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1.   Introduction and definitions

The theory of the basic and the fractional quantum calculus, that is, the basic (or $q$-) calculus and the fractional basic (or $q$-) calculus, play important roles in many diverse areas of the mathematical, physical and engineering sciences (see, for example, ^{[10,15,33,45]}). Our main objective in this paper is to introduce and study some subclasses of the class of the normalized $p$-valently analytic functions in the open unit disk:

by applying the $q$-derivative operator in conjunction with the principle of subordination between analytic functions (see, for details, ^[8,30]).

We begin by denoting by $\mathcal{A}\left(p\right)$ the class of functions $f\left(z\right)$ of the form:

which are analytic and $p$-valent in the open unit disk $\mathbb{U}$. In particular, we write $\mathcal{A}\left(1\right) = :\mathcal{A}$.

A function $f\left(z\right) \in \mathcal{A}\left(p\right)$ is said to be in the class $\mathcal{S}_{p}^{\ast}\left(\alpha \right)$ of $p$-valently starlike functions of order $\alpha$ in $\mathbb{U}$ if and only if

Moreover, a function $f\left(z\right) \in \mathcal{A}\left(p\right)$ is said to be in the class $\mathcal{C}_{p}\left(\alpha \right)$ of $p$-valently convex functions of order $\alpha$ in $\mathbb{U}$ if and only if

The $p$-valent function classes $\mathcal{S}_{p}^{\ast}\left(\alpha \right)$ and $\mathcal{C}_{p}\left(\alpha \right)$ were studied by Owa ^[32], Aouf ^[2,3] and Aouf et al. ^[4,5]. From (1.2) and (1.3), it follows that

Let $\mathcal{P}$ denote the Carathéodory class of functions $\mathfrak p\left(z\right)$, analytic in $\mathbb{U}$, which are normalized by

such that $\Re\big(p\left(z\right)\big) > 0$.

Recently, Kanas and Wiśniowska ^[18,19] (see also ^[17,31]) introduced the conic domain $\Omega _{k}\left(k\geqq 0\right)$, which we recall here as follows:

or, equivalently,

By using the conic domain $\Omega_{k}$, Kanas and Wiśniowska ^[18,19] also introduced and studied the class $k$-$\mathcal{UCV}$ of $k$-uniformly convex functions in $\mathbb{U}$ as well as the corresponding class $k$-$\mathcal{ST}$ of $k$-starlike functions in $\mathbb{U}$. For fixed $k$, $\Omega _{k}$ represents the conic region bounded successively by the imaginary axis when $k = 0$. For $k = 1$, the domain $\Omega_{k}$ represents a parabola. For $1 < k < \infty$, the domain $\Omega_{k}$ represents the right branch of a hyperbola. And, for $k > 1$, the domain $\Omega_{k}$ represents an ellipse. For these conic regions, the following function plays the role of the extremal function:

with

where $\kappa$ is so chosen that

Here ${\text K}(\kappa)$ is Legendre's complete elliptic integral of the first kind and

that is, ${\text K}^{\prime }\left(\kappa\right)$ is the complementary integral of ${\text K}\left(\kappa\right)$ (see, for example, [48,p. 326,Eq 9.4 (209)]).

We now recall the definitions and concept details of the basic (or $q$-) calculus, which are used in this paper (see, for details, ^[13,14,45]; see also ^{[1,6,7,11,34,38,39,42,54,59]}). Throughout the paper, unless otherwise mentioned, we suppose that $0 < q < 1$ and

Definition 1. The $q$-number $\left[\lambda\right]_{q}$ is defined by

so that

.

Definition 2. For functions given by (1.1), the $q$-derivative (or the $q$-difference) operator $D_{q}$ of a function $f$ is defined by

provided that $f^{\prime}\left(0\right)$ exists.

We note from Definition 2 that

for a function $f$ which is differentiable in a given subset of $\mathbb{C}$. It is readily deduced from (1.1) and (1.8) that

We remark in passing that, in the above-cited recently-published survey-cum-expository review article, 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 or superfluous (see, for details, [42,p. 340]).

Making use of the $q$-derivative operator $D_{q}$ given by (1.6), we introduce the subclass $\mathcal{S}_{q, p}^{\ast}\left(\alpha \right)$ of $p$-valently $q$-starlike functions of order $\alpha$ in $\mathbb{U}$ and the subclass $\mathcal{C}_{q, p}\left(\alpha\right)$ of $p$-valently $q$-convex functions of order $\alpha$ in $\mathbb{U}$ as follows (see ^[54]):

and

respectively. From (1.10) and (1.11), it follows that

For the simpler classes $\mathcal{S}_{q, p}^{\ast}$ and $\mathcal{C}_{q, p}^{\ast}$ of $p$-valently $q$-starlike functions in $\mathbb{U}$ and $p$-valently $q$-convex functions in $\mathbb{U}$, respectively, we have write

Obviously, in the limit when $q\rightarrow 1{-}$, the function classes $\mathcal{S}_{q, p}^{\ast}\left(\alpha\right)$ and $\mathcal{C}_{q, p}\left(\alpha \right)$ reduce to the familiar function classes $\mathcal{S}_{p}^{\ast}\left(\alpha \right)$ and $\mathcal{C}_{p}\left(\alpha \right)$, respectively.

Definition 3. A function $f\in \mathcal{A}\left(p\right)$ is said to belong to the class $\mathcal{S}_{q, p}^{\ast}$ of $p$-valently $q$-starlike functions in $\mathbb{U}$ if

In the limit when $q\rightarrow 1{-}$, the closed disk

becomes the right-half plane and the class $\mathcal{S}_{q, p}^{\ast}$ of $p$-valently $q$-starlike functions in $\mathbb{U}$ reduces to the familiar class $\mathcal{S}_{p}^{\ast}$ of $p$-valently starlike functions with respect to the origin ($z = 0$). Equivalently, by using the principle of subordination between analytic functions, we can rewrite the condition (1.13) as follows (see ^[58]):

We note that $\mathcal{S}_{q, 1}^{\ast} = \mathcal{S}_{q}^{\ast}$ (see ^[12,41]).

Definition 4. (see ^[50]) A function $\mathfrak p\left(z\right)$ given by (1.5) is said to be in the class $k$-$\mathcal{P}_{q}$ if and only if

where $p_{k}\left(z\right)$ is given by (1.6).

Geometrically, the function $p\in k$-$\mathcal{P}_{q}$ takes on all values from the domain $\Omega_{k, q}$ ($k\geqq 0$) which is defined as follows:

The domain $\Omega_{k, q}$ represents a generalized conic region which was introduced and studied earlier by Srivastava et al. (see, for example, ^[43,50]). It reduces, in the limit when $q\rightarrow 1{-}$, to the conic domain $\Omega_{k}$ studied by Kanas and Wiśniowska ^[18]. We note the following simpler cases.

(1) $k$-$\mathcal{P}_{q}\subseteq \mathcal{P}\left(\frac{2k}{2k+1+q}\right)$, where $\mathcal{P}\left(\frac{2k}{2k+1+q}\right)$ is the familiar class of functions with real part greater than $\frac{2k}{2k+1+q}$;

(2) $\lim_{q\rightarrow 1{-}}\{k$-$\mathcal{P}_{q}\} = \mathcal{P}\left(p_{k}\left(z\right) \right)$, where $\mathcal{P}\big(p_{k}(z)\big)$ is the known class introduced by Kanas and Wiśniowska ^[18];

(3) $\lim_{q\rightarrow 1{-}}\{0$-$\mathcal{P}_{q}\} = \mathcal{P}$, where $\mathcal{P}$ is Carathéodory class of analytic functions with positive real part.

Definition 5. A function $f\in \mathcal{A}\left(p\right)$ is said to be in the class $k$-$\mathcal{ST}_{q, p}$ if and only if

or, equivalently,

The folowing special cases are worth mentioning here.

(1) $k$-$\mathcal{ST}_{q, 1} = k$-$\mathcal{ST}_{q}$, where $k$-$\mathcal{ST}_{q}$ is the function class introduced and studied by Srivastava et al. ^[50] and Zhang et al. ^[60] with $\gamma = 1$;

(2) $0$-$\mathcal{ST}_{q, p} = \mathcal{S}_{q, p}$;

(3) $\lim_{q\rightarrow 1}\{k$-$\mathcal{ST}_{q, p}\} = k$-$\mathcal{ST}_{p}$, where $k$-$\mathcal{ST}_{p}$ is the class of $p$-valently uniformly starlike functions;

(4) $\lim_{q\rightarrow 1}\{0$-$\mathcal{ST}_{q, p}\} = \mathcal{S}_{p}$, where $\mathcal{S}_{p}^{\ast }$ is the class of $p$-valently starlike functions;

(5) $0$-$\mathcal{ST}_{q, 1} = \mathcal{S}_{q}^{\ast }$, where $\mathcal{S}_{q}^{\ast}$ (see ^[12,41]);

(6) $\lim_{q\rightarrow 1}\{k$-$\mathcal{ST}_{q, 1}\} = k$-$\mathcal{ST}$, where $k$-$\mathcal{ST}$ is a function class introduced and studied by Kanas and Wiśniowska ^[19];

(7) $\lim_{q\rightarrow 1}\{0$-$\mathcal{ST}_{q, 1}\} = \mathcal{S}^{\ast }$, where $\mathcal{S}^{\ast}$ is the familiar class of starlike functions in $\mathbb{U}$.

Definition 6. By using the idea of Alexander's theorem ^[9], the class $k$-$\mathcal{UCV}_{q, p}$ can be defined in the following way:

In this paper, we investigate a number of useful properties including coefficient estimates and the Fekete-Szegö inequalities for the function classes $k$-$\mathcal{ST}_{q, p}$ and $k$-$\mathcal{UCV}_{q, p}$, which are introduced above. Various corollaries and consequences of most of our results are connected with earlier works related to the field of investigation here.

2.   A set of Lemmas

In order to establish our main results, we need the following lemmas.

Lemma 1. (see ^[16]) Let $0\leqq k < \infty$ be fixed and let $p_{k}$ be defined by $(1.6)$. If

then

and

where

and $t\in \left(0, 1\right)$ is so chosen that

${\text K}\left(t\right)$ being Legendre's complete elliptic integral of the first kind.

Lemma 2. Let $0\leqq k < \infty$ be fixed and suppose that

where $p_{k}\left(z\right)$ be defined by $(1.6)$. Then

where $Q_{1}$ and $Q_{2}$ are given by $(2.1)$ and $(2.2),$ respectively.

Proof. By using (1.6) in (2.3), we can easily derive (2.4).

Lemma 3. (see ^[26]) Let the function $h$ given by

be analytic in $\mathbb{U}$ and satisfy $\Re\big(h(z)\big) > 0$ for $z$ in $\mathbb{U}$. Then the following sharp estimate holds true$:$

The result is sharp for the functions given by

Lemma 4. (see ^[26]) If the function $h$ is given by

then

When $\nu > 1,$ the equality holds true if and only if

or one of its rotations. If $0 < \nu < 1,$ then the equality holds true if and only if

or one of its rotations. If $\nu = 0,$ the equality holds true if and only if

or one of its rotations. If $\nu = 1,$ the equality holds true if and only if the function $h$ is the reciprocal of one of the functions such that equality holds true in the case when $\nu = 0$.

The above upper bound is sharp and it can be improved as follows when $0 < \nu < 1$$:$

and

3.   Main results

We assume throughout our discussion that, unless otherwise stated, $0\leqq k < \infty$, $0 < q < 1$, $p\in \mathbb{N}$, $Q_{1}$ is given by (2.1), $Q_{2}$ is given by (2.2) and $z\in \mathbb{U}$.

Theorem 1. If a function $f\in \mathcal{A}\left(p\right)$ is of the form $(1.1)$ and satisfies the following condition$:$

then the function $f\in k$-$\mathcal{ST}_{q, p}$.

Proof. Suppose that the inequality (3.1) holds true. Then it suffices to show that

In fact, we have

The last expression is bounded by $1$ if (3.1) holds true. This completes the proof of Theorem 1.

Corollary 1. If $f\left(z\right) \in k$-$\mathcal{ST}_{q, p},$ then

The result is sharp for the function $f(z)$ given by

Remark 1. Putting $p = 1$ Theorem 1, we obtain the following result which corrects a result of Srivastava et al. [50,Theorem 3.1].

Corollary 2. (see Srivastava et al. [50,Theorem 3.1]) If a function $f\in \mathcal{A}$ is of the form $(1.1)$ $($with $p = 1)$ and satisfies the following condition$:$

then the function $f\in k$-$\mathcal{ST}_{q}$.

Letting $q\rightarrow 1{-}$ in Theorem 1, we obtain the following known result [29,Theorem 1] with

Corollary 3. If a function $f\in \mathcal{A}\left(p\right)$ is of the form $(1.1)$ and satisfies the following condition$:$

then the function $f\in k$-$\mathcal{ST}_{p}$.

Remark 2. Putting $p = 1$ in Corollary 3, we obtain the result obtained by Kanas and Wiśniowska [19,Theorem 2.3].

By using Theorem 1 and (1.18), we obtain the following result.

Theorem 2. If a function $f\in \mathcal{A}\left(p\right)$ is of the form $(1.1)$ and satisfies the following condition$:$

then the function $f\in k$-$\mathcal{UCV}_{q, p}$.

Remark 3. Putting $p = 1$ Theorem 1, we obtain the following result which corrects the result of Srivastava et al. [50,Theorem 3.3].

Corollary 4. (see Srivastava et al. [50,Theorem 3.3]) If a function $f\in \mathcal{A}$ is of the form (1.1) (with $p = 1$) and satisfies the following condition$:$

then the function $f\in k$-$\mathcal{UCV}_{q}$.

Letting $q\rightarrow 1{-}$ in Theorem 2, we obtain the following corollary (see [29,Theorem 1]) with

Corollary 5. If a function $f\in \mathcal{A}\left(p\right)$ is of the form $(1.1)$ and satisfies the following condition$:$

then the function $f\in k$-$\mathcal{UCV}_{p}$.

Remark 4. Putting $p = 1$ in Corollary 5, we obtain the following corollary which corrects the result of Kanas and Wiśniowska [18,Theorem 3.3].

Corollary 6. If a function $f\in \mathcal{A}$ is of the form $(1.1)$ $($with $p = 1)$ and satisfies the following condition$:$

then the function $f\in k$-$\mathcal{UCV}$.

Theorem 3. If $f\in k$-$\mathcal{ST}_{q, p},$ then

and$,$ for all $n = 3, 4, 5, \cdots,$

Proof. Suppose that

where

Eq (3.4) can be written as follows:

which implies that

Comparing the coefficients of $z^{n+p-1}$ on both sides of (3.5), we obtain

By taking the moduli on both sides and then applying the following coefficient estimates (see ^[50]):

we find that

We now apply the principle of mathematical induction on (3.6). Indeed, for $n = 2$, we have

which shows that the result is true for $n = 2$. Next, for $n = 3$ in (3.7), we get

By using (3.7), we obtain

which is true for $n = 3$. Let us assume that (3.3) is true for $n = t\; \; (t\in\mathbb{N})$, that is,

Let us consider

Therefore, the result is true for $n = t+1$. Consequently, by the principle of mathematical induction, we have proved that the result holds true for all $n\; \; \left(n\in \mathbb{N}\setminus\{1\}\right)$. This completes the proof of Theorem 3.

Similarly, we can prove the following result.

Theorem 4. If $f\in k$-$\mathcal{UCV}_{q, p}$ and is of form $(1.1),$ then

and$,$ for all $n = 3, 4, 5, \cdots,$

Putting $p = 1$ in Theorems 3 and 4, we obtain the following corollaries.

Corollary 7. If $f\in k$-$\mathcal{ST}_{q},$ then

and$,$ for all $n = 3, 4, 5, \cdots,$

Corollary 8. If $f\in k$-$\mathcal{UCV}_{q},$ then

and$,$ for all $n = 3, 4, 5, \cdots,$

Theorem 5. Let $f\in k$-$\mathcal{ST}_{q, p}$. Then $f\left(\mathbb{U}\right)$ contains an open disk of the radius given by

Proof. Let $w_{0}\neq 0$ be a complex number such that $f\left(z\right) \neq w_{0}$ for $z\in \mathbb{U}$. Then

Since $f_{1}$ is univalent, so

Now, using Theorem 3, we have

Hence

This completes the proof of Theorem 5.

Theorem 6. Let the function $f\in k$-$\mathcal{ST}_{q, p}$ be of the form $(1.1)$. Then$,$ for a complex number $\mu,$

The result is sharp.

Proof. If $f\in k$-$\mathcal{ST}_{q, p}$, we have

From the definition of the differential subordination, we know that

where $w\left(z\right)$ is a Schwarz function with $w\left(0\right) = 0$ and $\left\vert w\left(z\right) \right\vert < 1$ for $z\in\mathbb{U}$.

Let $h\in \mathcal{P}$ be a function defined by

This gives

and

Using (3.10) in (3.9), we obtain

and

Now, for any complex number $\mu$, we have

Then (3.11) can be written as follows:

where

Finally, by taking the moduli on both sides and using Lemma 4, we obtain the required result. The sharpness of (3.8) follows from the sharpness of (2.5). Our demonstration of Theorem 6 is thus completed.

Similarly, we can prove the following theorem.

Theorem 7. Let the function $f\in k$-$\mathcal{UCV}_{q, p}$ be of the form $(1.1)$. Then$,$ for a complex number $\mu,$

The result is sharp.

Putting $p = 1$ in Theorems 6 and 7, we obtain the following corollaries.

Corollary 9. Let the function $f\in k$-$\mathcal{ST}_{q}$ be of the form $(1.1)$ $($with $p = 1)$. Then$,$ for a complex number $\mu,$

The result is sharp.

Corollary 10. Let the function $f\in k$-$\mathcal{UCV}_{q}$ be of the form $(1.1)$ $($with $p = 1)$. Then$,$ for a complex number $\mu,$

The result is sharp.

Theorem 8. Let

and

If the function $f$ given by $(1.1)$ belongs to the class $k$-$\mathcal{ST}_{q, p},$ then

Furthermore$,$ if $\sigma _{1}\leqq \mu \leqq \sigma _{3},$ then

If $\sigma _{3}\leqq \mu \leqq \sigma _{2},$ then

Proof. Applying Lemma 4 to (3.12) and (3.13), respectively, we can derive the results asserted by Theorem 8.

Putting $p = 1$ in Theorem 8, we obtain the following result.

Corollary 11. Let

and

If the function $f$ given by $(1.1)$ $($with $p = 1)$ belongs to the class $k$-$\mathcal{ST}_{q},$ then

Furthermore$,$ if $\sigma _{4}\leqq \mu \leqq \sigma _{6},$ then

If $\sigma _{3}\leqq \mu \leqq \sigma _{2},$ then

Similarly, we can prove the following result.

Theorem 9. Let

and

If the function $f$ given by $(1.1)$ belongs to the class $k$-$\mathcal{UCV}_{q, p},$ then

Furthermore$,$ if $\eta _{1}\leqq \mu \leqq \eta _{3},$ then

If $\eta _{3}\leqq \mu \leqq \eta _{2},$ then

Putting $p = 1$ in Theorem 9, we obtain the following result.

Corollary 12. Let

and

If the function $f$ given by $(1.1)$ $($with $p = 1)$ belongs to the class $k$-$\mathcal{UCV}_{q},$ then

Furthermore$,$ if $\eta _{4}\leqq \mu \leqq \eta _{6},$ then

If $\eta _{3}\leqq \mu \leqq \eta _{2},$ then

4.   Concluding remarks and observations

In our present investigation, we have applied the concept of the basic (or $q$-) calculus and a generalized conic domain, which was introduced and studied earlier by Srivastava et al. (see, for example, ^[43,50]). By using this concept, we have defined two subclasses of normalized multivalent functions which map the open unit disk:

onto this generalized conic domain. We have derived a number of useful properties including (for example) the coefficient estimates and the Fekete-Szegö inequalities for each of these multivalent function classes. Our results are connected with those in several earlier works which are related to this field of Geometric Function Theory of Complex Analysis.

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, [48,pp. 350-351]. Moreover, as we remarked in the introductory Section 1 above, in the recently-published survey-cum-expository review article by Srivastava ^[42], 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 or superfluous (see, for details, [42,p. 340]). This observation by Srivastava ^[42] will indeed apply to any attempt to produce the rather straightforward $(p, q)$-variations of the results which we have presented in this paper.

In conclusion, with a view mainly to encouraging and motivating further researches on applications of the basic (or $q$-) analysis and the basic (or $q$-) calculus in Geometric Function Theory of Complex Analysis along the lines of our present investigation, we choose to cite a number of recently-published works (see, for details, ^{[25,47,51,53,56]} on the Fekete-Szegö problem; see also ^{[20,21,22,23,24,27,28,35,36,37,40,44,46,49,52,55,57]} dealing with various aspects of the usages of the $q$-derivative operator and some other operators in Geometric Function Theory of Complex Analysis). Indeed, as it is expected, each of these publications contains references to many earlier works which would offer further incentive and motivation for considering some of these worthwhile lines of future researches.

Conflicts of interest:

The authors declare no conflicts of interest.