Some Applications of Analytic Functions Associated with q-Fractional Operator

Abstract

This paper introduces a new fractional operator by using the concepts of fractional q-calculus and q-Mittag-Leffler functions. With this fractional operator, Janowski functions are generalized and studied regarding their certain geometric characteristics. It also establishes the solution of the complex Briot–Bouquet differential equation by using the newly defined operator.

1. Introduction and Preliminaries

Let the class of all analytic functions h in the open unit disk $\mathcal{U}=\{\eta \in \mathbb{C}:\left|\eta \right|<1\}$ be denoted by $\mathcal{A}$ and the Taylor series expansion of $h\in \mathcal{A}$ be given as

$$h\left(\eta \right)=\eta +\sum _{n=2}^{\infty }{a}_n{\eta }^n.$$

(1)

The subset of $\mathcal{A}$ consisting of all univalent functions is denoted by $\mathcal{S}$. The well-known subclasses of the class $\mathcal{S}$ are convex $\left(\mathcal{C}\right)$ and starlike $\left({\mathcal{S}}^{\ast }\right)$ functions. Another important class $\mathcal{P}$ of analytic functions p maps the open unit disk onto the right half plane with $p\left(0\right)=1$ and $Rep\left(\eta \right)>0$ for $\eta \in \mathcal{U}$.

The class $\mathcal{P}[\mathcal{L},\mathcal{M}]$ was introduced by Janowski [1]. The function f with $f\left(0\right)=1$ is said to belong to the class $\mathcal{P}[\mathcal{L},\mathcal{M}]$ if and only if

$$f\left(\eta \right)=\frac{(\mathcal{L}+1)p\left(\eta \right)-(\mathcal{L}-1)}{(\mathcal{M}+1)p\left(\eta \right)-(\mathcal{M}-1)},-1\le \mathcal{M}<\mathcal{L}\le 1.$$

The function h is subordinate to an analytic function $\psi$ written $h\left(\eta \right)\prec \psi \left(\eta \right)$ if there exists a function $u\left(\eta \right)$$(\eta \in \mathcal{U})$ such that $u\left(0\right)=0$ and $\left|u\right(\eta \left)\right|<1,$ such that

$$h\left(\eta \right)=\psi \left(u\right(\eta \left)\right).$$

Let $h\left(\eta \right)$ be majorized by $\psi \left(\eta \right)$, written as

$$(h\ll \psi )\iff h\left(\eta \right)=u\left(\eta \right)\psi \left(\eta \right),\eta \in \mathcal{U}.$$

It is noted that majorization is closely related to the concept of quasi-subordination between analytic functions.

The convolution of two analytic functions h and $\psi$ (denoted by $h\ast \psi )$ is defined as

$$(h\ast \psi )\left(\eta \right)=\sum _{n=0}^{\infty }{a}_n{b}_n{\eta }^n=(\psi \ast h)\left(\eta \right),$$

where h is given by (1) and

$$\psi \left(\eta \right)=\sum _{n=0}^{\infty }{b}_n{\eta }^n,(\eta \in \mathcal{U}).$$

The theory of basic and fractional quantum calculus plays an important role in many diverse areas of mathematical, physical and engineering sciences. The various types of fractional differential equations play an important role not only in mathematics but also in physics, control systems, dynamical systems and engineering to create the mathematical modeling of many physical phenomena (see, for example, [2,3,4,5,6]).

Fractional calculus is a vast and growing subject of interest for mathematicians and physicist. The theory of fractional calculus has been applied to the theory of analytic functions. Fractional differential equations are emerging as a new and famous branch of applied mathematics that is being used for many mathematical models in science and engineering. In fact, fractional differential equations are viewed as an alternative model to nonlinear differential equations (see, for example, [7,8,9,10]).

The operators are used to define different subclasses of analytic functions and to solve fractional algebraic differential equations (see [11,12]). In order to investigate several subclasses of class $\mathcal{A}$, the q-calculus as well as the fractional q-calculus have been used as an important tool. A firm footing of the usage of q-calculus in the context of geometric function theory was provided by Srivastava in the book [13]. Recently, in [14], Srivastava and Bansal studied a certain family of q-Mittag-Leffler functions and found sufficient conditions for a function to belong to the family of close-to-convex functions. In the survey-cum-expository article [15], Srivastava provided the operators of basic (or q-) calculus and fractional q-calculus and discussed their applications to the geometric function theory of complex analysis. For more recent work on analytic functions, see [16,17,18,19] and the references therein.

Now, we give some basic definitions of fractional q-calculus, which help us to define new classes of functions.

Definition 1.

For $a,$$q\in \mathbb{C},$ the q-shifted factorial ${(a,q)}_n$ is defined by

$${(a,q)}_n=\prod _{k=0}^{n-1}\left(1-a{q}^k\right),\left(n\in \mathbb{N}\right).$$

(2)

If $a\ne q^{-m},$$\left(m\in {\mathbb{N}}_0\right),$ then we may define:

$${(a,q)}_{\infty }=\prod _{k=0}^{\infty }\left(1-a{q}^k\right),\left(a\in \mathbb{C}\mathrm{and}\left|q\right|<1\right),$$

(3)

when $a\ne 0$ and $\left|q\right|\ge 1,$ then ${(a,q)}_{\infty }$ diverges. So, whenever we use ${(a,q)}_{\infty }$, then $\left|q\right|<1$ is assumed.

Remark 1.

It is noted that when $q\to 1-$ in ${(a,q)}_n$, then (2) reduces to Pochhammer symbol ${\left(a\right)}_n$ defined by

$${\left(a\right)}_n=\begin{array}{cc}a(a+1)...(a+n-1)& \mathrm{if}n\in \mathbb{N}.\end{array}$$

If $n=0,$ then ${\left(a\right)}_n=1.$

Definition 2.

The q-Gamma function is defined in terms of ${(a,q)}_n$ in (2) as follows:

$${\Gamma }_q\left(a\right)=\frac{{\left(1-q\right)}^{1-\alpha }{(q,q)}_{\infty }}{{(q^a,q)}_{\infty }},\left(0<q<1\right)$$

or

$${(q^a,q)}_n=\frac{\left(1-q^n\right){\Gamma }_q(a+n)}{{\Gamma }_q\left(a\right)},\left(n\in \mathbb{N}\right),$$

and q-factorial ${\left[n\right]}_q!$ is defined by:

$${\left[n\right]}_q!=\prod _{k=1}^n\left[k\right],(n\in \mathbb{N}).$$

(4)

If $n=0,$ then ${\left[n\right]}_q!=1.$

Definition 3

([20]). For $0<q<1$ and $h\in \mathcal{A},$ the q-derivative operator $\left(D_q\right)$ is defined as

$$D_{q}h\left(\eta \right)=\frac{h\left(\eta \right)-h\left(q\eta \right)}{(1-q)\eta },\eta \in \mathcal{U}.$$

(5)

From (1) and (5), we get the following series of the form

$$D_{q}h\left(\eta \right)=1+\sum _{n=2}^{\infty }{\left[n\right]}_q{a}_n{\eta }^{n-1},$$

where

$${\left[n\right]}_q=\frac{1-q^n}{1-q}=1+q+q^2+...+q^{n-1}.$$

For $n\in \mathbb{N}$ and $\eta \in \mathcal{U},$ we have

$$D_q{\eta }^n={\left[n\right]}_q{\eta }^{n-1},D_q\left(\sum _{n=1}^{\infty }{a}_n{\eta }^n\right)=\sum _{n=1}^{\infty }{\left[n\right]}_q{a}_n{\eta }^{n-1}.$$

Note that

$$\underset{q\to 1-}{lim}{D}_{q}h\left(\eta \right)=h^{{}^{\prime }}\left(\eta \right).$$

Definition 4.

The q-integral of a function $h\left(t\right)$ is defined by

$${\int }_0^{\eta }h\left(t\right)d_q\left(t\right)=\eta (1-q)\sum _{n=0}^{\infty }{q}^{n}h\left(\eta q^n\right).$$

In article [21,22], Mittag-Leffler introduced the Mittag-Leffler function ${\mathcal{H}}_{\alpha }\left(\eta \right)$ as:

$${\mathcal{H}}_{\alpha }\left(\eta \right)=\sum _{n=0}^{\infty }\frac{1}{\Gamma (\alpha n+1)}{\eta }^n,\left(\alpha \in \mathbb{C},Re\left(\alpha \right)>0\right),$$

and the function ${\mathcal{H}}_{\alpha ,\beta }\left(\eta \right)$ introduced in [23], having the series representation

$${\mathcal{H}}_{\alpha ,\beta }\left(\eta \right)=\sum _{n=0}^{\infty }\frac{1}{\Gamma (\alpha n+\beta )}{\eta }^n,\left(\alpha ,\beta \in \mathbb{C},Re\left(\alpha \right)>0,Re\left(\beta \right)>0\right).$$

In 2014, Sharma et al. [24] generalized the idea of the Mittag-Leffler function by using q-calculus as follows:

$${\mathcal{H}}_{\alpha ,\beta }(\eta ,q)=\sum _{n=0}^{\infty }\frac{1}{{\Gamma }_q(\alpha n+\beta )}{\eta }^n,$$

(6)

and the normalization of q-Mittag-Leffler function ${\mathcal{T}}_{\alpha ,\beta }^q\left(\eta \right)$ is given by

$$\begin{array}{ccc}\hfill {\mathcal{T}}_{\alpha ,\beta }^q\left(\eta \right)& =& \eta {\Gamma }_q\left(\beta \right){\mathcal{H}}_{\alpha ,\beta }\left(\eta \right)\hfill \\ & & \hfill \\ & =& \eta +\sum _{n=2}^{\infty }\frac{{\Gamma }_q\left(\beta \right)}{{\Gamma }_q(\alpha \left(n-1\right)+\beta )}{\eta }^n,\hfill \end{array}$$

(7)

where, $\eta \in \mathcal{U},$$Re\alpha >0$, $\beta \in \mathbb{C}\backslash \left\{0,-1,-2,\dots \right\}.$

Note that the q-Mittag-Leffler function is the special case of the q-Fox–Wright function (see, [25,26,27]).

For $\delta >0,$ the fractional q-integral operator (see [28]) given as

$$I_q^{\delta }h\left(\eta \right)=I_q^{-\delta }h\left(\eta \right)=\frac{1}{{\Gamma }_q\left(\delta \right)}{\int }_0^{\eta }{\left(\eta -tq\right)}_{\delta -1}h\left(t\right)d_q\left(t\right),$$

(8)

where the q-binomial function ${\left(\eta -tq\right)}_{\delta -1}$ is defined by

$${\left(\eta -tq\right)}_{\delta -1}={\eta }^{\delta -1}{}_1{\mathsf{\Phi }}_0\left(q^{-\delta +1},-,q,t{q}^{\delta }/\eta \right).$$

The representation of series ${}_1{\mathsf{\Phi }}_0$ is given by

$${}_1{\mathsf{\Phi }}_0\left(a,-,q,\eta \right)=1+\sum _{n=1}^{\infty }\frac{{\left(a,q\right)}_n}{{\left(q,q\right)}_n}{\eta }^n,\left(\left|q\right|<1,\left|\eta \right|<1\right)$$

The last equality is called q-binomial theorem (see [29]). The series ${}_1{\mathsf{\Phi }}_0\left(a,-,q,\eta \right)$ is single-valued when $\left|arg\left(\eta \right)\right|<\pi$ and $\left|\eta \right|<1$, and so the function ${\left(\eta -tq\right)}_{\delta -1}$ in (8) is single-valued when $\left|arg(-t{q}^{\delta }/\eta )\right|<\pi ,\left|(t{q}^{\delta }/\eta )\right|<\pi$ and $\left|arg\left(\eta \right)\right|<\pi .$

Indeed, for a complex-valued function $h\left(\eta \right)$, the fractional q-derivative (or the q-difference) operator $D_q$ is given by Definition 5 below, which is defined as follows (see, for example, Purohit and Raina [28]; see also Srivastava [15]).

Definition 5.

For analytic function $h\left(\eta \right),$ the fractional q-derivative operator $D_q$ of order δ is defined by

$$\begin{array}{ccc}\hfill D_{q}h\left(\eta \right)& =& D_q{I}_q^{1-\delta }h\left(\eta \right)\hfill \\ & =& \frac{1}{{\Gamma }_q\left(1-\delta \right)}{D}_q{\int }_0^{\eta }{\left(\eta -tq\right)}_{-\delta }h\left(t\right)d_q\left(t\right),\left(0\le \delta <1\right).\hfill \end{array}$$

Definition 6.

For m being the smallest integer. The extended fractional q-derivative $D_{q,\eta }^{\delta }$ of order δ is defined by

$$D_q^{\delta }h\left(\eta \right)=D_q^m{I}_q^{m-\delta }h\left(\eta \right).$$

(9)

We find from (9) that

$$D_q^{\delta }{\eta }^n=\frac{{\Gamma }_q\left(n+1\right)}{{\Gamma }_q\left(n+1-\delta \right)}{\eta }^{n-\delta },(0\le \delta ,n>-1).$$

Note that: When $-\infty <\delta <0,$ then $D_q^{\delta }$ represents a fractional q-integral of $h\left(\eta \right)$ of order $\delta$, and for $0\le \delta <2,$ then $D_q^{\delta }$ represents a fractional q-derivative of $h\left(\eta \right)$ of order $\delta$.

Definition 7

([30]). Selvakumaran et al. defined the q-differintegral operator ${\mathsf{\Omega }}_q^{\delta }:\mathcal{A}\to \mathcal{A}$ as follows:

$$\begin{array}{ccc}\hfill {\mathsf{\Omega }}_q^{\delta }h\left(\eta \right)& =& \frac{{\Gamma }_q\left(2-\delta \right)}{{\Gamma }_q\left(2\right)}{\eta }^{\delta }{D}_q^{\delta }h\left(\eta \right)\hfill \\ & =& \eta +\sum _{n=2}^{\infty }\frac{{\Gamma }_q\left(2-\delta \right){\Gamma }_q\left(n+1\right)}{{\Gamma }_q\left(2\right){\Gamma }_q\left(n+1-\delta \right)}{a}_n{\eta }^n,\eta \in \mathcal{U},\hfill \end{array}$$

(10)

here,

$$\delta <2,0<q<1.$$

Now, by using the technique of convolution on (7) and (10), we define a new type of q-differintegral operator ${\mathfrak{D}}_q(\alpha ,\beta ,\delta ):\mathcal{A}\to \mathcal{A},$ as follows:

Definition 8.

For $h\in \mathcal{A},$ the q-differintegral operator ${\mathfrak{D}}_q(\alpha ,\beta ,\delta )$ of a function $h\left(\eta \right)$ is defined by

$$\left({\mathfrak{D}}_q(\alpha ,\beta ,\delta )h\right)\left(\eta \right)=\eta +\sum _{n=2}^{\infty }Q(q,\delta ,\alpha ,\beta )a_n{\eta }^n,$$

(11)

where,

$$Q(q,\delta ,\alpha ,\beta )=\left(\frac{{\Gamma }_q\left(2-\delta \right){\Gamma }_q\left(n+1\right)}{{\Gamma }_q\left(2\right){\Gamma }_q\left(n+1-\delta \right)}\right)\left(\frac{{\Gamma }_q\left(\beta \right)}{{\Gamma }_q(\alpha \left(n-1\right)+\beta )}\right)$$

(12)

and

$$\delta <2,0<q<1,Re\alpha >0,\beta \in \mathbb{C}\backslash \left\{0,-1,-2,\dots \right\}\mathrm{and}\eta \in \mathcal{U}.$$

Clearly

$${\mathfrak{D}}_q(0,1,0)h\left(\eta \right)=h\left(\eta \right).$$

Definition 9.

Let $h\in \mathcal{A},$ a new extended form of the linear multiplier fractional q-differintegral operator, be defined as

$$\begin{array}{ccc}\hfill {\mathsf{\Lambda }}_{\alpha ,0}^{\beta ,0}\left(0,q\right)h\left(\eta \right)& =& h\left(\eta \right)\hfill \\ \hfill {\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,1}\left(m,q\right)& =& \left(1-\lambda \right){\mathfrak{D}}_q(\alpha ,\beta ,\delta )h\left(\eta \right)+\lambda \eta {\left({\mathfrak{D}}_q(\alpha ,\beta ,\delta )h\left(\eta \right)\right)}^{{}^{\prime }},\hfill \\ & & ⋮\hfill \\ \hfill {\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)& =& {\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,1}\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta -1}\left(m,q\right)h\left(\eta \right)\right).\hfill \end{array}$$

(13)

It is seen from (11) that, for $h\left(\eta \right)$ given in (1), we have

$${\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)=\eta +\sum _{n=2}^{\infty }{A}_q\left(\lambda ,\delta ,\alpha ,\beta ,m,n\right)a_n{\eta }^n,$$

(14)

where

$$A_q\left(\lambda ,\delta ,\alpha ,\beta ,m,n\right)={\left(Q(q,\delta ,\alpha ,\beta )\left(1-\lambda +{\left[n\right]}_q\lambda \right)\right)}^m,$$

$$(\delta <2,m\in \mathbb{N},\lambda \ge 0,0<q<1,Re\alpha >0,\beta \in \mathbb{C}\backslash \left\{0,-1,-2,\dots \right\}),\eta \in \mathcal{U}.$$

and $Q(q,\delta ,\alpha ,\beta )$ is given by (12).

1. For $\alpha =0,$ and $\beta =1$, then (14) reduces to the operator introduced in [30].

2. When $\delta =0$, $\alpha =0,\beta =1$ and $q\to 1-$, then (14) reduces to the operator introduced by Al-Oboudi [31].

3. When $\delta =0,$$\lambda =1,$$\alpha =0,$$\beta =1$, then (14) reduces to the Salagean q-differential operator defined in [32].

4. When $\delta =0,$$\lambda =1,$$\alpha =0,$$\beta =1$, and $q\to 1-,$ then (14) reduces to the operator given by the Salagean differential operator defined in [33].

Now, considering the above-defined operator ${\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right),$ we define two new subclasses of analytic functions and find some interesting and (potentially) useful properties for these functions. It is also noted that the results presented here are general enough to reduce to yield many simpler ones.

Definition 10.

Let the function h given in (1) belongs to the class ${\mathcal{S}}_q^{\ast }(\alpha ,\beta ,\sigma ,\lambda ,\delta )$ if and only if

$${\mathcal{S}}_q^{\ast }(\sigma ,\alpha ,\beta ,\lambda ,\delta )=\left\{h\in \mathcal{A}:\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}\prec \sigma \left(\eta \right),\sigma \left(0\right)=1\right\}.$$

(1). For $\delta =0,\lambda =1,\alpha =0,\beta =1$ and $m=0,$ then

$${\mathcal{S}}_q^{\ast }(\sigma ,\alpha ,\beta ,\lambda ,\delta )={\mathcal{S}}^{\ast }\left(\sigma \right).$$

(2). For $\delta =0,\lambda =1,\alpha =0,\beta =1$ and $m=0,$ then (see [34,35])

$${\mathcal{S}}_q^{\ast }(\sigma ,\alpha ,\beta ,\lambda ,\delta )={\mathcal{S}}^{\ast }\left(\sigma \right),\mathrm{and}\sigma \left(\eta \right)=\frac{1+\mathcal{L}\eta }{1+\mathcal{M}\eta }.$$

(3). For $\delta =0,\lambda =1,\alpha =0,\beta =1$ and $m=0,$ then (see [36])

$${\mathcal{S}}_q^{\ast }(\sigma ,\alpha ,\beta ,\lambda ,\delta )={\mathcal{S}}^{\ast }\left(\sigma \right),\mathrm{and}\sigma \left(\eta \right)=\frac{2}{1+e^{-\eta }}.$$

Definition 11.

Let the function h given in (1) belong to the class ${\mathcal{J}}_{q,\lambda }^{\delta ,m}(\alpha ,\beta ,\mathcal{L},\mathcal{M},b)$ if and only if

$${\mathcal{J}}_{q,\lambda }^{\delta ,m}(\alpha ,\beta ,\mathcal{L},\mathcal{M},b)=1+\frac{1}{b}\left(\frac{2\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)-{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h(-\eta )}\right)\prec \frac{1+\mathcal{L}\eta }{1+\mathcal{M}\eta }.$$

(1). For $q\to 1-,\delta =0,\lambda =1,\alpha =0,\beta =1$ and $m=0,$ then (see [37])

$${\mathcal{J}}_{q,\lambda }^{\delta ,m}(\alpha ,\beta ,\mathcal{L},\mathcal{M},b)=\mathcal{J}(\mathcal{L},\mathcal{M},b).$$

(2). For $q\to 1-,\mathcal{M}=0,\delta =0,\lambda =1,\alpha =0,\beta =1$ and $m=0,$ then (see [38])

$${\mathcal{J}}_{q,\lambda }^{\delta ,m}(\alpha ,\beta ,\mathcal{L},\mathcal{M},b)=\mathcal{J}(\mathcal{L},b).$$

(3). For $q\to 1-,\mathcal{L}=1,\mathcal{M}=-1,b=2,\delta =0,\lambda =1,\alpha =0,\beta =1$ and $m=0,$ then (see [39])

$${\mathcal{J}}_{q,\lambda }^{\delta ,m}(\alpha ,\beta ,\mathcal{L},\mathcal{M},b)=\mathcal{J}.$$

2. A Set of Lemmas

The following lemma is necessary to prove our main results.

Lemma 1

([40]). Let $\mathcal{G}(h,n)$ be a class of analytic functions defined as

$$\mathcal{G}(h,n)=\left\{h:h\left(\eta \right)=\varrho +{\varrho }_n{\eta }^n+{\varrho }_{n+1}{\eta }^{n+1}+\dots \right\},$$

(15)

where, $\varrho \in \mathbb{C}$ and a positive integer $n.$

(i) For real numbers,

$$Re\left(h\left(\eta \right)+l\eta h^{{}^{\prime }}\left(\eta \right)\right)>0\Rightarrow Re\left(h\left(\eta \right)\right)>0.$$

(16)

and $l>0$ and $h\in \mathcal{G}(1,n)$; then, for $\delta >0$, $k>0,$ and $k=k(l,\delta ,n)$,

$$h\left(\eta \right)+l\eta h^{{}^{\prime }}\left(\eta \right)\prec {\left(\frac{1+\eta }{1-\eta }\right)}^k\Rightarrow h\left(\eta \right)\prec {\left(\frac{1+\eta }{1-\eta }\right)}^{\delta }.$$

(17)

(ii) Let $d\in [0,1)$ and $h\in \mathcal{G}(1,n)$; then, there is a fixed real number $l>0,$ so that

$$Re\left(h^2\left(\eta \right)+2h\left(\eta \right).\eta h^{{}^{\prime }}\left(\eta \right)\right)>d\Rightarrow Re\left(h\left(\eta \right)\right)>l.$$

(18)

(iii) Let $h\in \mathcal{G}(h,n)$ and $Reh>0$; then,

$$Re\left(h\left(\eta \right)+\eta h^{{}^{\prime }}\left(\eta \right)+{\eta }^2{h}^{{}^{″}}\left(\eta \right)\right)>0$$

(19)

or for $\varphi :\mathcal{U}\to \mathbb{R},$ such that

$$Re\left(h\left(\eta \right)+\varphi \left(\eta \right)\frac{\eta h^{{}^{\prime }}\left(\eta \right)}{h\left(\eta \right)}\right)>0,$$

(20)

then

$$Re\left(h\left(\eta \right)\right)>0.$$

(21)

3. Main Results

Theorem 1.

Let $h\in \mathcal{A}$ be given in (1). If one of the inequalities from the following inequalities is considered,

(i) ${\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)$ is of bounded turning function.

(ii)

$${\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{\prime }\prec {\left(\frac{1+\eta }{1-\eta }\right)}^k,k>0.$$

(iii)

$$Re\left({\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{\prime }\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)/\eta \right)\right)>\frac{d}{2},d\in [0,1).$$

(iv)

$$Re\left(\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{″}-{\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}+2\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)/\eta \right)\right)>0.$$

(v)

$$Re\left({\left(\eta {\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}/\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)+2\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)/\eta \right)\right)>1.$$

Then,

$$\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)/\eta \right)\in \mathcal{P}\left(\nu \right),\mathrm{for}\mathrm{some}\nu \in [0,1).$$

Proof. 

Let

$$p\left(\eta \right)=\frac{1}{\eta }{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right),\eta \in \mathcal{U}.$$

(22)

Then, after some calculations, we have

$$\eta p^{{}^{\prime }}\left(\eta \right)+p\left(\eta \right)={\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}.$$

(23)

In virtue of the first inequality, we find ${\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)$ is the bounding turning function, which leads to

$$Re\left(\eta p^{{}^{\prime }}\left(\eta \right)+p\left(\eta \right)\right)>0.$$

So, using Lemma 1(i) implies that

$$Re\left(p\left(\eta \right)\right)>0.$$

Hence, the first part of the theorem is completed. Consequently, the second part is confirmed. In light of Lemma 1(i), we fixed a real number $l>0,$ such that $k=k\left(l\right),$ and

$$\frac{1}{\eta }{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\prec {\left(\frac{1+\eta }{1-\eta }\right)}^l.$$

(24)

Therefore, (24) implies that

$$Re\left(\frac{1}{\eta }{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)>\nu ,\nu \in [0,1).$$

(25)

Suppose that

$$\begin{array}{ccc}& & Re\left(p^2\left(\eta \right)+2p\left(\eta \right).\eta p^{{}^{\prime }}\left(\eta \right)\right)\hfill \\ & =& 2Re\left(\frac{1}{\eta }{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\left({\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}-\frac{1}{2\eta }{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)\right)>d.\hfill \end{array}$$

(26)

According to Lemma 1(ii), there exists a fixed real number $l>0$ that satisfies

$$Re\left(p\left(\eta \right)\right)>l$$

and

$$p\left(\eta \right)=\frac{1}{\eta }{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\in \mathcal{P}\left(\nu \right).$$

(27)

Now from (26), we find that

$$Re{\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}>0.$$

Taking the derivative of (22), we have

$$\begin{array}{ccc}& & Re\left(p\left(\eta \right)+\eta p^{{}^{\prime }}\left(\eta \right)+{\eta }^2{p}^{{}^{″}}\left(\eta \right)\right)\hfill \\ & =& Re\left(\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{″}}-{\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}+2\left(\frac{1}{\eta }{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)\right)>0.\hfill \end{array}$$

Hence, Lemma 1(ii) implies

$$Re\left(\frac{1}{\eta }{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)>0.$$

Again, taking the logarithmic differentiation of (22) yields

$$\begin{array}{ccc}& & Re\left(p\left(\eta \right)+\frac{\eta p^{{}^{\prime }}\left(\eta \right)}{p\left(\eta \right)}+{\eta }^2{p}^{{}^{\prime }}\left(\eta \right)\right)\hfill \\ & =& Re\left(\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}+2\left(\frac{1}{\eta }{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)-1\right)>0.\hfill \end{array}$$

(28)

Hence, from Lemma 1(iii) and $\varphi \left(\eta \right)=1,$ we have

$$Re\left(\frac{1}{\eta }{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)>0.$$

□

Upper bounds of the operator  ${\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right).$

Theorem 2.

Suppose that $h\in {\mathcal{S}}_q^{\ast }(\sigma ,\alpha ,\beta ,\lambda ,\delta ),$ where $\sigma \left(\eta \right)$ is convex in $\mathcal{U}.$ Then,

$${\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\prec \eta exp{\int }_0^{\eta }\frac{\sigma \left(\mathsf{\Psi }\right(\xi \left)\right)-1}{\xi }d\xi ,$$

(29)

where $\mathsf{\Psi }\left(\eta \right)$ is analytic in $\mathcal{U}$ having the conditions $\mathsf{\Psi }\left(0\right)=0$ and $\left|\mathsf{\Psi }\left(\eta \right)\right|<1.$ Furthermore, for $\left|\eta \right|=\rho ,$ we have

$$exp{\int }_0^1\frac{\sigma \left(\mathsf{\Psi }\right(-\xi \left)\right)-1}{\rho }d\rho \le \left|\frac{1}{\eta }{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right|\le exp{\int }_0^1\frac{\sigma \left(\mathsf{\Psi }\right(\xi \left)\right)-1}{\rho }d\rho .$$

Proof. 

By the hypothesis, we get

$$\begin{array}{ccc}\hfill \frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}& \prec & \sigma \left(\eta \right)\hfill \\ \hfill \frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}& =& \sigma \left(\mathsf{\Psi }\right(\eta \left)\right),\eta \in \mathcal{U}.\hfill \end{array}$$

and

$$\frac{{\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}-\frac{1}{\eta }=\frac{\sigma \left(\mathsf{\Psi }\right(\eta \left)\right)-1}{\eta }.$$

(30)

Consequently, integrating (30), we get

$$log\left(\frac{1}{\eta }{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)={\int }_0^{\eta }\frac{\sigma \left(\mathsf{\Psi }\right(u)-1}{u}du.$$

(31)

By the definition of subordination, we get

$${\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\prec \eta exp{\int }_0^{\eta }\frac{\sigma \left(\mathsf{\Psi }\right(u)-1}{u}du.$$

Hence, (29) is proved. Since $\sigma \left(\eta \right)$ plots the disk $0<\left|\eta \right|<\xi <1$ onto a reign, which is convex and symmetric with respect to real axis, that is,

$$\sigma (-\xi \left|\eta \right|)\le Re\left\{\sigma \left(\mathsf{\Psi }\right(\xi \eta )\right\}\le \sigma \left(\xi \left|\eta \right|\right)(0<\xi <1,\eta \in \mathcal{U}),$$

then we have the inequalities

$$\sigma (-\xi )\le \sigma (-\xi \left|\eta \right|),\sigma \left(\xi \left|\eta \right|\right)\le \sigma \left(\xi \right).$$

Consequently, we get

$${\int }_0^1\frac{\sigma \left(\mathsf{\Psi }(-\xi \left|\eta \right|)\right)-1}{\xi }d\xi \le Re{\int }_0^1\frac{\sigma \left(\mathsf{\Psi }\right(\xi \left)\right)-1}{\xi }d\xi \le {\int }_0^1\frac{\sigma \left(\mathsf{\Psi }\left(\xi \left|\eta \right|\right)\right)-1}{\xi }d\xi .$$

In view of Equation (31), we obtain the general log inequality

$${\int }_0^1\frac{\sigma \left(\mathsf{\Psi }(-\xi \left|\eta \right|)\right)-1}{\xi }d\xi \le log\left|\frac{1}{\eta }{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right|\le {\int }_0^1\frac{\sigma \left(\mathsf{\Psi }\left(\xi \left|\eta \right|\right)\right)-1}{\xi }d\xi .$$

This implies that

$$exp{\int }_0^1\frac{\sigma \left(\mathsf{\Psi }(-\xi \left|\eta \right|)\right)-1}{\xi }d\xi \le \left|\frac{1}{\eta }{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right|\le exp{\int }_0^1\frac{\sigma \left(\mathsf{\Psi }\left(\xi \left|\eta \right|\right)\right)-1}{\xi }d\xi .$$

Hence, we have

$$exp{\int }_0^1\frac{\sigma \left(\mathsf{\Psi }\right(-\xi \left)\right)-1}{\xi }d\xi \le \left|\frac{1}{\eta }{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right|\le exp{\int }_0^1\frac{\sigma \left(\mathsf{\Psi }\right(\xi \left)\right)-1}{\xi }d\xi .$$

□

If $\sigma$ is convex univalent and $\sigma \left(0\right)=1$, then we find a condition on h to be in the class ${\mathcal{S}}_q^{\ast }(\sigma ,\alpha ,\beta ,\lambda ,\delta ).$

Theorem 3.

If$h\in \mathcal{A}$ satisfy the subordination condition

$$\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}\left(2+\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{″}}}{{\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}\right)-\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}\prec \sigma \left(\eta \right),$$

then, $h\in {\mathcal{S}}_q^{\ast }(\sigma ,\alpha ,\beta ,\lambda ,\delta ).$

Proof. 

Let

$$p\left(\eta \right)=\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}$$

and $p\left(\eta \right)=1$, we have

$$p\left(\eta \right)+p\left(\eta \right).\left(\eta p^{{}^{\prime }}\left(\eta \right)\right)\prec \sigma \left(\eta \right),$$

then,

$$\begin{array}{ccc}& & p\left(\eta \right)+p\left(\eta \right).\left(\eta p^{{}^{\prime }}\left(\eta \right)\right)\hfill \\ & =& \frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}\times \left(\begin{array}{c}2+\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{″}}}{{\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}\\ -\left(\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}\right)\end{array}\right)\prec \sigma \left(\eta \right).\hfill \end{array}$$

This implies that

$$p\left(\eta \right)=\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}\prec \sigma \left(\eta \right),$$

that is

$$h\in {\mathcal{S}}_q^{\ast }(\sigma ,\alpha ,\beta ,\lambda ,\delta ).$$

□

Corollary 1.

Let the assumption of Theorem 3 hold. Then,

$$\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}\times \left(1+\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{″}}}{{\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}\right)-\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}\ll {\sigma }^{{}^{\prime }}\left(\eta \right).$$

Proof. 

Let

$$p\left(\eta \right)=\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}.$$

According to the Theorem 3, we have

$$\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}\prec \sigma \left(\eta \right),$$

where, $\sigma \in \mathbb{C}.$ Then, by a result given in [41], we get $p^{{}^{\prime }}\left(\eta \right)\ll {\sigma }^{{}^{\prime }}\left(\eta \right).$

$$p^{{}^{\prime }}\left(\eta \right)=\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}\left(1+\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{″}}}{{\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}\right)-\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)};$$

hence, we obtained the required result. □

For the next result, we set a function $\sigma \left(\eta \right)=e^{\in \eta },$$1<\left|\in \right|\le \frac{\pi }{2}.$

Theorem 4.

If$h\in \mathcal{A},$ satisfy the inequality

$$1+\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{″}}}{{\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}\prec e^{\in \eta }.$$

Then

$$h\in {\mathcal{S}}_q^{\ast }(\alpha ,\beta ,\lambda ,\delta ,e^{\in \eta }).$$

Proof. 

Let

$$p\left(\eta \right)=\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}.$$

Then, a simple calculation gives

$$\begin{array}{ccc}& & p\left(\eta \right)+\frac{\eta p^{{}^{\prime }}\left(\eta \right)}{p\left(\eta \right)}\hfill \\ & =& \frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}\hfill \\ & & +\frac{\left(\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}\right)\left(1+\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{″}}}{{\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}-\left(\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}\right)\right)}{\frac{\eta D_q{\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}}\hfill \\ & =& 1+\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{″}}}{{\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}\prec e^{\in \eta }.\hfill \end{array}$$

This implies that (see [40] p. 123)

$$p\left(\eta \right)=\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}\prec e^{\in \eta }$$

that is,

$$h\in {\mathcal{S}}_q^{\ast }(\alpha ,\beta ,\lambda ,\delta ,e^{\in \eta }).$$

□

Theorem 5.

If$h\in {\mathcal{J}}_{q,\lambda }^{\delta ,m}(\alpha ,\beta ,\mathcal{L},\mathcal{M},b),$ then the function

$$\mathcal{B}\left(\eta \right)=\frac{1}{2}\left(h\left(\eta \right)-h(-\eta )\right)$$

satisfies

$$1+\frac{1}{b}\left(\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)\mathcal{B}\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)\mathcal{B}\left(\eta \right)}\right)\prec \frac{1+\mathcal{L}\eta }{1+\mathcal{M}\eta },$$

$$Re\left(\frac{\eta {\left(\mathcal{B}\left(\eta \right)\right)}^{{}^{\prime }}}{\mathcal{B}\left(\eta \right)}\right)\ge \frac{1-{\vartheta }^2}{1+{\vartheta }^2},\left|\eta \right|=\vartheta <1.$$

Proof. 

Let $h\in {\mathcal{J}}_{q,\lambda }^{\delta ,m}(\alpha ,\beta ,\mathcal{L},\mathcal{M},b)$; then, the function $J\left(\eta \right)$ can be written as

$$\begin{array}{ccc}\hfill b\left(J\right(\eta )-1)& =& \frac{2\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)-{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h(-\eta )},\hfill \\ \hfill b\left(J\right(-\eta )-1)& =& \frac{2\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h(-\eta )\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h(-\eta )-{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}.\hfill \end{array}$$

This confirm that

$$1+\frac{1}{b}\left(\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)\mathcal{B}\left(\eta \right)\right)}^{{}^{\prime }}}{\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)\mathcal{B}\left(\eta \right)\right)}-1\right)=\frac{J\left(\eta \right)+J(-\eta )}{2}.$$

However, J satisfies

$$J\left(\eta \right)\prec \frac{1+\mathcal{L}\eta }{1+\mathcal{M}\eta },$$

which is univalent; then, we get

$$1+\frac{1}{b}\left(\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)\mathcal{B}\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)\mathcal{B}\left(\eta \right)}-1\right)\prec \frac{1+\mathcal{L}\eta }{1+\mathcal{M}\eta }.$$

In addition, $\mathcal{B}\left(\eta \right)$ is starlike in $\mathcal{U}$, which implies that

$$\phi \left(\eta \right)=\frac{\eta {\left(\mathcal{B}\left(\eta \right)\right)}^{{}^{\prime }}}{\mathcal{B}\left(\eta \right)}\prec \frac{1-{\eta }^2}{1+{\eta }^2}.$$

Hence, the Schwarz function $u\left(\eta \right)\in \mathcal{U},\left|u\left(\eta \right)\right|\le \left|\eta \right|<1,$$u\left(0\right)=0$ gets

$$\phi \left(\eta \right)=\frac{1-u{\left(\eta \right)}^2}{1+u{\left(\eta \right)}^2},$$

which leads us to

$$u{\left(\zeta \right)}^2=\frac{1-\phi \left(\zeta \right)}{1+\phi \left(\zeta \right)},\zeta \in \mathcal{U},\left|\zeta \right|=r<1.$$

A simple calculation yields

$$\left|\frac{1-\phi \left(\zeta \right)}{1+\phi \left(\zeta \right)}\right|={\left|u\left(\zeta \right)\right|}^2\le {\left|\zeta \right|}^2.$$

Therefore, we get the following inequalities:

$$\begin{array}{ccc}\hfill {\left|\phi \left(\zeta \right)-\frac{1+{\left|\zeta \right|}^4}{1-{\left|\zeta \right|}^4}\right|}^2& \le & \frac{4{\left|\zeta \right|}^4}{{\left(1-{\left|\zeta \right|}^4\right)}^2},\hfill \\ \hfill \left|\phi \left(\zeta \right)-\frac{1+{\left|\zeta \right|}^4}{1-{\left|\zeta \right|}^4}\right|& \le & \frac{2{\left|\zeta \right|}^2}{\left(1-{\left|\zeta \right|}^4\right)}.\hfill \end{array}$$

Thus, we have

$$Re\left(\frac{\eta {\left(\mathcal{B}\left(\eta \right)\right)}^{{}^{\prime }}}{\mathcal{B}\left(\eta \right)}\right)\ge \frac{1-{\vartheta }^2}{1+{\vartheta }^2},\left|\zeta \right|=\vartheta <1.$$

Hence, the proof is completed. □

Example 1.

Let

$$\begin{array}{ccc}\hfill \frac{\eta h^{{}^{\prime }}\left(\eta \right)}{h\left(\eta \right)}& =& \frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}\hfill \\ \hfill {\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)& =& \frac{\eta }{q{\left(1-\eta \right)}^2},h\in \mathcal{A}.\hfill \end{array}$$

Then, the solution of $\frac{\eta h^{{}^{\prime }}\left(\eta \right)}{h\left(\eta \right)}=\frac{1+\eta }{1-\eta }$ is formulated as follows:

$${\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)=\frac{\eta }{q{\left(1-\eta \right)}^2}.$$

Applications

The solution of the complex Briot–Bouquet (BB) differential equation is established in [40]. We produce a presentation of our results in complex BB differential equations, and the class of BB differential equations is a link of differential equations whose consequences are visible in the complex plane. Recently, the complex modelings of phenomena in nature and society have been the object of several investigations based on the methods originally developed in a physical context. Ibrahim [10] studied various types of fractional differential equations in the complex domain, such as the Cauchy equation, the diffusion equation and telegraph equations. The study of first ODEs specifies new transcendental special functions as follows:

$$\begin{array}{ccc}\hfill \varsigma h\left(\eta \right)+\left(1-\varsigma \right)\frac{\eta h^{{}^{\prime }}\left(\eta \right)}{h\left(\eta \right)}& =& \phi \left(\eta \right),\hfill \\ \hfill \phi \left(0\right)& =& h\left(0\right),\varsigma \in \left[0,1\right].\hfill \end{array}$$

(32)

In [40], many new applications of these equations in Geometric Function Theory have been discussed.

Now, we investigate (32) by using the operator ${\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)$ and find its solutions by applying the subordination relations. The operator ${\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)$ propagates the complex BB differential equation as follows:

$$\varsigma {\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)+\left(1-\varsigma \right)\left(\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}\right)=\phi \left(\eta \right),$$

(33)

where,

$$\phi \left(0\right)=h\left(0\right),\eta \in \mathcal{U}.$$

A trivial solution of (33) is given when $\varsigma =1$; our investigation concerns the case with $h\in \mathcal{A}$ and $\varsigma =0$.

In Theorem 6, we investigated the subordination condition and distortion bounds for a class of complex fractional derivatives.

Theorem 6.

Let we have Equation (33) with $\varsigma =0$. If $\sigma \left(\eta \right)$ is convex in $U.$ Then

$${\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\prec \eta exp{\int }_0^{\eta }\frac{\sigma \left(\mathsf{\Psi }\right(\xi \left)\right)-1}{\xi }d\xi ,$$

(34)

where $\mathsf{\Psi }\left(\eta \right)$ is analytic in $\mathcal{U}$ and $\mathsf{\Psi }\left(0\right)=0$ and $\left|\mathsf{\Psi }\left(\eta \right)\right|<1.$

Proof. 

Take all the assumptions of Equation (33), and $h\left(\eta \right)\in \mathcal{A}$. Then, we have

$$Re\left(\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}\right)>0$$

$$\begin{array}{ccc}& \iff & Re\left(\frac{\eta +\sum _{n=2}^{\infty }n\left(A_q\left(\lambda ,\delta ,\alpha ,\beta ,m,n\right)\right)a_n{\eta }^n}{\eta +\sum _{n=2}^{\infty }{A}_q\left(\lambda ,\delta ,\alpha ,\beta ,m,n\right)a_n{\eta }^n}\right)>0\hfill \\ & \iff & Re\left(\frac{\eta +\sum _{n=2}^{\infty }n\left(A_q\left(\lambda ,\delta ,\alpha ,\beta ,m,n\right)\right)a_n{\eta }^{n-1}}{\eta +\sum _{n=2}^{\infty }{A}_q\left(\lambda ,\delta ,\alpha ,\beta ,m,n\right)a_n{\eta }^{n-1}}\right)>0\hfill \\ & \iff & Re\left(\frac{\eta +\sum _{n=2}^{\infty }n\left(A_q\left(\lambda ,\delta ,\alpha ,\beta ,m,n\right)\right)a_n}{\eta +\sum _{n=2}^{\infty }{A}_q\left(\lambda ,\delta ,\alpha ,\beta ,m,n\right)a_n}\right)>0,\eta \to 1^{+}\hfill \\ & \iff & Re\left(\eta +\sum _{n=2}^{\infty }n\left(A_q\left(\lambda ,\delta ,\alpha ,\beta ,m,n\right)\right)a_n\right)>0.\hfill \end{array}$$

Moreover, by the definition of ${\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)$, we indicate that ${\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(0\right)=0.$ Consequently,

$$\frac{\eta {\left({\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)\right)}^{{}^{\prime }}}{{\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)}\in \mathcal{P}⟹h\left(\eta \right)\in {\mathcal{S}}_q^{\ast }(\sigma ,\alpha ,\beta ,\lambda ,\delta ).$$

Hence, in the light of Theorem 2, the result given in (34) is completed. □

4. Conclusions

We considered fractional q-differential operator and q-Mittag-Leffler functions and defined a new operator ${\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)$. We investigated new subclasses of univalent functions associated with the operator ${\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)h\left(\eta \right)$ in the open unit disk and discussed some geometric properties of this newly defined operator. By using the BB equation and involving the newly defined operator ${\mathsf{\Lambda }}_{\alpha ,\lambda }^{\beta ,\delta }\left(m,q\right)$, we investigated its solution.