Real positive solutions of operator equations $AX = C$ and $XB = D$

1.   Introduction, definitions and motivation

Let $\mathcal{A}$ denote the class of functions that are analytic in the open unit disc $U = \left \{ z\in \mathbb{C} :\left \vert z\right \vert < 1\right \}$ and let $\mathcal{A}^{0}$ be the subclass of $\mathcal{A}$ consisting of functions $h$ with the normalization $h(0) = h^{^{\prime }}(0)-1 = 0$ and has the following series expansion,

A continuous function $f = u+iv$ is a complex valued harmonic function defined in $U$, where $u$ and $v$ are real harmonic functions in $U$. We can write $f(z) = h+\overline{g}$ where $h$ and $g$ are analytic in $U$ (see ^[2]). We call $h$ the analytic part and $g$ the co-analytic part of $f$, where $h\in \mathcal{A}^{0}$ is given by (1.1) and $g\in \mathcal{A}$ has the following power series expansion (see, for details, ^[4,5,10]):

A necessary and sufficient condition for $f$ to be locally univalent and sense preserving in $U$ is that $\left \vert h^{^{\prime }}(z)\right \vert > \left \vert g^{^{\prime }}(z)\right \vert$ in $U.$

We denote by $\mathcal{S}^{\ast }\mathcal{H}$ the class of functions $f = h+ \overline{g}$, that are harmonic univalent and sense preserving in $U$ and satisfies the normalization conditions. For $f = h+\overline{g}\in \mathcal{S} ^{\ast }\mathcal{H}$, where $h(z)$ and $g(z)$ are given in (1.1) and (1.2). Note that $\mathcal{S}^{\ast }\mathcal{H}$ reduces to $\mathcal{S} ^{\ast }$, the class of normalized analytic univalent functions if the co-analytic part of $f \; = h+\overline{g}$ is identically zero. Also, $\mathcal{SH}$ is the subclass of $\mathcal{S}^{\ast }\mathcal{H}$ consisting of functions $f$ that map $U = \{z:\left \vert z\right \vert < 1\}$ onto starlike domain (see ^[27]).

The fractional $q$-calculus is the $q$-extension of the ordinary fractional calculus and dates back to early 20th century. The theory of $q$-calculus operators are used in various areas of science such as ordinary fractional calculus, optimal control, $q$-difference and $q$-integral equations, and also in the Geometric Function Theory of complex analysis. Initially in 1908, Jackson ^[7] defined the $q$-analogue of derivative and integral operator as well as provided some of their applications. Further in ^[6] Ismail et al. gave the idea of\ $q$-extension of\ class of $q$-starlike functions after that Srivastava ^[28] studied $q$-calculus in the context of univalent functions theory. Kanas and Raducanu ^[14] introduced the $q$-analogue of Ruscheweyh differential operator and Srivastava in ^[30] studied $q$-starlike functions related with generalized conic domain$.$ By using the concept of convolution Srivastava et al. ^[31] introduced $q$-analogue of Noor integral operator and studied some of its applications, also Srivastava et al. also published a set of articles in which they concentrated class of $q$-starlike functions from different aspects (see ^{[32,33,35,36]}). Additionally, a recently published survey-cum-expository review article by Srivastava ^[29] is potentially useful for researchers and scholars working on these topics. For some more recent investigation about $q$-calculus we may refer to ^{[20,22,23,37,38,39]}.

The theory of symmetric $q$-calculus has been applied to many areas of mathematics and physics such as fractional calculus and quantum physics. The symmetric $q$-calculus has proven to be valuable in a few areas, specially in quantum mechanics ^[3,24]. Recently in ^[13] Kanas et al. defined a symmetric operator of $q$-derivative and introduced a new family of univalent functions. Furthermore Shahid et al. ^[21] used the concepts of symmetric $q$-calculus operator theory and defined symmetric conic domains. But research on symmetric $q$-calculus in connection with Geometric Function Theory and especially harmonic univalent functions is fairly new and not much is published on this topic.

Jahangiri in ^[9] applied certain $q$-calculus operators to complex harmonic functions, while Porwal and Gupta discussed applications of $q$ -calculus to harmonic univalent functions in ^[25]. Srivastava ^[34] defined the $q$-analogue of derivative operator as well as provided some of its applications to complex harmonic functions. In this article, first time we apply symmetric $q$-calculus theory in order to define symmetric Salagean $q$-differential operator to complex harmonic functions and introduce a new class of harmonic univalent functions.

For better understanding of the article, we recall some concept details and definitions of the symmetric $q$-difference calculus. We suppose throughout in this paper that $0 < q < 1$ and that

Definition 1.1. For $n\in \mathbb{N},$ the symmetric $q$-number is defined by:

We note that the symmetric $q$-number do not reduce to the $q$-number, and frequently occurs in the study of $q$-deformed quantum mechanical simple harmonic oscillator (see ^[1]).

Definition 1.2. For any $n\in \mathbb{Z} ^{+}\cup \left \{ 0\right \},$ the symmetric $q$-number shift factorial is defined by:

Note That

Definition 1.3. The symmetric $q$-derivative ($q$-difference) operator of symmetric $q$ -calculus operated on the function $h \; ($^[12]) is defined by

and

We can observe that

A successive application of the symmetric $q$-derivative ($q$-difference) operator of symmetric $q$-calculus as defined in (1.3) leads to symmetric Salagean $q$-differential operator which is define as:

Definition 1.4. The symmetric Salagean $q$-differential operator of $h$ is defined by

where $m$ is a positive integer and the operator $\ast$ stands for the Hadamard product or convolution of two analytic power series. The operator $\widetilde{D}_{q}^{m}h(z)$ is called symmetric Salagean $q$-differential operator.

Note that

which is the famous Salagean operator defined in ^[26].

It is the aim of this article to define the symmetric $q$-derivative ($q$ -difference) operator by using symmetric $q$-calculus on the complex functions that are harmonic in $U$ and obtain sharp coefficient bounds, distortion theorems and covering results.

Definition 1.5. We define the symmetric Salagean $q$-differential operator for harmonic function $f = h+\overline{g}$ as follows:

where

Remark 1.6. It is easy to see that, for $q\rightarrow 1-,$ we obtain symmetric Salagean differential operator for harmonic functions $f = h+\overline{g}.$

Definition 1.7. For $0\leq \alpha < 1,$ let $\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right)$ denote the family of harmonic functions $f = h+\overline{g}$ which satisfy the condition

where$\widetilde{\text{ }D}_{q}^{m}f(z)$ is given by (1.4). Further, denote by $\overline{\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) }$ the subclass of $\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right)$ consisting of harmonic functions $f = h+\overline{g}$ so that $h$ and $g$ are of the form

where $a_{n}, b_{n}\geq 0.$

2.   Main results

In the following theorem we shall determine coefficient bounds for harmonic functions belonging to the classes $\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right)$ and $\overline{\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) }.$

Theorem 2.1. For $0\leq \alpha < 1$ and $f = h+\overline{g}$, let

where $h$ and $g$ are respectively given by (1.1) and (1.2). Then

(i) $f$ is harmonic univalent in $U$ and $f\in \widetilde{\mathcal{H}} _{q}^{m}\left(\alpha \right)$ if the inequality (2.1) holds.

(ii) $f$ is harmonic univalent in $U$ and $f\in \overline{\widetilde{ \mathcal{H}}_{q}^{m}\left(\alpha \right) }$ if and only if the inequality (2.1) holds.

The equality in (2.1) occurs for harmonic function

where

Proof. For part (i): First we need to show that $f = h+\overline{g}$ is locally univalent and orientation-preserving in $U$. It suffices to show that the second complex dilatation $w$ of $f$ satisfies $\left \vert w\right \vert = \left \vert g^{^{\prime }}/h^{^{\prime }}\right \vert < 1$ in $U.$ This is the case since for $z = re^{i\theta }\in U$. We have

in $U$ which implies as $q\rightarrow 1^{-}$ that $\left \vert h^{^{\prime }}(z)\right \vert > \left \vert g^{^{\prime }}(z)\right \vert$ in $U$ that is the function $f$ is locally univalent and sense-preserving in $U.$ To show that $f = h+\overline{g}$ is univalent in $U$ we use an argument that is due to author ^[8]. Suppose $z_{1}$ and $z_{2}$ are in $U$ so that $z_{1}\neq z_{2}.$ Since $U$ is simply connected and convex, we have $z(t) = \left(1-t\right) z_{1}+tz_{2}\in U$ for $0\leq t\leq 1.$ Then for $z_{1}-z_{2}\neq 0,$ we can write

On the other hand, we observe that

Therefore, $f = h+\overline{g}$ is univalent in $U$. It remains to show that the inequality (1.5) holds if the coefficients of the univalent harmonic function $f = h+\overline{g}$ satisfy the condition (2.1). In other words, for $0\leq \alpha < 1,$ we need to show that

Using the fact that $\Re(w)\geq \alpha$ if and only if $\left \vert 1-\alpha +w\right \vert \geq \left \vert 1+\alpha -w\right \vert,$ it suffices to show that

Substituting for

and

In the left hand side of the inequality (2.2), we obtain

This last expression is non-negative by (2.1), and this completes the proof.

For part (ii): Since $\overline{\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) }$ is subset of $\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right)$, we only need to prove the "only if" part of the theorem. Let $f\in \overline{\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) }.$ Then by the required condition (1.5), we have

This must hold for all values of $z$ in $U$. So, upon choosing the values of $z$ on the positive real axis where $0\leq z = r < 1$, we have

If the condition (2.1) does not hold, then the numerator in (2.3) is negative for $r$ sufficiently close to $1$. Hence there exists $z_{0} = r_{0}$ in $(0, 1)$ for which the left hand side of the inequality (2.3) is negative. This contradicts the required condition that $f\in \overline{\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) }$ and so the proof is complete.

Example 2.2. The function $f = h+\overline{g}$ given by

where

belonging to the class $\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right),$ for $\delta > -2, \; 0\leq \alpha < 1, \; q\in \left(0, 1\right), \; \epsilon _{n}\in \mathbb{C}, \; \left \vert \epsilon _{n}\right \vert = 1.$ Because, we know that

For $q\rightarrow 1-,$ then Theorem 2.1 reduces to following known results ([11,Theorems 1 and 2]).

Corollary 2.3. For $0\leq \alpha < 1$ and $f = h+\overline{g}$, let

where $h$ and $g$ are, respectively, given by (1.1) and (1.2). Then

(i) $f$ is harmonic univalent in $U$ and $f\in \mathcal{H}^{m}\left(\alpha \right)$ if the inequality (2.4) holds.

(ii) $f$ is harmonic univalent in $U$ and $f\in \overline{\mathcal{H} ^{m}\left(\alpha \right) }$ if and only if the inequality (2.4) holds.

The equality in (2.4) occurs for harmonic functions

where

The closed convex hull of $\overline{\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) }$ denoted by $clco\overline{\widetilde{\mathcal{H}} _{q}^{m}\left(\alpha \right) },$ is the smallest closed set containing $\overline{\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) },$ it is the intersection of all closed convex sets containg $\overline{\widetilde{ \mathcal{H}}_{q}^{m}\left(\alpha \right) },$ In the next theorem we determine the extreme points of the closed convex hull of $\overline{ \widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right).}$

Theorem 2.4. If the functions $f = h+\overline{g}\in clco\overline{\widetilde{ \mathcal{H}}_{q}^{m}\left(\alpha \right) }$ if and only if

In particular, the extreme points of $\overline{\widetilde{\mathcal{H}} _{q}^{m}\left(\alpha \right) }$ are $h_{n}$ and $g_{n}.$

Proof. For the functions of the form (2.5), we have

This yields

and so $f = h+\overline{g}\in clco\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right).$ Conversely, let $f = h+\overline{g}\in clco\widetilde{\mathcal{H}} _{q}^{m}\left(\alpha \right).$ Then by setting

where

We obtain the functions of the form (2.5) as required.

Remark 2.5. For $q\rightarrow 1-,$ then Theorem 2.4 reduces to the known results proved in ^[11].

Finally, we give the following distortion bounds which yields a covering result for $\overline{\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) }.$

Theorem 2.6. If $f = h+\overline{g}\in \overline{\widetilde{\mathcal{H}} _{q}^{m}\left(\alpha \right) },$ then for $\left \vert z\right \vert = r < 1,$ we have the distortion bounds

The bounds given in (2.6) and (2.7) for the function $f(z) = h+ \overline{g}$ of the form (1.6) also hold for functions of the form (1.1), (1.2), if the coefficient condition (2.1) is satisfied. Let the

and

for

show that the bounds given in Theorem 2.6 are sharp.

Proof. We will only prove the inequality (2.7) of the Theorem 2.6. The arguments for the inequality $\left(2.6\right)$ is similar and so we omit it. Let $f\in \overline{\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) }.$ Taking the absolute value of $f(z),$ we obtain

The following covering result follows the inequality (2.7) in Theorem (2.6).

Theorem 2.7. If $f\in \overline{\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) },$ then for $\left \vert z\right \vert = r < 1,$ we have

Proof. Using the inequality (2.6) of Theorem 2.6 and letting $r\rightarrow 1$, it follows that

3.   Conclusions

Research on symmetric $q$-calculus in connection with Geometric Function Theory and especially harmonic univalent functions is fairly new and not much is published on this topic. In this paper we have made use of the symmetric quantum (or $q$-) calculus to defined and investigated new classes of harmonic univalent functions by using newly defined symmetric Salagean $q$-differential operator for complex harmonic functions and obtained sharp coefficient bounds, distortion theorems and covering results. Furthermore, we also highlighted some known consequence of our main results.

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, [28,p.351–352] and [29,p.328]). Moreover, in this recently-published survey-cum-expository review article by Srivastava ^[29], 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, [29,p.340], see also ^{[15,16,17,18,19]}). This observation by Srivastava ^[29] 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.

Conflict of interest

The authors declare no conflict of interest.