On q-Convex Functions Defined by the q-Ruscheweyh Derivative Operator in Conic Regions

Abstract

The core objective of this article is to introduce and investigate a new class of convex functions associated with the conic domain defined by the Ruscheweyh q-differential operator. Many interesting properties such as sufficiency criteria, coefficient bounds, partial sums, and radius of convexity of order for the functions of the said class are investigated here.

1. Introduction

Quantum calculus has emerged as one of the most vibrant areas of research in recent years. Researchers have discussed and found its applications in numerous dimensions, such as hypergeometric series, complex analysis, and applied physics. It has developed techniques to be used in -calculus, time scales, partitions, and continued fractions. Jackson, for the first time, in the beginning of the 20th century, introduced quantum calculus, where he developed and standardized it. For more details about quantum calculus, see [1–14]. To make a good pace and understanding of the results presented in this article, we are going to give below some primary definitions and relevant details of quantum calculus. Suppose represents the class of holomorphic functions of typein open unit disk and normalized by the conditions and . Moreover, represents the class of all functions in which are univalent in ; see [15].

A domain is starlike with respect to a point if all possible lines which are confined by two points, connecting to any other point, lie entirely within . Correspondingly, a domain is convex if all possible lines which are obtained by connecting any two points in lie thoroughly within . More clearly, we can say that if the domain is starlike with respect to each of its points in , then it is convex. If is starlike for with respect to the origin, then it is called a starlike function, whereas if is convex, then it is called a convex function. The class of all convex functions is represented by , and the class of all starlike functions is represented by . Analytically, these are defined as follows:

For , suppose that and are subclasses of consisting of -starlike functions and -convex functions, respectively, defined analytically as follows:

For , the class and the class . Moreover, the following two classes are closely related with their functions defined, respectively.

Note that and . The partial sum of the function , denoted by , is the polynomial, defined by

Generally, lower bounds on ratios such as or have been found to be sharp only when , but Silverman determined sharpness ; see [16, 17]. He investigated that lower bounds are strictly increasing functions of . In the present article, by using Silverman’s technique [16], we will find the function’s ratio having Taylor series (1) to its sequence of partial sums when the coefficients of are sufficiently small to fulfill the necessary and sufficient condition. In more details to clarify, we will find sharp lower bounds for , , , and . Indeed, we will use the familiar result, i.e., , , if and only if satisfies . Unless otherwise stated, we will presume that has form (1) and that its sequence of partial sums is represented by (5).

For , Ravichandran gave the sharp radius of starlike and convex functions of order with form (1) whose Taylor series coefficients satisfy the conditions , and ; or for .

Consider that and are holomorphic functions in with and , , so that ; will be subordinated by and denoted by . If is holomorphic, then iff and .

For two holomorphic functionsthe Hadamard product of and is defined as

We will define some notations and concepts of quantum calculus which are to be used in this article. All results can be found in [2, 3, 18]. For , , we see the classical -theory begins with the -extension of the positive numbers. The expressionproposes that we define the -generalization of , which is also called the -bracket of , given asand the -generalization of the factorial which is called -factorial given by

The -difference operator for is defined asand we can see that, for and ,

For , the -analogue of the Ruscheweyh differential operator is defined aswhereandwhere is a Pochhammer symbol, which is defined as follows:

From (13), it is clear that

It follows that , and the Ruscheweyh -differential operator converts into the Ruscheweyh differential operator ; for more details, see [19]. Using (13),

If , then

Definition 1. The function will lie in the class if and only ifwhereFor more details, see [20–24]. If , then it is shown in [25] that, from (46), one can have

Definition 2. A function will lie in the class , , , if and only ifor equivalently,

For more details about the above classes and conic domain, we refer the readers to [20, 25–28]. Using the -Ruscheweyh differential operator, we now define the following more general class of functions associated with the conic domain defined by Janowski functions.

Definition 3. A function will lie in the class , , if and only ifor equivalently,

The above defined class generalizes many known classes which can be obtained by setting suitable particular values to the parameters as follows.

Special cases:(1) the well-known class of -uniformly Janowski convex functions, introduced by Noor and Malik [27](2), the well-known class of Janowski convex functions, introduced by Janowski [20](3), see [29](4) see [15]

Lemma 1 (see [30]). Let be subordinate to . If is holomorphic in and is convex, then

2. Main Results

Theorem 1. A function with form (1) will lie in class , , , if it satisfies the conditionwhereand

Proof. Suppose that (28) holds; then, it is enough to show thatWe considerThe last expression is bounded above by 1 ifwhich reduces toThis finalizes the proof. □

For and , we have the following known result, proved in [27].

Corollary 1. A function with form (1) will lie in class , , , if it satisfies the condition

For , , and , we have the following known result, proved in [29].

Corollary 2. A function with form (1) will lie in class , , , if it satisfies the condition

Theorem 2. Let , , , and be of form (1); then, for ,where is defined by (15).

Proof. By the definition for , we havewhereIf , thenNow, if , then by (27) and (39), we haveNow, from (38), we haveLet , and using the Cauchy product formula, we obtainThis implies thatComparison of coefficients of gives usorUsing (41), we haveorNow, we prove thatFor this, we use the induction technique. For , we have from (46),orFor , we have from (46),From (37), we haveLet the assumption be true for . From (46), we haveFrom (37), we haveBy the induction hypothesis,Multiplying both sides by we haveThat is,Hence, the consequence is true for . Therefore, using mathematical induction, we have proved that (37) is true , . □

For and , we have the following known result, proved in [27].

Corollary 3. Let , , , and be of form (1); then, for ,

For , , and , we have the following known result, proved in [29].

Corollary 4. Let , , , and be of form (1); then, for ,

Using the already proven results of Silverman [16] and Silvia [17] on partial sums of holomorphic functions, we will find the fraction of (1) to its sequence of partial sums when the function has coefficients small enough to satisfy condition (28). We will investigate sharp lower bounds for , , , and in the class .

Theorem 3. If , thenwhere is defined by (29) and . The extremal functiongives the sharp result.

Proof. Define a function :and this will reduce toWe haveNow,ifIt is sufficient to show that the left hand side of (28) is bounded above by ifThis leads to the following expression:To ensure that the function defined by (62) gives the sharp outcome, we note that, for ,

Theorem 4. If , thenwhere is defined by (29) and . The result (71) is sharp with the function given by (62).

Proof. Define the function :This will becomeThis implies thatNow,ifIt would be enough to show that the left side of (28) is bounded above by ifwhich leads to the following expression:orConsequently, the equality holds for the extreme function given by (62). □

We now turn to fractions related to the derivatives.

Theorem 5. If , thenwhere is defined by (29) and . The result (80) is sharp with the function given by (62).

Proof. Define the function :and this will reduce toNow, we haveThis implies thatNow,ifIt would be enough to show that the left side of (28) is bounded above by ifwhich leads to the following expression:The result (80) is sharp with respect to the function given by (62). □

Theorem 6. If , thenwhere is defined by (29) and . The result (89) is sharp with respect to the function given by (62).

Proof. Define the function :This will becomeThis leads us towhich reduces toNow,ifIt is sufficient to show that the left hand side of (28) is bounded above by ifwhich leads to the following expression:The result (89) is sharp with respect to the function given by (62). □

In the next theorem, we will find the radii of starlikeness for the class .

Theorem 7. Let . Then, is a convex function of order in , wherewhere is defined by (29) and .

Proof. Let . Then, by the theorem,where is defined by (29) and . For we need to show thatthat is,Thus, ifAccording to theorem (99), inequality (102) will be true ifSolving (103) for , we obtainSetting in (104), we havewhich is the required result. □

3. Conclusion

In this article, we have applied the -Ruscheweyh differential operator to define and study a new class of -convex functions associated with the conic domain. This class generalizes the classes , , , , and which have been defined and studied earlier. This fact has been illustrated above with details and proper referencing. The results presented include sufficiency criteria related to Taylor series coefficients, the coefficient bounds, and the ratios of partial sums to their infinite sum for functions of the class .

Data Availability

No data were used in this article.

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 and approved the final manuscript.