Research Article | Open Access
Some Coefficient Inequalities of q-Starlike Functions Associated with Conic Domain Defined by q-Derivative
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.
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.  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  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 . Noor et al.  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. , Ramachandran et al. , Altinkaya et al. , Bulut , and Mahmood and Sokół  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  introduced the concept of conic domain by defining uniformly convex functions and then in 2000, they defined the corresponding starlike functions; see . 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  that from (8), one can have
This class was then generalized by Noor and Malik  and its generalization was introduced by using the concept of Janowski functions. The detail of Janowski functions may be seen from . 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.  and , the well-known class of Janowski k-starlike functions, introduced by Noor and Malik .
2. Preliminary Results
We need the following lemmas to prove our main results.
Lemma 3 (see ). Let be subordinate to If is univalent in and is convex, then
Lemma 4 (see ). 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.
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. .
Corollary 8. Let the function be of the form (5); thenwhere .
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 . 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