Sufficiency Criterion for A Subfamily of Meromorphic Multivalent Functions of Reciprocal Order with Respect to Symmetric Points

Abstract

In the present research paper, our aim is to introduce a new subfamily of meromorphic p-valent (multivalent) functions. Moreover, we investigate sufficiency criterion for such defined family.

1. Introduction

Let the notation ${\Omega }_p$ be the family of meromorphic p-valent functions f that are holomorphic (analytic) in the region of punctured disk $\mathbb{E}=\left\{z\in \mathbb{C}:0<\left|z\right|<1\right\}$ and obeying the following normalization

$$f\left(z\right)=\frac{1}{z^p}+\sum _{j=1}^{\infty }{a}_{j+p}{z}^{j+p}\left(z\in \mathbb{E}\right).$$

(1)

In particular ${\Omega }_1=\Omega$, the familiar set of meromorphic functions. Further, the symbol ${\mathcal{MS}}^{*}$ represents the set of meromorphic starlike functions which is a subfamily of $\Omega$ and is given by

$${\mathcal{MS}}^{*}=\left\{f:f\left(z\right)\in \Omega \mathrm{and}\Re \left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)<0\left(z\in \mathbb{E}\right)\right\}.$$

Two points p and $p^{\prime }$ are said to be symmetrical with respect to o if $o^{\prime }$ is the midpoint of the line segment $p{p}^{\prime }$. This idea was further nourished in [1,2] by introducing the family ${\mathcal{MS}}_s^{*}$ which is defined in set builder form as;

$${\mathcal{MS}}_s^{*}=\left\{f:f\left(z\right)\in \Omega \mathrm{and}\Re \left(\frac{-2z{f}^{\prime }\left(z\right)}{f\left(-z\right)-f\left(z\right)}\right)<0\left(z\in \mathbb{E}\right)\right\}.$$

Now, for $-1\le t<s\le 1$ with $s\ne 0\ne t,$ $0<\xi <1,$ $\lambda$ is real with $\left|\lambda \right|<\frac{\pi }{2}$ and $p\in \mathbb{N},$ we introduce a subfamily of ${\Omega }_p$ consisting of all meromorphic p-valent functions of reciprocal order $\xi$, denoted by ${\mathcal{NS}}_p^{\lambda }\left(s,t,\xi \right)$, and is defined by

$${\mathcal{NS}}_p^{\lambda }\left(s,t,\xi \right)=\left\{f:f\left(z\right)\in {\Omega }_p\mathrm{and}\Re \left(e^{-i\lambda }\frac{p{s}^p{t}^p}{s^p-t^p}\frac{f\left(sz\right)-f\left(tz\right)}{z{f}^{\prime }\left(z\right)}\right)>\xi cos\lambda \left(z\in \mathbb{E}\right)\right\}.$$

We note that for $p=s=1$ and $t=-1$, the class ${\mathcal{NS}}_p^{\lambda }\left(s,t,\xi \right)$ reduces to the class ${\mathcal{NS}}_1^{\lambda }\left(1,-1,\xi \right)={\mathcal{NS}}_{*}^{\lambda }\left(\xi \right)$ and is represented by

$${\mathcal{NS}}_{*}^{\lambda }\left(\xi \right)=\left\{f:f\left(z\right)\in \Omega \mathrm{and}\Re \left(e^{-i\lambda }\frac{f\left(-z\right)-f\left(z\right)}{2z{f}^{\prime }\left(z\right)}\right)>\xi cos\lambda \left(z\in \mathbb{E}\right)\right\}.$$

For detail of the related topics, see the work of Al-Amiri and Mocanu [3], Rosihan and Ravichandran [4], Aouf and Hossen [5], Arif [6], Goyal and Prajapat [7], Joshi and Srivastava [8], Liu and Srivastava [9], Raina and Srivastava [10], Sun et al. [11], Shi et al. [12] and Owa et al. [13], see also [14,15,16].

For simplicity and ignoring the repetition, we state here the constraints on each parameter as $0<\xi <1,$ $-1\le t<s\le 1$ with $s\ne 0\ne t,$ $\lambda$ is real with $\left|\lambda \right|<\frac{\pi }{2}$ and $p\in \mathbb{N}.$

We need to mention the following lemmas which will use in the main results.

Lemma 1.

“Let $H\subset \mathbb{C}$ and let $\Phi :{\mathbb{C}}^2\times {\mathbb{E}}^{*}\to \mathbb{C}$ be a mapping satisfying $\Phi \left(ia,b:z\right)\notin$ H for $a,b\in \mathbb{R}$ such that $b\le -n\frac{1+a^2}{2}.$ If $p\left(z\right)=1+c_n{z}^n+\cdots$ is regular in ${\mathbb{E}}^{*}$ and $\Phi \left(p\left(z\right),z{p}^{\prime }\left(z\right):z\right)\in H$ ∀ $z\in {\mathbb{E}}^{*}$, then $\Re \left(p\left(z\right)\right)>0.$”

Lemma 2.

“Let $p\left(z\right)=1+c_{1}z+\cdots$ be regular in ${\mathbb{E}}^{*}$ and η be regular and starlike univalent in ${\mathbb{E}}^{*}$ with $\eta \left(0\right)=0.$ If $z{p}^{\prime }\left(z\right)\prec \eta \left(z\right),$ then

$$p\left(z\right)\prec 1+\underset{0}{\overset{z}{\int }}\frac{\eta \left(t\right)}{t}dt.$$

This result is the best possible.”

2. Sufficiency Criterion for the Family ${\mathcal{NS}}_p^{\lambda }\left(s,t,\xi \right)$

In this section, we investigate the sufficiency criterion for any meromorphic p-valent functions belonging to the introduced family ${\mathcal{NS}}_p^{\lambda }\left(s,t,\xi \right)$:

Now, we obtain the necessary and sufficient condition for the p-valent function f to be in the family ${\mathcal{NS}}_p^{\lambda }\left(s,t,\xi \right)$ as follows:

Theorem 1.

Let the function $f\left(z\right)$ be the member of the family ${\Omega }_p.$ Then

$$f\left(z\right)\in {\mathcal{NS}}_p^{\lambda }\left(s,t,\xi \right)\iff \left|\frac{e^{i\lambda }}{\mathcal{G}\left(z\right)}-\frac{1}{2\xi cos\lambda }\right|<\frac{1}{2\xi cos\lambda },$$

(2)

where

$$\mathcal{G}\left(z\right)=\frac{p{s}^p{t}^p}{\left(s^p-t^p\right)}\frac{f\left(sz\right)-f\left(tz\right)}{z{f}^{\prime }\left(z\right)}.$$

(3)

Proof. 

Suppose that inequality (2) holds. Then, we have

$$\begin{array}{ccc}\hfill \left|\frac{2\xi cos\lambda -e^{-i\lambda }\mathcal{G}\left(z\right)}{2\xi cos\lambda e^{-i\lambda }\mathcal{G}\left(z\right)}\right|& <& \frac{1}{2\xi cos\lambda }\hfill \\ & \iff & {\left|\frac{2\xi cos\lambda -e^{-i\lambda }\mathcal{G}\left(z\right)}{2\xi cos\lambda e^{-i\lambda }\mathcal{G}\left(z\right)}\right|}^2<\frac{1}{4{\xi }^2{cos}^2\lambda }\hfill \\ & \iff & \left(2\xi cos\lambda -e^{-i\lambda }\mathcal{G}\left(z\right)\right)\left(\overline{2\xi cos\lambda -e^{-i\lambda }\mathcal{G}\left(z\right)}\right)<\left(e^{i\lambda }\overline{\mathcal{G}\left(z\right)}\right)e^{-i\lambda }\mathcal{G}\left(z\right)\hfill \\ & \iff & 4{\xi }^2{cos}^2\lambda -2\xi cos\lambda \left(e^{i\lambda }\overline{\mathcal{G}\left(z\right)}+e^{-i\lambda }\mathcal{G}\left(z\right)\right)<0\hfill \\ & \iff & 2\xi cos\lambda -2\Re \left(e^{-i\lambda }\mathcal{G}\left(z\right)\right)<0\hfill \\ & \iff & \Re \left(e^{-i\lambda }\mathcal{G}\left(z\right)\right)>\xi cos\lambda ,\hfill \end{array}$$

and hence the result follows. □

Next, we investigate the sufficient condition for the p-valent function f to be in the family ${\mathcal{NS}}_p^{\lambda }\left(s,t,\xi \right)$ in the following theorem:

Theorem 2.

If $f\left(z\right)$ belongs to the family ${\Omega }_p$ of meromorphic p-valent functions and obeying

$$\sum _{n=p+1}^{\infty }\left|\left(\frac{s^n-t^n}{s^p-t^p}{s}^p{t}^p-\frac{n\beta cos\lambda }{p}{e}^{i\lambda }\right)\right|\left|a_n\right|<\frac{1}{2}\left(1-\left|1-2\beta cos\lambda e^{i\lambda }\right|\right),$$

(4)

then $f\left(z\right)\in {\mathcal{NS}}_p^{\lambda }\left(s,t,\xi \right).$

Proof. 

To prove the required result we only need to show that

$$\left|\frac{2e^{i\lambda }\xi cos\lambda z{f}^{\prime }\left(z\right)/p-\frac{s^p{t}^p}{\left(t^p-s^p\right)}\left(f\left(tz\right)-f\left(sz\right)\right)}{\frac{s^p{t}^p}{\left(t^p-s^p\right)}\left(f\left(tz\right)-f\left(sz\right)\right)}\right|<1.$$

(5)

Now consider the left hand side of (5), we get

$$\begin{array}{ccc}\hfill LHS& =& \left|\frac{2e^{i\lambda }\xi cos\lambda z{f}^{\prime }\left(z\right)/p-\frac{s^p{t}^p}{\left(t^p-s^p\right)}\left(f\left(tz\right)-f\left(sz\right)\right)}{\frac{s^p{t}^p}{\left(t^p-s^p\right)}\left(f\left(tz\right)-f\left(sz\right)\right)}\right|\hfill \\ & =& \left|\frac{\left(2e^{i\lambda }\xi cos\lambda -1\right)+{\displaystyle \sum _{n=p+1}^{\infty }}\left(\frac{s^n-t^n}{s^p-t^p}{s}^p{t}^p-\frac{2n\xi cos\lambda }{p}{e}^{i\lambda }\right)a_n{z}^{n+p}}{1+{\displaystyle \sum _{n=p+1}^{\infty }}\left(\frac{s^n-t^n}{s^p-t^p}\right)s^p{t}^p{a}_n{z}^{n+p}}\right|\hfill \\ & \le & \frac{\left|2e^{i\lambda }\xi cos\lambda -1\right|+{\displaystyle \sum _{n=p+1}^{\infty }}\left|\left(\frac{s^n-t^n}{s^p-t^p}{s}^p{t}^p-2\beta cos\lambda e^{i\lambda }\frac{n}{p}\right)\right|\left|a_n\right|\left|z^{n+p}\right|}{1-{\displaystyle \sum _{n=p+1}^{\infty }}\left|\left(\frac{s^n-t^n}{s^p-t^p}\right)s^p{t}^p\right|\left|a_n\right|\left|z^{n+p}\right|}\hfill \\ & \le & \frac{\left|2e^{i\lambda }\xi cos\lambda -1\right|+{\displaystyle \sum _{n=p+1}^{\infty }}\left|\left(\frac{s^n-t^n}{s^p-t^p}{s}^p{t}^p-2\beta cos\lambda e^{i\lambda }\frac{n}{p}\right)\right|\left|a_n\right|}{1-{\displaystyle \sum _{n=p+1}^{\infty }}\left|\left(\frac{s^n-t^n}{s^p-t^p}\right)s^p{t}^p\right|\left|a_n\right|}.\hfill \end{array}$$

By virtue of inequality (4), we at once get the desired result. □

Also, we obtain another sufficient condition for the p-valent function f to be in the family ${\mathcal{NS}}_p^{\lambda }\left(s,t,\xi \right)$ by using Lemma 1, in the following theorem:

Theorem 3.

If $f\left(z\right)\in {\Omega }_p$ satisfies

$$\Re \left\{e^{-i\lambda }\left(\alpha z\frac{{\mathcal{G}}^{\prime }\left(z\right)}{\mathcal{G}\left(z\right)}+1\right)\mathcal{G}\left(z\right)\right\}>\beta cos\lambda -\frac{n}{2}\left(\left(1-\beta \right)\alpha cos\lambda \right),$$

then $f\left(z\right)\in {\mathcal{NS}}_p^{\lambda }\left(s,t,\xi \right),$ where $\mathcal{G}\left(z\right)$ is defined in Equation (3).

Proof. 

Let we choose the function $q\left(z\right)$ by

$$q\left(z\right)=\frac{e^{-i\lambda }\mathcal{G}\left(z\right)-\beta cos\lambda +isin\lambda }{\left(1-\beta \right)cos\lambda },$$

(6)

then Equation (6) shows that $q\left(z\right)$ is holomorphic in $\mathbb{E}$ and also normalized by $q\left(0\right)=1.$

From Equation (6), we can easily obtain that

$$e^{-i\lambda }\mathcal{G}\left(z\right)\left(1+\alpha z\frac{{\mathcal{G}}^{\prime }\left(z\right)}{\mathcal{G}\left(z\right)}\right)=\Phi \left(q\left(z\right),z{q}^{\prime }\left(z\right),z\right),$$

where

$$\Phi \left(q\left(z\right),z{q}^{\prime }\left(z\right),z\right)=\left[\left(1-\beta \right)\alpha z{q}^{\prime }\left(z\right)+\left(1-\beta \right)q\left(z\right)+\beta \right]cos\lambda -isin\lambda .$$

Now for all $a,b\in \mathbb{R}$ satisfying $2y\le -n\left(1+a^2\right),$ we have

$$\begin{array}{ccc}\hfill \Re \left\{\Phi \left(ia,b,z\right)\right\}& \le & \beta cos\lambda -\frac{n}{2}\left(1+a^2\right)\left(1-\beta \right)\alpha cos\lambda \hfill \\ & \le & \beta cos\lambda -\frac{n}{2}\left(1-\beta \right)\alpha cos\lambda .\hfill \end{array}$$

Now, let us define a set as

$$H=\left\{\zeta :\Re \left(\zeta \right)>\beta cos\lambda -\frac{n}{2}\left(\left(1-\beta \right)\alpha cos\lambda \right)\right\},$$

then, we see that $\Phi \left(ia,b,z\right)\notin H$ and $\Phi \left(q\left(z\right),z{q}^{\prime }\left(z\right),z\right)\in H.$ Therefore, by using Lemma 1, we obtain that ℜ $\left(q\left(z\right)\right)>0.$ □

Further, in the next theorem, we obtain the sufficient condition for the p-valent function f to be in the family ${\mathcal{NS}}_p^{\lambda }\left(s,t,\xi \right)$ by using Lemma 2.

Theorem 4.

If $f\left(z\right)$ is a member of the family ${\Omega }_p$ of meromorphic p-valent functions and satisfies

$$\left|\frac{e^{i\lambda }}{\mathcal{G}\left(z\right)}\left(\frac{z{\mathcal{G}}^{\prime }\left(z\right)}{\mathcal{G}\left(z\right)}\right)\right|<\frac{1}{\beta cos\lambda }-1,$$

(7)

then $f\left(z\right)\in {\mathcal{NS}}_p^{\lambda }\left(s,t,\xi \right),$ where $\mathcal{G}\left(z\right)$ is given by Equation (3).

Proof. 

In order to prove the required result, we need to define the following function

$$q\left(z\right)cos\lambda =e^{-i\lambda }\mathcal{G}\left(z\right)+isin\lambda ,$$

then, Equation (6) shows that th function $q\left(z\right)$ is holomorphic in $\mathbb{E}$ and also normalized by $q\left(0\right)=1.$

Now, by routine computations, we get

$$\frac{z{q}^{\prime }\left(z\right)}{q\left(z\right)-itan\lambda }=\frac{z{\mathcal{G}}^{\prime }\left(z\right)}{\mathcal{G}\left(z\right)}.$$

Now, let us consider $z{\left(\frac{1}{q\left(z\right)cos\lambda -isin\lambda }\right)}^{\prime }$ and then by using inequality (7), we have

$$\left|z{\left(\frac{1}{q\left(z\right)cos\lambda -isin\lambda }\right)}^{\prime }\right|=\left|\frac{e^{i\lambda }}{\mathcal{G}\left(z\right)}\left(\frac{z{\mathcal{G}}^{\prime }\left(z\right)}{\mathcal{G}\left(z\right)}\right)\right|<\frac{1}{\beta cos\lambda }-1,$$

therefore

$$z{\left(\frac{1}{q\left(z\right)cos\lambda -isin\lambda }\right)}^{\prime }\prec \frac{\left(1-\beta cos\lambda \right)z}{\beta cos\lambda }.$$

Using Lemma 2, we have

$$\frac{1}{\left(q\left(z\right)-itan\lambda \right)cos\lambda }\prec 1+\frac{\left(1-\beta cos\lambda \right)}{\beta cos\lambda }z,$$

equivalently

$$\left(q\left(z\right)-itan\lambda \right)cos\lambda \prec \frac{\beta cos\lambda }{\beta cos\lambda +\left(1-\beta cos\lambda \right)z}=H\left(z\right)\left(say\right).$$

(8)

After simplifications, we get

$$1+\Re \left(\frac{z{H}^{″}\left(z\right)}{H^{\prime }\left(z\right)}\right)=2\beta cos\lambda -1>0,for\frac{1}{2}<\beta <1.$$

The region $H\left(\mathbb{E}\right)$ shows that it is symmetric about the real axis and also $H\left(z\right)$ is convex. Hence

$$\Re \left(\mathcal{G}\left(z\right)\right)\ge H\left(1\right)>0,$$

or

$$\Re \left(q\left(z\right)cos\lambda -isin\lambda \right)>\beta cos\lambda ,$$

or

$$\Re \left(e^{-i\lambda }\mathcal{G}\left(z\right)\right)>\beta cos\lambda ,for\frac{1}{2}<\beta <1.$$

 □

Finally, we investigate the sufficient condition for the p-valent function f to be in the family ${\mathcal{NS}}_p^{\lambda }\left(s,t,\xi \right)$ in the following theorem:

Theorem 5.

If $f\left(z\right)\in {\Omega }_p$ satisfies

$$\left|{\left(\frac{2\beta cos\lambda e^{i\lambda }}{\mathcal{G}\left(z\right)}-1\right)}^{\prime }\right|\le \eta {\left|z\right|}^{\gamma },for0<\eta \le \gamma +1,$$

(9)

then $f\left(z\right)\in {\mathcal{NS}}_p^{\lambda }\left(s,t,\xi \right),$ where $\mathcal{G}\left(z\right)$ is defined in Equation (3).

Proof. 

Let us put

$$G\left(z\right)=z\left(\frac{2\beta cos\lambda e^{i\lambda }}{\mathcal{G}\left(z\right)}-1\right).$$

Then $G\left(0\right)=0$ and $G\left(z\right)$ is analytic in $\mathbb{E}$. Using inequality (9), we can write

$$\left|{\left(\frac{G\left(z\right)}{z}\right)}^{\prime }\right|=\left|{\left(\frac{2\beta cos\lambda e^{i\lambda }}{\mathcal{G}\left(z\right)}-1\right)}^{\prime }\right|\le \eta {\left|z\right|}^{\gamma }.$$

Now,

$$\left|\left(\frac{G\left(z\right)}{z}\right)\right|=\left|\underset{0}{\overset{z}{\int }}{\left(\frac{G\left(t\right)}{t}\right)}^{\prime }dt\right|\le \underset{0}{\overset{\left|z\right|}{\int }}\left|{\left(\frac{G\left(t\right)}{t}\right)}^{\prime }\right|dt\le \underset{0}{\overset{\left|z\right|}{\int }}\eta {\left|t\right|}^{\gamma }dt=\frac{\eta {\left|z\right|}^{\gamma +1}}{\gamma +1}<1,$$

and this implies that

$$\left|\frac{2\beta cos\lambda e^{i\lambda }}{\mathcal{G}\left(z\right)}-1\right|<1.$$

Now by using Theorem 1, we get the result which we needed. □

3. Conclusions

In our results, a new subfamily of meromorphic p-valent (multivalent) functions were introduced. Further, various sufficient conditions for meromorphic p-valent functions belonging to these subfamilies were obtained and investigated.