Meromorphic harmonic univalent functions related with generalized (p, q)-post quantum calculus operators

1.   Introduction and preliminaries

An analytic function $s: \mathbb{U} = \{z:|z| < 1\}\rightarrow\mathbb{C}$ is subordinate to an analytic function $t: \mathbb{U}\rightarrow\mathbb{C}$ and write $s(z)\prec t(z)$, if there exists a complex value function $\omega$ which maps $\mathbb{U}$ into itself with $\omega(0) = 0\; \; \hbox{and}\; \; |\omega(z)| < 1\; \; (z\in\mathbb{U})$ such that $s(z) = t(\omega(z))\; (z\in\mathbb{U})$. Furthermore, if the function $t$ is univalent in $\mathbb{U}$, then we have the following equivalence (see ^[1]):

Let $\mathcal{A}$ define the class of functions $f$ that are analytic in the open unit disc $\mathbb{U}$ of the form

The theory of ($p, q$)-calculus (or post quantum calculus ^[2]) operators are used in various areas of science and also in geometric function theory. For $0 < q\leq p\leq1$ and $f\in\mathcal{A}$, Chakrabarti and Jagannathan ^[2] defined the ($p, q$)-derivative operator $D_{p, q}:\mathcal{A}\rightarrow \mathcal{A}$ by

where

and

From (1.1), we have

Next, we introduce the ($p, q$)-derivative operator in the class of meromorphic functions.

Suppose $\mathcal{M}$ be the class of functions $f$ that are meromorphic analytic in the punctured disk $\mathbb{U}^{\ast} = \mathbb{U}\setminus \{0\} = \{z:0 < |z| < 1\}$ of the form

Now, we define the ($p, q$)-post quantum derivative operator $\tilde{d}_{p, q}:\mathcal{M}\rightarrow \mathcal{M}$ by

Using (1.4) and (1.5), we have

where $0 < q\leq p\leq1$ and $[k]_{p, q}$ is defined by (1.3).

Let $\lambda\geq 0, 0 < q\leq p\leq1, \; m\in \mathbb{N}_{0} = \mathbb{N}\cup\{0\}$ and $f(z)\in\mathcal{M},$ we introduce the generalized $(p, q)$-post quantum calculus operator $\mathfrak{\widetilde{D}}_{p, q}^{m, \lambda}:\mathcal{M}\rightarrow \mathcal{M}$ as follows,

and in general,

After a simple calculation, we can obtain the following conclusion

where $[k]_{p, q}$ is defined by (1.3). For simple of notation, we let

Obviously, for $\lambda = 0$, the operator $\mathfrak{\widetilde{D}}_{p, q}^{m, 0}f(z) = L^{m}_{p, q}f(z)$ reduces to the ($p, q$)-Sǎlǎgean operator ^[3].

A complex valued harmonic function $f$ in a simply connected domain $\mathbb{D}\subset \mathbb{C}$ has the canonical representation $f = h+\overline{g}$, where $h$ and $g$ are analytic in $\mathbb{D}$ and $g(z_0) = 0$ for some prescribed point $z_0\in\mathbb{D}$. A necessary and sufficient condition for $f$ to be locally univalent and sense preserving in $\mathbb{D}$ is that $|h'(z)| > |g'(z)|$ in $\mathbb{D}$ (see ^[4,5]).

Denote by $\mathcal{M}_H$ the class of meromorphic univalent and harmonic functions $f$ that are sense preserving in $\mathbb{U}^{\ast}$ and have the following form

where $h(z)$ and $g(z)$ are analytic in $\mathbb{U}^{\ast}$ and $\mathbb{U}$ respectively. The class $\mathcal{M}_H$ was studied in ^[6,7,8,9,10].

Let $\lambda\geq0, 0 < q\leq p\leq1, m\in \mathbb{N}_{0}$ and $f\in \mathcal{M}_H$, we now define the operator $\mathfrak{\widetilde{D}}_{p, q}^{m, \lambda}: \mathcal{M}_H \rightarrow \mathcal{M}_H$ as

where

with $\omega_{k}(\lambda, p, q)$ defined by (1.11).

Assume that $F$ be fixed meromorphic harmonic function given by

For $f$ given by (1.12) and $F$ given by (1.15), we define the convolution (or Hadamard product) of $F$ and $f$ by

Also, we denote by $\mathcal{T}(\mathcal{T}\subset\mathcal{M}_H)$ the class of meromorphic harmonic functions $f$ of the following form

Throughout this paper, we shall assume $\lambda\geq 0, 0 < q\leq p\leq1, m\in \mathbb{N}_{0}\;$ and $-1\leq B < A\leq1$.

Let

be harmonic in $\mathbb{U}^{\ast}$ with $u_{k} > 0$ and $v_{k} > 0.$

Taking

Now, using the operator $\mathfrak{\widetilde{D}}_{p, q}^{m, \lambda}$ and subordination relationship, we define the following two classes.

Definition 1. Let the function $f\in \mathcal{M}_H$ of the form (1.12). The function $f\in \mathcal{M}_\phi^{p, q}(\lambda, m, A, B)$ if and only if

and also the function $f\in \mathcal{K}^{p, q}_{\phi}(\lambda, m, A, B)$ if and only if

where

with $\omega_{k}(\lambda; p, q)$ given by (1.11).

We let

and

The classes $\mathcal{M}_\phi^{p, q}(\lambda, m, A, B)$ and $\mathcal{K}_\phi^{p, q}(\lambda, m, A, B)$ reduce to the well-known subclasses of $\mathcal{M}_H$ as well as many new ones. For example, let $\phi(z) = \frac{1}{z}+\sum_{k = 1}^{\infty}(z^{k}+\bar{z}^{k})$, we have

and

where $\gamma\in[0, 1)$.

The classes $MHS^\ast(\gamma)$ and $MCH(\gamma)$ were studied by Jahangiri ^[9].

In particular, the classes $MHS^{\ast}(0) = MHS^{\ast}$ (Meromorphically harmonic starlike functions) and $MCH(0) = MCH$ (Meromorphically harmonic convex functions) were studied by Jahangiri and Silverman ^[10].

In this paper, the sufficient and necessary conditions of coefficients are discussed. As what we have hoped, distortion estimates, extreme points and convolution properties for the above-defined classes are also obtained.

2.   Basic properties

First of all, we provide the sufficient conditions of coefficients for the classes defined in Definition 1.

Theorem 1. Let $f = h+\overline{g}$ be given by (1.12) and $\omega_{k}(\lambda; p, q)$ given by (1.11).

(i) The sufficient condition for $f$ to be sense-preserving and meromorphic harmonic univalent in $\mathbb{U}^\ast$ and $f\in \mathcal{M}_{\phi}^{p, q}(\lambda, m, A, B)$ is

where

(ii) The sufficient condition for $f$ to be sense-preserving and meromorphic harmonic univalent in $\mathbb{U}^\ast$ and $f\in \mathcal{K}_{\phi}^{p, q}(\lambda, m, A, B)$ is

where $\xi_{k, m}(p, q)$ and $\mu_{k, m}(p, q)$ are given by (2.2).

Proof. (i) For $0 < |z_{1}|\leq| z_{2}| < 1$, we obtain

which proves univalence. Note that $f$ is sense-preserving harmonic in $\mathbb{U}^{\ast}$. This is because

Next, we show that if the inequality (2.1) holds, then the required condition (1.19) is satisfied.

By means of Definition 1 and relationship of subordination, the function $f\in \mathcal{M}_{\phi}^{p, q}(\lambda, m, A, B)$ iff there exists an analytic function $\varpi(z)$ satisfying $\varpi(0) = 0, \; |\varpi(z)| < 1\; (z\in \mathbb{U})$ such that

or equivalently

We only need to show that

Letting

Therefore, from (2.1) we get

Hence, we complete the proof of (i). Also, applying the same method as (i), we can obtain (ii).

The harmonic univalent function

where $\sum_{k = 1}^{\infty}(|x_{k}|+|y_{k}|) = 1,$ shows that the coefficient bound given by (2.1) is sharp.

Theorem 2. Let $f = h+\overline{g}$ be given by (1.17). Then

(i) $f\in\widetilde{\mathcal{M}}_\phi^{p, q}(\lambda, m, A, B)$ iff (2.1) holds true.

(ii) $f\in\widetilde{\mathcal{K}}_{\phi}^{p, q}(\lambda, m, A, B)$ iff (2.3) holds true.

Proof. (i) It appears from (1.17) that $\widetilde{\mathcal{M}}_{\phi}^{p, q}(\lambda, m, A, B)\subset \mathcal{M}_{\phi}^{p, q}(\lambda, m, A, B)$. In view of Theorem 1, it is straightforward to show that if $f\in \widetilde{\mathcal{M}}_{\phi}^{p, q}(\lambda, m, A, B)$, then (2.1) holds true. Next, we use the method in ^[11] to prove.

Let $f\in \widetilde{\mathcal{M}}_\phi^{p, q}(\lambda, m, A, B)$, then it satisfies (1.19) or equivalently

From (2.7), we get

which holds for all $z\in\mathbb{U}^{\ast}$. Setting $z = r\; (0 < r < 1)$ in (2.8), we get

Thus, from (2.9) we have

where $\xi_{k, m}(p, q)$ and $\mu_{k, m}(p, q)$ are given by (2.2).

Putting

For the series $\sum_{k = 1}^\infty[\xi_{k, m}(p, q)|a_k|+\mu_{k, m}(p, q)|b_k|]$, $\{S_{n}\}$ is the nondecreasing sequence of partial sums of it. Moreover, by (2.10) it is bounded by 1. Therefore, it is convergent and

Thus, we get the inequality (2.1). Similarly, it is easy to prove (ii) of Theorem 2.

Clearly, from Theorem 2, we have

Next, we give the extreme points of these classes.

Theorem 3. Let $X_{k}\geq 0, Y_{k}\geq 0, \sum_{k = 0}^\infty X_{k}+\sum_{k = 1}^\infty Y_{k} = 1, \xi_{k, m}(p, q)$ and $\mu_{k, m}(p, q)$ be given by (2.2).

(i) If $f\in \widetilde{\mathcal{M}}_{\phi}^{p, q}(\lambda, m, A, B)$. Then $f\in clco\widetilde{\mathcal{M}}_{\phi}^{p, q}(\lambda, m, A, B)$ iff

where

(ii) If $f\in \widetilde{\mathcal{K}}_{\phi}^{p, q}(\lambda, m, A, B)$. Then $f\in clco\widetilde{\mathcal{K}}_{\phi}^{p, q}(\lambda, m, A, B)$ iff the condition (2.12) holds and

Proof. From (2.12) we get

Since $0\leq X_{k}\leq 1\; (k = 0, 1, 2, \cdots)$, we obtain

It follows, from (i) of Theorem 2, that $f\in \widetilde{\mathcal{M}}_\phi^{p, q}(\lambda, m, A, B).$

Conversely, if $f\in \widetilde{\mathcal{M}}_\phi^{p, q}(\lambda, m, A, B)$, then

Putting $X_{k} = \xi_{k, m}(p, q)|a_{k}|, Y_{k} = \mu_{k, m}(p, q)|b_{k}|$ and $X_0 = 1-\sum_{k = 1}^\infty X_{k}-\sum_{k = 1}^\infty Y_{k}\geq 0$, we obtain

Thus $f$ can be expressed in the form of (2.12). The remainder of the proof is analogous to (i) in Theorem 3 and so we omit.

Next, using Theorem 2, we proceed to discuss the distortion theorems for functions of these classes.

Theorem 4. Let $f = h+\overline{g}$ be of the form (1.17), $|z| = r\in (0, 1)$, $\xi_{k, m}(p, q)$ and $\mu_{k, m}(p, q)$ are defined by (2.2), $\{\xi_{k, m}(p, q)\}$ and $\{\mu_{k, m}(p, q)\}$ are non-decreasing sequences. If $f\in \widetilde{\mathcal{M}}_{\phi}^{p, q}(\lambda, m, A, B)$, then

Proof. For $f\in \widetilde{\mathcal{M}}_{\phi}^{p, q}(\lambda, m, A, B)$, using Theorem 2 and (2.1), we have

and

The result is sharp and the extremal function is

So, we complete the proof of Theorem 4.

By virtue of Theorem 4, we obtain the following covering result.

Theorem 5. Let $\xi_{k, m}(p, q)$ and $\mu_{k, m}(p, q)$ be given by (2.2). If $f\in \widetilde{\mathcal{M}}_{\phi}^{p, q}(\lambda, m, A, B)$, then

Theorem 6. The classes $\widetilde{\mathcal{M}}_{\phi}^{p, q}(\lambda, m, A, B)$ and $\widetilde{\mathcal{K}}_{\phi}^{p, q}(\lambda, m, A, B)$ are closed under convex combinations.

Remark 1. By taking the special values of the parameters $\lambda, p, q, m, A, B$ and $\phi$ in Theorems 1-6, it is easy to show the corresponding results for the classes $MHS^{\ast}(\gamma)$ and $MCH(\gamma)$ which are defined in Section 1.

Especially, let $\lambda = 0, p = q = m = 1, A = 1-2\gamma, B = -1, 0\leq \gamma < 1$ and $\phi(z) = \frac{1}{z}+\sum_{k = 1}^{\infty}(z^{k}+\bar{z}^{k})$ in Theorem 2, we can obtain the results of Theorems 1 and 7 in ^[9].

Corollary 1. Let $f = h+\overline{g}$ be given by (1.17). Then

(i) $f\in MHS^*(\gamma)$ iff

(ii) $f\in MCH^*(\gamma)$ iff

3.   Convolution properties

Next, in order to obtain the convolution properties of functions belonging to the classes $\widetilde{\mathcal{M}}_\phi^{p, q}(\lambda, m, A, B)$ and $\widetilde{\mathcal{K}}_\phi^{p, q}(\lambda, m, A, B)$, we now introduce a new class of harmonic functions.

Definition 2. Let $\delta\geq 0$, the function $f = h+\overline{g}$ of the form (1.17) belongs to the class $\widetilde{\mathcal{L}}_\phi^{\delta, p, q}(\lambda, m, A, B)$ if and only if

where $\xi_{k, m}(p, q)$ and $\mu_{k, m}(p, q)$ are defined by (2.2).

Obviously, for any positive integer $\delta$, we have the following inclusion relation:

Let the harmonic functions $f_{t}\; (t = 1, 2, \cdots, \rho)$ and $F_{l}\; (l = 1, 2, \cdots, \eta)$ of the following form

and

We define the Hadamard product (or convolution) of $f_{t}$ and $F_{\ell}$ by

where $t = 1, 2, \cdots, \rho$ and $l = 1, 2, \cdots, \eta.$

Using Theorem 2, we obtain the following results.

Theorem 7. Let $f_{t}$ of the form (3.3) be in the class $\widetilde{\mathcal{K}}_\phi^{p, q}(\lambda, m, A, B)(t = 1, 2, \cdots, \rho)$ and $F_{l}$ of the form (3.4) be in the class $\widetilde{\mathcal{M}}_\phi^{p, q}(\lambda, m, A, B)(l = 1, 2, \cdots, \eta)$. Then the Hadamard product $(f_{1}\ast f_{2}\ast\cdots\ast f_{\rho}\ast F_{1}\ast F_{2}\ast \cdots \ast F_{\eta})(z)$ belongs to the class $\widetilde{\mathcal{L}}_\phi^{\delta, p, q}(\lambda, m, A, B)$, where $\delta = 2\rho+\eta-1.$

Proof. Using the method in ^[8] to prove the theorem. Putting

From (3.6) we have

According to Definition 2, we only need to show that

where $\xi_{k, m}(p, q)$ and $\mu_{k, m}(p, q)$ are defined by (2.2).

For $f_{t}\in\widetilde{\mathcal{K}}_\phi^{p, q}(\lambda, m, A, B)$, we obtain

for every $t = 1, 2, \cdots, \rho.$ Therefore

Further, by $\xi_{k, m}(p, q)\geq k$ and $\mu_{k, m}(p, q)\geq k$, we have

Also, since $F_{l}\in\widetilde{\mathcal{M}}_\phi^{p, q}(\lambda, m, A, B)$, we have

Hence we obtain

Using (3.11) for $t = 1, 2, \cdots, \rho$, (3.13) for $l = 1, 2, \cdots, \eta-1$ and (3.12) for $l = \eta$, we obtain

and therefore $\chi(z)\in \widetilde{\mathcal{L}}_\phi^{\delta, p, q}(\lambda, m, A, B), \; \delta = 2\rho+\eta-1.$ We note that the required estimate can also be obtained by using (3.11) for $t = 1, 2, \cdots, \eta-1;$ (3.13) for $l = 1, 2, \cdots, \eta$ and (3.9) for $t = \rho.$

Taking into account the Hadamard product of functions $f_{1}\ast f_{2}\ast\cdots\ast f_{\rho}$ only, in the proof of Theorem 3.3, and using (3.11) for $t = 1, 2, \ldots, \rho-1;$ and relation (3.9) for $t = \rho$, we are led to

Corollary 2. Let the functions $f_{t}$ defined by (3.3) be in the class $\widetilde{\mathcal{K}}_\phi^{p, q}(\lambda, m, A, B)$ for every $t = 1, 2, \ldots, \rho.$ Then the Hadamard product $(f_{1}\ast f_{2}\ast\cdots\ast f_{\rho})(z)$ belongs to the class $\widetilde{\mathcal{L}}_\phi^{2\rho-1, p, q}(\lambda, m, A, B).$

Also, taking into account the Hadamard product of functions $F_{1}\ast F_{2}\ast\cdots\ast F_{\eta}$ only, in the proof of Theorem 3.3, and using (3.13) for $l = 1, 2, \ldots, \eta-1;$ and relation (3.12) for $l = \eta$, we are led to

Corollary 3. Let the functions $F_{m, l}$ defined by (3.4) be in the class $\widetilde{\mathcal{M}}_\phi^{p, q}(\lambda, m, A, B)$ for every $l = 1, 2, \ldots, \eta$. Then the Hadamard product $(F_{1}\ast F_{2}\ast \cdots \ast F_{\eta})(z)$ belongs to the class $\widetilde{\mathcal{L}}_\phi^{\eta-1, p, q}(\lambda, m, A, B).$

Remark 2. For different choices of the parameters $\lambda, p, q, m, A, B$ and $\phi$ in Theorem 7, we can deduce some new results for each of the following univalent harmonic function classes $MHS^{\ast}(\gamma)$ and $MCH(\gamma)$ which are defined in Section 1.

Acknowledgments

This work was supported by Natural Science Foundation of Inner Mongolia Autonomous Region of China (Grant No. 2019MS01023; Grant No. 2020MS01011; Grant No. 2018MS01026) and Research Program of science and technology at Universities of Inner Mongolia Autonomous Region (Grant No. NJZZ19209; Grant No. NJZY20198).

Conflict of interest

The authors agree with the contents of the manuscript, and there is no conflict of interest among the authors.