Abstract
Recently, there is a rapid increase of research in the area of Quantum calculus (known as -calculus) due to its widespread applications in many areas of study, such as geometric functions theory. To this end, using the concept of -conic domains of Janowski type as well as - calculus, new subclasses of analytic functions are introduced. This family of functions extends the notion of -convex and quasi-convex functions. Furthermore, a coefficient inequality, sufficiency criteria, and covering results for these novel classes are derived. Besides, some remarkable consequences of our investigation are highlighted.
1. Introduction
Recently, there is a rapid increase in the area of Quantum calculus (known as q-calculus) due to its widespread applications in many areas of study such as geometry functions theory (GFT), combinatorial analysis, Lie theory, mechanical engineering, cosmology, and statistics. The concept of -integral was first introduced and studied by Jackson et al. [6] at the beginning of the twentieth century.
The development of the concept of -calculus in GFT had its history from the work of Ismail et al. [5], where the notion of -starlike functions was extensively studied. As such, many subclasses of univalent functions correlated with -calculus have been on increase (see [1, 14, 17, 20, 23–25, 27, 28]). In recent times, various family of - extension of starlike functions, which are connected to Janowski functions in the open unit disc , were initiated and examined from many different viewpoints and perspective (see [17, 20, 28]).
In an attempt to generalize the notion of uniformly closed-to-convex functions considered by Goodman [3], Kanas and Wisniowska [8–10, 13] and Kanas and Srivastava [12] introduced the conic domain and studied the classes and of -uniformly convex and starlike functions. Furthermore, Noor and Malik [22], using the concept of Janowski class, extended the domain to . In the latest article by Mahmood et al. [17], the importance of -calculus was used to improve the Noor–Malik conic domains to . Using this domain, they examined the coefficient inequalities associated with the class of -uniformly starlike functions. Afterward, the same coefficient problems were also explored for the classes of -uniformly -convex, close-to-convex and quasi-convex functions by Naeem et al. [20].
Motivated by these recent articles [15, 17, 20, 28], our aim is to introduce the novel classes and consisting of -uniformly -alpha convex and quasi-convex functions. We study the coefficient inequalities associated with these classes and some other related properties. Some relevant consequences of our results which were studied in previous work show the significance of our investigation.
2. Materials and Methods
Now, we give some useful preliminaries which are necessary for our study.
Let be the class of normalized analytic functions in with
Let , and be the subclasses of consisting functions that are univalent, convex, starlike, quasi-convex, and close-to-convex functions, respectively. The function of form (1) is subordinate to the analytic function (written as ) of the formif there exists a Schwarz function in such that
Let . Then, class (see [7]) of function satisfies the subordination conditionor equivalently,where (class of functions with positive real part). For , and , the class reduces to the class , the class of functions whose real part is greater than .
The conic domains of Janowski type introduced by Noor and Malik [22] are defined as follows:
Geometrical interpretation of and its effect on were also demonstrated in [22]. The class represents the class of all functions that maps onto . Equivalently, a function belongs to if and only ifwhere has its definition in [10, 11] and given bywhere and is chosen such that , is Legendre’s complete elliptic integral of the first kind, and is the complementary integral of
Definition 1 (see [2]). Let . Then, the -number is given asand the -derivative of a complex valued function in is given byFrom the above explanation, it is easy to see that, for given by (1),
Definition 2 (see [28]). An analytic function in belongs to if and only if the conditionis satisfied, where and .
Definition 3 (see [17]). An analytic function in belongs to if and only if the subordination conditionis satisfied, where is given by (8). Equivalently, if and only if conformally maps onto the domain defined bywhereEquivalently,We note that(i), where is given by(ii)As , the class becomes the class and [22].(iii)When and , the class reduces to the class and [10].
Definition 4 (see [20]). Let of form (2) be in and . Then, if and only ifwhere is given by (8).
Inspired by the above recent mentioned work, we announce the following novel classes of analytic functions.
Definition 5. Let . Then, if and only ifEquivalently, if and only if
Definition 6. Let . Then, if and only if there exists an analytic function such thatEquivalently, if and only ifWe note the following special cases:(i)When , the classes and reduce to the classes [21] and [18], respectively.(ii)When and , the classes and cut down to the classes [22] and [16].(iii)When and , the classes and scale down to the classes [22] and [16].(iv)When and , the classes and diminish, respectively, to those classes of functions considered in [17, 20].(v)When in Definition 2, the class becomes the class -starlike functions of Janowski type recently explored by Srivastava et al. [28].To effectively establish our findings, the following set of lemmas is required.
3. A Set of lemmas
Lemma 1 (see [4]). Let and given by (8) be of the form . Then,where and is chosen such that , where is Legendre’s complete elliptic integral of the first kind.
Lemma 2 (see [17]). If , then, for any real number ,when or , the equality holds if and only if or one of its rotations. If , then the equality holds if and only if or one of its rotations. If , then the equality holds if and only ifor one of its rotations. If , then the equality holds if and only if is the reciprocal of one of the functions such that equality holds for the case . Although the above upper bound is sharp, when it can be improved as follows:
Lemma 3 (see [20]). Let be of form (2) and . Then,where is defined by (23).
4. Results and Discussion
We now turn our attention to the main results of this article.
4.1. Sufficient Conditions
Theorem 1. A function of form (1) belongs to the class if it satisfies the conditionwhere
Proof. Suppose condition (29) holds. Then, we need to prove thatTherefore,We haveSimilarly,From inequality (32) and equations (33) and (34), we arrive atThe last inequality is bounded by 1 if (29) is satisfied. This completes the proof.
As in Theorem 1, we are led to Theorem 1 in [21].
Corollary 1. A function of type (1) is in if it satisfies the conditionwhere
Setting in Theorem 1, we obtain a variance version of the result presented in [17].
Corollary 2. If having representation (1) satisfies the conditionthen .
If we choose and allow in the above theorem, our investigation comes down to Theorem 1 in [22].
Corollary 3. A function demonstrated by (1) is in the class if it satisfies the condition
If we choose and allow in the above theorem, our investigation comes down to Theorem 1 in [22].
Corollary 4 (see [11]). If satisfies the conditionthen .
Theorem 2. Let . Then, if inequality (42) is satisfied,where is given by (23).
Proof. Assuming (42) holds, then it suffices to establish thatFollowing the same process in the proof of Theorem 1, we havewhere we have used Lemma 3. Thus, the last inequality is bounded by 1 if (42) is satisfied. Hence, we complete the proof.
As in Theorem 2, we obtain the similar result proved in [18].
Corollary 5. Let . Then, ifholds.
For in Theorem 2, we get the result established by Naeem et al. [20].
Corollary 6. Let . Then, if inequality (46) is satisfied,
For and as in Theorem 2, we obtain the following results established in [16].
Corollary 7. Let . Then, ifis satisfied.
Corollary 8. Let . Then, ifis satisfied.
4.2. Fekete Szegö Inequality
Theorem 3. Let . Then, for any real number , we havewhereand are given by (23) and (24). This result cannot be improved. Then, implies there exists a Schwarz function such that
Using the representation for in Lemma 3 and the relationship between and , we can write
Thus,
However,
On comparing the coefficients of and of (53) and (54), we obtain
Now, for a real number , we havewhere
Hence, the result follows from Lemma 2.
If and , then Theorem 3 reduces to Theorem 10 in [17].
Corollary 9. Let and be of form (1). Then, for any real number , we havewherewhere
The result is sharp.
Proof. The proof is straightforward from Theorem 3 and Lemma 1.
If in Theorem 3, the case is contained in the following corollary.
Corollary 10. Let , and of form (1) belongs to . Then, for a real number , we have the following sharp inequality:wherewhere
The result is sharp.
Theorem 3 becomes Theorem 1 in [28] when .
Corollary 11. Let of the series representation (1) be in . Then, we have the sharp inequality:where
4.3. Covering Theorems
Theorem 4. The range of every univalent functions contains the disc:
Proof. From the proof of Theorem 3, we can see thatSince the Koebe one-quater theorem asserted that each omitted value of the univalent function of form (1) satisfiesThus, we have the result.
Corollary 12. The range of every univalent function contains the disc:
Theorem 5. The range of every univalent function contains the same disc given by (66).
Proof. Let be a Schwarz function. We note first in Theorem 3 thatwhereSince , then for some , we havewhich in turn impliesIt is easy to see thatTherefore, comparing the coefficients of of (73) and applying (74), we obtainsuch thatNow, proceeding the same way as in the proof of Theorem 4, we have the required result.
5. Conclusion
Using the concept of -calculus, we have introduced some new subclasses of analytic functions in the unit disc related to Janowski class of functions. In addition, sufficient conditions, Fekete–Szegö inequality as well as covering results for functions belonging to these new classes were established. Consequently, many remarkable special cases of our findings which were studied in the previous work were obtained [19, 26].
Data Availability
No data were used to support the findings of the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank the Rector of COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research and academic environment. Maslina Darus was supported by the Grant GUP-2019-032.