Some Coefficient Inequalities of q-Starlike Functions Associated with Conic Domain Defined by q-Derivative

Abstract

This article deals with q-starlike functions associated with conic domains, defined by Janowski functions. It generalizes the recent study of q-starlike functions while associating it with the conic domains. Certain renowned coefficient inequalities in connection with the previously known ones have been included in this work.

1. Introduction

Quantum calculus or q-calculus is none other than a version of classical calculus because in this, we do not take limits. We consider derivatives as differences whereas antiderivatives as sums. The q-derivative of a complex valued function , defined in the domain , is given as follows. where This implies the following. provided that the function is differentiable in domain The function has Maclaurin’s series representation where For more details about q-derivatives, we refer the reader to [1–14].

We denote by the class of functions which are analytic in the open unit disc and are of the formLet denote the class of all functions in which are univalent in . Also let and be the subclasses of consisting of all functions which map onto a star shaped with respect to origin and convex domains, respectively. A function is said to be subordinate to a function , written symbolically as , if there exists a function with , such that for .

The credit of systematic initiation of q-calculus goes to Jackson [13, 14] who introduced and gave the early definitions of q-derivatives and q-integrals. One of the very early contributions of usage of Q-calculus in Geometric Function Theory was made by Ismail et al. [12] who defined the generalized version of class of starlike functions. He named his newly introduced class as class of q-starlike functions since he used q-derivatives in defining it. It took a long while in further development in this direction but it proved to be a good comeback when Anastassiu and Gal [4, 5] presented their work on complex operators with their respective q-generalizations. These are known as q-Picard and q-Gauss-Weierstrass singular integral operators. Continuing this work, Srivastava [15] laid a strong foundation of application of Q-calculus in Geometric Function Theory by using basic q-hypergeometric functions. Another series of contributions was made by Aral and Gupta [6–8] who used q-beta functions to define the q-Baskakov Durrmeyer operator. They also derived a number of geometric results with their q-extensions. In the similar manner, many q-calculus operators including integral and derivative in fractional form have been used to define and analyze a number of subclasses of analytic functions. Inspired by the research carried out in q-calculus, Aldweby and Darus [1, 2] defined the q-operators by using the concept of convolution of analytic functions which are normalized. Moreover, they discussed the geometrical structure of these defined operators in the analytic functions which involve q-version of hypergeometric functions in compact disc. Several useful results related to the q-version of class of close to convex functions were proved by Sahoo and Sharma [16]. Noor et al. [17] gave the research a new direction from application point of view and derived integral inequalities for relative harmonic preinvex functions. Very recently, many researchers of Geometric Function Theory like Noor et al. [18], Ramachandran et al. [19], Altinkaya et al. [3], Bulut [9], and Mahmood and Sokół [20] have contributed to the development of results in the background of q-calculus. The work on q-polynomials and (p,q)-polynomials also contributed remarkably to the field of q-calculus; see [10, 11].

In 1999, Kanas and Wiśniowska [21] introduced the concept of conic domain by defining uniformly convex functions and then in 2000, they defined the corresponding starlike functions; see [22]. The class of starlike functions is defined as follows.

A function is said to be in the class if and only if or equivalently wherewhere , , , is chosen such that , is the Legendre’s complete elliptic integral of the first kind, and is complementary integral of ; for more detail, see [21–25]. If , then it is shown in [26] that from (8), one can have

This class was then generalized by Noor and Malik [27] and its generalization was introduced by using the concept of Janowski functions. The detail of Janowski functions may be seen from [28]. The class is defined as follows.

A function is said to be in the class , , if and only if or equivalently Motivated by the above classes, we now define the following more general class of q-starlike functions associated with conic domain defined by Janowski functions.

Definition 1. A function is said to be in the class , if and only ifwhere is defined by (8), , and

Geometrically, the function takes all values from the domain , which is defined as

Definition 2. A function is said to be in the class , , , if and only ifor equivalentlyIt is noted that , the well-known class of q-starlike functions, introduced by Srivastava et al. [29] and , the well-known class of Janowski k-starlike functions, introduced by Noor and Malik [27].

2. Preliminary Results

We need the following lemmas to prove our main results.

Lemma 3 (see [30]). Let be subordinate to If is univalent in and is convex, then

Lemma 4 (see [31]). If is a function with positive real part in , then, for any real number , When or , the equality holds if and only if is or one of its rotations. If , then, the equality holds if and only if or one of its rotations. If , the equality holds if and only if or one of its rotations. If , then, the equality holds if and only if is reciprocal of one of the functions such that equality holds in the case of . Although the above upper bound is sharp, when , it can be improved as follows: and

3. Main Results

Theorem 5. A function and of the form (5) is in the class , if it satisfies the conditionwhere and

Proof. Assuming that (21) holds, then it suffices to show thatand we considerThe last expression is bounded above by 1 ifwhich reduces to and this completes the proof.

For , Theorem 5 reduces to the following result, proved by Noor and Malik [27].

Corollary 6. A function and of the form (5) is in the class , if it satisfies the conditionwhere .

Theorem 7. Let the function be of the form (5); thenThis result is sharp.

Proof. By definition, for , we havewhereIf , thenNow if , then by (16) and (29), we getNow from (28), we have which implies that This implies that Comparison of coefficients of gives us which reduces to By using (31), we getNext, we need to show thatFor that, we use the principle of mathematical induction. For , we find from (37) that which results also from (27). Now for , we find from (37) that which also follows from (27). Now let inequality (38) be true for . We find from (37) thatOn the other hand, from (27), we have By the induction hypothesis, we haveMultiplying both sides of (43) by , we have That is, which shows that inequality (38) is true for , and hence the required result.

For , The above result reduces to the following result, proved by Srivastava et al. [29].

Corollary 8. Let the function be of the form (5); thenwhere .

For , Theorem 7 reduces to the following result, proved by Noor and Malik [27].

Corollary 9. Let the function be of the form (5); thenwhere .

Theorem 10. Let , and be of the form (5). Then for real number , we havewhereand

Proof. For and of the form , we consider where is such that and It follows easily that This implies thatIf , then it follows from relation (15) that we have That is, there exists a function with and such thatwhere Now if , then from (53), one may have where ,, and are given by and ; see [26]. Using these, the above series reduces toUsing this, (55) becomesIf and , thenFrom (59) and (60), comparison of coefficients of and givesandNow for a real number , we consider where Applying Lemma 4, we get the required result. Inequality (48) is sharp and equality holds for or when is or one of its rotations, where is defined such that If , then, the equality holds for the function or one of its rotations, where is defined such that If , the equality holds for the function or one of its rotations, where is defined such that where If , then, the equality holds for , which is such that is reciprocal of one of the function such that equality holds in the case of , where and are defined by (49) and (50), respectively.

For , the above result reduces to the following result, proved by Srivastava et al. [29].

Corollary 11. Let , and be of the form (5). Then for real number , we havewhere and Inequality (68) is sharp.

For , Theorem 10 reduces to the following result.

Corollary 12. Let , and be of the form (5). Then for real number , we havewhere and Inequality (71) result is sharp.

Theorem 13. Let , and be of the form (5). Then for real number , we havewhere and Inequality (74) is sharp.

Proof. The use of function instead of in the proof of Theorem 10 and following the similar method as followed in Theorem 10 will lead us to the required result.

For , the theorem reduces to the following form.

Corollary 14. Let , and be of the form (5). Then, for a real number ,This result is sharp.

Now we consider the inverse function , defined as , , and we find the following coefficient bound for inverse functions.

Theorem 15. Let , , and Then,

Proof. Since , it is easy to see that By using (61) and (62), one can have andFrom (59) and (60), comparison of givesNow, from (81) and (82), one can have and where and Application of the bounds and (see Lemma 4 for and ) gives Lastly, (83) reduces towhere andApplying the bounds , see [32], and , see [31], to the right hand side of (88) and using the fact that , we have and this completes the proof.

Theorem 16. Let , , and Then for real number , we have where and

Proof. The proof follows directly from (81), (82), and Lemma 4.

Following the similar method as proceeded in Theorems 15 and 16, we have the following results for

Theorem 17. Let , , and Then,

Theorem 18. Let , , and Then for real number , we have where and

4. Conclusion

We have generalized the concept of q-starlike functions by associating them with conic domain. The concept of q-derivatives is used to define class of certain analytic functions. The certain renowned coefficient inequalities are found for these functions and their inverse functions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Authors’ Contributions

All authors contributed equally to and approved the final manuscript.

Acknowledgments

This research is partially supported by Sarhad University of Science and IT, Ring Road, Peshawar, Pakistan. The authors are grateful to their institutions’ heads for the conducive research environment.