\underset{0}{\overset{\zeta }{\int }}{\displaystyle \frac{h\left(t\right)}{{(\zeta -t)}^{\sigma }}}dt,0\le \tau <1,$$

(4)

where the multiplicity of ${(\zeta -t)}^{1-\sigma }$ and ${(\zeta -t)}^{-\sigma }$ is removed by requiring $log(\zeta -t)$ to be real when $\zeta -t>0.$

Definition 2.

The fractional derivative of h of order $m+\sigma$ is

$${\mathcal{D}}_{\zeta }^{m+\sigma }h\left(\zeta \right)={\displaystyle \frac{d^m}{d{\zeta }^m}}{\mathcal{D}}_{\zeta }^{\sigma }h\left(\zeta \right),0\le \sigma <1;m\in {\mathbb{N}}_0.$$

(5)

With the aid of the above definitions, and their known extensions involving fractional derivatives and fractional integrals, Srivastava and Owa [17] introduced the operator

$${\Theta }^{\varrho }:\mathcal{A}\to \mathcal{A}$$

defined by

$${\Theta }^{\rho }h\left(\zeta \right)=\Gamma (2-\varrho ){\zeta }^{\rho }{\mathcal{D}}_{\zeta }^{\rho }h\left(\zeta \right)=\zeta +\sum _{n=2}^{\infty }\Phi (n,\rho )d_n{\zeta }^n,$$

where

$$\Phi (n,\rho )={\displaystyle \frac{\Gamma (n+1)\Gamma (2-\rho )}{\Gamma (n+1-\rho )}}$$

and $\rho \in \mathbb{R};\rho \ne 2,3,4,\cdots$. For $h\in \mathcal{A}$ and various choices of $\rho ,$ we obtain different operators

$${\Theta }^{0}h\left(\zeta \right)=h\left(\zeta \right)=\zeta +\sum _{n=2}^{\infty }{d}_n{\zeta }^n$$

(6)

$${\Theta }^{1}h\left(\zeta \right)=\zeta h^{\prime }\left(\zeta \right)=\zeta +\sum _{n=2}^{\infty }n{d}_n{\zeta }^n$$

(7)

$${\Theta }^{j}h\left(\zeta \right)=\Theta \left({\Theta }^{j-1}h\left(\zeta \right)\right)=\zeta +\sum _{n=2}^{\infty }{n}^j{d}_n{\zeta }^n,j=1,2,3,\cdots$$

(8)

commonly known as Salagean operators (Salagean [18]) by taking $j=-k,k\in \mathbb{N}$. It includes a Libera–Bernardi integral operator [19]. Throughout the article, we look at Caputo’s definition (Caplinger and Causey [20]) of the fractional-order derivative assumed to be

$${\mathcal{D}}^{\alpha }h\left(\zeta \right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}\underset{a}{\overset{n}{\int }}{\displaystyle \frac{h^{\left(t\right)}\left(\tau \right)}{{(n-\tau )}^{\alpha +1-t}}}d\tau$$

(9)

where $n-1<\Re \left(\alpha \right)\le n,n\in \mathbb{N},$ and the parameter $\alpha$ is allowed to be real or even complex. $\alpha$ is the initial value of the function $h.$ The operator that follows was recently defined by Salah and Darus in [21].

$${\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)={\displaystyle \frac{\Gamma (2+\upsilon -\rho )}{\Gamma (\upsilon -\rho )}}{\zeta }^{\rho -\upsilon }{\int }_0^{\zeta }{\displaystyle \frac{{\Theta }^{\upsilon }h\left(t\right)}{{(\zeta -t)}^{\rho +1-\upsilon }}}dt$$

(10)

where $\upsilon$ (real number) and $(\upsilon -1<\rho <\upsilon <2).$ With simple, straightforward computations for $h\in \mathcal{A}$, we obtain the following (see also [22]):

$$\begin{array}{ccc}\hfill {\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)& =& \zeta +\sum _{n=2}^{\infty }{\displaystyle \frac{\Gamma (2+\upsilon -\rho )\Gamma (2-\upsilon ){(\Gamma (n+1))}^2}{\Gamma (n-\upsilon +1)\Gamma (n+\upsilon -\rho +1)}}{d}_n{\zeta }^n,\zeta \in {\mathbb{U}}_d\hfill \\ & =& \zeta +\sum _{n=2}^{\infty }{\mathfrak{M}}_n{d}_n{\zeta }^n,\zeta \in \mathbb{E}\hfill \end{array}$$

(11)

where

$${\mathfrak{M}}_n={\displaystyle \frac{\Gamma (2+\upsilon -\rho )\Gamma (2-\upsilon ){(\Gamma (n+1))}^2}{\Gamma (n-\upsilon +1)\Gamma (n+\upsilon -\rho +1)}}.$$

(12)

Further, note that ${\mathcal{C}}_0^{0}h\left(\zeta \right)=h\left(\zeta \right)$ and ${\mathcal{C}}_1^{1}h\left(\zeta \right)=\zeta h^{\prime }\left(\zeta \right).$

Motivated by earlier works on certain classes of bi-univalent functions with bounded boundary rotation [23,24,25,26,27] and also references cited therein and application of Caputo-type fractional derivatives to bi-univalent functions [22,28], three new subclasses of bi-univalent functions associated with bounded boundary rotations in open unit disk $\mathbb{E}$ are introduced and investigated. For these new classes, the initial coefficient estimates and the Fekete–Szegö inequality are obtained. Moreover, certain special cases are stated as corollaries without proof.

2. Initial Coefficient Estimates and the Fekete–Szegö Inequality for $\mathit{h}\in {\mathit{R}}_{\mathbf{\Sigma }}^{\mathit{\upsilon },\mathit{\rho }}(\mathit{\lambda },\mathit{\vartheta },\mathit{\beta })$

Definition 3.

Let $h\in \mathcal{A}$ be given by (1) and $h\in R_{\Sigma }^{\upsilon ,\rho }(\lambda ,\vartheta ,\beta )$ if the following conditions

$$(1-\lambda ){\displaystyle \frac{{\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)}{\zeta }}+\lambda {\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{\prime }\in {\mathcal{P}}_{\vartheta }\left(\beta \right)$$

and

$$(1-\lambda ){\displaystyle \frac{{\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)}{\omega }}+\lambda {\left({\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)\right)}^{\prime }\in {\mathcal{P}}_{\vartheta }\left(\beta \right),$$

hold where $0\le \lambda \le 1,$ $2\le \vartheta \le 4,$ $0\le \beta <1,\upsilon -1<\rho <\upsilon <2$ and the function $\gamma \left(\omega \right)$ is as defined by (2).

Example 1.

Consider the function $h:\mathbb{E}⟶\mathbb{C}$ defined by

$${\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)=\zeta +{\displaystyle \frac{{\mathfrak{M}}_2}{4}}{\zeta }^2.$$

Clearly the function is $h\in \mathcal{S}.$ Now consider

$${\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{\prime }=1+{\mathfrak{M}}_2{\displaystyle \frac{\zeta }{2}}$$

and

$${\displaystyle \frac{{\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)}{\zeta }}=1+{\mathfrak{M}}_2{\displaystyle \frac{\zeta }{4}}.$$

Then

$$(1-\lambda ){\displaystyle \frac{{\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)}{\zeta }}+\lambda {\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{\prime }=1+{\mathfrak{M}}_2{\displaystyle \frac{1+\lambda }{4}}\zeta .$$

Now, if $\zeta =e^{i\theta }$, then

$$\Re \left((1-\lambda ){\displaystyle \frac{{\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)}{\zeta }}+\lambda {\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{\prime }\right)=1+{\mathfrak{M}}_2{\displaystyle \frac{1+\lambda }{4}}cos\theta .$$

$${\int }_0^{2\pi }\Re \left((1-\lambda ){\displaystyle \frac{{\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)}{\zeta }}+\lambda {\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{\prime }\right)d\theta ={\int }_0^{2\pi }\left(1+{\mathfrak{M}}_2{\displaystyle \frac{1+\lambda }{4}}cos\theta \right)=2\pi \le \vartheta \pi .$$

Hence, the function is $h\in R_{\Sigma }^{\upsilon ,\rho }(\lambda ,\vartheta ,\beta ).$

Remark 1.

Fixing $\lambda =1$ in Definition 3, we have $R_{\Sigma }^{\upsilon ,\rho }(1,\vartheta ,\beta )\equiv I_{\Sigma }^{\upsilon ,\rho }(\vartheta ,\beta ).$ That is, $h\in \Sigma$ is given by (1) and $h\in I_{\Sigma }^{\upsilon ,\rho }(\vartheta ,\beta )$ if

$${\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{\prime }\in {\mathcal{P}}_{\vartheta }\left(\beta \right)$$

and

$${\left({\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)\right)}^{\prime }\in {\mathcal{P}}_{\vartheta }\left(\beta \right),$$

hold where $2\le \vartheta \le 4,$ $0\le \beta <1,\upsilon -1<\rho <\upsilon <2$ and the function $\gamma \left(\omega \right)$ is as defined by (2).

Remark 2.

Assuming $\lambda =0$ in Definition 3, we have $R_{\Sigma }^{\upsilon ,\rho }(0,\vartheta ,\beta )\equiv J_{\Sigma }^{\upsilon ,\rho }(\vartheta ,\beta ).$ That is, function $h\in \Sigma$ is given by (1) and $h\in J_{\Sigma }^{\upsilon ,\rho }(\vartheta ,\beta )$ if

$${\displaystyle \frac{{\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)}{\zeta }}\in {\mathcal{P}}_{\vartheta }\left(\beta \right)$$

and

$${\displaystyle \frac{{\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)}{\omega }}\in {\mathcal{P}}_{\vartheta }\left(\beta \right),$$

hold where $2\le \vartheta \le 4,$ $0\le \beta <1,\upsilon -1<\rho <\upsilon <2$ and $\gamma \left(\omega \right)$ is as given by (2).

For the sake of brevity, unless otherwise stated, we let $2\le \vartheta \le 4$ and $0\le \beta <1$ and

$${\mathfrak{M}}_2={\displaystyle \frac{\Gamma (2+\upsilon -\rho )\Gamma (2-\upsilon ){(\Gamma \left(3\right))}^2}{\Gamma (3-\upsilon )\Gamma (3+\upsilon -\rho )}}.$$

(13)

and

$${\mathfrak{M}}_3={\displaystyle \frac{\Gamma (2+\upsilon -\rho )\Gamma (2-\upsilon ){(\Gamma \left(4\right))}^2}{\Gamma (4-\upsilon )\Gamma (4+\upsilon -\rho )}}.$$

(14)

Theorem 1.

Let $h\in R_{\Sigma }^{\upsilon ,\rho }(\lambda ,\vartheta ,\beta )$ be of the form (1). Then,

$$|d_n|\le \left\{\begin{array}{cc}\sqrt{{\displaystyle \frac{\vartheta (1-\beta )}{{\mathfrak{M}}_3(1+2\lambda )}}}\hfill & \mathit{for}n=2,\hfill \\ {\displaystyle \frac{\vartheta (1-\beta )}{{\mathfrak{M}}_3(1+2\lambda )}}\hfill & \mathit{for}n=3\hfill \end{array}\right..$$

(15)

For any $\aleph \in \mathbb{R},$

$$|d_3-\aleph d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\vartheta (1-\beta )(1-\aleph )}{2(1+2\lambda ){\mathfrak{M}}_3}}\hfill & \mathit{for}\aleph <0,\hfill \\ {\displaystyle \frac{\vartheta (1-\beta )}{(1+2\lambda ){\mathfrak{M}}_3}}\hfill & \mathit{for}0\le \aleph \le 2,\hfill \\ {\displaystyle \frac{\vartheta (1-\beta )(\aleph -1)}{2(1+2\lambda ){\mathfrak{M}}_3}}\hfill & \mathit{for}\aleph >2,\hfill \end{array}\right.$$

(16)

Proof. 

As $h\in R_{\Sigma }^{\upsilon ,\rho }(\lambda ,\vartheta ,\beta )$, from Definition 3,

$$(1-\lambda ){\displaystyle \frac{{\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)}{\zeta }}+\lambda {\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{\prime }=U\left(\zeta \right)$$

(17)

and

$$(1-\lambda ){\displaystyle \frac{{\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)}{\omega }}+\lambda {\left({\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)\right)}^{\prime }=V\left(\omega \right),$$

(18)

where U and V are analytic functions in ${\mathcal{P}}_{\vartheta }\left(\beta \right)$ given by

$$U\left(\zeta \right)=1+u_1\zeta +u_2{\zeta }^2+u_3{\zeta }^3+\cdots$$

(19)

and

$$V\left(\omega \right)=1+v_1\omega +v_2{\omega }^2+v_3{\omega }^3+\cdots .$$

(20)

Comparing the coefficients by using (17)–(20), we have

$$(1+\lambda ){\mathfrak{M}}_2{d}_2=u_1,$$

(21)

$$(1+2\lambda ){\mathfrak{M}}_3{d}_3=u_2,$$

(22)

$$-(1+\lambda ){\mathfrak{M}}_2{d}_2=v_1$$

(23)

and

$$2(1+2\lambda ){\mathfrak{M}}_3{d}_2^2-(1+2\lambda ){\mathfrak{M}}_3{d}_3=v_2.$$

(24)

Adding (22) and (24), we have

$$2(1+2\lambda ){\mathfrak{M}}_3{d}_2^2=u_2+v_2.$$

(25)

Now, by using Lemma 1, in (25), we have

$$|d_2{|}^2\le {\displaystyle \frac{\vartheta (1-\beta )}{{\mathfrak{M}}_3(1+2\lambda )}}.$$

(26)

Equation (26) gives the bound of $|d_2|$ given in (15). Now, by using Lemma 1, in (22), we obtain

$$(1+2\lambda ){\mathfrak{M}}_3\left|d_3\right|\le \vartheta (1-\beta )$$

(27)

which gives the bound of $|d_3|$ as in (15). Now fix $\aleph \in \mathbb{R}$, and, by using (22) and (25), we have

$$d_3-\aleph d_2^2={\displaystyle \frac{(2-\aleph )u_2-\aleph v_2}{2(1+2\lambda ){\mathfrak{M}}_3}}.$$

(28)

By appling Lemma 1, in (28), we obtain

$$|d_3-\aleph d_2^2|\le {\displaystyle \frac{\vartheta (1-\beta )\left[\right|2-\aleph |+|\aleph \left|\right]}{2(1+2\lambda ){\mathfrak{M}}_3}}$$

(29)

as given in (16). This completes the proof of Theorem 1. □

3. Initial Coefficient Estimates and the Fekete–Szegö Inequality for $\mathit{h}\mathbf{\in }{\mathit{N}}_{\mathbf{\Sigma }}^{\mathit{\upsilon }\mathbf{,}\mathit{\rho }}\mathbf{(}\mathit{\sigma }\mathbf{,}\mathit{\vartheta }\mathbf{,}\mathit{\beta }\mathbf{)}$

Definition 4.

Let $h\in \mathcal{A}$ be given by (1), which is in $N_{\Sigma }^{\upsilon ,\rho }(\sigma ,\vartheta ,\beta )$ if the conditions

$${\displaystyle \frac{\zeta {\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{\prime }}{{\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)}}+\sigma {\displaystyle \frac{{\zeta }^2{\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{″}}{{\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)}}\in {\mathcal{P}}_{\vartheta }\left(\beta \right)$$

and

$${\displaystyle \frac{\omega {\left({\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)\right)}^{\prime }}{{\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)}}+\sigma {\displaystyle \frac{{\omega }^2{\left({\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)\right)}^{″}}{{\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)}}\in {\mathcal{P}}_{\vartheta }\left(\beta \right),$$

hold where $\sigma \ge 0,$ $2\le \vartheta \le 4,$ $0\le \beta <1,\upsilon -1<\rho <\upsilon <2$ and the function $\gamma \left(\omega \right)$ is as defined by (2).

Remark 3.

If $\sigma =0$ in Definition 4, we have $N_{\Sigma }^{\upsilon ,\rho }(0,\vartheta ,\beta )\equiv {\mathcal{I}}_{\Sigma }^{\upsilon ,\rho }(\vartheta ,\beta ).$ That is, $h\in \Sigma$ is given by (1) and $h\in {\mathcal{I}}_{\Sigma }^{\upsilon ,\rho }(\vartheta ,\beta )$ if

$${\displaystyle \frac{\zeta {\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{\prime }}{{\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)}}\in {\mathcal{P}}_{\vartheta }\left(\beta \right)$$

and

$${\displaystyle \frac{\omega {\left({\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)\right)}^{\prime }}{{\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)}}\in {\mathcal{P}}_{\vartheta }\left(\beta \right),$$

where $2\le \vartheta \le 4,$ $0\le \beta <1,\upsilon -1<\rho <\upsilon <2$ and $\gamma \left(\omega \right)$ is assumed as in (2).

Theorem 2.

If $h\in N_{\Sigma }^{\upsilon ,\rho }(\sigma ,\vartheta ,\beta )$ is as in (1), then

$$|d_2|\le \sqrt{{\displaystyle \frac{\vartheta (1-\beta )}{2(1+3\sigma ){\mathfrak{M}}_3-(1+2\sigma ){\mathfrak{M}}_2^2}}}$$

(30)

and

$$|d_3|\le {\displaystyle \frac{\vartheta (1-\beta )}{2(1+3\sigma ){\mathfrak{M}}_3-(1+2\sigma ){\mathfrak{M}}_2^2}}.$$

(31)

For any $\aleph \in \mathbb{R},$

$$|d_3-\aleph d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\vartheta (1-\beta )(1-\aleph )}{2(1+3\sigma ){\mathfrak{M}}_3-(1+2\sigma ){\mathfrak{M}}_2^2}}:\hfill & \aleph <\Theta ,\hfill \\ {\displaystyle \frac{\vartheta (1-\beta )}{2(1+3\sigma ){\mathfrak{M}}_3}}:\hfill & \Theta \le \aleph \le 2-\Theta ,\hfill \\ {\displaystyle \frac{\vartheta (1-\beta )(\aleph -1)}{2(1+3\sigma ){\mathfrak{M}}_3-(1+2\sigma ){\mathfrak{M}}_2^2}}:\hfill & \aleph >2-\Theta ,\hfill \end{array}\right.$$

(32)

where

$$\Theta ={\displaystyle \frac{(1+2\sigma ){\mathfrak{M}}_2^2}{2(1+3\sigma ){\mathfrak{M}}_3}}.$$

Proof. 

As $h\in N_{\Sigma }^{\upsilon ,\rho }(\sigma ,\vartheta ,\beta )$, we have

$${\displaystyle \frac{\zeta {\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{\prime }}{{\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)}}+\sigma {\displaystyle \frac{{\zeta }^2{\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{″}}{{\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)}}=U\left(\zeta \right)$$

(33)

and

$${\displaystyle \frac{\omega {\left({\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)\right)}^{\prime }}{{\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)}}+\sigma {\displaystyle \frac{{\omega }^2{\left({\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)\right)}^{″}}{{\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)}}=V\left(\omega \right),$$

(34)

where $U,V\in {\mathcal{P}}_{\vartheta }\left(\beta \right)$ are analytic functions given by (19) and (20). Comparing the coefficients by using (19), (20), (33) and (34), we have

$$(1+2\sigma ){\mathfrak{M}}_2{d}_2=u_1,$$

(35)

$$2(1+3\sigma ){\mathfrak{M}}_3{d}_3-(1+2\sigma ){\mathfrak{M}}_2^2{d}_2^2=u_2,$$

(36)

$$-(1+2\sigma ){\mathfrak{M}}_2{d}_2=v_1,$$

(37)

and

$$[4(1+3\sigma ){\mathfrak{M}}_3-(1+2\sigma ){\mathfrak{M}}_2^2]d_2^2-2(1+3\sigma ){\mathfrak{M}}_3{d}_3=v_2.$$

(38)

Adding (36) and (38), we have

$$2[2(1+3\sigma ){\mathfrak{M}}_3-(1+2\sigma ){\mathfrak{M}}_2^2]d_2^2=u_2+v_2.$$

(39)

Now, by using Lemma 1, in (39), we have

$$|d_2{|}^2\le {\displaystyle \frac{\vartheta (1-\beta )}{2(1+3\sigma ){\mathfrak{M}}_3-(1+2\sigma ){\mathfrak{M}}_2^2}}.$$

(40)

which gives $|d_2|$, given in (30). Again, from (36) and (38), we have

$$4(1+3\sigma ){\mathfrak{M}}_3{d}_3-4(1+3\sigma ){\mathfrak{M}}_2^2{d}_2^2=u_2-v_2.$$

(41)

Now, using (39) in (41), we have

$$4(1+3\sigma ){\mathfrak{M}}_3{d}_3={\displaystyle \frac{[4(1+3\sigma ){\mathfrak{M}}_3-(1+2\sigma ){\mathfrak{M}}_2^2]u_2+(1+2\sigma ){\mathfrak{M}}_2^2{v}_2}{2(1+3\sigma ){\mathfrak{M}}_3-(1+2\sigma ){\mathfrak{M}}_2^2}}.$$

(42)

Now, by using Lemma 1, in (42), we have

$$[2(1+3\sigma ){\mathfrak{M}}_3-(1+2\sigma ){\mathfrak{M}}_2^2]d_3=\vartheta (1-\beta ).$$

(43)

which gives $|d_3|$ as given in (31). Now, by fixing $\aleph \in \mathbb{R}$ and by using (39) and (42), we have

$$d_3-\aleph d_2^2={\displaystyle \frac{[4(1+3\sigma ){\mathfrak{M}}_3-(1+2\sigma ){\mathfrak{M}}_2^2-2(1+3\sigma ){\mathfrak{M}}_3\aleph ]u_2+[(1+2\sigma ){\mathfrak{M}}_2^2-2(1+3\sigma ){\mathfrak{M}}_3\aleph ]v_2}{4(1+3\sigma ){\mathfrak{M}}_3[2(1+3\sigma ){\mathfrak{M}}_3-(1+2\sigma ){\mathfrak{M}}_2^2]}}.$$

(44)

Now, by applying Lemma 1, in (44), we have

$$|d_3-\aleph d_2^2|\le {\displaystyle \frac{\vartheta (1-\beta )\left[\right|4(1+3\sigma ){\mathfrak{M}}_3-(1+2\sigma ){\mathfrak{M}}_2^2-2(1+3\sigma ){\mathfrak{M}}_3\aleph |+|(1+2\sigma ){\mathfrak{M}}_2^2-2(1+3\sigma ){\mathfrak{M}}_3\aleph \left|\right]}{4(1+3\sigma ){\mathfrak{M}}_3[2(1+3\sigma ){\mathfrak{M}}_3-(1+2\sigma ){\mathfrak{M}}_2^2]}}.$$

(45)

Equation (45) gives the bound of $|d_3-\aleph d_2^2|$ given in (32), finishing Theorem 2. □

4. Initial Coefficient Estimates and the Fekete–Szegö Inequality for $\mathit{h}\mathbf{\in }{\mathit{M}}_{\mathbf{\Sigma }}^{\mathit{\upsilon }\mathbf{,}\mathit{\rho }}\mathbf{(}\mathit{\eta }\mathbf{,}\mathit{\vartheta }\mathbf{,}\mathit{\beta }\mathbf{)}$

Definition 5.

Let $h\in \mathcal{A}$ be given by (1) and for $0\le \eta \le 1,$ $2\le \vartheta \le 4,$ $0\le \beta <1,\upsilon -1<\rho <\upsilon <2$, and, if $h\in {\mathcal{M}}_{\Sigma }^{\upsilon ,\rho }(\eta ,\vartheta ,\beta )$, then

$$(1-\eta ){\displaystyle \frac{\zeta {\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{\prime }}{{\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)}}+\eta \left(1+{\displaystyle \frac{\zeta {\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{″}}{{\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{\prime }}}\right)\in {\mathcal{P}}_{\vartheta }\left(\beta \right)$$

and

$$(1-\eta ){\displaystyle \frac{\omega {\left({\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)\right)}^{\prime }}{{\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)}}+\eta \left(1+{\displaystyle \frac{\omega {\left({\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)\right)}^{″}}{{\left({\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)\right)}^{\prime }}}\right)\in {\mathcal{P}}_{\vartheta }\left(\beta \right),$$

are satisfied.

Remark 4.

If $\eta =0$ in Definition 5, we obtain ${\mathcal{M}}_{\Sigma }^{\upsilon ,\rho }(0,\vartheta ,\beta )\equiv {\mathcal{S}}_{\Sigma }^{\upsilon ,\rho }(\vartheta ,\beta )\equiv {\mathcal{I}}_{\Sigma }^{\upsilon ,\rho }(\vartheta ,\beta )$ by Remark 3.

Remark 5.

If $\eta =1$ in Definition 5 and we have ${\mathcal{M}}_{\Sigma }^{\upsilon ,\rho }(1,\vartheta ,\beta )\equiv {\mathcal{C}}_{\Sigma }^{\upsilon ,\rho }(\eta ,\vartheta ,\beta ),$ then $h\in {\mathcal{C}}_{\Sigma }^{\upsilon ,\rho }(\eta ,\vartheta ,\beta )$ is

$$1+{\displaystyle \frac{\zeta {\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{″}}{{\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{\prime }}}\in {\mathcal{P}}_{\vartheta }\left(\beta \right)$$

and

$$1+{\displaystyle \frac{\omega {\left({\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)\right)}^{″}}{{\left({\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)\right)}^{\prime }}}\in {\mathcal{P}}_{\vartheta }\left(\beta \right),$$

where $2\le \vartheta \le 4,$ $0\le \beta <1,\upsilon -1<\rho <\upsilon <2$ and $\gamma \left(\omega \right)$ is as in (2).

Theorem 3.

If $h\in {\mathcal{M}}_{\Sigma }^{\upsilon ,\rho }(\eta ,\vartheta ,\beta )$ is given in the form (1), then

$$|d_2|\le \sqrt{{\displaystyle \frac{\vartheta (1-\beta )}{2(1+2\eta ){\mathfrak{M}}_3-(1+3\eta ){\mathfrak{M}}_2^2}}}$$

(46)

and

$$|d_3|\le {\displaystyle \frac{\vartheta (1-\beta )}{2(1+2\eta ){\mathfrak{M}}_3-(1+3\eta ){\mathfrak{M}}_2^2}}.$$

(47)

For any $\aleph \in \mathbb{R},$

$$|d_3-\aleph d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\vartheta (1-\beta )(1-\aleph )}{2(1+2\eta ){\mathfrak{M}}_3-(1+3\eta ){\mathfrak{M}}_2^2}}:\hfill & \aleph <\pounds ,\hfill \\ {\displaystyle \frac{\vartheta (1-\beta )}{2(1+2\eta ){\mathfrak{M}}_3}}:\hfill & \pounds \le \aleph \le 2-\pounds ,\hfill \\ {\displaystyle \frac{\vartheta (1-\beta )(\aleph -1)}{2(1+2\eta ){\mathfrak{M}}_3-(1+3\eta ){\mathfrak{M}}_2^2}}:\hfill & \aleph >2-\pounds ,\hfill \end{array}\right.$$

(48)

where

$$\pounds ={\displaystyle \frac{(1+3\eta ){\mathfrak{M}}_2^2}{2(1+2\eta ){\mathfrak{M}}_3}}.$$

Proof. 

As $h\in {\mathcal{M}}_{\Sigma }^{\upsilon ,\rho }(\eta ,\vartheta ,\beta )$, we have

$$(1-\eta ){\displaystyle \frac{\zeta {\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{\prime }}{{\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)}}+\eta \left(1+{\displaystyle \frac{\zeta {\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{″}}{{\left({\mathcal{C}}_{\rho }^{\upsilon }h\left(\zeta \right)\right)}^{\prime }}}\right)=U\left(\zeta \right)$$

(49)

and

$$(1-\eta ){\displaystyle \frac{\omega {\left({\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)\right)}^{\prime }}{{\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)}}+\eta \left(1+{\displaystyle \frac{\omega {\left({\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)\right)}^{″}}{{\left({\mathcal{C}}_{\rho }^{\upsilon }\gamma \left(\omega \right)\right)}^{\prime }}}\right)=V\left(\omega \right),$$

(50)

where $U,V$ are analytic and, in ${\mathcal{P}}_{\vartheta }\left(\beta \right)$, given by (19) and (20). Comparing the coefficients by using (19), (20), (49) and (50), we have

$$(1+\eta ){\mathfrak{M}}_2{d}_2=u_1,$$

(51)

$$2(1+2\eta ){\mathfrak{M}}_3{d}_3-(1+3\eta ){\mathfrak{M}}_2^2{d}_2^2=u_2,$$

(52)

$$-(1+\eta ){\mathfrak{M}}_2{d}_2=v_1$$

(53)

and

$$[4(1+2\eta ){\mathfrak{M}}_3-(1+3\eta ){\mathfrak{M}}_2^2]d_2^2-2(1+2\eta ){\mathfrak{M}}_3{d}_3=v_2.$$

(54)

Adding (52) and (54), we have

$$2[2(1+2\eta ){\mathfrak{M}}_3-(1+3\eta ){\mathfrak{M}}_2^2]d_2^2=u_2+v_2.$$

(55)

Now, by using Lemma 1, in (55), we have

$$|d_2{|}^2\le {\displaystyle \frac{\vartheta (1-\delta )}{2(1+2\eta ){\mathfrak{M}}_3-(1+3\eta ){\mathfrak{M}}_2^2}}.$$

(56)

which gives the bound of $|d_2|$ given in (46). Again, from (52) and (54), we have

$$4(1+2\eta ){\mathfrak{M}}_3{d}_3-4(1+2\eta ){\mathfrak{M}}_3{d}_2^2=u_2-v_2.$$

(57)

Now, using (55) in (57), we have

$$4(1+2\eta ){\mathfrak{M}}_3{d}_3={\displaystyle \frac{[4(1+2\eta ){\mathfrak{M}}_3-(1+3\eta ){\mathfrak{M}}_2^2]u_2+(1+3\eta ){\mathfrak{M}}_2^2{v}_2}{2(1+2\eta ){\mathfrak{M}}_3-(1+3\eta ){\mathfrak{M}}_2^2}}.$$

(58)

Now, by using Lemma 1, in (58), we have

$$[2(1+2\eta ){\mathfrak{M}}_3-(1+3\eta ){\mathfrak{M}}_2^2]d_3=\vartheta (1-\beta ).$$

(59)

which gives the bound of $|d_3|$ given in (47). Now, fixing $\aleph \in \mathbb{R}$ and by using (55) and (58), we have

$$d_3-\aleph d_2^2={\displaystyle \frac{[4(1+2\eta ){\mathfrak{M}}_3-(1+3\eta ){\mathfrak{M}}_2^2-2(1+2\eta ){\mathfrak{M}}_3\aleph ]u_2+[(1+3\eta ){\mathfrak{M}}_2^2-2(1+2\eta )\aleph ]v_2}{4(1+2\eta ){\mathfrak{M}}_3[2(1+2\eta ){\mathfrak{M}}_3-(1+3\eta ){\mathfrak{M}}_2^2]}}.$$

(60)

Now, by using Lemma 1, in (60), we have

$$|d_3-\aleph d_2^2|\le {\displaystyle \frac{\vartheta (1-\beta )\left[\right|4(1+2\eta ){\mathfrak{M}}_3-(1+3\eta ){\mathfrak{M}}_2^2-2(1+2\eta ){\mathfrak{M}}_3\aleph |+|(1+3\eta ){\mathfrak{M}}_2^2-2(1+2\eta )\aleph \left|\right]}{4(1+2\eta ){\mathfrak{M}}_3[2(1+2\eta ){\mathfrak{M}}_3-(1+3\eta ){\mathfrak{M}}_2^2]}}.$$

(61)

which gives the bound of $|d_3-\aleph d_2^2|$ given in (48) which completes the proof of Theorem 3. □

5. Corollaries and Consequences

For the choices of $\lambda =1$ and $\lambda =0$ in Theorem 1, we obtain the following corollaries, namely, Corollary 1 and Corollary 2, respectively.

Corollary 1.

If $h\in I_{\Sigma }^{\upsilon ,\rho }(\vartheta ,\beta )$ is given in the form (1), then

$$|d_2|\le \sqrt{{\displaystyle \frac{\vartheta (1-\beta )}{3{\mathfrak{M}}_3}}}$$

and

$$|d_3|\le {\displaystyle \frac{\vartheta (1-\beta )}{3{\mathfrak{M}}_3}}.$$

For any $\aleph \in \mathbb{R},$

$$|d_3-\aleph d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\vartheta (1-\beta )(1-\aleph )}{6{\mathfrak{M}}_3}}\hfill & \mathrm{for}\aleph <0,\hfill \\ {\displaystyle \frac{\vartheta (1-\beta )}{3{\mathfrak{M}}_3}}\hfill & \mathrm{for}0\le \aleph \le 2,\hfill \\ {\displaystyle \frac{\vartheta (1-\beta )(\aleph -1)}{6{\mathfrak{M}}_3}}\hfill & \mathrm{for}\aleph >2,\hfill \end{array}\right.$$

(62)

Corollary 2.

If $h\in J_{\Sigma }^{\upsilon ,\rho }(\vartheta ,\beta )$ is of the form (1), then

$$|d_2|\le \sqrt{{\displaystyle \frac{\vartheta (1-\beta )}{{\mathfrak{M}}_3}}}$$

and

$$|d_3|\le {\displaystyle \frac{\vartheta (1-\beta )}{{\mathfrak{M}}_3}}.$$

For any $\aleph \in \mathbb{R},$

$$|h_3-\aleph h_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\vartheta (1-\beta )(1-\aleph )}{2{\mathfrak{M}}_3}}\hfill & \mathrm{for}\aleph <0,\hfill \\ {\displaystyle \frac{\vartheta (1-\beta )}{{\mathfrak{M}}_3}}\hfill & \mathrm{for}0\le \aleph \le 2,\hfill \\ {\displaystyle \frac{\vartheta (1-\beta )(\aleph -1)}{2{\mathfrak{M}}_3}}\hfill & \mathrm{for}\aleph >2,\hfill \end{array}\right.$$

For the selection of $\sigma =0$ in Theorem 2, we obtain Corollary 3.

Corollary 3.

If $h\in {\mathcal{I}}_{\Sigma }^{\upsilon ,\rho }(\vartheta ,\beta )$ is represented in the form (1), then

$$|d_2|\le \sqrt{{\displaystyle \frac{\vartheta (1-\beta )}{2{\mathfrak{M}}_3-{\mathfrak{M}}_2^2}}}$$

(63)

and

$$|d_3|\le {\displaystyle \frac{\vartheta (1-\beta )}{2{\mathfrak{M}}_3-{\mathfrak{M}}_2^2}}.$$

(64)

For any $\aleph \in \mathbb{R},$

$$|d_3-\aleph d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\vartheta (1-\beta )(1-\aleph )}{2{\mathfrak{M}}_3-{\mathfrak{M}}_2^2}}:\hfill & \aleph <{\Theta }^{*},\hfill \\ {\displaystyle \frac{\vartheta (1-\beta )}{2{\mathfrak{M}}_3}}:\hfill & {\Theta }^{*}\le \aleph \le 2-{\Theta }^{*},\hfill \\ {\displaystyle \frac{\vartheta (1-\beta )(\aleph -1)}{2{\mathfrak{M}}_3-{\mathfrak{M}}_2^2}}:\hfill & \aleph >2-{\Theta }^{*},\hfill \end{array}\right.$$

(65)

where

$${\Theta }^{*}={\displaystyle \frac{{\mathfrak{M}}_2^2}{2{\mathfrak{M}}_3}}.$$

For the choices $\eta =0$and $\eta =1$in Theorem 3, we obtain Corollary 3 and Corollary 4.

Corollary 4.

If $h\in {\mathcal{C}}_{\Sigma }^{\upsilon ,\rho }(\vartheta ,\beta )$ is of the form (1), then

$$|d_2|\le \sqrt{{\displaystyle \frac{\vartheta (1-\beta )}{6{\mathfrak{M}}_3-4{\mathfrak{M}}_2^2}}}$$

and

$$|d_3|\le {\displaystyle \frac{\vartheta (1-\beta )}{6{\mathfrak{M}}_3-4{\mathfrak{M}}_2^2}}.$$

For any $\aleph \in \mathbb{R},$ then

$$|d_3-\aleph d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\vartheta (1-\beta )(1-\aleph )}{6{\mathfrak{M}}_3-4{\mathfrak{M}}_2^2}}:\hfill & \aleph <{\pounds }^{**},\hfill \\ {\displaystyle \frac{\vartheta (1-\beta )}{6{\mathfrak{M}}_3}}:\hfill & {\pounds }^{**}\le \aleph \le 2-{\pounds }^{**},\hfill \\ {\displaystyle \frac{\vartheta (1-\beta )(\aleph -1)}{6{\mathfrak{M}}_3-4{\mathfrak{M}}_2^2}}:\hfill & \aleph >2-{\pounds }^{**},\hfill \end{array}\right.$$

(66)

where

$${\pounds }^{**}={\displaystyle \frac{4{\mathfrak{M}}_2^2}{6{\mathfrak{M}}_3}}.$$

6. Conclusions

Three new subclasses of bi-univalent functions related to bounded boundary rotation are presented in this work based on Caputo-type fractional derivatives. The first two initial non-sharp Taylor–Maclaurin coefficient bounds are obtained for these new function classes. Additionally, for these new functions classes, the renowned Fekete–Szegö inequality is also derived. When compared to the previously published results in the literature, some enhanced outcomes are also mentioned. Since there are many differential operators in the literature, other operators can also be used for these classes considered here or associated subclasses in $\mathcal{S}.$ Apart from this, more corollaries can be stated for the choice of parameters involved in Caputo-type fractional derivatives (including many fractional derivatives and integral operators).