Monotonicity, convexity and inequalities involving zero-balanced Gaussian hypergeometric function

1.   Introduction and preliminaries

The hypothesis of fractional calculus, which is primarily due to its proven developments in many branches of science and engineering over the past three decades or so, has gained considerable prominence and recognition in recent times. Most of the theory of fractional calculus is specifically based upon the familiar Riemann-Liouville fractional derivative (or integral). The fractional $q$-calculus is the extension of the ordinary fractional calculus in the $q$-theory. Recently, there was notable increase of articles written in the area of the $q$-calculus due to significant applications of the $q$-calculus in mathematics, statistics and physics. For more details, interested readers may refer to the books of ^{[1,2,4,5,6,8,9,14]} on the subject.

Indeed the fourth author of this article with Raina in ^[11] have used the fractional $q$-calculus operators in investigating certain classes of functions which are analytic in the open disk $\mathbb{D}$. Recently, many authors have introduced new classes of analytic functions using $q$-calculus operators. For some recent investigations on the classes of analytic functions defined by using $q$-calculus operators and related topics, we refer the reader to ^{[12,13,17,18]} and the references cited therein. In the present paper, we aim at introducing a new class of Bazilevič functions involving the fractional $q$-calculus operators, which is analytic in the open unit disk. Certain interesting subordination results are also derived for the functions belonging to this class.

We first give various definitions and notations in $q$-calculus which are useful to understand the subject of this paper.

For any complex number $\alpha$, the $q$-shifted factorials are defined as

and in terms of the basic analogue of the gamma function

where the $q$-gamma function is defined by

If $|q| < 1$, the definition (1.1) remains meaningful for $n = \infty$ as a convergent infinite product

In view of the relation

we observe that the $q$-shifted factorial (1.1) reduces to the familiar Pochhammer symbol $(\alpha)_n$, where $(\alpha)_n = \alpha(\alpha+1)\cdots(\alpha+n-1).$

Also, the $q$-derivative and $q$-integral of a function on a subset of $\mathbb{C}$ are, respectively, given by (see ^[6] pp.19-22)

and

Therefore, the $q$-derivative of $f(z) = z^n$, where $n$ is a positive integer is given by

where

and is called the $q$-analogue of $n$. As $q \rightarrow1$, we have $[n]_q = q^{n-1}+\cdots +1 \rightarrow 1+ \cdots +1 = n.$ The $q$-derivative of $f(z) = \ln z$ is given by

We now define the fractional $q$-calculus operators of a complex-valued function $f(z)$, which were recently studied by Purohit and Raina ^[11].

Definition 1.1. (Fractional $q$-integral operator) The fractional $q$-integral operator $I_{q, z}^\delta$ of a function $f (z)$ of order $\delta$ is defined by

where $f(z)$ is analytic in a simply connected region of the $z$-plane containing the origin and the $q$-binomial function $(z-tq)_{\delta-1}$ is given by

The series $_{1}\Phi_0[\delta; -;q, z]$ is single valued when $|\arg(z)| < \pi$ and $|z| < 1$ (see for details ^[6], pp.104-106), therefore, the function $(z-tq)_{\delta-1}$ in (1.4) is single valued when $|\arg(-tq^\delta/z)| < \pi$, $|tq^\delta/z| < 1$ and $|\arg(z)| < \pi$.

Definition 1.2. (Fractional $q$-derivative operator) The fractional $q$-derivative operator $D_{q, z}^\delta$ of a function $f(z)$ of order $\delta$ is defined by

where $f(z)$ is suitably constrained and the multiplicity of $(z-tq)_{-\delta}$ is removed as in Definition 1.1.

Definition 1.3. (Extended fractional $q$-derivative operator) Under the hypotheses of Definition 1.2, the fractional $q$-derivative for a function $f (z)$ of order $\delta$ is defined by

where $m-1\leq\delta < 1, \, \, m\in\mathbb{N}_0 = \mathbb{N}\cup\{0\},$ and $\mathbb{N}$ denotes the set of natural numbers.

Remark 1.1. It can be easily seen from Definition 1.2 that

2.   The class $\mathcal{B}_{q, k}^{m}(\lambda, \delta, \gamma; \phi)$

Let $\mathcal{H}(a, n)$ denote the class of functions $f(z)$ of the form

which are analytic in the unit disk $\mathbb{D} = \{z\in\mathbb{C}: \, |z| < 1\}$. In particular, let $\mathcal{A}$ be the subclass of $\mathcal{H}(0, 1)$ containing functions of the form

We denote by $\mathcal{S}, \, \mathcal{S}^*, \, \mathcal{C}$ and $\mathcal{K}$, the classes of all functions in $\mathcal{A}$ which are, respectively, univalent, starlike, convex and close-to-convex in $\mathbb{D}$. Let $f(z)$ and $g(z)$ be analytic in $\mathbb{D}$. A function $f$ is subordinate to $g$ in $\mathbb{D}$, if there exists an analytic function $w(z)$ in $\mathbb{D}$ with

such that

We denote this subordination by $f(z)\prec g(z).$ Further, if the function $g(z)$ is univalent in $\mathbb{D}$, then $f(z)\prec g(z)\quad (z\in \mathbb{D})\Longleftrightarrow f(0) = g(0)$ and $f(\mathbb{D})\subset g(\mathbb{D})$.

Let $k$ be a positive integer and let $\varepsilon_k = \exp(\frac{2\pi i}{k})$. For $f\in \mathcal{A}$, let

The function $f$ is said to be starlike with respect to $k$-symmetric points if it satisfies

We denote by $S_s^{(k)}$ the subclass of $\mathcal{A}$ consisting of all functions starlike with respect to $k$-symmetric points in $\mathbb{D}$. The class $S_s^{(2)}$ was introduced and studied by Sakaguchi ^[15]. We also note that different subclasses of $S_s^{(k)}$ can be obtained by replacing condition (2.4) by

where $h(z)$ is a given convex function, with $h(0) = 1$ and $\Re\{ h(z)\} > 0$.

Using $D^{\delta}_{q, z}$, we define a fractional $q$-differintegral operator $\Omega^\delta_{q, z}:\mathcal{A}\longrightarrow\mathcal{A}$, as follows:

where $D^{\delta}_{q, z}\, f(z)$ in (2.5) represents, respectively, a fractional $q$-integral of $f(z)$ of order $\delta$ when $-\infty < \delta < 0$ and a fractional $q$-derivative of $f(z)$ of order $\delta$ when $0\leq\delta < 2$. Here we note that $\Omega^0_{q, z}f(z) = f(z).$

We now define a linear multiplier fractional $q$-differintegral operator $\mathscr{D}^{\delta, m}_{q, \lambda}$ as follows:

If $f(z)$ is given by (2.2), then by (2.6), we have

It can be seen that, by specializing the parameters the operator $\mathscr{D}_{q, \lambda}^{\delta, m}~~$ reduces to many known and new integral and differential operators. In particular, when $\delta = 0\,$ the operator $\mathscr{D}_{q, \lambda}^{\delta, m}~~$ reduces to the operator introduced by Al-Oboudi ^[3] and for $\delta = 0, \, \lambda = 1$ it reduces to the operator introduced by Sălăgean ^[16].

Throughout this paper, we assume that

Clearly, for $k = 1$, we have

Let $\mathcal{P}$ denote the class of analytic functions $h(z)$ with $h(0) = 1$, which are convex and univalent in $\mathbb{D}$ and for which $\Re\{ h(z)\} > 0, \quad (z \in \mathbb{D}).$

In the present article, a new subclass $\mathcal{B}_{q, k}^{m}(\lambda, \delta, \gamma; \phi)$ of analytic functions, using the linear multiplier $q$-fractional differintegral operator $\mathscr{D}_{q, \lambda}^{\delta, m}~~$ defined by (2.6) is introduced. It is interesting to note that the class $\mathcal{B}_{q, k}^{m}(\lambda, \delta, \gamma; \phi)$ is a generalization of the class of Bazilevič functions.

Definition 2.1. A function $f\in \mathcal{A}$ is said to be in the class $\mathcal{B}_{q, k}^{m}(\lambda, \delta, \gamma; \phi)$ if and only if

where $\gamma\in [0, \, \infty), \, \phi\in\mathcal{P}$, $f_{q, k}^{m}~~ (\lambda, \delta; z)$ and $g_{q, k}^{m}~~ (\lambda, \delta; z)\neq0$ for all $z\in\mathbb{D}\setminus\{0\}$ are defined as in (2.7).

It is easy to see to verify the following relations: If $q\rightarrow 1^-$, then

$1.$$\mathcal{B}_{q, 1}^{0}(\lambda, \delta, 0;\frac{1+z}{1-z}) = \mathcal{S}^*$;

$2.$$\mathcal{B}_{q, k}^{0}(\lambda, \delta, 0;\frac{1+z}{1-z}) = \mathcal{S}^{(k)}_s$;

$3.$$\mathcal{B}_{q, 1}^{0}(\lambda, \delta, 1;\frac{1+z}{1-z}) = \mathcal{K}$;

$4.$$\mathcal{B}_{q, 1}^{0}(\lambda, \delta, \gamma; \frac{1+z}{1-z}) = \mathcal{B}(\gamma),$ the class of Bazilevič functions of type $\gamma$.

Now, we derive some sufficient conditions for functions belonging to the class $\mathcal{B}_{q, k}^{m}(\lambda, \delta, \gamma; \phi)$. In order to prove our results we need the following results.

Lemma 2.1. $($^[10], see also ^[7]$)$ Let $h$ be convex in $\mathbb{D}$, with $h(0) = a, \beta\neq 0 \, {and}\, \, \Re\, (\beta)\geq 0$. If $g \in \mathcal{H}(a, n)$ and

then

where

The function $\phi$ is convex and is the best (a, n)-dominant.

Lemma 2.2. $($^[10], see also ^[19]$)$ Let $h$ be starlike in $\mathbb{D}$, with $h(0) = 0.$ If $g \in \mathcal{H}(a, n)$ satisfies

then

The function $\phi$ is convex and is the best (a, n)-dominant.

3.   Subordination problems for the class $\mathcal{B}_{q, k}^{m}(\lambda, \delta, \gamma; \phi)$

In this section, we establish some sufficient conditions of subordination for analytic functions defined above involving the $q$-differential operator.

Theorem 3.1. Let $f, \, g\in\mathcal{A}$ with $f_{q, k}^{m}~~ (\lambda, \delta; z)$ and $g_{q, k}^{m}~~ (\lambda, \delta; z)\neq0$ for all $z\in\mathbb{D}\setminus\{0\}$ and let $h$ be convex in $\mathbb{D}$, with $h(0) = 1 \, {and}\, \, \Re\{ h(z)\} > 0$. Further suppose that $g\in\mathcal{S}^*$ and

Then

where

and the function $q(z)$ is the best dominant.

Proof. Let ${p(z) = \frac{z\, \big(\mathscr{D}_{q, \lambda}^{\delta, m}~~f(z)\big)'}{[f_{q, k}^{m}~~ (\lambda, \delta; z)]^{1-\gamma}~~[g_{q, k}^{m}~~ (\lambda, \delta; z)]^{\gamma}}\quad\quad (z\in\mathbb{D}; z\neq0).}$

Then $p(z)\neq 0$ in $\mathbb{D}.$ Further, $p(z)\in \mathcal{H}(1, 1)$. Differentiating (3.3) we obtain

If we let

then $p_1$ is analytic and $p_1(0) = 1$. From (3.4) we obtain

and

Since $h$ is convex, we have

which implies $\Re (p_1(z)) > 0$ (^[10], Theorem 3.2a) and therefore, $Q(z)$ is convex and univalent. We now set $P(z) = p^2(z).$ Then $P(z)\in \mathcal{H}(1, 1)$ with $P(z)\neq 0$ in $\mathbb{D}.$ By logarithmic differentiation we have,

Therefore, by (3.1) we have

Now, by Lemma 2.1 with $\beta = 1$, we deduce that

and $Q$ is the best dominant of (3.5). Since $\Re\{ h(z)\} > 0$ and $Q(z)\prec h(z)$ we also have $\Re\{ Q(z)\} > 0.$ Hence, the univalence of $Q$ implies the univalence of $q(z) = \sqrt{Q(z)},$ and

which implies that $p(z)\prec q(z)$. Since $Q$ is the best dominant of (3.5), we deduce that $q(z)$ is the best dominant of (3.2). This completes the proof.

Corollary 3.2. Let $f, \, g\in\mathcal{A}$ with $f_{q, k}^{m}~~ (\lambda, \delta; z)$ and $g_{q, k}^{m}~~ (\lambda, \delta; z)\neq0$ for all $z\in\mathbb{D}\setminus\{0\}$. If $g\in\mathcal{S}^*$ and $\Re\big(\Psi(z)\big) > \alpha, \quad (0\leq \alpha < 1),$ where

then ${\quad\quad\quad \Re\bigg\{\frac{z\, \big(\mathscr{D}_{q, \lambda}^{\delta, m}~~f(z)\big)'}{[f_{q, k}^{m}~~ (\lambda, \delta; z)]^{1-\gamma}~~[g_{q, k}^{m}~~ (\lambda, \delta; z)]^{\gamma}}\bigg\}>\mu(\alpha), }$

where $\mu(\alpha) = \big[2(1-\alpha){\ln2}+(2\alpha-1)\big]^{\frac{1}{2}}$, and this result is sharp.

Proof. Let $h(z) = \frac{1+(2\alpha-1)z }{1+z}\quad$ with $\quad 0\leq \alpha <1$. Then from Theorem 3.1, it follows that $Q(z)$ is convex and $\Re\{ Q(z)\}>0.$ Also we have,

This completes the proof the corollary.

By setting $m = 0$ in Corollary 3.2, we have the following corollary.

Corollary 3.3. Let $f, \, g\in\mathcal{A}$ with $f'(z), \, \, f_{k}(z)$ and $g_{k}(z)\neq0$ for all $z\in\mathbb{D}\setminus\{0\}$. If $g\in\mathcal{S}^*$ and

then

where $\mu(\alpha) = \big[2(1-\alpha)\, \ln2+(2\alpha-1)\big]^{\frac{1}{2}}$, and this result is sharp.

Also, taking $\gamma = 0$ in Corollary 3.3, we obtain

Corollary 3.4. Let $f\in \mathcal{A}$ with $f'(z)$ and $f_k(z)\neq 0$ for all $z\in \mathbb{D}\backslash\{0\}$. If

then

where $\mu(\alpha) = \big[2(1-\alpha)\, \ln2+(2\alpha-1)\big]^{\frac{1}{2}}$, and this result is sharp.

Further, setting $\gamma = 1$ in Corollary 3.3, we obtain the following corollary.

Corollary 3.5. Let $f, \, g\in \mathcal{A}$ with $f'(z)$ and $g_k(z)\neq 0$ for all $z\in \mathbb{D}\backslash\{0\}$. If $g\in\mathcal{S}^*$ and

then

where $\mu(\alpha) = \big[2(1-\alpha)\ln2+(2\alpha-1)\big]^{\frac{1}{2}}$, and this result is sharp.

Next, we prove the following result.

Theorem 3.6. Let $f, \, g\in\mathcal{A}$ with $f_{q, k}^{m}~~ (\lambda, \delta; z)$ and $g_{q, k}^{m}~~ (\lambda, \delta; z)\neq0$ for all $z\in\mathbb{D}\setminus\{0\}$. Further suppose that $h$ is starlike with $h(0) = 0$, in the unit disk $\mathbb{D}, \, \, g\in\mathcal{S}^*$ and

Then

where $\phi$ is convex and is the best dominant.

Proof. Let $p(z) = \frac{z\, \big(\mathscr{D}_{q, \lambda}^{\delta, m}~~f(z)\big)'}{[f_{q, k}^{m}~~ (\lambda, \delta; z)]^{1-\gamma}~~[g_{q, k}^{m}~~ (\lambda, \delta; z)]^{\gamma}}\quad\quad (z\in\mathbb{D}; z\neq0)$.

Since $p(z)\in \mathcal{H}(1, 1)$ with $p(z)\neq 0$ in $\mathbb{D}$, we can define the analytic function $P(z)\equiv \ln p(z)$. Clearly $P(z) \in \mathcal{H}(0, 1)$. Now by logarithmic differentiation we have,

Thus we have

Now by Lemma 2.2, we deduce that

This completes the proof of Theorem 3.6.

By putting $m = 0$ in Theorem 3.6, we arrive at the following corollary.

Corollary 3.7. Let $f, \, g\in\mathcal{A}$ with $f'(z), \, \, f_{k}(z)$ and $g_{k}(z)\neq0$ for all $z\in\mathbb{D}\setminus\{0\}$. Further suppose that $h$ is starlike with $h(0) = 0$, in the unit disk $\mathbb{D}, \, \, g\in\mathcal{S}^*$ and

Then

where $\phi$ is convex and is the best dominant.

Also, specializing $\gamma = 0$ and $\gamma = 1$ in Corollary 3.7, we obtain the following interesting two corollaries as given below.

Corollary 3.8. Let $f, \, g\in\mathcal{A}$ with $f'(z)$ and $f_{k}(z)\neq0$ for all $z\in\mathbb{D}\setminus\{0\}$. Further suppose that $h$ is starlike with $h(0) = 0$, in the unit disk $\mathbb{D}$ and

Then

where $\phi$ is convex and is the best dominant.

Corollary 3.9. Let $f, \, g\in\mathcal{A}$ with $f'(z)$ and $g_{k}(z)\neq0$ for all $z\in\mathbb{D}\setminus\{0\}$. Further suppose that $h$ is starlike with $h(0) = 0$, in the unit disk $\mathbb{D}, \, \, g\in\mathcal{S}^*$ and

Then

where $\phi$ is convex and is the best dominant.

4.   Conclusions

By suitably choosing the parameters, one can further easily obtain a number of inequalities from the main results stated in Theorem 3.1 and Theorem 3.6. Moreover, the class $\mathcal{B}_{q, k}^{m}(\lambda, \delta, \gamma; \phi)$ defined in this article can also be used in the investigation of various geometric properties like, the coefficient estimates, distortion bounds, radii of starlikeness, convexity and close to convexity etc. in the unit disk.

Conflict of interest

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the referee(s) for his/her suggestions that essentially corrected few of the conditions in the theorem. The research of corresponding author (Huo Tang) was partly supported by the Natural Science Foundation of the People's Republic of China under Grant 11561001, the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region under Grant NJYT-18-A14, the Natural Science Foundation of Inner Mongolia of the People's Republic of China under Grant 2018MS01026, the Natural Science Foundation of Chifeng of Inner Mongolia, the Program for Key Laboratory Construction of Chifeng University (No.CFXYZD202004) and The Research and Innovation Team of Complex Analysis and Nonlinear Dynamic Systems of Chifeng University (No.cfxykycxtd202005) and the Youth Science Foundation of Chifeng University (No.cfxyqn202133). The work of third-named author is supported by a grant from the Science and Engineering Research Board, Government of India under Mathematical Research Impact Centric Support of Department of Science and Technology (DST)(vide ref: MTR/2017/000607).