Some applications of q-difference operator involving a family of meromorphic harmonic functions

Abstract

In this paper, we establish certain new subclasses of meromorphic harmonic functions using the principles of q-derivative operator. We obtain new criteria of sense preserving and univalency. We also address other important aspects, such as distortion limits, preservation of convolution, and convexity limitations. Additionally, with the help of sufficiency criteria, we estimate sharp bounds of the real parts of the ratios of meromorphic harmonic functions to their sequences of partial sums.

1 Introduction and definitions

Univalent harmonic functions are a new research area that was initially developed by Clunie and Sheil-Small [15]; see also [40]. The significance of such functions is attributed to their usage in the analysis of minimal surfaces and in problems relevant to applied mathematics. Hengartner and Schober [18] introduced and analyzed some specific types of harmonic functions in the region $\widetilde{\mathfrak{D}}\mathbb{=} \{ z\in \mathbb{C} : \vert z \vert >1 \}$. They proved that a harmonic complex-valued sense-preserving univalent mapping f defined in $\widetilde{\mathfrak{D}}$ and obeying $f ( \infty ) =\infty$ must satisfy the following representation:

$$\begin{aligned} f ( z ) =\mathfrak{G}_{1} ( z ) + \overline{\mathfrak{G}_{2} ( z ) }+A\log \vert z \vert , \end{aligned}$$

(1.1)

where

$$\begin{aligned} \mathfrak{G}_{1} ( z ) =\mu _{1}z+\sum _{n=1}^{\infty }a_{n}z^{-n}\quad\text{and}\quad\mathfrak{G}_{2} ( z ) =\mu _{2} \overline{z}+\sum_{n=1}^{\infty }b_{n} \overline{z}^{-n} \end{aligned}$$

with $0\leq \vert \mu _{2} \vert < \vert \mu _{1} \vert$ and $A\in \mathbb{C}$. In 1999, Jahangiri and Silverman [26] gave adequate coefficient criteria for functions of type (1.1) to be univalent. They also provided necessary and sufficient coefficient criteria within certain constraints for functions to be harmonic and starlike. Using this idea, the authors of [24] contributed a certain family of harmonic close-to-convex functions involving the Alexander integral transform. In 2000, Jahangiri [22] and Murugusundaramoorthy [35, 36] analyzed the families of meromorphic harmonic function in $\widetilde{\mathfrak{D}}$. In [12, 14] the authors used the technique developed by Zou and his coauthors in [55] to examine the natures of meromorphic harmonic starlike functions with respect to symmetrical conjugate points in the punctured disc $\mathfrak{D}^{\ast }\mathbb{=} \{ z\in \mathbb{C} :0< \vert z \vert <1 \} =\mathfrak{D} \backslash \{0\}$. Particularly, in [14] a sharp approximation of the coefficients and a structural description of these functions are also determined. To understand the basics in a more clear way, we denote by $\mathcal{H}$ the family of harmonic functions f that can be represented in the series form

$$\begin{aligned} f ( z ) =\mathfrak{h} ( z ) + \overline{\mathfrak{g} ( z ) }=\frac{1}{z}+ \sum_{n=1}^{\infty } \bigl( a_{n}z^{n}+b_{n} \overline{z}^{n} \bigr)\quad \bigl( z\in \mathfrak{D}^{\ast } \bigr) , \end{aligned}$$

(1.2)

where $\mathfrak{h}$ and $\mathfrak{g}$ are holomorphic functions in $\mathfrak{D}^{\ast }$ and $\mathfrak{D}$ of the form

$$\begin{aligned} \mathfrak{h} ( z ) =\frac{1}{z}+\sum_{n=1}^{\infty }a_{n}z^{n} \bigl( z\in \mathfrak{D}^{\ast } \bigr) \quad\text{and}\quad \mathfrak{g} ( z ) = \sum_{n=1}^{\infty }b_{n}z^{n} \quad( z\in \mathfrak{D} ) \end{aligned}$$

(1.3)

and

$$\begin{aligned} \vert a_{n} \vert \geq 1,\qquad \vert b_{n} \vert \geq 1\quad ( n=2,3,\ldots ) . \end{aligned}$$

Also, let us denote by $\mathcal{M}_{\mathcal{H}}$ the set of complex-valued functions $f\in \mathcal{H}$ that are sense preserving and univalent in $\mathfrak{D}^{\ast }$. Clearly, if $\mathfrak{g} ( z ) \equiv 0$ $( z\in \mathfrak{D} )$, then $\mathcal{M}_{\mathcal{H}}$ matches with the collection $\mathcal{M}$ of holomorphic univalent normalized functions in $\mathfrak{D}$. The above foundational papers opened a new door for the researchers to add some input in this area of function theory. In this regard, we consider the collections of meromorphic harmonic starlike and meromorphic harmonic convex functions in $\mathfrak{D}^{\ast }$

$$\begin{aligned} \mathcal{MS}_{\mathcal{H}}^{\ast }= \biggl\{ f\in \mathcal{M}_{ \mathcal{H}}:-\frac{\mathcal{D}_{\mathcal{H}}f ( z ) }{f ( z ) } \prec \frac{1+z}{1-z} \bigl( z\in \mathfrak{D}^{\ast } \bigr) \biggr\} \end{aligned}$$

and

$$\begin{aligned} \mathcal{MS}_{\mathcal{H}}^{c}= \biggl\{ f\in \mathcal{M}_{\mathcal{H}}:-\frac{\mathcal{D}_{\mathcal{H}} ( \mathcal{D}_{\mathcal{H}}f ( z ) ) }{\mathcal{D}_{\mathcal{H}}f ( z ) } \prec \frac{1+z}{1-z} \bigl( z\in \mathfrak{D}^{\ast } \bigr) \biggr\} , \end{aligned}$$

where the notation ≺ shows the familiar subordination between the holomorphic functions, and

$$\begin{aligned} \mathcal{D}_{\mathcal{H}}f ( z ) =z\mathfrak{h}^{\prime } ( z ) - \overline{z\mathfrak{g}^{\prime } ( z ) }. \end{aligned}$$

Furthermore, many subfamilies of meromorphic harmonic functions have also been established by some well-known researchers; for example, see Bostanci [11], Bostanci and Öztürk [13], Öztürk and Bostanci [38], Wang et al. [54], Al-dweby and Darus [3], Al-Shaqsi and Darus [4], Ponnusamy and Rajasekaran [39], Ahuja and Jahangiri [2], Al-Zkeri and Al-Oboudi [5], Stephen et al. [53], and Khan et al. [32].

The investigation of q-calculus (q stands for quantum) fascinated and inspired many scholars due its use in various areas of the quantitative sciences. Jackson [20, 21] was among the key contributors of all the scientists who introduced and developed the q-calculus theory. Just like q-calculus was used in other mathematical sciences, the formulations of this idea are commonly used to examine the existence of various structures of function theory. The first paper in which a link was established between certain geometric nature of the analytic functions and the q-derivative operator is due to the authors [19]. For the usage of q-calculus in function theory, a solid and comprehensive foundation is given by Srivastava [43]. After this development, many researchers introduced and studied some useful operators in q-analog with applications of convolution concepts. For example, Kanas and Răducanu [27] established the q-differential operator and then examined the behavior of this operator in function theory. For more applications of this operator, see [1, 7, 17]. This operator was generalized further for multivalent analytic functions by Arif et al. [8] and later studied in [30, 41, 51]. Analogous to q-differential operator Arif et al. [9] and Khan et al. [33] contributed the integral operators for analytic and multivalent functions, respectively. Similarly, in [6] the authors developed and analyzed operators in q-analog for meromorphic functions. Also, see the survey-type paper [44] on quantum calculus and its applications. In 2021, Srivastava, Arif, and Raza [46] introduced and studied a generalized convolution q-derivative operator for meromorphic harmonic functions. Using these operators, many researchers contributed some good papers in this direction in geometric function theory; see [16, 23, 25, 28, 29, 31, 37, 45, 50, 52].

Definition 1.1

Let $q\in ] 0,1 [$. Then the q-analog derivative of f is

$$\begin{aligned} D_{q}f ( z ) = \frac{f ( z ) -f ( qz ) }{z ( 1-q ) } \quad( z\in \mathfrak{D} ) . \end{aligned}$$

(1.4)

See also [10, 48, 49], and [47] for some recent applications of the q-difference operators in the theory of q-series and q-polynomials.

By means of (1.2) and (1.4) we obtain

$$\begin{aligned} D_{q}f ( z ) &=D_{q}\mathfrak{h} ( z ) + \overline{D_{q}\mathfrak{g} ( z ) } \\ &=-\frac{1}{qz^{2}}+\sum_{n=1}^{\infty } [ n ] _{q}a_{n}z^{n-1}+\sum _{n=1}^{\infty } [ n ] _{q}\overline{b_{n}z^{n-1}} \quad\text{for }n\in \mathbb{N}, \end{aligned}$$

(1.5)

where

$$\begin{aligned}{} [ n ] _{q}=\frac{1-q^{n}}{1-q}=1+\sum_{k=1}^{n-1}q^{k}, \quad\text{and}\quad [ 0 ] _{q}=0. \end{aligned}$$

To prevent repetition, we will assume, unless otherwise stated, that

$$\begin{aligned} -1\leq M< L\leq 1\quad\text{and}\quad q\in ] 0,1 [ . \end{aligned}$$

Definition 1.2

By $\mathcal{MS}_{\mathcal{H}}^{\ast } ( q,L,M )$ we denote the set of functions $f\in \mathcal{M}_{\mathcal{H}}$ such that

$$\begin{aligned} - \frac{q\mathcal{D}_{\mathcal{H}}^{q}f ( z ) }{f ( z ) }\prec \frac{1+Lz}{1+Mz} \quad\bigl( z\in \mathfrak{D}^{\ast } \bigr), \end{aligned}$$

where

$$\begin{aligned} \mathcal{D}_{\mathcal{H}}^{q}f ( z ) =zD_{q}\mathfrak{h} ( z ) -\overline{zD_{q}\mathfrak{g} ( z ) .} \end{aligned}$$

Similarly, we denote

$$\begin{aligned} \mathcal{MS}_{\mathcal{H}}^{c} ( q,L,M ) := \bigl\{ f\in \mathcal{M}_{\mathcal{H}}:\mathcal{D}_{\mathcal{H}}^{q}f ( z ) \in \mathcal{MS}_{\mathcal{H}}^{\ast } ( q,L,M ) \bigl( z \in \mathfrak{D}^{\ast } \bigr) \bigr\} . \end{aligned}$$

In this paper, we learn some nice properties for the currently established families including distortion limits, univalency criteria, partial-sum problems, sufficiency criteria, convexity conditions, and preserving convolutions.

2 Necessary and sufficient conditions

Theorem 2.1

If $f\in \mathcal{H}$ is described by the series of the form (1.2) and if

$$\begin{aligned} \sum_{n=1}^{\infty } \bigl( \rho _{n} \vert a_{n} \vert + \sigma _{n} \vert b_{n} \vert \bigr) \leq L-M, \end{aligned}$$

(2.1)

then $f\in \mathcal{MS}_{\mathcal{H}}^{\ast } ( q,L,M )$ with

$$\begin{aligned} &\rho _{n} = \bigl\vert \bigl( q[n]_{q}+1 \bigr) \bigr\vert + \bigl\vert \bigl( Mq[n]_{q}+L \bigr) \bigr\vert , \end{aligned}$$

(2.2)

$$\begin{aligned} &\sigma _{n} = \bigl\vert \bigl( q[n]_{q}-1 \bigr) \bigr\vert + \bigl\vert \bigl( Mq[n]_{q}-L \bigr) \bigr\vert . \end{aligned}$$

(2.3)

Proof

If $f ( z ) =\frac{1}{z}$, then we have $\mathfrak{h} ( z ) =\frac{1}{z}$ and $\mathfrak{g} ( z ) =0$. This implies that

$$\begin{aligned} \bigl\vert \mathfrak{h}^{\prime } ( z ) \bigr\vert - \bigl\vert \mathfrak{g}^{\prime } ( z ) \bigr\vert >0. \end{aligned}$$

Hence by the result of Lewy [34] the function f in $\mathfrak{D}^{\ast }$ is locally univalent and orientation-preserving. Now we show that f is univalent in $\mathfrak{D}^{\ast }$. Let $z_{1},z_{2}\in \mathfrak{D}^{\ast }$ with $z_{1}\neq z_{2}$. Then

$$\begin{aligned} \bigl\vert f ( z_{1} ) -f ( z_{2} ) \bigr\vert = \biggl\vert \frac{1}{z_{1}}-\frac{1}{z_{2}} \biggr\vert = \frac{ \vert z_{2}-z_{1} \vert }{ \vert z_{1}z_{2} \vert }>0. \end{aligned}$$

To show that $f\in \mathcal{MS}_{\mathcal{H}}^{\ast } ( q,L,M )$, we have to establish that

$$\begin{aligned} \biggl\vert \frac{q\mathcal{D}_{\mathcal{H}}^{q}f ( z ) +f ( z ) }{Lf ( z ) +Mq\mathcal{D}_{\mathcal{H}}^{q}f ( z ) } \biggr\vert < 1. \end{aligned}$$

It is easy to find that $q\mathcal{D}_{\mathcal{H}}^{q}f ( z ) =-\frac{1}{z}$ and $L-M>0$. This indicates that

$$\begin{aligned} \biggl\vert \frac{q\mathcal{D}_{\mathcal{H}}^{q}f ( z ) +f ( z ) }{Lf ( z ) +Mq\mathcal{D}_{\mathcal{H}}^{q}f ( z ) } \biggr\vert = \biggl\vert \frac{-\frac{1}{z}+\frac{1}{z}}{L-M} \biggr\vert =0< 1. \end{aligned}$$

Hence $f\in \mathcal{MS}_{\mathcal{H}}^{\ast } ( q,L,M )$. Now let $f\in \mathcal{H}$ have be of the form (1.2), and let us choose $n\geq 1$ such that $a_{n}\neq 0$ or $b_{n}\neq 0$. Also, by using

$$\begin{aligned} q [ n ] _{q}=q \Biggl( 1+\sum_{k=1}^{n-1}q^{k} \Biggr) >q \quad\text{for }0< q< 1 \end{aligned}$$

we have

$$\begin{aligned} \frac{\sigma _{n}}{L-M} &=\frac{ \vert ( q[n]_{q}-1 ) \vert + \vert ( Mq[n]_{q}-L ) \vert }{L-M} \\ &>\frac{ \vert ( q-n ) \vert + \vert ( Mq-Ln ) \vert }{L-M}= \frac{ ( n-q ) + ( Ln-Mq ) }{L-M} \\ &>\frac{ ( n-1 ) + ( Ln-M ) }{L-M}= \frac{ ( 1+L ) n- ( 1+M ) }{L-M} \\ &>\frac{ ( 1+L ) n- ( 1+M ) n}{L-M}=n \quad\text{for all }n\geq 1. \end{aligned}$$

Similarly, $\frac{\rho _{n}}{L-M}\geq n$ for $n\geq 1$. Thus using (2.1) together with the above evidence, we get

$$\begin{aligned} \sum_{n=1}^{\infty } \bigl( n \vert a_{n} \vert +n \vert b_{n} \vert \bigr) \leq 1, \end{aligned}$$

(2.4)

and therefore

$$\begin{aligned} \bigl\vert \mathfrak{h}^{\prime } ( z ) \bigr\vert - \bigl\vert \mathfrak{g}^{\prime } ( z ) \bigr\vert & \geq \frac{1}{ \vert z \vert ^{2}}-\sum _{n=1}^{\infty }n \vert a_{n} \vert \vert z \vert ^{n-1}-\sum_{n=1}^{\infty }n \vert b_{n} \vert \vert z \vert ^{n-1} \\ &\geq \frac{1}{ \vert z \vert ^{2}} \Biggl( 1- \vert z \vert \sum _{n=1}^{\infty } \bigl( n \vert a_{n} \vert +n \vert b_{n} \vert \bigr) \Biggr) \\ &\geq \frac{1}{ \vert z \vert ^{2}} \Biggl( 1- \frac{ \vert z \vert }{L-M}\sum _{n=1}^{\infty } \bigl( \rho _{n} \vert a_{n} \vert +\sigma _{n} \vert b_{n} \vert \bigr) \Biggr) \\ &\geq \frac{1}{ \vert z \vert ^{2}} \bigl( 1- \vert z \vert \bigr) >0\quad \bigl( z\in \mathfrak{D}^{\ast } \bigr) . \end{aligned}$$

Therefore by Lewy’s result [34] the function f in $\mathfrak{D}^{\ast }$ is sense-preserving and locally univalent. Moreover, if $z_{1,}z_{2}\in \mathfrak{D}^{\ast }$ with$z_{1}\neq z_{2}$, then

$$\begin{aligned} \biggl\vert \frac{z_{1}^{n}-z_{2}^{n}}{z_{1}-z_{2}} \biggr\vert = \sum _{k=1}^{n} \vert z_{1} \vert ^{k-1} \vert z_{2} \vert ^{k-1}\leq n\quad\text{for }n\geq 2{.} \end{aligned}$$

Hence by (2.4) we have

$$\begin{aligned} \bigl\vert f ( z_{1} ) -f ( z_{2} ) \bigr\vert &\geq \bigl\vert \mathfrak{h} ( z_{1} ) - \mathfrak{h} ( z_{2} ) \bigr\vert - \bigl\vert \mathfrak{g} ( z_{1} ) -\mathfrak{g} ( z_{2} ) \bigr\vert \\ &= \Biggl\vert \frac{1}{z_{1}}-\frac{1}{z_{2}}-\sum _{n=1}^{\infty }a_{n} \bigl( z_{1}^{n}-z_{2}^{n} \bigr) \Biggr\vert - \Biggl\vert \sum_{n=1}^{ \infty }b_{n} \bigl( \overline{z_{1}^{n}-z_{2}^{n}} \bigr) \Biggr\vert \\ &\geq \vert z_{1}-z_{2} \vert \Biggl( \frac{1}{ \vert z_{1}z_{2} \vert }-\sum_{n=1}^{\infty } \bigl( n \vert a_{n} \vert +n \vert b_{n} \vert \bigr) \Biggr) \\ &\geq \vert z_{1}-z_{2} \vert \biggl( \frac{1}{ \vert z_{1}z_{2} \vert }-1 \biggr) >0. \end{aligned}$$

This shows that f is univalent in $\mathfrak{D}^{\ast }$, and thus $f\in \mathcal{M}_{\mathcal{H}}$. Therefore $f\in \mathcal{MS}_{\mathcal{H}}^{\ast } ( q,L,M )$ if and only if there exists a holomorphic function u with $u ( 0 ) =0$ and $\vert u ( z ) \vert <1$ such that

$$\begin{aligned} - \frac{q\mathcal{D}_{\mathcal{H}}^{q}f ( z ) }{f ( z ) }=\frac{1+Lu ( z ) }{1+Mu ( z ) } \end{aligned}$$

or, alternatively,

$$\begin{aligned} \biggl\vert \frac{q\mathcal{D}_{\mathcal{H}}^{q}f ( z ) +f ( z ) }{Lf ( z ) +Mq\mathcal{D}_{\mathcal{H}}^{q}f ( z ) } \biggr\vert < 1. \end{aligned}$$

(2.5)

To prove (2.5), it suffices to show that

$$\begin{aligned} \bigl\vert q\mathcal{D}_{\mathcal{H}}^{q}f ( z ) +f ( z ) \bigr\vert - \bigl\vert Lf ( z ) +Mq \mathcal{D}_{\mathcal{H}}^{q}f ( z ) \bigr\vert < 0 \end{aligned}$$

for $z\in \mathfrak{D}$. Putting $\vert z \vert =r$ $( 0< r<1 )$, we attain

$$\begin{aligned} & \bigl\vert q\mathcal{D}_{\mathcal{H}}^{q}f ( z ) +f ( z ) \bigr\vert - \bigl\vert Lf ( z ) +Mq \mathcal{D}_{\mathcal{H}}^{q}f ( z ) \bigr\vert \\ &\quad\leq \Biggl\vert \sum_{n=1}^{\infty } \bigl( q[n]_{q}+1 \bigr) a_{n}z^{n}- \sum _{n=1}^{\infty } \bigl( q[n]_{q}-1 \bigr) \overline{b_{n}z^{n}} \Biggr\vert \\ &\qquad{}- \Biggl\vert \frac{ ( L-M ) }{z}+\sum_{n=1}^{\infty } \bigl( L+Mq[n]_{q} \bigr) a_{n}z^{n}-\sum _{n=1}^{\infty } \bigl( Mq[n]_{q}-L \bigr) \overline{b_{n}z^{n}} \Biggr\vert \\ &\quad\leq \Biggl\{ \sum_{n=1}^{\infty } \bigl\vert \bigl( q[n]_{q}+1 \bigr) \bigr\vert \vert a_{n} \vert +\sum_{n=1}^{ \infty } \bigl\vert \bigl( q[n]_{q}-1 \bigr) \bigr\vert \vert b_{n} \vert \Biggr\} \\ &\qquad{}-\frac{1}{r} \Biggl\{ \bigl\vert ( L-M ) \bigr\vert - \sum _{n=1}^{\infty } \bigl\vert \bigl( Mq[n]_{q}+L \bigr) \bigr\vert \vert a_{n} \vert -\sum _{n=1}^{\infty } \bigl\vert \bigl( Mq[n]_{q}-L \bigr) \bigr\vert \vert b_{n} \vert \Biggr\} \\ &\quad\leq \frac{1}{r} \Biggl\{ - \vert L-M \vert +\sum _{n=1}^{ \infty } \bigl( \bigl\vert \bigl( q[n]_{q}+1 \bigr) \bigr\vert + \bigl\vert \bigl( Mq[n]_{q}+L \bigr) \bigr\vert \bigr) \vert a_{n} \vert \\ &\qquad{} +\sum_{n=1}^{\infty } \bigl( \bigl\vert \bigl( q[n]_{q}-1 \bigr) \bigr\vert + \bigl\vert \bigl( Mq[n]_{q}-L \bigr) \bigr\vert \bigr) \vert b_{n} \vert \Biggr\} \\ &\quad\leq \frac{1}{r} \Biggl\{ - ( L-M ) +\sum_{n=1}^{\infty } \bigl( \rho _{n} \vert a_{n} \vert +\sigma _{n} \vert b_{n} \vert \bigr) \Biggr\} \end{aligned}$$

due inequality (2.1). Thus $f\in \mathcal{MS}_{\mathcal{H}}^{\ast } ( q,L,M )$. □

By substituting specific values of the parameters included in this result we obtain the following corollaries.

Corollary 2.2

Let $f\in \mathcal{H}$ be of the form (1.2). If

$$\begin{aligned} \sum_{n=1}^{\infty } \bigl( \rho _{n} \vert a_{n} \vert + \sigma _{n} \vert b_{n} \vert \bigr) \leq ( 1+q ) \end{aligned}$$

with

$$\begin{aligned} &\rho _{n} = \bigl\vert \bigl( q[n]_{q}+1 \bigr) \bigr\vert + \bigl\vert \bigl( q^{2}[n]_{q}-1 \bigr) \bigr\vert , \\ &\sigma _{n} = \bigl\vert \bigl( q[n]_{q}-1 \bigr) \bigr\vert + \bigl\vert \bigl( q^{2}[n]_{q}+1 \bigr) \bigr\vert , \end{aligned}$$

then $f\in \mathcal{MS}_{\mathcal{H}}^{\ast } ( q,1,-q )$

Proof

The result is obtained by setting $L=1$ and $M=-q$ in the last theorem. □

Corollary 2.3

Let $f\in \mathcal{H}$ be given in (1.2). If

$$\begin{aligned} \sum_{n=1}^{\infty }n \bigl( \vert a_{n} \vert + \vert b_{n} \vert \bigr) \leq 1, \end{aligned}$$

then $f\in \mathcal{MS}_{\mathcal{H}}^{\ast } ( 1,1,-1 )$.

Proof

Taking the limit as $q\rightarrow 1-$ in the above corollary, we get the needed result. □

Influenced by Silverman’s paper [42], the set $\vartheta ^{\lambda }$, $\lambda \in \{ 0,1 \}$, of functions $f\in \mathcal{H}$ of type (1.2) is now described as

$$\begin{aligned} a_{n}=- \vert a_{n} \vert ,\qquad b_{n}= ( -1 ) ^{\lambda } \vert b_{n} \vert \quad\bigl(\text{for }n\in \mathbb{N} \backslash \{1\} \bigr) . \end{aligned}$$

Hence (1.2) and (1.3) give $f ( z ) =\mathfrak{h} ( z ) + \overline{\mathfrak{g} ( z ) }$ with

$$\begin{aligned} \mathfrak{h} ( z ) =\frac{1}{z}-\sum_{n=1}^{\infty } \vert a_{n} \vert z^{n},\qquad \overline{\mathfrak{g} ( z ) }= ( -1 ) ^{\lambda }\sum_{n=1}^{\infty } \vert b_{n} \vert \overline{z^{n}} \quad( z\in \mathfrak{D} ) . \end{aligned}$$

(2.6)

Using the above facts, we now define the families

$$\begin{aligned} &\mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M ) =\vartheta ^{0} \cap \mathcal{MS}_{\mathcal{H}}^{\ast } ( q,L,M ) , \\ &\mathcal{MS}_{\mathcal{H}_{\vartheta }}^{c} ( q,L,M ) =\vartheta ^{1} \cap \mathcal{MS}_{\mathcal{H}}^{c} ( q,L,M ) . \end{aligned}$$

Let us now prove that condition (2.1) is also appropriate for $f\in \mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast }$.

Theorem 2.4

Let $f\in \vartheta ^{0}$ have expansion (2.6). Then $f\in \mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$ if and only if (2.1) is true.

Proof

To achieve the result, it is sufficient to determine that $f\in \mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$ validates relationship (2.1). Let $f\in \mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$. Then inequality (2.5) holds, that is, for $z\in \mathfrak{D}^{\ast }$,

$$\begin{aligned} \biggl\vert \frac{\sum_{n=1}^{\infty } ( q [ n ] _{q}+1 ) a_{n}z^{n}+\sum_{n=1}^{\infty } ( q [ n ] _{q}-1 ) \overline{b_{n}z^{n}}}{\frac{L-M}{z}-\sum_{n=1}^{\infty } ( qM [ n ] _{q}+L ) a_{n}z^{n}+\sum_{n=1}^{\infty } ( qM [ n ] _{q}-L ) \overline{b_{n}z^{n}}} \biggr\vert < 1. \end{aligned}$$

Setting $z=r$ $( r\in ( 0,1 ) )$, we obtain

$$\begin{aligned} \frac{\sum_{n=1}^{\infty } ( \vert ( q [ n ] _{q}+1 ) \vert \vert a_{n} \vert + \vert ( q [ n ] _{q}-1 ) \vert \vert b_{n} \vert ) r^{n+1}}{ ( L-M ) -\sum_{n=1}^{\infty } ( \vert ( qM [ n ] _{q}+L ) \vert \vert a_{n} \vert + \vert ( qM [ n ] _{q}-L ) \vert \vert b_{n} \vert ) r^{n+1}}< 1. \end{aligned}$$

(2.7)

Obviously, in case of $r\in ( 0,1 )$, the left-hand side denominator of (2.7) cannot be zero. In addition, this is positive when $r=0$. Thus, using (2.7), we get

$$\begin{aligned} \sum_{n=1}^{\infty } \bigl( \rho _{n} \vert a_{n} \vert + \sigma _{n} \vert b_{n} \vert \bigr) r^{n+1}\leq ( L-M )\quad ( 0\leq r< 1 ) . \end{aligned}$$

(2.8)

It is straightforward that the partial-sum sequence $\{ S_{n} \}$ attached with the series $\sum_{n=1}^{\infty } ( \rho _{n} \vert a_{n} \vert + \sigma _{n} \vert b_{n} \vert )$ is nondecreasing sequence, and by (2.8) it is bounded by $( L-M )$. So $\{ S_{n} \}$ is a convergent sequence, and

$$\begin{aligned} \sum_{n=1}^{\infty } \bigl( \rho _{n} \vert a_{n} \vert + \sigma _{n} \vert b_{n} \vert \bigr) r^{n+1}=\lim_{n \rightarrow \infty }S_{n}\leq ( L-M ) , \end{aligned}$$

which gives assumption (2.1). □

Example 2.5

Let us choose the function

$$\begin{aligned} T(z)=\frac{1}{z}-\sum_{n=1}^{\infty } \biggl( \frac{L-M}{\rho _{n}}\frac{1}{2^{n}}z^{n}+ \frac{L-M}{\sigma _{n}}\frac{1}{2^{n}} \overline{z^{n}} \biggr)\quad \bigl( z\in \mathfrak{D}^{\ast } \bigr) . \end{aligned}$$

Then we easily get

$$\begin{aligned} \sum_{n=1}^{\infty } \bigl( \rho _{n} \vert a_{n} \vert +\sigma _{n} \vert b_{n} \vert \bigr) = \sum_{n=1}^{\infty } \frac{1}{2^{n-1}} ( L-M ) = ( L-M ) . \end{aligned}$$

Thus $T\in \mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$.

By using the above-mentioned theorem along with the notion of class $\mathcal{MS}_{\mathcal{H}}^{c} ( q,L,M )$ we can easily derive the following results.

Corollary 2.6

Let $f\in \mathcal{H}$ be written in the form of Taylor expansion (1.2). If

$$\begin{aligned} \sum_{n=1}^{\infty } [ n ] _{q} \bigl( \rho _{n} \vert a_{n} \vert +\sigma _{n} \vert b_{n} \vert \bigr) \leq ( L-M ) , \end{aligned}$$

(2.9)

then $f\in \mathcal{MS}_{\mathcal{H}}^{c} ( q,L,M )$.

Proof

From inequality (2.9), Theorem 2.1, and Alexander-type relation

$$\begin{aligned} f \in \mathcal{MS}_{\mathcal{H}}^{c} ( q,L,M )\quad \Longleftrightarrow\quad \mathcal{D}_{\mathcal{H}}^{q}f \in \mathcal{MS}_{ \mathcal{H}}^{\ast } ( q,L,M ) \end{aligned}$$

(2.10)

we easily get the desired result. □

Corollary 2.7

Let $f\in \vartheta ^{1}$ be written in the series form (2.6). Then $f\in \mathcal{MS}_{\mathcal{H}_{\vartheta }}^{c} ( q,L,M )$ if and only if inequality (2.9) is fulfilled.

Proof

Using relationship (2.10) and Theorem 2.4, we get the desired result. □

3 Investigation of partial-sum problems

In this section, we investigate problems of partial sums of certain meromorphic harmonic functions belonging to $\mathcal{MS}_{\mathcal{H}}^{\ast } ( q,L,M )$. We produce some new findings that connect the meromorphic harmonic functions with their partial-sum sequences. Let $f=\mathfrak{h}+$ $\overline{\mathfrak{g}}$ with $\mathfrak{h}$ and $\mathfrak{g}$ given in (1.3). Then the partial-sum sequences of f are specified by

$$\begin{aligned} &\mathcal{M}_{\mathcal{S}_{t}} ( f ) =\frac{1}{z}+\sum _{n=1}^{t}a_{n}z^{n}+\sum _{n=1}^{\infty }b_{n} \overline{z}^{n}:=\mathcal{M}_{\mathcal{S}_{t}} ( \mathfrak{h} ) + \overline{\mathfrak{g}}, \\ &\mathcal{M}_{\mathcal{S}_{l}} ( f ) =\frac{1}{z}+\sum _{n=1}^{\infty }a_{n}z^{n}+\sum _{n=1}^{l}b_{n}\overline{z}^{n}:=\mathcal{M}_{\mathcal{S}_{l}} ( \overline{\mathfrak{g}} ) +\mathfrak{h}, \\ &\mathcal{M}_{\mathcal{S}_{t,l}} ( f ) =\frac{1}{z}+\sum _{n=1}^{t}a_{n}z^{n}+\sum _{n=1}^{l}b_{n} \overline{z}^{n}:= \mathcal{M}_{\mathcal{S}_{t}} ( \mathfrak{h} ) + \mathcal{M}_{ \mathcal{S}_{l}} ( \overline{\mathfrak{g}} ). \end{aligned}$$

Now we find sharp lower bounds for

$$\begin{aligned} \operatorname{Re} \biggl( \frac{f ( z ) }{\mathcal{M}_{\mathcal{S}_{t}} ( f ) } \biggr) \quad\text{and}\quad \operatorname{Re} \biggl( \frac{\mathcal{M}_{\mathcal{S}_{t}} ( f ) }{f ( z ) } \biggr) ,\qquad \operatorname{Re} \biggl( \frac{f ( z ) }{\mathcal{M}_{\mathcal{S}_{l}} ( f ) } \biggr) \quad\text{and}\quad \operatorname{Re} \biggl( \frac{\mathcal{M}_{\mathcal{S}_{l}} ( f ) }{f ( z ) } \biggr) , \end{aligned}$$

and

$$\begin{aligned} \operatorname{Re} \biggl( \frac{f ( z ) }{\mathcal{M}_{\mathcal{S}_{t,l}} ( f ) } \biggr) \quad\text{and}\quad \operatorname{Re} \biggl( \frac{\mathcal{M}_{\mathcal{S}_{t,l}} ( f ) }{f ( z ) } \biggr) . \end{aligned}$$

Theorem 3.1

Let f have the form (1.2). If f fulfills (2.1), then

$$\begin{aligned} \operatorname{Re} \biggl( \frac{f ( z ) }{\mathcal{M}_{\mathcal{S}_{t}} ( f ) } \biggr) \geq \frac{\mathcal{I}_{t+1}-L+M}{\mathcal{I}_{t+1}} \end{aligned}$$

(3.1)

and

$$\begin{aligned} \operatorname{Re} \biggl( \frac{\mathcal{M}_{\mathcal{S}_{t}} ( f ) }{f ( z ) } \biggr) \geq \frac{\mathcal{I}_{t+1}}{\mathcal{I}_{t+1}-L+M}, \end{aligned}$$

(3.2)

where

$$\begin{aligned} \mathcal{I}_{n}=\min ( \rho _{n},\sigma _{n} ) \end{aligned}$$

(3.3)

and

$$\begin{aligned} \mathcal{I}_{n}\geq \textstyle\begin{cases} L-M & \textit{for } n=1,2,\ldots ,t, \\ \mathcal{I}_{t+1} & \textit{for } n=t+1,\ldots .\end{cases}\displaystyle \end{aligned}$$

(3.4)

The findings above are best suited for the function

$$\begin{aligned} f ( z ) =\frac{1}{z}+\frac{L-M}{\mathcal{I}_{t+1}}z^{t+1}, \end{aligned}$$

(3.5)

where $\mathcal{I}_{t+1}$ is given by (3.4).

Proof

Let us represent

$$\begin{aligned} \Theta _{1} ( z ) &=\frac{\mathcal{I}_{t+1}}{L-M} \biggl\{ \frac{f ( z ) }{\mathcal{M}_{\mathcal{S}_{t}} ( f ) }- \biggl( 1-\frac{L-M}{\mathcal{I}_{t+1}} \biggr) \biggr\} \\ &=1+ \frac{\frac{\mathcal{I}_{t+1}}{L-M}\sum_{n=1}^{\infty }a_{n}z^{n}}{\frac{1}{z}+\sum_{n=1}^{t}a_{n}z^{n}+\sum_{n=1}^{\infty }b_{n}\overline{z}^{n}}. \end{aligned}$$

Inequality (3.1) will be acquired if we are able to show that Re$\{ \Theta _{1} ( z ) \} >0$, and for this, we required to conclude that

$$\begin{aligned} \biggl\vert \frac{\Theta _{1} ( z ) -1}{\Theta _{1} ( z ) +1} \biggr\vert \leq 1. \end{aligned}$$

Alternatively, we have the following inequalities:

$$\begin{aligned} \biggl\vert \frac{\Theta _{1} ( z ) -1}{\Theta _{1} ( z ) +1} \biggr\vert \leq \frac{\frac{\mathcal{I}_{t+1}}{L-M}\sum_{n=t+1}^{\infty } \vert a_{n} \vert }{2-2 ( \sum_{n=1}^{t} \vert a_{n} \vert +\sum_{n=1}^{\infty } \vert b_{n} \vert ) -\frac{\mathcal{I}_{t+1}}{L-M}\sum_{n=t+1}^{\infty } \vert a_{n} \vert } \leq 1 \end{aligned}$$

if and only if

$$\begin{aligned} \sum_{n=1}^{t} \vert a_{n} \vert +\sum_{n=1}^{\infty } \vert b_{n} \vert +\frac{\mathcal{I}_{t+1}}{L-M}\sum_{n=t+1}^{ \infty } \vert a_{n} \vert \leq 1. \end{aligned}$$

(3.6)

From (2.1) we have that it suffices to guarantee that the left-hand side of (3.6) is bounded above by

$$\begin{aligned} \sum_{n=1}^{\infty }\frac{\mathcal{I}_{n}}{L-M} \vert a_{n} \vert +\sum_{n=1}^{\infty } \frac{\mathcal{I}_{n}}{L-M} \vert b_{n} \vert , \end{aligned}$$

which is exactly equivalent to

$$\begin{aligned} \sum_{n=1}^{t}\frac{\mathcal{I}_{n}-L+M}{L-M} \vert a_{n} \vert +\sum_{n=1}^{\infty } \frac{\mathcal{I}_{n}-L+M}{L-M} \vert b_{n} \vert +\sum _{n=t+1}^{\infty } \frac{\mathcal{I}_{n}-\mathcal{I}_{t+1}}{L-M} \vert a_{n} \vert \geq 0, \end{aligned}$$

and this is true because of (3.4). We observe that the function

$$\begin{aligned} f ( z ) =\frac{1}{z}+\frac{L-M}{\mathcal{I}_{t+1}}z^{t+1} \end{aligned}$$

offers the best possible outcome. We see for $z=re^{i\frac{\pi }{t}}$ that

$$\begin{aligned} \frac{f ( z ) }{\mathcal{M}_{\mathcal{S}_{t}} ( f ) }=1+\frac{L-M}{\mathcal{I}_{t+1}}z^{t+2}\rightarrow 1- \frac{L-M}{\mathcal{I}_{t+1}}r^{t+2}=\frac{\mathcal{I}_{t+1}-L+M}{\mathcal{I}_{t+1}}. \end{aligned}$$

To examine (3.2), let us write

$$\begin{aligned} \Theta _{2} ( z ) &=\frac{\mathcal{I}_{t+1}+L-M}{L-M} \biggl\{ \frac{\mathcal{M}_{\mathcal{S}_{t}} ( f ) }{f ( z ) }- \biggl( 1-\frac{L-M}{\mathcal{I}_{t+1}+L-M} \biggr) \biggr\} \\ &=1- \frac{\frac{\mathcal{I}_{t+1}+L-M}{L-M}\sum_{n=t+1}^{\infty }a_{n}z^{n}}{\frac{1}{z}+\sum_{n=1}^{\infty }a_{n}z^{n}+\sum_{n=1}^{\infty }b_{n}\overline{z}^{n}}. \end{aligned}$$

Then

$$\begin{aligned} \biggl\vert \frac{\Theta _{2} ( z ) -1}{\Theta _{2} ( z ) +1} \biggr\vert \leq \frac{\frac{\mathcal{I}_{t+1}+L-M}{L-M}\sum_{n=t+1}^{\infty } \vert a_{n} \vert }{2-2 ( \sum_{n=1}^{t} \vert a_{n} \vert +\sum_{n=1}^{\infty } \vert b_{n} \vert ) -\frac{\mathcal{I}_{t+1}+L-M}{L-M}\sum_{n=t+1}^{\infty } \vert a_{n} \vert }\leq 1 \end{aligned}$$

if and only if

$$\begin{aligned} \sum_{n=1}^{t} \vert a_{n} \vert +\sum_{n=1}^{\infty } \vert b_{n} \vert +\frac{\mathcal{I}_{t+1}}{L-M}\sum_{n=t+1}^{ \infty } \vert a_{n} \vert \leq 1. \end{aligned}$$

(3.7)

Inequality (3.7) is valid if the left-hand side of this inequality is bounded above by

$$\begin{aligned} \sum_{n=1}^{\infty }\frac{\mathcal{I}_{n}}{L-M} \vert a_{n} \vert +\sum_{n=1}^{\infty } \frac{\mathcal{I}_{n}}{L-M} \vert b_{n} \vert , \end{aligned}$$

and thus the proof is accomplished by using (2.1). □

Theorem 3.2

Let $f=\mathfrak{h}+$ $\overline{\mathfrak{g}}$, where $\mathfrak{h}$ and $\mathfrak{g}$ are given by (1.3). If f fulfills (2.1), then

$$\begin{aligned} \operatorname{Re} \biggl( \frac{f ( z ) }{\mathcal{M}_{\mathcal{S}_{l}} ( f ) } \biggr) \geq \frac{\mathcal{I}_{l+1}-L+M}{\mathcal{I}_{l+1}} \end{aligned}$$

(3.8)

and

$$\begin{aligned} \operatorname{Re} \biggl( \frac{\mathcal{M}_{\mathcal{S}_{l}} ( f ) }{f ( z ) } \biggr) \geq \frac{\mathcal{I}_{l+1}}{\mathcal{I}_{l+1}-L+M}, \end{aligned}$$

(3.9)

where $\mathcal{I}_{n}$ is given by (3.3), and

$$\begin{aligned} \mathcal{I}_{n}\geq \textstyle\begin{cases} L-M & \textit{for } n=1,2,\dots ,l, \\ \mathcal{I}_{l+1} & \textit{for }n=l+1,\dots .\end{cases}\displaystyle \end{aligned}$$

(3.10)

The equalities are achieved by considering the function

$$\begin{aligned} f ( z ) =\frac{1}{z}+\frac{L-M}{\mathcal{I}_{t+1}} \overline{z}^{l+1}. \end{aligned}$$

(3.11)

Proof

The proof for this specific outcome is similar to that of Theorem 3.1 and is thus omitted. □

Theorem 3.3

Let $f=\mathfrak{h}+\overline{\mathfrak{g}}$ have the power series form (1.3). If f meets inequality (2.1), then

$$\begin{aligned} \operatorname{Re} \biggl( \frac{f ( z ) }{\mathcal{M}_{\mathcal{S}_{t,l}} ( f ) } \biggr) \geq \frac{\mathcal{I}_{t+1}- ( L-M ) }{\mathcal{I}_{t+1}} \vadjust{\goodbreak} \end{aligned}$$

(3.12)

and

$$\begin{aligned} \operatorname{Re} \biggl( \frac{\mathcal{M}_{\mathcal{S}_{t,l}} ( f ) }{f ( z ) } \biggr) \geq \frac{\mathcal{I}_{t+1}}{\mathcal{I}_{t+1}+ ( L-M ) }, \end{aligned}$$

(3.13)

where $\mathcal{I}_{n}$ is given by (3.3). The equalities are easily achieved by using (3.5).

Proof

To establish (3.12), let us consider

$$\begin{aligned} \Theta _{3} ( z ) &=\frac{\mathcal{I}_{t+1}}{L-M} \biggl\{ \frac{f ( z ) }{\mathcal{S}_{t,l} ( f ) }- \biggl( 1- \frac{L-M}{\mathcal{I}_{t+1}} \biggr) \biggr\} \\ &=1+ \frac{\frac{\mathcal{I}_{t+1}}{L-M} ( \sum_{n=t+1}^{\infty }a_{n}z^{n}+\sum_{n=l+1}^{\infty }b_{n}\overline{z}^{n} ) }{\frac{1}{z}+\sum_{n=1}^{t}a_{n}z^{n}+\sum_{n=1}^{l}b_{n}\overline{z}^{n}}. \end{aligned}$$

Therefore, to show inequality (3.12), it is sufficient to prove the inequality

$$\begin{aligned} \biggl\vert \frac{\Theta _{3} ( z ) -1}{\Theta _{3} ( z ) +1} \biggr\vert \leq 1. \end{aligned}$$

Now recalling the left-hand side of the above-mentioned inequality, by easy calculations we get

$$\begin{aligned} \biggl\vert \frac{\Theta _{3} ( z ) -1}{\Theta _{3} ( z ) +1} \biggr\vert \leq \frac{\frac{\mathcal{I}_{t+1}}{L-M}\sum_{n=t+1}^{\infty } \vert a_{n} \vert }{2-2 ( \sum_{n=1}^{t} \vert a_{n} \vert +\sum_{n=1}^{l} \vert b_{n} \vert ) -\frac{\mathcal{I}_{t+1}}{L-M} ( \sum_{n=t+1}^{\infty } \vert a_{n} \vert +\sum_{n=l+1}^{\infty } \vert b_{n} \vert ) }. \end{aligned}$$

Since we observe that from (2.1) that the denominator of the last inequality is positive. The right-hand side of the last inequality is also constrained by one if and only if the following inequality is maintained:

$$\begin{aligned} \sum_{n=1}^{t} \vert a_{n} \vert +\sum_{n=1}^{l} \vert b_{n} \vert +\frac{\mathcal{I}_{t+1}}{L-M} \Biggl( \sum _{n=t+1}^{ \infty } \vert a_{n} \vert +\sum _{n=t+1}^{\infty } \vert b_{n} \vert \Biggr) \leq 1. \end{aligned}$$

(3.14)

Eventually, to verify inequality (3.12), it suffices to show that the left-hand side of (3.14) is bounded above by

$$\begin{aligned} \sum_{n=1}^{\infty }\frac{\mathcal{I}_{n}}{L-M} \vert a_{n} \vert +\sum_{n=1}^{\infty } \frac{\mathcal{I}_{n}}{L-M} \vert b_{n} \vert , \end{aligned}$$

which is further analogous to

$$\begin{aligned} &\sum_{n=1}^{t}\frac{\mathcal{I}_{n}- ( L-M ) }{L-M} \vert a_{n} \vert +\sum_{n=1}^{l} \frac{\mathcal{I}_{n}- ( L-M ) }{L-M} \vert b_{n} \vert +\sum _{n=l+1}^{\infty } \frac{\mathcal{I}_{n}-\mathcal{I}_{t+1}}{L-M} \Biggl( \sum _{n=t+1}^{\infty } \vert a_{n} \vert +\sum _{n=t+1}^{\infty } \vert b_{n} \vert \Biggr)\\ &\quad \geq 0, \end{aligned}$$

and this is true due to (3.10). Now let us choose

$$\begin{aligned} f ( z ) =\frac{1}{z}+\frac{L-M}{\mathcal{I}_{t+1}}z^{t+1}, \end{aligned}$$

which delivers a sharp result. We observe that for $z=re^{i\frac{\pi }{t}}$,

$$\begin{aligned} \frac{f ( z ) }{\mathcal{M}_{\mathcal{S}_{t}} ( f ) }=1+\frac{L-M}{\mathcal{I}_{t+1}}z^{t+2}\rightarrow 1- \frac{L-M}{\mathcal{I}_{t+1}}r^{t+2} ( r\rightarrow 1 ) . \end{aligned}$$

Similarly, we obtain (3.9). □

Theorem 3.4

Let $f=\mathfrak{h}+\overline{\mathfrak{g}}$, where $\mathfrak{h}$ and $\mathfrak{g}$ are expressed by (1.3). If f meets (2.1), then

$$\begin{aligned} \operatorname{Re} \biggl( \frac{f ( z ) }{\mathcal{M}_{\mathcal{S}_{t,l}} ( f ) } \biggr) \geq \frac{\mathcal{I}_{l+1}- ( L-M ) }{\mathcal{I}_{l+1}} \end{aligned}$$

and

$$\begin{aligned} \operatorname{Re} \biggl( \frac{\mathcal{M}_{\mathcal{S}_{t,l}} ( f ) }{f ( z ) } \biggr) \geq \frac{\mathcal{I}_{l+1}}{\mathcal{I}_{l+1}+ ( L-M ) }, \end{aligned}$$

where $\mathcal{I}_{n}$ is given by (3.3). The equalities are obtained for the function stated in (3.11).

Proof

The proof is very similar to that of Theorem 3.3 and is therefore omitted. □

4 Further properties of the class $\mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$

Theorem 4.1

If $f\in \mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$, then for $\vert z \vert =r$,

$$\begin{aligned} \bigl\vert f ( z ) \bigr\vert \leq \frac{1}{r}+ \frac{L-M}{\sigma _{1}}r \end{aligned}$$

(4.1)

and

$$\begin{aligned} \bigl\vert f ( z ) \bigr\vert \geq \frac{1}{r}- \frac{L-M}{\sigma _{1}}r. \end{aligned}$$

(4.2)

Proof

Let $f=\mathfrak{h}+\overline{\mathfrak{g}}$ $\in \mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$ with $\mathfrak{h}$ and $\mathfrak{g}$ of the series form (1.3). Then by Theorem 2.4 we have

$$\begin{aligned} \bigl\vert f ( z ) \bigr\vert &\leq \frac{1}{ \vert z \vert }+ \sum _{n=1}^{\infty } \bigl( \vert a_{n} \vert + \vert b_{n} \vert \bigr) \bigl\vert z^{n} \bigr\vert \\ &\leq \frac{1}{ \vert z \vert }+\frac{1}{\sigma _{1}} \vert z \vert \sum _{n=1}^{\infty } \bigl( \rho _{n} \vert a_{n} \vert +\sigma _{n} \vert b_{n} \vert \bigr) \\ &\leq \frac{1}{ \vert z \vert }+\frac{L-M}{\sigma _{1}} \vert z \vert . \end{aligned}$$

This completes the proof of (4.1). By similar arguments we easily obtain (4.2). □

Theorem 4.2

A function $f\in \mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$ if and only if

$$\begin{aligned} f ( z ) =\sum_{n=1}^{\infty } ( \mathfrak{X}_{n} \mathfrak{h}_{n}+ \mathfrak{Y}_{n}\mathfrak{g}_{n} ) , \end{aligned}$$

(4.3)

where

$$\begin{aligned} &\mathfrak{h} ( z ) =\frac{1}{z}, \\ &\mathfrak{h}_{n} ( z ) =\frac{1}{z}-\frac{L-M}{\rho _{n}}z^{n} \quad\textit{for } n\in \mathbb{N}, \\ &\mathfrak{g}_{n} ( z ) =\frac{1}{z}- \frac{L-M}{\sigma _{n}}\overline{z}^{n} \quad \textit{for } n\in \mathbb{N}, \end{aligned}$$

and $\mathfrak{X}_{n}\mathfrak{,}$ $\mathfrak{Y}_{n}$ ≥0 for $n\in \mathbb{N}$ are such that

$$\begin{aligned} \sum_{n=1}^{\infty } ( \mathfrak{X}_{n}+ \mathfrak{Y}_{n} ) =1. \end{aligned}$$

(4.4)

In particular, the points $\{ \mathfrak{h}_{n} \}$, $\{ \mathfrak{g}_{n} \}$ are called the extreme points of the closed convex hull of the set $\mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$ denoted by $clco\mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$.

Proof

Let f be specified by (4.3). Then from (4.4) we get

$$\begin{aligned} f ( z ) =\frac{1}{z}-\sum_{n=1}^{\infty } \biggl( \frac{L-M}{\rho _{n}}\mathfrak{X}_{n}z^{n}+ \frac{L-M}{\sigma _{n}}\mathfrak{Y}_{n} \overline{z}^{n} \biggr) , \end{aligned}$$

which by Theorem 2.4 indicates that $f\in \mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$, since

$$\begin{aligned} \sum_{n=1}^{\infty } \biggl( \frac{\rho _{n}}{L-M} \frac{L-M}{\rho _{n}}\mathfrak{X}_{n}+\frac{\sigma _{n}}{L-M} \frac{L-M}{\sigma _{n}} \mathfrak{Y}_{n} \biggr) =\sum _{n=1}^{\infty } ( \mathfrak{X}_{n}+ \mathfrak{Y}_{n} ) =1. \end{aligned}$$

Thus $f\in clco\mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$. For the converse part, let $f=\mathfrak{h}+\overline{\mathfrak{g}}$ $\in \mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$. Put

$$\begin{aligned} \mathfrak{X}_{n}=\frac{\rho _{n}}{L-M} \vert a_{n} \vert ,\qquad \mathfrak{Y}_{n}=\frac{\sigma _{n}}{L-M} \vert b_{n} \vert . \end{aligned}$$

Then utilizing (4.4) together with the hypothesis, we have

$$\begin{aligned} f ( z ) &=\frac{1}{z}+\sum_{n=1}^{\infty }a_{n}z^{n}+ \sum_{n=1}^{\infty }\overline{b_{n}z^{n}} \\ &=\frac{1}{z}-\sum_{n=1}^{\infty } \vert a_{n} \vert z^{n}- \sum_{n=1}^{\infty } \vert b_{n} \vert \overline{z}^{n} \\ &=\frac{1}{z}-\sum_{n=1}^{\infty } \mathfrak{X}_{n} \frac{L-M}{\rho _{n}}z^{n}-\sum _{n=1}^{\infty }\mathfrak{Y}_{n} \frac{L-M}{\sigma _{n}} \overline{z}^{n} \\ &=\frac{1}{z}-\sum_{n=1}^{\infty } \mathfrak{X}_{n} \biggl\{ \frac{1}{z}-\mathfrak{h}_{n} \biggr\} -\sum_{n=1}^{\infty } \mathfrak{Y}_{n} \biggl\{ \frac{1}{z}-\mathfrak{g}_{n} \biggr\} \\ &= \Biggl( 1-\sum_{n=1}^{\infty } ( \mathfrak{X}_{n}+ \mathfrak{Y}_{n} ) \Biggr) \frac{1}{z}+\sum_{n=1}^{\infty } \bigl\{ \mathfrak{X}_{n}\mathfrak{h}_{n} ( z ) + \mathfrak{Y}_{n}\mathfrak{g}_{n} ( z ) \bigr\} \\ &=\sum_{n=1}^{\infty } \bigl\{ \mathfrak{X}_{n}\mathfrak{h}_{n} ( z ) +\mathfrak{Y}_{n} \mathfrak{g}_{n} ( z ) \bigr\} , \end{aligned}$$

which is the needed form (4.3). Thus the proof of Theorem 4.2 is completed. □

Theorem 4.3

Let $f_{1},f_{2}$ $\in \mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$. Then $f_{1}\ast f_{2}\in \mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$.

Proof

Let

$$\begin{aligned} f_{1} ( z ) =\frac{1}{z}-\sum_{n=1}^{\infty } \vert a_{n} \vert z^{n}- \vert b_{n} \vert \overline{z}^{n} \end{aligned}$$

and

$$\begin{aligned} f_{2} ( z ) =\frac{1}{z}-\sum_{n=1}^{\infty } \vert A_{n} \vert z^{n}- \vert B_{n} \vert \overline{z}^{n}. \end{aligned}$$

Then

$$\begin{aligned} ( f_{1}\ast f_{2} ) ( z ) =\frac{1}{z}+ \sum_{n=1}^{\infty } \vert A_{n} \vert \vert a_{n} \vert z^{n}- \vert B_{n} \vert \vert b_{n} \vert \overline{z}^{n}. \end{aligned}$$

Now if $f_{2}$ $\in \mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$, then by Theorem 2.4 we have $\vert A_{n} \vert \leq 1$ and $\vert B_{n} \vert \leq 1$. Thus

$$\begin{aligned} \frac{1}{L-M}\sum_{n=1}^{\infty } \bigl( \rho _{n} \vert A_{n} \vert \vert a_{n} \vert +\sigma _{n} \vert B_{n} \vert \vert b_{n} \vert \bigr) \leq \frac{1}{L-M}\sum _{n=1}^{\infty } \bigl( \rho _{n} \vert a_{n} \vert + \sigma _{n} \vert b_{n} \vert \bigr) \leq 1. \end{aligned}$$

By Theorem 2.4 this gives that $f_{1}\ast f_{2}\in \mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$. □

Theorem 4.4

The family $\mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$ is closed by a convex combination.

Proof

For $k\in \mathbb{N}$, let $f_{k}\in \mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$ be represented by

$$\begin{aligned} f_{k} ( z ) =\frac{1}{z}-\sum_{n=1}^{\infty } \vert a_{n,k} \vert z^{n}-\sum _{n=1}^{\infty } \vert b_{n,k} \vert \overline{z}^{n}. \end{aligned}$$

Then by (2.1) we have

$$\begin{aligned} \sum_{n=1}^{\infty } \biggl\{ \frac{\rho _{n} \vert a_{n,k} \vert +\sigma _{n} \vert b_{n,k} \vert }{L-M} \biggr\} \leq 1. \end{aligned}$$

For $\sum_{k=1}^{\infty }\xi _{k}=1$, $0\leq \xi _{k}<1$, the convex combination of $f_{k}$ is

$$\begin{aligned} \sum_{k=1}^{\infty }\xi _{k}f_{k} ( z ) =\frac{1}{z}-\sum_{n=1}^{\infty } \Biggl( \sum_{k=1}^{\infty }\xi _{k} \vert a_{n,k} \vert \Biggr) z^{n}-\sum _{n=1}^{\infty } \Biggl( \sum_{k=1}^{\infty } \xi _{k} \vert b_{n,k} \vert \Biggr) \overline{z}^{n}. \end{aligned}$$

Then by Theorem 2.4 we can write

$$\begin{aligned} \sum_{n=1}^{\infty } \Biggl( \sum _{k=1}^{\infty }\rho _{n}\xi _{k} \vert a_{n,k} \vert +\sum_{k=1}^{\infty } \sigma _{n}\xi _{k} \vert b_{n,k} \vert \Biggr) &\leq \sum_{k=1}^{\infty } \xi _{k} \Biggl\{ \sum_{n=1}^{\infty }\rho _{n} \vert a_{n,k} \vert +\sum_{n=1}^{\infty } \sigma _{n} \vert b_{n,k} \vert \Biggr\} \\ &\leq ( L-M ) \sum_{k=1}^{\infty }\xi _{k}=L-M, \end{aligned}$$

and so $\sum_{k=1}^{\infty }\xi _{k}f_{k} ( z ) \in \mathcal{MS}_{\mathcal{H}_{\vartheta }}^{\ast } ( q,L,M )$. □

5 Conclusion

Utilizing the principles of quantum calculus, we have added some new subfamilies of meromorphic harmonic mappings linked to a circular domain. We learned also certain important problems for the newly specified function families, namely necessary and sufficient conditions, problems for partial sums, distortion limits, convexity conditions, and convolution preserving. For these families, other problems, such as topological properties, fundamental mean inequality, and their implications are open problems for the scholars to investigate.

As pointed out in the survey-cum-expository review paper by Srivastava [44, p. 340], any attempt to produce the so-called $(p,q)$-variation of the q-results, which we have presented in this paper, will be trivial and inconsequential because the additional parameter p is obviously redundant or superfluous.