Convolution properties of meromorphically harmonic functions defined by a generalized convolution $q$-derivative operator

1.   Introduction and definitions

A new research area, which was initially developed by Clunie and Sheil-Small ^[14], is the subject of univalent harmonic functions (see also ^[31,38]). The significance of such functions is attributed to their usages in the analysis of minimal surfaces as well as in problems relevant to applied mathematics. Hengartner and Schober ^[21] introduced and analyzed some specific types of harmonic functions which are given in the region $\widetilde{\mathbb{U}}$ in the complex $z$-plane $\mathbb{C}$, which is given by

Hengartner and Schober ^[21] proved, among other things, that a harmonic complex-valued and sense preserving univalent mapping $f,$ defined in $\widetilde{\mathbb{U}}$ and such that $f\left(\infty \right) = \infty,$ must satisfy the following representation:

with $Q\in \mathbb{C}$ and, for $0\leqq \left\vert \eta_{2}\right\vert < \left\vert \eta_{1}\right\vert,$

In the year 1999, Jahangiri and Silverman ^[27] gave adequate coefficient criteria for which functions of the type $\left(1.1 \right)$ are univalent. They also provided necessary and sufficient coefficient criteria within certain constraints for functions to be harmonic and starlike, also see ^[39]. Later, Jahangiri ^[25] and Murugusundaramoorthy ^[33,34] analyzed the families of meromorphically harmonic function in $\widetilde{\mathbb{U}}.$ The authors in ^[12,13] used the technique developed by Zou and his co-authors in ^[47] to examine the nature of meromorphically harmonic starlike functions with respect to symmetrical conjugate points in the punctured unit disk $\mathbb{U}^{\ast }$ given by

Particularly, in ^[13], a sharp approximation of the coefficient and a structural description of these functions were also determined. To understand the basics in a more clear way, let us represent the symbol $\mathcal{H}$ by the family of harmonic functions $f$ which have the series form given by

where the functions $\lambda$ and $\mu$ are holomorphic in $\mathbb{U} ^{\ast }$ and $\mathbb{U}$, respectively, with the following series forms:

and

We denote the family of all complex-valued functions $f\in \mathcal{H}$ by $\mathcal{M}_{\mathcal{H}}$, which are sense-preserving and univalent in $\mathbb{U}^{\ast }$. Indeed, if

then

In the view of this point, the function class $\mathcal{M}_{\mathcal{H}}$ coincides with the class $\mathcal{M}$ of normalized holomorphic univalent functions in $\mathbb{U}^{\ast }.$ The above-mentioned papers have obviously opened up a new door for researchers to investigate further inputs in this area of geometric function theory. In this regard, we consider the collections of meromorphically harmonic-starlike and meromorphically harmonic-convex functions in $\mathbb{U}^{\ast }$ given, respectively, by

and

where

Now we give the definition of weak subordination in $\mathbb{U}$. For some details about subordinations for hamonic mappings, we refer ^[15,16].

Acomplex valued function $f$ in $\mathbb{U}$ is said to be a weakly subordinate to a complex valued function $g$ in $\mathbb{U}$ and written as $f\left(z\right) \preceq g\left(z\right)$ or simply as $f\preceq g$ if $f\left(\infty \right) = g\left(\infty \right)$ and $f\left(\mathbb{U} \right) \subset g\left(\mathbb{U}\right).$ If $f\preceq g$ and $g$ is univalent in $\mathbb{U}$, then we consider a complex valued function $w\left(z\right) = g^{-1}\left(f\left(z\right) \right), \; z\in \mathbb{U}$ which maps $\mathbb{U}$ onto itself with $w\left(\infty \right) = \infty.$ Conversely if $w\left(z\right) = g^{-1}\left(f\left(z\right) \right)$ in $\mathbb{U}$ and maps $\mathbb{U}$ onto itself with $w\left(\infty \right) = \infty,$ then $f\preceq g.$

This can be written in the following equivalence.

Lemma 1.1. ^[17] A complex valued function $f$ in $\mathbb{U}$ is weakly subordinate to a complex valued function $g$ in $\mathbb{U}$ if and only if there exists a complex valued function $w$ which maps $\mathbb{U}$ onto itself with $w\left(\infty \right) = \infty$ such that $f\left(z\right) = g\left(w\left(z\right) \right), \; z\in \mathbb{U}$.

Many sub-families of meromorphically harmonic functions have also been introduced and investigated by several earlier researchers (see, for example, the works of Bostanci ^[10], Bostanci and Öztürk ^[11], Öztürk and Bostanci ^[35], Wang et al. ^[46], Al-Dweby and Darus ^[3], Al-Shaqsi and Darus ^[4], Ponnusamy and Rajasekaran ^[37], Ahuja and Jahangiri ^[2], Al-Zkeri and Al-Oboudi ^[5], Jahangiri et al. ^[26] and Stephen et al. ^[45]).

2.   The generalized $q$-derivative operator

The theory of the Hadamard product (or convolution) is incredibly essential in the solution of many function-theoretic problems and, as a result of these realities, this technique becomes a significant part of the area of our study. The objective of this section is to examine the properties and consequences of the convolution for the two newly-defined families of meromorphically harmonic functions. For two functions $g_{1}, \ g_{2}\in \mathcal{M}$ with their series expansions as follows:

the Hadamard product (or convolution), which is denoted by $\left(g_{1}\ast g_{2}\right) (z),$ is defined by

The potentially useful properties, which are mentioned below, can only be true if $g\in \mathcal{M}$:

We now consider a function $\mathfrak{N}_{q} \; ($with $q\in \mathbb{C})$ of the following form:

where, as usual, the $q$-number $[\nu ]_{q}\; \; (\nu \in \mathbb{C})$ is given by

where $\mathbb{N}$ denotes the set of natural numbers. Then, clearly, the function $\mathfrak{N}_{q}$ is meromorphically starlike for all complex numbers $q$ such that $\left\vert q\right\vert < 1.$ Also, one can easily see that, if $q\rightarrow 1-,$ then the $\mathfrak{N}_{q}$ reduces as follows:

Moreover, the $q$-derivative (or $q$-difference) operator $D_{q}$ of a function $f$ defined on a subset of the complex space $\mathbb{C}$ is given by (see ^[23,24])

provided that the first-order derivative of the function $f(z)$ at $z = 0$ exists.

Now, by using the function $\mathfrak{N}_{q}$, we define the generalized convolution $q$-derivative operator $\mathfrak{D}_{q}$ for meromorphically functions $f\in \mathcal{M}$ by

We observe that, by taking $q = 1$ in the above $q$-derivative operator $\mathfrak{D}_{q}$, we achieve the ordinary derivative operator $\frac{d}{dz}$. Also, for $0 < q < 1$, we attain Jackson's $q$-derivative (or $q$-difference) $D_{q}$ of the function $f$ which is defined above by (2.2).

The $q$-derivative (or $q$-difference) operator $D_{q}$ has fascinated and inspired many researchers due mainly to its use in various areas of the mathematical and physical sciences. Although the first article in which a link was established between geometric nature of analytic functions associate with the $q$-derivative operator $D_{q}$ was initiated in ^[22] in 1990, yet the usage of $q$-calculus in geometric function theory as well as a solid and comprehensive foundation was given in 1989 in a book chapter by Srivastava ^[41]. After this development, many researchers introduced and studied various useful operators in $q$-calculus together with the applications of the associated convolution concepts. For example, Kanas and Rǎducanu ^[28] studied the $q$-derivative operator $D_{q}$ and examined its behavior in geometric function theory. The operator $D_{q}$ was generalized for multivalent analytic functions by Arif et al ^[9]. Analogous to these $q$-derivative operators, Arif et al. ^[8] and Khan et al. ^[30] contributed by introducing the $q$-integral operators for analytic and multivalent functions. Similarly, in the authors in ^[6] developed and analyzed some analogues of the $q$-derivative operators for meromorphically functions. Very recently, a survay-cum-expository review article on the subject of quantum (or $q$-) calculus and its various applications in geometric function theory was published by Srivastava ^[40] (see also ^{[1,7,18,19,20,29,32,36,42,43,44]}).

We next define an operator for the function $f\in \mathcal{M}_{\mathcal{H}}$ as follows. Let $\mathcal{D}_{\mathcal{H}}^{q, \tau }:\mathcal{M}_{\mathcal{H} }\rightarrow \mathcal{M}_{\mathcal{H}}$ be a linear operator defined for a function $f = \lambda +\overline{\mu }\in \mathcal{M}_{\mathcal{H}}$ by

where the convolution $q$-derivative operator $\mathfrak{D}_{q}$ is given by (2.3).

Making use of the operator $\mathcal{D}_{\mathcal{H}}^{q, \tau }f\left(z\right),$ we now introduce two families $\mathcal{MS}_{\mathcal{H}}\left[ q, L, M\right]$ and $\mathcal{MK}_{\mathcal{H}}\left[ q, L, M\right]$ of functions of the Janowski type for $\left\vert q\right\vert \leqq 1$ and $-1\leqq M\leqq L\leqq 1$, which are defined below:

and

respectively. We note the following special cases:

(ⅰ) Taking $L = 1-2\xi$ and $M = -1$ in $\mathcal{ \ MS}_{\mathcal{H}}\left[ q, L, M\right]$ and $\mathcal{MK}_{\mathcal{H}} \left[ q, L, M\right],$ we have

and

where

and

(ⅱ). By setting $L = \left(1-2\xi \right) \beta$ and $M = -\beta$ in $\mathcal{MS}_{\mathcal{H}}\left[ q, L, M\right]$ and $\mathcal{MK}_{\mathcal{H}}\left[ q, L, M\right]$ with $0\leqq \xi < 1$ and $0\leqq \beta < 1,$ we get the following function classes:

and

where

and

We observe also that

and

where

and

From the definitions of the newly-introduced classes, we have

In this paper, we investigate a number of convolution properties and several coefficient estimates for functions in the classes $\mathcal{MS}_{\mathcal{H} }\left[q, L, M\right]$ and $\mathcal{MK}_{\mathcal{H}}\left[ q, L, M\right]$, which are associated with the generalized convolution $q$-derivative operator $\mathcal{D}_{\mathcal{H}}^{q, \tau}.$

Unless otherwise mentioned, we assume throughout this paper that

3.   Convolution properties

Our first result in this section, which is asserted by Theorem 3.1 below, provides a necessary and sufficient condition for a given function to be in the class $\mathcal{MS}_{\mathcal{H}}\left[q, L, M\right]$.

Theorem 3.1. A function $f$ defined by $\left(1.2\right)$ is in the class $\mathcal{MS}_{\mathcal{H}}\left[q, L, M\right]$ if and only if

for all $\rho$ given by

and also for $\rho = 0.$

Proof. Let $f\in \mathcal{MS}_{\mathcal{H}}\left[q, L, M\right]$ have the series form $\left(1.2\right).$ Then, from the concept of weak subordination, a function $u$ exists, with the restrictions that $u\left(\infty\right) = \infty$ and $\left \vert u\left(z\right) \right \vert < 1$, such that

which is equivalent to the following assertion:

or

Since

together with following identities:

and

We find from $\left(3.4\right)$ that

which leads to $\left(3.1\right)$ and the necessary part of the proof of Theorem $3.1$ is completed.

Conversely, we suppose that $\left(3.1\right)$ holds true for $\rho = 0$. Then it follows that $zf\left(z\right) \neq 0$ for all $z\in \mathbb{U}$. Consequently, the function $\Phi \left(z\right)$ given by

is regular at $z_{0} = 0$, with $\Phi \left(0\right) = 1$. It was shown that the assumption $\left(3.1\right)$ is equivalent to $\left(3.3\right)$, so we obtain

Let we now put

Then the relation $\left(3.5\right)$ shows that $\Phi \left(\mathbb{U} \right) \cap \Psi \left(\partial \mathbb{U}\right) = \emptyset$. Thus, clearly, $\Phi \left(\mathbb{U}\right)$ is connected to $\mathbb{C\setminus }\Psi \left(\partial \mathbb{U}\right)$. Hence we have

which, together with the univalence of $\Psi$, shows that $\Phi \left(z\right) \preceq \Psi \left(z\right)$ in terms of the subordination in $\left(3.2 \right)$, that is, $f\in \mathcal{MS}_{\mathcal{H}}^{q}\left[L, M\right]$. This completes the proof of Theorem 3.1.

Upon letting $q\rightarrow 1-$ in Theorem $3.1$, we obtain the following result.

Corollary 3.2. A function $f$ defined by $\left(1.2\right)$ is in the class $\mathcal{MS}_{\mathcal{H}}\left[L, M\right]$ if and only if

for all $\rho$ given by

and also for $\rho = 0.$

Putting $L = 1-2\xi \left(0\leqq \xi < 1\right)$ and $M = -1$ in Theorem $3.1$, we deduce the following result.

Corollary 3.3. A function $f$ defined by $\left(1.2\right)$ is in the class $\mathcal{MS}_{\mathcal{H}}\left(q, \xi \right)$ if and only if

for all $\vartheta$ given by

and also for $\vartheta = 0.$

By letting $q\rightarrow 1-$ in Corollary $3.3$, we have the following result.

Corollary 3.4. A function $f$ defined by $\left(1.2\right)$ is in the class $\mathcal{MS}_{\mathcal{H}}\left(\xi \right)$ if and only if

for all $\vartheta$ given by

and also for $\vartheta = 0.$

Our next result in this section, which is asserted by Theorem 3.5 below, provides a necessary and sufficient condition for a given function to be in the class $\mathcal{MK}_{\mathcal{H}}\left[ q, L, M\right]$.

Theorem 3.5. A function $f$ defined by $\left(1.2\right)$ is in the class $\mathcal{MK}_{\mathcal{H}}\left[ q, L, M\right]$ if and only if

for all $\rho$ given by

and also for $\rho = 0.$

Proof. First of all, we suppose that the function $f$ defined by $\left(1.2\right)$ is in the class $\mathcal{MK}_{\mathcal{H}}\left[q, L, M\right]$ if it satisfies the condition $\left(2.6\right)$ or, equivalently,

By setting

we note that

We also recall the following identity:

and the fact that

Hence, clearly, the result asserted by Theorem 3.1, would follow by using the above relations in conjunction with Theorem $3.1$.

If we let $q\rightarrow 1-$ in Theorem $3.5$, we obtain the following result.

Corollary 3.6. A function $f$ defined by $\left(1.2\right)$ is in the class $\mathcal{MK}_{\mathcal{H}}\left[L, M\right]$ if and only if

for all $\rho$ given by

and also for $\rho = 0.$

Putting $L = 1-2\xi \; \left(0\leqq \xi < 1\right)$ and $M = -1$ in Theorem $3.5$, we obtain the following corollary.

Corollary 3.7. A function $f$ defined by $\left(1.2\right)$ is in the class $\mathcal{MK}_{\mathcal{H}}\left(q, \xi \right) \; \left(0\leqq \xi < 1\right)$ if and only if

for all $\vartheta$ given by

and also for $\vartheta = 0.$

Letting $q\rightarrow 1-$ in Corollary $3.7$, we obtain the following result.

Corollary 3.8. A function $f$ defined by $\left(1.2\right)$ is in the class $\mathcal{MK}_{\mathcal{H}}\left(\xi \right) \; \left(0\leqq \xi < 1\right)$ if and only if

for all $\vartheta$ given by

and also for $\vartheta = 0.$

Theorem 3.9 below provides a necessary and sufficient condition for a given function to be in the class $\mathcal{MS}_{\mathcal{H}}\left[ q, L, M \right]$.

Theorem 3.9. A function $f$ defined by $\left(1.2\right)$ is in the class $\mathcal{MS}_{\mathcal{H}}\left[ q, L, M\right]$ if and only if

Proof. From Theorem $3.1$, we know that $f\in \mathcal{MS}_{\mathcal{H}} \left[q, L, M\right]$ if and only if

for all $\rho$ given by

and also for $\rho = 0.$

The left-hand side of $\left(3.7\right)$ can be written as follows:

This evidently completes our proof of the result asserted by Theorem 3.9.

Upon letting $q\rightarrow 1-$ in Theorem $3.9$, we obtain the following result.

Corollary 3.10. A function $f$ defined by $\left(1.2\right)$ is in the class $\mathcal{MS}_{\mathcal{H}}\left[ L, M\right]$ if and only if

Putting $L = 1-2\xi \; \left(0\leqq \xi < 1\right)$ and $M = -1$ in Theorem $3.9$, we deduce the following result.

Corollary 3.11. A function $f$ defined by $\left(1.2\right)$ is in the class $\mathcal{MS}_{\mathcal{H}}\left(q, \xi \right)$ if and only if

Taking $q\rightarrow 1-$ in Corollary $3.11$, we obtain the following result.

Corollary 3.12. A function $f$ defined by $\left(1.2\right)$ is in the class $\mathcal{MS}_{\mathcal{H}}\left(\xi \right)$ if and only if

Theorem 3.13. A function $f$ defined by $\left(1.2\right)$ is in the class $\mathcal{M}\mathcal{K}_{\mathcal{H}}\left[ q, L, M\right]$ if and only if

Taking $q\rightarrow 1-$ in Theorem $3.13$, we obtain the following result.

Corollary 3.14. A function $f$ defined by $\left(1.2\right)$, is in the class $\mathcal{M}\mathcal{K}_{\mathcal{H}}\left[ L, M\right]$ if and only if

Putting $L = 1-2\xi, \; \left(0\leqq \xi < 1\right)$ and $M = -1$ in Theorem $3.13$, we obtain the following result.

Corollary 3.15. A function $f$ defined by $\left(1.2\right)$ is in the class $\mathcal{M}\mathcal{K}_{\mathcal{H}}\left(q, \xi \right)$ if and only if

Letting $q\rightarrow 1-$ in Corollary $3.15$, we obtain the following result.

Corollary 3.16. A function $f$ defined by $\left(1.2\right)$ is in the class $\mathcal{M}\mathcal{K}_{\mathcal{H}}\left(\xi \right)$ if and only if

4.   Coefficient estimates and inclusion relations

We now determine the coefficient estimates and inclusion relations for functions belonging to the classes $\mathcal{MS}_{\mathcal{H}}\left[ q, L, M \right]$ and $\mathcal{MK}_{\mathcal{H}}\left[ q, L, M\right]$.

Theorem 4.1. If a function $f$ defined by $\left(1.2\right)$ satisfies the following inequality$:$

then $f\in \mathcal{MS}_{\mathcal{H}}\left[ q, L, M\right].$

Proof. From $\left(3.6\right)$, we have

which completes the proof of Theorem 4.1.

By letting $q\rightarrow 1-$ in Theorem $4.1$, we obtain the following result.

Corollary 4.2. If a function $f$ defined by $\left(1.2\right)$ satisfies the following inequality$:$

then $f\in \mathcal{MS}_{\mathcal{H}}\left[ L, M\right].$

Putting $L = 1-2\xi \; \left(0\leqq \xi < 1\right)$ and $M = -1$ in Theorem $4.1$, we obtain the following result.

Corollary 4.3. If the function $f$ defined by $\left(1.2\right)$ satisfies the following inequality$:$

then $f\in \mathcal{MS}_{\mathcal{H}}\left(q, \xi \right).$

Letting $q\rightarrow 1-$ in Corollary $4.3$, we obtain the following result.

Corollary 4.4. If a function $f$ defined by $\left(1.2\right)$ satisfies the following inequality$:$

then $f\in \mathcal{MS}_{\mathcal{H}}\left(\xi \right).$

Similarly, we can prove the next result (Theorem 4.5 below).

Theorem 4.5. If the function $f$ defined by $\left(1.2\right)$ satisfies the following inequality$:$

then $f\in \mathcal{MK}_{\mathcal{H}}\left[ q, L, M\right].$

Upon letting $q\rightarrow 1-$ in Theorem $4.5$, we obtain the following result.

Corollary 4.6. If the function $f$ defined by $\left(1.2\right)$ satisfies the following inequality$:$

then $f\in \mathcal{MK}_{\mathcal{H}}\left[L, M\right].$

Putting $L = 1-2\xi \; \left(0\leqq \xi < 1\right)$ and $M = -1$ in Theorem $4.5,$ we obtain the following corollary.

Corollary 4.7. If the function $f$ defined by $\left(1.2\right)$ satisfies the following inequality$:$

then $f\in \mathcal{MK}_{\mathcal{H}}\left(q, \xi \right).$

If we let $q\rightarrow 1-$ in Corollary $4.7$, we obtain the following result.

Corollary 4.8. If the function $f$ defined by $\left(1.2\right)$ satisfies the following inequality$:$

then $f\in \mathcal{MK}_{\mathcal{H}}\left(\xi \right).$

5.   Conclusions

In this paper, we have introduced the generalized convolution $q$-derivative operator $\mathfrak{D}_{q}$, which is defined by

where $q\in \mathbb{C}$ and $\left\vert q\right\vert \leqq 1$. By letting $q\rightarrow 1-,$ this generalized convolution $q$-operator $\mathfrak{D} _{q}$ takes the form of the ordinary derivative. Also, for $0 < q < 1$, we obtain the $q$-analog of derivative operator. By applying this operator, we have defined a corresponding operator for meromorphically harmonic functions and introduced some subclasses of meromorphically harmonic starlike and meromorphically harmonic convex functions and have studied a number of properties and results for functions belonging to each of these function classes.

Basic (or $q$-) series and basic (or $q$-) polynomials, especially the basic (or $q$-) hypergeometric functions and basic (or $q$-) hypergeometric polynomials, are applicable particularly in several diverse areas (see, for example, [41,pp. 351–352] and [40,p. 328]). Moreover, in this recently-published survey-cum-expository review article by Srivastava ^[40], the so-called $(p, q)$-calculus was exposed to be a rather trivial and inconsequential variation of the classical $q$-calculus, the additional parameter $p$ being redundant (see, for details, [40,p. 340]). This observation by Srivastava ^[40] will indeed apply also to any attempt to produce the rather straightforward $(p, q)$-variations of the results which we have presented in this paper.

Conflicts of interest

The authors declare no conflicts of interest.