Some Properties of Janowski Symmetrical Functions

Abstract

In our present work, the concepts of symmetrical functions and the concept of spirallike Janowski functions are combined to define a new class of analytic functions. We give a structural formula for functions in ${\mathcal{S}}^{\eta ,\mu }(F,H,\lambda )$, a representation theorem, the radius of starlikeness estimates, covering and distortion theorems and integral mean inequalities are obtained.

1. Introduction

Let $\mathbb{K}=\left\{t\in \mathbb{C}:\left|t\right|<1\right\}$ be the open disk and $\mathbb{F}$ be the family of functions that are analytic in $\mathbb{K}$ which has the form

$$h\left(t\right)=t+\sum _{v=2}^{\infty }{a}_v{t}^v.$$

(1)

Suppose that $\tilde{\mathcal{S}}$ denote the subfamily of all functions that are univalent in $\mathbb{K}$.

For given two functions h and g analytic in $\mathbb{K}$, we say that the function h is subordinate to g in $\mathbb{K}$ and write $h\prec g$, if there exists a Schwarz function $s\in \Omega$, where

$$\Omega :=\{s:s(0)=0,|s\left(t\right)|<1,t\in \mathbb{K}\},$$

(2)

such that $h\left(t\right)=g\left(s\right(t\left)\right)$, $t\in \mathbb{K}$. If g is univalent in $\mathbb{K}$, then

$$h\prec g\iff h\left(0\right)=g\left(0\right)\mathrm{and}h\left(\mathbb{K}\right)\subset g\left(\mathbb{K}\right).$$

In order to define new classes, we first recall the notions of Janowski $\lambda$-Spirallike functions and $(\eta ,\mu )$-symmetric points.

Nehari [1] introduced the class $\mathcal{P}$ of analytic positive real parts functions that has the power series

$$p\left(t\right)=1+{\rho }_{1}t+{\rho }_2{t}^2+{\rho }_3{t}^3+\dots ,t\in \mathbb{K}.$$

The class $\mathcal{P}[F,H]$ was introduced by Janowski in [2] as:

$$p\in \mathcal{P}[F,H]\iff p\left(t\right)={\displaystyle \frac{1+Fs\left(t\right)}{1+Hs\left(t\right)}}\mathrm{for}-1\le H<F\le 1,s\left(t\right)\in \Omega .$$

In 1933, Spa$\breve{\mathrm{c}}$ek [3] introduced the class of $\lambda$-spirallike functions $\mathcal{S}\left(\lambda \right)$ as follows

$$h\in \mathcal{S}\left(\lambda \right)\iff Re\left\{e^{i\lambda }\frac{t{h}^{\prime }\left(t\right)}{h\left(t\right)}\right\}\ge 0,\left|\lambda \right|<\frac{\pi }{2}\mathrm{for}\mathrm{all}t\in \mathbb{K}.$$

Let us fix $\mu \in \mathbb{N}$ and let $\epsilon =e^{\frac{2\pi i}{\mu }}$, a subset $\mathbf{D}$ of the complex space is called $\mu$- fold symmetric set if $\epsilon \mathbf{D}=\mathbf{D}$. A function h is said to be $\mu$-symmetrical function if $h\left(\epsilon t\right)=\epsilon h\left(t\right)\mathrm{for}\mathrm{all}t\in \mathbf{D}.$

Liczberski and Polubinski [4] constructed the notion of $(\eta ,\mu )$-symmetrical functions for any integer $(\mu \ge 2\mathrm{and}\eta =0,1,2,\dots ,\mu -1)$. If $\mathbf{D}$ is $\mu$-fold symmetric domain, then a function $h:\mathbf{D}\to \mathbb{C}$ is $(\eta ,\mu )$-symmetrical if $h\left(\epsilon t\right)={\epsilon }^{\eta }h\left(t\right),\mathrm{for}\mathrm{each}t\in \mathbf{D}.$ We denote by ${\mathbb{F}}_{\mu }^{\eta }$ for the class of all $(\eta ,\mu )$-symmetrical functions note that ${\mathbb{F}}_2^0$, ${\mathbb{F}}_2^1$ and ${\mathbb{F}}_{\mu }^1$ are the classes of even, odd and $\mu$—symmetrical functions, respectively. We observe that $\mathbb{K}$ is $\mu$-fold symmetric. We use the below unique decomposition [4] of every mappings $h:\mathbb{K}\mapsto \mathbb{C}$ by

$$h\left(t\right)=\sum _{\eta =0}^{\mu -1}{h}_{\eta ,\mu }\left(t\right),\mathrm{where}{h}_{\eta ,\mu }\left(t\right)={\mu }^{-1}\sum _{r=0}^{\mu -1}{\epsilon }^{-r\eta }h\left({\epsilon }^{r}t\right),t\in \mathbb{K}.$$

(3)

Equivalently, (3) may be written as:

$$h_{\eta ,\mu }\left(t\right)=\sum _{v=1}^{\infty }{a}_v{\alpha }_{\eta }^v{t}^v,a_1=1,$$

(4)

where

$${\alpha }_{\eta }^v=\frac{1}{\mu }\sum _{r=0}^{\mu -1}{\epsilon }^{(v-\eta )r}=\left\{\begin{array}{cc}1,\hfill & v=l\mu +\eta ;\hfill \\ 0,\hfill & v\ne l\mu +\eta ;\hfill \end{array}\right.$$

(5)

and

$$(\mu =2,3,\dots ,\eta =0,1,2,\dots ,\mu -1,l\in \mathbb{N}).$$

Recently, several interesting subclasses of $(\eta ,\mu )$-symmetrical functions were introduced and investigated; see, for example, [5,6,7]. However, by motivation of the above work, we introduce and study a new subclass as follows:

Definition 1.

For $-1\le H<F\le 1 and \left|\lambda \right|<\frac{\pi }{2}$, let ${\mathcal{S}}^{\eta ,\mu }(F,H,\lambda )$ denote the class consisting of functions $h\in \mathbb{F}$ and satisfy the condition

$$\frac{t{h}^{\prime }\left(t\right)}{h_{\eta ,\mu }\left(t\right)}\prec cos\left(\lambda \right)p\left(t\right)+isin\left(\lambda \right),p\left(t\right)\in \mathcal{P}[F,H],t\in \mathbb{K},$$

where $h_{\eta ,\mu }$ is defined in (3).

For various choices of $\eta ,\mu ,F,H$ and $\lambda$, Definition 1 yields several subclasses of $\mathbb{F}$, as ${\mathcal{S}}^{1,1}(F,H,\lambda ):=\mathcal{S}(F,H,\lambda )$ motivated by Polato glu, et al. [8], ${\mathcal{S}}^{1,\mu }(F,H,0):={\mathcal{S}}^{\mu }(F,H)$ introduced by y Kwon and Sim in [9], ${\mathcal{S}}^{\eta ,\mu }(F,H,0):={\mathcal{S}}^{\eta ,\mu }(F,H)$ introduced by Alsarari and Latha in [10], ${\mathcal{S}}^{1,\mu }(1,-1,0):={\mathcal{S}}_{\mu }$ defined by Sakaguchi [11], ${\mathcal{S}}_{1,\mu }^0(F,H,\alpha ):={\mathcal{S}}_{\mu }(F,K,\alpha )$ introduced by Al sarari, Latha and Darus [12], the class ${\mathcal{S}}^{1,1}(F,H,0):=\mathcal{S}[F,H]$ reduce to the class of Janowski [2], etc.

For the goal of this investigation, we consider the concepts of Janowski $\lambda$-spirallike and $(\eta ,\mu )$-symmetrical functions to define a new subclass of analytic symmetrical functions and investigate a structural formula for functions in our class a representation theorem, radius of starlikeness estimates, covering, distortion theorems and the right integral mean inequalities.

2. A Set of Lemmas

In order to establish our main results, we need the following results.

Lemma 1

([2]). Let $p\in \mathcal{P}[F,H]$, then

$$\frac{1-Fr}{1-Hr}\le \left|p\left(t\right)\right|\le \frac{1+Fr}{1+Hr},\left|t\right|\le r<1.$$

Lemma 2

([8]). If $h\in {\mathcal{S}}^{\lambda }(F,H)$, then

$$h\left(t\right)=\left\{\begin{array}{ccc}t{\left(1+Hs\left(t\right)\right)}^{\frac{(F-H)cos\left(\lambda \right)e^{-i\lambda }}{H}}\hfill & if\hfill & H\ne 0\hfill \\ texp[Fcos\left(\lambda \right)e^{-i\lambda }s\left(t\right)]\hfill & if\hfill & H=0\hfill \end{array}\right.$$

(6)

for some $s\in \Omega$.

Lemma 3

([8]). If $h\in {\mathcal{S}}^{\lambda }(F,H)$, then

$$\left.\begin{array}{ccc}{G}_1(r,F,H)\hfill & if\hfill & H\ne 0\hfill \\ r{e}^{Fcos\left(\lambda \right)r}\hfill & if\hfill & H=0\hfill \end{array}\right\}\le \left|h\left(t\right)\right|\le \left\{\begin{array}{ccc}{G}_2(r,F,H)\hfill & if\hfill & H\ne 0\hfill \\ r{e}^{-Fcos\left(\lambda \right)r}\hfill & if\hfill & H=0,\hfill \end{array}\right.$$

where

$$G_1(r,F,H)=r{(1-Hr)}^{\frac{(F-H)cos\left(\lambda \right)(cos(\lambda )+1)}{2H}}{(1+Hr)}^{\frac{(F-H)cos\left(\lambda \right)(cos(\lambda )-1)}{2H}},$$

(7)

$$G_2(r,F,H)=r{(1-Hr)}^{\frac{(F-H)cos\left(\lambda \right)(cos(\lambda )-1)}{2H}}{(1+Hr)}^{\frac{(F-H)cos\left(\lambda \right)(cos(\lambda )+1)}{2H}}.$$

(8)

3. Main Results

Theorem 1.

A function h is in the class ${\mathcal{S}}^{\eta ,\mu }(F,H,\lambda )$ if and only if

$$h\left(t\right)={\int }_0^t\alpha \left(\omega \right)q\left(\omega \right)d\omega ,$$

(9)

where $q\left(\omega \right)=exp\left\{{\int }_0^{\omega }\frac{1}{\mu u}\left({\sum }_{v=0}^{\mu -1}\alpha \left({\epsilon }^{v}u\right)-\mu \right)du\right\}$ and α is given by (12).

Proof. 

For an arbitrary function $h\in {\mathcal{S}}^{\eta ,\mu }(F,H,\lambda )$, we have

$${\displaystyle \frac{t{h}^{\prime }\left(t\right)}{h_{\eta ,\mu }\left(t\right)}}=[cos\left(\lambda \right)p\left(t\right)+isin\left(\lambda \right)]e^{-i\lambda },p\in \mathcal{P}[F,H].$$

(10)

Replacing t by ${\epsilon }^{v}t$ in (10) we obtain

$$\frac{{\epsilon }^{v(1-\eta )}t{h}^{\prime }\left({\epsilon }^{v}t\right)}{h_{\eta ,\mu }\left(t\right)}=\alpha \left({\epsilon }^{v}t\right),$$

(11)

where

$$\alpha \left(t\right)=[cos\left(\lambda \right)p\left(t\right)+isin\left(\lambda \right)]e^{-i\lambda },p\in \mathcal{P}[F,H].$$

(12)

From (10) and (11), we obtain

$$h^{\prime }\left({\epsilon }^{v}t\right)=\alpha \left({\epsilon }^{v}t\right)\frac{{\epsilon }^{v(\eta -1)}{h}^{\prime }\left(t\right)}{\alpha \left(t\right)}.$$

(13)

By differentiation (10), we have

$$h_{\eta ,\mu }^{\prime }\left(t\right)=\frac{t{h}^{″}\left(t\right)+h^{\prime }\left(t\right)}{\alpha \left(t\right)}-t{h}^{\prime }\left(t\right)\frac{{\alpha }^{\prime }\left(t\right)}{{\alpha }^2\left(t\right)}.$$

(14)

From (5) and (13), we obtain

$$h_{\eta ,\mu }^{\prime }\left(t\right)=\frac{1}{\mu }\frac{h^{\prime }\left(t\right)}{\alpha \left(t\right)}\sum _{v=0}^{\mu -1}\alpha \left({\epsilon }^{v}t\right),$$

(15)

from (14) and (15), we have

$$\frac{h^{″}\left(t\right)}{h^{\prime }\left(t\right)}=\frac{{\alpha }^{\prime }\left(t\right)}{\alpha \left(t\right)}+\frac{1}{\mu t}\left(\sum _{v=0}^{\mu -1}\alpha \left({\epsilon }^{v}t\right)-\mu \right).$$

Integrating the above, we obtain

$$h\left(t\right)={\int }_0^t\alpha \left(\omega \right)q\left(\omega \right)d\omega .$$

Hence, the necessity condition is proven. As proof of the sufficiency of (9), assume that (9) is given with p is in the class $\mathcal{P}[F,H]$. Let h specified by (9) is of course in $\mathbb{F}$ and $h\left(0\right)=0$ and $h^{\prime }\left(0\right)=1$. The next identity may be checked by differentiation

$$tq\left(t\right)={\int }_0^{{\epsilon }^{v}t}\left[\frac{1}{\mu }\sum _{v=0}^{\mu -1}{\epsilon }^{-\eta v}\alpha \left(\omega \right)q\left(\omega \right)\right]d\omega ,$$

(16)

for $q,\alpha$ are supplied by (9) and (12), and by (9), we obtain

$$h^{\prime }\left(t\right)=\alpha \left(t\right)q\left(t\right),$$

(17)

which shows that $h^{\prime }\ne 0$ in $\mathbb{K}$.

From (9), given that $\epsilon$ is the root of unity, we find that

$$h_{\eta ,\mu }\left(t\right)={\int }_0^{{\epsilon }^{v}t}\left[\frac{1}{\mu }\sum _{v=0}^{\mu -1}{\epsilon }^{-\eta v}\alpha \left(\omega \right)q\left(\omega \right)\right]d\omega .$$

(18)

Using (16)–(18) we arrive the result

$$h_{\eta ,\mu }\left(t\right)=\frac{t{h}^{\prime }\left(t\right)}{\alpha \left(t\right)},$$

which thus proves the sufficiency of (9). □

Remark 1.

For special choices of $\eta ,\mu ,F$ and H we obtain the structural formula for classes derived earlier [13,14].

Corollary 1

([14]). A function h is in the class ${\mathcal{SSP}}_{\mathcal{N}}$ if and only if

$$h\left(t\right)={\int }_0^t\alpha \left(\omega \right)exp\left\{q\left(\omega \right)\right\}d\omega ,$$

where $q\left(\omega \right)={\int }_0^{\omega }\frac{1}{\mathcal{N}u}\left({\sum }_{v=0}^{\mu -1}\alpha \left({\epsilon }^{v}u\right)-\mathcal{N}\right)du.$

Theorem 2.

If $h\in {\mathcal{S}}^{\eta ,\mu }(F,H,\lambda )$, then

$$h_{\eta ,\mu }\left(t\right)=\left\{\begin{array}{ccc}t{\left(1+Hs\left(t\right)\right)}^{\frac{(F-H)cos\left(\lambda \right)e^{-i\lambda }}{H}}\hfill & if\hfill & H\ne 0\hfill \\ texp[Fcos\left(\lambda \right)e^{-i\lambda }s\left(t\right)]\hfill & if\hfill & H=0,\hfill \end{array}\right.$$

(19)

where $h_{\eta ,\mu }$ is given by (3).

Proof. 

Let $h\in {\mathcal{S}}^{\eta ,\mu }(F,H,\lambda )$, we can obtain

$$\frac{t{h}^{\prime }\left(t\right)}{h_{\eta ,\mu }\left(t\right)}\prec \frac{[cos\left(\lambda \right)(1+Ft)+isin\left(\lambda \right)(1+Ht)]e^{-i\lambda }}{Ht+1}.$$

(20)

Substituting t by ${\epsilon }^{v}t$ in above, it follows

$$\begin{array}{ccc}\hfill \frac{{\epsilon }^{v}t{h}^{\prime }\left({\epsilon }^{v}t\right)}{h_{\eta ,\mu }\left({\epsilon }^{v}t\right)}& \prec & \frac{[cos\left(\lambda \right)(1+Ft{\epsilon }^v)+isin\left(\lambda \right)(1+Ht{\epsilon }^v)]e^{-i\lambda }}{Ht{\epsilon }^v+1}\hfill \\ & \prec & \frac{[cos\left(\lambda \right)(1+Ft)+isin\left(\lambda \right)(1+Ht)]e^{-i\lambda }}{Ht+1},\hfill \end{array}$$

and hence,

$$\frac{{\epsilon }^{v-v\eta }t{h}^{\prime }\left({\epsilon }^{v}t\right)}{h_{\eta ,\mu }^{\prime }\left(t\right)}\prec \frac{[cos\left(\lambda \right)(1+Ft)+isin\left(\lambda \right)(1+Ht)]e^{-i\lambda }}{Ht+1}.$$

(21)

For $v=0,1,\dots ,\mu -1$ in (21) and given that $\mathcal{P}[F,H]$ is a convex set, we obtain

$$\frac{t\frac{1}{\mu }{\sum }_{v=0}^{\mu -1}{\epsilon }^{v-v\eta }{h}^{\prime }\left({\epsilon }^{v}t\right)}{h_{\eta ,\mu }\left(t\right)}\prec \frac{[cos\left(\lambda \right)(1+Ft)+isin\left(\lambda \right)(1+Ht)]e^{-i\lambda }}{Ht+1},$$

or

$$\frac{t{h}_{\eta ,\mu }^{\prime }\left(t\right)}{h_{\eta ,\mu }\left(t\right)}\prec \frac{[cos\left(\lambda \right)(1+Ft)+isin\left(\lambda \right)(1+Ht)]e^{-i\lambda }}{Ht+1},$$

(22)

that is, $h_{\eta ,\mu }\in {\mathcal{S}}^{\lambda }(F,H)$, and using Lemma 2, to obtain (19). □

Corollary 2.

Let $h\in {\mathcal{S}}^{\eta ,\mu }(F,H,\lambda )$, with $-1\le H<F\le 1$ and $\left|\lambda \right|<\frac{\pi }{2}$. Then,

$$h\left(t\right)=\left\{\begin{array}{ccc}{\displaystyle {\int }_0^t\frac{1+\gamma \tilde{s}\left(\varphi \right)}{1+H\tilde{s}\left(\varphi \right)}{(1+Hs\left(\varphi \right))}^{\frac{e^{-i\lambda }(F-H)cos\left(\lambda \right)}{H}}d\varphi }\hfill & if\hfill & H\ne 0\hfill \\ {\displaystyle {\int }_0^t{e}^{\{cos\left(\lambda \right)s\left(\varphi \right)F{e}^{-i\lambda }\}}[cos\left(\lambda \right)F\tilde{s}\left(\varphi \right)(e^{-i\lambda }+1)]d\varphi }\hfill & if\hfill & H=0,\hfill \end{array}\right.$$

where $s,\tilde{s}\in \Omega$ and γ is given by (23).

Proof. 

Suppose that $h\in {\mathcal{S}}^{\eta ,\mu }(F,H,\lambda )$, then for some $\tilde{s}\in \Omega$ and

$$\frac{e^{i\lambda }\frac{t{h}^{\prime }\left(t\right)}{h_{\eta ,\mu }\left(t\right)}-isin\left(\lambda \right)}{cos\left(\lambda \right)}=\frac{1+F\tilde{s}\left(t\right)}{1+H\tilde{s}\left(t\right)},t\in \mathbb{K}.$$

By using Theorem 2 in the above relation, we obtain

$$h^{\prime }\left(t\right)=\left\{\begin{array}{ccc}{\displaystyle \frac{1+\gamma \tilde{s}\left(t\right)}{1+H\tilde{s}\left(t\right)}{(1+Hs\left(t\right))}^{\frac{e^{-i\lambda }(F-H)cos\left(\lambda \right)}{H}}}\hfill & \mathrm{if}\hfill & H\ne 0\hfill \\ {\displaystyle e^{e^{-i\lambda }Fcos\left(\lambda \right)s\left(t\right)}[1+e^{-i\lambda }Fcos\left(\lambda \right)\tilde{s}\left(t\right)]}\hfill & \mathrm{if}\hfill & H=0,\hfill \end{array}\right.$$

where

$$\gamma =[Fcos\left(\lambda \right)+iHsin\left(\lambda \right)]e^{-i\lambda }.$$

(23)

Integrating the aforementioned relationships yields our conclusion. □

Putting $\eta =\mu =1$, Corollary 2 yields the corresponding results found in [8].

Corollary 3

([8]). If $h\in {\mathcal{S}}^{\lambda }(F,H)$, then

$$h\left(t\right)=\left\{\begin{array}{ccc}t{\left(1+Hs\left(t\right)\right)}^{\frac{(F-H)cos\left(\lambda \right)e^{-i\lambda }}{H}}\hfill & if\hfill & H\ne 0\hfill \\ texp[Fcos\left(\lambda \right)e^{-i\lambda }s\left(t\right)]\hfill & if\hfill & H=0\hfill \end{array}\right.$$

for some $s\in \Omega$.

In the following theorem, we will discuss the radii of star-likeness.

Theorem 3.

The radii of star-likeness of the class ${\mathcal{S}}^{\eta ,\mu }(F,H,\lambda )$ is

$$r=\left\{\begin{array}{ccc}{\displaystyle \frac{2}{(F-H)cos\left(\lambda \right)+\sqrt{{(F-H)}^2{cos}^2\left(\lambda \right)+4\left\{FH{cos}^2\left(\lambda \right)+H^2{sin}^2\left(\lambda \right)\right\}}}}\hfill & if\hfill & H\ne 0\hfill \\ {\displaystyle \frac{1}{Fcos\left(\lambda \right)}}\hfill & if\hfill & H=0.\hfill \end{array}\right.$$

This radius is sharp for the extremal function

$$h_{\eta ,\mu }\left(t\right)=\left\{\begin{array}{c}t{(1+Ht)}^{\frac{(F-H)cos\left(\lambda \right)e^{-i\lambda }}{H}},H\ne 0\hfill \\ t{e}^{Fcos\left(\lambda \right)e^{-i\lambda }t},H=0.\hfill \end{array}\right.$$

Proof. 

Since

$$\frac{{\displaystyle \frac{e^{i\lambda }t{h}^{\prime }\left(t\right)}{h_{\eta ,\psi }\left(t\right)}}-isin\left(\lambda \right)}{cos\left(\lambda \right)}=p\left(t\right),p\in \mathcal{P}[F,H],$$

(24)

then, we have

$$\left|p\left(t\right)-\frac{1-FH{r}^2}{1-H^2{r}^2}\right|\le \frac{(F-H)r}{1-H^2{r}^2}.$$

(25)

Using (24) in (25) and after straightforward calculations, we get

$$\begin{array}{ccc}& & \left.\begin{array}{ccc}\frac{1-(F-H)cos\left(\lambda \right)r-\left\{FH{cos}^2\left(\lambda \right)+H^2{sin}^2\left(\lambda \right)\right\}{r}^2}{1-H^2{r}^2}\hfill & \mathrm{if}\hfill & H\ne 0\hfill \\ 1-Fcos\left(\lambda \right)r\hfill & \mathrm{if}\hfill & H=0\hfill \end{array}\right\}\le Re\left\{t\frac{h^{\prime }\left(t\right)}{h_{\eta ,\mu }\left(t\right)}\right\}\hfill \\ & & \le \left\{\begin{array}{ccc}\frac{1+(F-H)cos\left(\lambda \right)r-\left\{FH{cos}^2\left(\lambda \right)+H^2{sin}^2\left(\lambda \right)\right\}{r}^2}{1-H^2{r}^2}\hfill & \mathrm{if}\hfill & H\ne 0\hfill \\ Fcos\left(\lambda \right)r+1\hfill & \mathrm{if}\hfill & H=0,\hfill \end{array}\right.\hfill \end{array}$$

where $\left|r\right|\le r<1$. This theorem is true, as shown by the inequalities mentioned above. □

Remark 2.

• We obtain $r=1$, if we put $\lambda =0,\mu =-1=-\eta$.
• We obtain $r=\frac{1}{cos\left(\lambda \right)+|sin(\lambda \left)\right|}$, if we put $\mu =-1=-\eta$.

Additionally, we observe that if we add more special values to $\alpha ,F,H,\eta$ and μ, We determine the subclass’s starlikeness radius of $\mathcal{S}\left(\lambda \right)$ (see [15]).

Corollary 4.

If $h\in {\mathcal{S}}^{\eta ,\mu }(F,H,\lambda )$, then

$$J(F,H,\lambda ,-r,1)\le \left|\frac{t{h}^{\prime }\left(t\right)}{h_{\eta ,\mu }\left(t\right)}\right|\le J(F,H,\lambda ,r,2)$$

(26)

where

$$J(F,H,\lambda ,r,n)=\left\{\begin{array}{ccc}\frac{(1+Fr)cos\left(\lambda \right)+{(-1)}^n(1+Hr)|sin\left(\lambda \right)|}{1+Hr}\hfill & if\hfill & H\ne 0\hfill \\ cos\left(\lambda \right)(1+Fr)+{(-1)}^n|sin\left(\lambda \right)|\hfill & if\hfill & H=0\hfill \end{array}\right.$$

and $\left|t\right|\le r<1$.

Proof. 

For a function $h\in {\mathcal{S}}^{\eta ,\mu }(F,H,\lambda )$, we have

$$\frac{1}{cos\left(\lambda \right)}\left\{{\displaystyle \frac{e^{i\lambda }t{h}^{\prime }\left(t\right)}{h_{\eta ,\mu }\left(t\right)}}-isin\left(\lambda \right)\right\}=p\left(t\right),p\in \mathcal{P}[F,H].$$

Using Lemma 1 after doing the sample calculations we obtain (26). □

Theorem 4.

If $h\in {\mathcal{S}}^{\eta ,\mu }(F,H,\lambda )$, then

$$\mathbb{J}(G_1,\lambda ,r,1)\le \left|h^{\prime }\left(t\right)\right|\le \mathbb{J}(G_2,\lambda ,r,2)$$

where

$$\mathbb{J}(G_n,\lambda ,r,n)=\left\{\begin{array}{ccc}{G}_n(r,F,H)\left[cos\left(\lambda \right){\displaystyle \frac{1+{(-1)}^{n}Fr}{1+{(-1)}^{n}Hr}}+{(-1)}^n|sin\left(\lambda \right)|\right]\hfill & if\hfill & H\ne 0\hfill \\ \left[cos\left(\lambda \right)\{1+{(-1)}^{n}Fr\}+{(-1)}^n|sin\left(\lambda \right)|\right]e^{\left[{(-1)}^{n}cos\left(\lambda \right)Fr\right]}\hfill & if\hfill & H=0,\hfill \end{array}\right.$$

where $G_1$ and $G_2$ are given by (7) and (8), respectively.

Proof. 

Supposing that $h\in {\mathcal{S}}^{\eta ,\mu }(F,H,\lambda )$, according to (22), we obtain that $h_{\eta ,\mu }\in {\mathcal{S}}^{\lambda }(F,H)$; then, we have to distinguish two cases:

Case 1

$H\ne 0$, using Lemmas 1 and 3, for $\left|t\right|\le r<1$, we obtain

$$G_1(r,F,H)\left[cos\left(\lambda \right){\displaystyle \frac{1-Fr}{1-Hr}}-|sin\left(\lambda \right)|\right]\le \left|h^{\prime }\left(t\right)\right|\le G_2(r,F,H)\left[cos\left(\lambda \right){\displaystyle \frac{1+Fr}{1+Hr}}+|sin\left(\lambda \right)|\right].$$

Case 2

$H=0$, there is $s\in \Omega$ such that $h_{\eta ,\mu }\left(t\right)=texp\left[cos\left(\lambda \right)e^{-i\lambda }Fs\left(t\right)\right]$, $\left|t\right|\le r<1$ and therefore

$$\begin{array}{ccc}& & \left|exp\left[cos\left(\lambda \right)e^{-i\lambda }Fs\left(t\right)\right]\right|\left[cos\left(\lambda \right)\{1-Fs(t\left)\right\}-|sin(\lambda \left)\right|\right]\le \left|h^{\prime }\left(t\right)\right|\hfill \\ & & \le \left[cos\left(\lambda \right)\{1+Fs(t\left)\right\}+|sin(\lambda \left)\right|\right]\left|exp\left[cos\left(\lambda \right)e^{-i\lambda }Fs\left(t\right)\right]\right|.\hfill \end{array}$$

Since

$$\left|exp\left[Fs\left(t\right)cos\left(\lambda \right)e^{-i\lambda }\right]\right|=exp\left[Fs\left(t\right)cos\left(\lambda \right)Re\left\{e^{-i\lambda }\right\}\right],t\in \mathbb{K},$$

using the same calculation as in the above, we obtain

$$exp\left[-Frcos\left(\lambda \right)\right]\le \left|exp\left[Fs\left(t\right)cos\left(\lambda \right)e^{-i\lambda }\right]\right|\le exp\left[Frcos\left(\lambda \right)\right].$$

Thus, (9) yields to

$$\begin{array}{ccc}& & \left[\{1-Fr\}cos\left(\lambda \right)-|sin(\lambda \left)\right|\right]exp\left[-Frcos\left(\lambda \right)\right]\hfill \\ & & \le |h^{\prime }\left(t\right)|\le \left[\{1+Fr\}cos\left(\lambda \right)+|sin(\lambda \left)\right|\right]exp\left[Frcos\left(\lambda \right)\right],\hfill \end{array}$$

(27)

for $\left|t\right|\le r<1$, thus completes the proof. □

Corollary 5.

If $h\in {\mathcal{S}}^{\eta ,\mu }(F,H,\lambda )$, then

$$\left|h\left(t\right)\right|\le \left\{\begin{array}{ccc}{\displaystyle {\int }_0^r\left[cos\left(\lambda \right){\displaystyle \frac{1+F\rho }{1+H\rho }}+|sin\left(\lambda \right)|\right]G_2(\rho ,F,H)d\rho }\hfill & if\hfill & H\ne 0\hfill \\ {\displaystyle {\int }_0^r\left[|sin(\lambda \left)\right|+cos\left(\lambda \right)[1+F\rho ]\right]e^{\left[\{cos(\lambda )+sin(\lambda \left)\right\}F\rho cos\left(\lambda \right)\right]}d\rho }\hfill & if\hfill & H=0,\hfill \end{array}\right.$$

where $G_2$ is given by (8) and for $\left|t\right|\le r<1$.

Proof. 

For $t\in \mathbb{K}$, note that a point on the short section that connects the origin has the form by integrating the function $h^{\prime }$ along this segment

$$\zeta =\rho e^{i\theta } \mathrm{where}, \rho \in [0,r] \mathrm{and} \theta =argt,\left|t\right|=r,$$

we obtain

$$h\left(t\right)={\int }_0^t{h}^{\prime }\left(\zeta \right)d\zeta ,t=r{e}^{i\theta }$$

and hence,

$$\left|h\left(t\right)\right|=\left|{\int }_0^r{h}^{\prime }\left(\rho e^{i\theta }\right)e^{i\theta }d\rho \right|\le {\int }_0^r\left|h^{\prime }\left(\rho e^{i\theta }\right)e^{i\theta }\right|d\rho .$$

For any function $h\in {\mathcal{S}}_{\eta ,\mu }^{\lambda }(F,H,\alpha )$, we have

$$\frac{1}{cos\left(\lambda \right)}\left[{\displaystyle \frac{e^{i\lambda }t{h}^{\prime }\left(t\right)}{h_{\eta ,\mu }\left(t\right)}}-isin\left(\lambda \right)\right]=p\left(t\right),p\in \mathcal{P}[F,H].$$

Using the previous theorem’s right-side inequalities and the aforementioned inequality, we must make the following two circumstances distinct:

(i) 

If $H\ne 0$, we have

$$|h^{\prime }\left(t\right)|\le \left[cos\left(\lambda \right){\displaystyle \frac{1+Fr}{1+Hr}}+|sin\left(\lambda \right)|\right]G_2(r,F,H),\left|t\right|\le r<1,$$

(28)

then

$$\begin{array}{ccc}\hfill \left|h\right(t\left)\right|& \le & {\int }_0^r\left|h^{\prime }\left(\rho e^{i\theta }\right)e^{i\theta }\right|d\rho \hfill \\ & \le & {\int }_0^r\left[cos\left(\lambda \right){\displaystyle \frac{1+F\rho }{1+H\rho }}+|sin\left(\lambda \right)|\right]G_2(\rho ,F,H)d\rho ,\hfill \end{array}$$

that is,

$$\left|h\left(t\right)\right|\le {\int }_0^r\left[cos\left(\lambda \right){\displaystyle \frac{1+F\rho }{1+H\rho }}+|sin\left(\lambda \right)|\right]G_2(\rho ,F,H)d\rho ,\left|t\right|\le r<1.$$

(ii) 

If $H=0$, then

$$\begin{array}{ccc}\hfill \left|h\right(t\left)\right|& \le & {\int }_0^r\left|h^{\prime }\left(\rho e^{i\theta }\right)e^{i\theta }\right|d\rho \hfill \\ & \le & {\int }_0^r\left[|sin(\lambda \left)\right|+cos\left(\lambda \right)[1+F\rho ]\right]e^{\left[\{cos(\lambda )+sin(\lambda \left)\right\}F\rho cos\left(\lambda \right)\right]}d\rho ,\hfill \end{array}$$

that is,

$$\left|h\left(t\right)\right|\le {\int }_0^r\left[|sin(\lambda \left)\right|+cos\left(\lambda \right)[1+F\rho ]\right]e^{\left[\{cos(\lambda )+sin(\lambda \left)\right\}F\rho cos\left(\lambda \right)\right]}d\rho ,\left|t\right|\le r<1.$$

□

4. Conclusions

This paper makes a modest effort to introduce the class ${\mathcal{S}}^{\eta ,\mu }(F,H,\lambda )$; this offers an intriguing changeover from spirallike Janowski-type functions, combining the concept of $(\eta ,\mu )$-symmetrical functions. We derived a structural formula for functions in our class A representation theorem, radius of starlikeness estimates, covering, distortion theorems and the right integral mean inequalities. These results will open up many new opportunities for research in this field and related fields. Using the generalized Janowski class and symmetric functions or using the symmetric $q$—derivative operator are applicable particularly in several diverse areas, such as subordination, inclusion, coefficients and operators of the Geometric Function Theory.