A generalized conic domain and its applications to certain subclasses of multivalent functions associated with the basic (or $q$-) calculus

In this paper, by using the concept of the basic (or q-) calculus and a generalized conic domain, we define two subclasses of normalized multivalent functions which map the open unit disk: U = {z : z ∈ C and |z| < 1} onto this generalized conic domain. We investigate 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.


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: U = {z : z ∈ C and |z| < 1} 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 A (p) the class of functions f (z) of the form: f (z) = z p + ∞ n=p+1 a n z n (p ∈ N := {1, 2, 3, · · · }) , The p-valent function classes S * p (α) and C p (α) were studied by Owa [32], Aouf [2,3] and Aouf et al. [4,5]. From (1.2) and (1.3), it follows that f (z) ∈ C p (α) ⇐⇒ z f (z) p ∈ S * p (α) . (1.4) Let P denote the Carathéodory class of functions p (z), analytic in U, which are normalized by such that p (z) > 0.
By using the conic domain Ω k , Kanas and Wiśniowska [18,19] also introduced and studied the class k-UCV of k-uniformly convex functions in U as well as the corresponding class k-ST of k-starlike functions in U. For fixed k, Ω k represents the conic region bounded successively by the imaginary axis when k = 0. For k = 1, the domain Ω k represents a parabola. For 1 < k < ∞, the domain Ω k represents the right branch of a hyperbola. And, for k > 1, the domain Ω k represents an ellipse. For these conic regions, the following function plays the role of the extremal function: where κ is so chosen that Here K(κ) is Legendre's complete elliptic integral of the first kind and that is, K (κ) is the complementary integral of K (κ) (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 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 (0) exists.
We note from Definition 2 that for a function f which is differentiable in a given subset of C. It is readily deduced from (1.1) and (1.8) that [n] q a n z n−1 . (1.9) 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 S * q,p (α) of pvalently q-starlike functions of order α in U and the subclass C q,p (α) of p-valently q-convex functions of order α in U as follows (see [54]): respectively. From (1.10) and (1.11), it follows that For the simpler classes S * q,p and C * q,p of p-valently q-starlike functions in U and p-valently q-convex functions in U, respectively, we have write S * q,p (0) =: S * q,p and C q,p (0) =: C q,p .
Obviously, in the limit when q → 1−, the function classes S * q,p (α) and C q,p (α) reduce to the familiar function classes S * p (α) and C p (α), respectively.

Definition 3.
A function f ∈ A (p) is said to belong to the class S * q,p of p-valently q-starlike functions in U if zD q f (z) (1. 13) In the limit when q → 1−, the closed disk becomes the right-half plane and the class S * q,p of p-valently q-starlike functions in U reduces to the familiar class S * p 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]): (1.14) We note that S * q,1 = S * q (see [12,41]). Definition 4. (see [50]) A function p (z) given by (1.5) is said to be in the class k-P q if and only if where p k (z) is given by (1.6).
Geometrically, the function p ∈ k-P q takes on all values from the domain Ω k,q (k 0) which is defined as follows: The domain Ω 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 → 1−, to the conic domain Ω k studied by Kanas and Wiśniowska [18]. We note the following simpler cases.

function class introduced and studied by Kanas and
Wiśniowska [19]; where S * is the familiar class of starlike functions in U. Definition 6. By using the idea of Alexander's theorem [9], the class k-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-ST q,p and k-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.

A set of Lemmas
In order to establish our main results, we need the following lemmas. Lemma 1. (see [16]) Let 0 k < ∞ be fixed and let p k be defined by (1.6). If and t ∈ (0, 1) is so chosen that K (t) being Legendre's complete elliptic integral of the first kind.
Lemma 2. Let 0 k < ∞ be fixed and suppose that where p k (z) be defined by (1.6). Then where Q 1 and Q 2 are given by (2.1) and (2.2), respectively.
Lemma 3. (see [26]) Let the function h given by c n z n ∈ P be analytic in U and satisfy h(z) > 0 for z in 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 or one of its rotations. If ν = 0, the equality holds true if and only if or one of its rotations. If ν = 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 ν = 0.
The above upper bound is sharp and it can be improved as follows when 0 < ν < 1:

Main results
We assume throughout our discussion that, unless otherwise stated, 0 k < ∞, 0 < q < 1, p ∈ N, Q 1 is given by (2.1), Q 2 is given by (2.2) and z ∈ U. Theorem 1. If a function f ∈ A (p) is of the form (1.1) and satisfies the following condition: then the function f ∈ k-ST q,p .
Proof. Suppose that the inequality (3.1) holds true. Then it suffices to show that In fact, we have [n] q − p q a n z n−p (q − 1) [n] q + 2 p q a n z n−p The last expression is bounded by 1 if (3.1) holds true. This completes the proof of Theorem 1.
The result is sharp for the function f (z) given by f (z) = z p + (1 + q) p q 2 (k + 1) [n] q − p q + q n + 2 p q − 1 z n (n p + 1) . then the function f ∈ k-ST q .

Corollary 3.
If a function f ∈ A (p) is of the form (1.1) and satisfies the following condition: then the function f ∈ k-ST p . If a function f ∈ A (p) is of the form (1.1) and satisfies the following condition: then the function f ∈ k-UCV q,p .  [n] q 2 (k + 1) [n] q − 1 + q n + 1 |a n | < (1 + q) , then the function f ∈ k-UCV q .
and, for all n = 3, 4, 5, · · · , Eq (3.4) can be written as follows: 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 ∈ 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 (n ∈ N \ {1}). This completes the proof of Theorem 3.
Similarly, we can prove the following result.
Proof. Let w 0 0 be a complex number such that f (z) w 0 for z ∈ 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 ∈ k-ST q,p be of the form (1.1). Then, for a complex number µ, a p+2 − µa 2 p+1 p q Q 1 2q p max 1, The result is sharp.
Proof. If f ∈ k-ST q,p , we have .
From the definition of the differential subordination, we know that where w (z) is a Schwarz function with w (0) = 0 and |w (z)| < 1 for z ∈ U.
Let h ∈ P be a function defined by This gives and p k,q (w (z)) = 1 + 1 + q 4 (3.10) Using (3.10) in (3.9), we obtain Now, for any complex number µ, 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 ∈ k-UCV q,p be of the form (1.1). Then, for a complex number µ, The result is sharp.
Putting p = 1 in Theorems 6 and 7, we obtain the following corollaries.

Corollary 9.
Let the function f ∈ k-ST q be of the form (1.1) (with p = 1). Then, for a complex number µ, The result is sharp.

Corollary 10.
Let the function f ∈ k-UCV q be of the form (1.1) (with p = 1). Then, for a complex number µ, The result is sharp.

Theorem 8.
Let If the function f given by (1.1) belongs to the class k-ST q,p , 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 If the function f given by (1.1) (with p = 1) belongs to the class k-ST q , then Furthermore, if σ 4 µ σ 6 , then If σ 3 µ σ 2 , then Similarly, we can prove the following result.
If the function f given by (1.1) belongs to the class k-UCV q,p , then If η 3 µ η 2 , then Putting p = 1 in Theorem 9, we obtain the following result.

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 [20-24, 27, 28, 35-37, 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.