Coefficient Inequalities for a Subclass of Symmetric <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5268pt" id="M1" height="12.3952pt" version="1.1" viewBox="-0.0657574 -7.8684 8.58866 12.3952" width="8.58866pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-114" d="M474 429L457 433C435 440 389 448 367 448C348 448 323 446 309 443C266 434 195 406 148 366C78 307 23 210 23 101C23 35 55 -12 92 -12C118 -12 146 1 196 35C247 70 311 130 346 173H348L281 -148C268 -211 256 -221 208 -229L191 -232L187 -257L433 -245L437 -219L411 -216C357 -210 350 -205 362 -140L427 207C447 315 461 381 474 429ZM387 387C379 337 363 262 355 236C318 180 201 57 142 57C126 57 112 81 112 128C112 205 150 321 220 376C244 395 280 403 312 403C345 403 370 396 387 387Z"/></g></svg> </span>-Starlike Functions Involving Certain Conic Domains

Khan, Mohammad Faisal; Khan, Nazar; Araci, Serkan; Khan, Shahid; Khan, Bilal

doi:https://doi.org/10.1155/2022/9446672

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Abstract Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 9446672 | https://doi.org/10.1155/2022/9446672

Coefficient Inequalities for a Subclass of Symmetric -Starlike Functions Involving Certain Conic Domains

Mohammad Faisal Khan,¹Nazar Khan,²Serkan Araci,³Shahid Khan,⁴and Bilal Khan⁵

Academic Editor: Xiaolong Qin

Received23 Jun 2022

Accepted17 Aug 2022

Published29 Sept 2022

Abstract

In this paper, we make use of a certain Ruscheweyh-type -differential operator to introduce and study a new subclass of -starlike symmetric functions, which are associated with conic domains and the well-known celebrated Janowski functions in . We then investigate many properties for the newly defined functions class, including for example coefficients inequalities, the Fekete–Szegö Problems, and a sufficient condition. There are also relevant connections between the results provided in this study and those in a number of other published articles on this subject.

1. Introduction, and Preliminaries

Let be a class of analytic functions where is the open unit disk and is given by the following equation:and let be those functions in the open unit disk which are normalized by the following equation:thus, we have the following series form for

Moreover, all normalized univalent functions in are contained in the set . For two given functions , we say that is subordinate to , written symbolically as , if there exist a Schwarz function , which is holomorphic in withso that

Moreover, if the function is univalent in , then the following equivalence hold true:

Let be the class of Carathedory function, an analytic function ifsuch that

Definition 1. A given function is said to be in the class ifJanowski [1] investigated the class of functions and found that if and only if there exist a function such that

Definition 2. A function of the form (3) be in the functions class if and only ifHistorically speaking, Kanas and Wiśniowska were the first (see [2], [3]), (see also [4]) who introduced and defined the class of -uniformly convex functions and -starlike functions subject to the conic domain , whereMoreover if , is fixed and then denote the conic region bounded by the imaginary axis, if , we have a parabola, if this domain represents the right branch of hyperbola and for an ellipse.
For these conic areas, the following functions serve as extremal functionswhereand is chosen such that . Here, is Legendre’s complete elliptic integral of first kind and , that is is the complementary integral of .
The function in [5] be given as follows:whereand .
The following is defined by Noor et al. [6], who combine the ideas of Janowski functions and conic regions.

Definition 3. A function from the functions class be in the functions class , ifwhere is defined by (13).
Geometrically for the domain , we have the followingor equivalently for , we have the following equation:The domain denote the conic type regions (see [6]).

Definition 4. (see [6]). An analytic function be in the class , ifThe angle at which domain rotates around the origin maps onto itself The domain is thus said to as -fold symmetric. In a function is said to be -fold symmetric if for every in Here, denote the -fold symmetric functions and symbolize the imaginary unit. For , the -symmetric functions are an extension of the concept of even, odd, and -symmetric functions. Several applications of the theory of -symmetric functions may be found in [7]. For , the functions is known to be -symmetric ifThe set of all -symmetric functions is denote by . First of all, if and , then is called even symmetric functions. Secondly, if and , then is called odd symmetric functions. Thirdly, if , then is called -symmetric functions.

Theorem 1 (see [7]). For mapping , there is just one series of -symmetrical functions exists that given as follows:

From equation (25), we can get the following equation:

thenwhere and

2. Some Basic Concepts of -Calculus

The usage of the quantum (or -) calculus in many diverse areas of mathematics and physics are quite significant. In the theory of Univalent functions, Srivastava [8] firstly apply the -calculus in order to put the foundation of a new direction for other researchers. Motivated by [8] of Srivastava, many researchers have worked on this direction. For example, the convolution theory, enable us to investigate various properties of analytic functions. Due to the large range of applications of -calculus and the importance of -operators instead of regular operators, many researchers have explored -calculus in depth, such as, Kanas and Reducanu [9], Muhammad and Sokol [10] and Noor et al. [11–15]. Also in [1–5, 9, 16–22], Ahmad et al. see also [21], have used the -derivative operator to define a new subclass -meromorphic starlike functions. They also developed some remarkable results for their defined classes of analytic functions. In addition, Srivastava [23] see also [8, 12–15, 23–26] recently published survey-cum-expository review paper this might be useful for researchers and scholars working on these subjects. For some recent and related study about -series, we may refer the interested readers to see [27–29].

Definition 5. (see [30]). Let and -integer , be defined as follows:

Definition 6. We define the -shifted factorial as follows:It could be seen thatIn general .

Definition 7 (see [20]). For an analytic function , the -deformation or -generalization of derivative is defined by the following equation:and

Definition 8. The generalized -Pochhammer symbol is given by the following equation:and -gamma function be given as follows:

Definition 9. (see [31]). A function - if and only ifwhereand is defined by (13).
Geometrically, we have the following equation:whereFor more detail (see [31]).

Definition 10. (see [9]). For , the Rusceweyh -differential operator be defined as follows:wherewithBy “” we mean convolution (or Hadamard product).
Moreover from (42), we have the following equation:andMaking use of (42) and (43), the power series of is given by the following equation:Similarly,where is given by (44) and is given by (49).
Note that,andWhen , Ruscheweyh -differential operator reduce to Ruscheweyh differential operator [14]. Also, note thatand

Definition 11. A given function is said to be in the functions class -, , , ifor equivalentlywhereEach of the following special case of the above-defined functions class - is worthy of note.(i)One can easily seen that where - is the functions classes studied by Al-Sarari and Latha (see [18]).(ii)If we set in Definition 11, we have class - introduced in [17].(iii)If we put we get the functions class - (see [30]).(iv)We see that where Kanas and Wisniowska [3] have studied the class -.(v)If we set in Definition 11, we get the class (see [24]).(vi)It can be easily seen that where the functions class is studied in [1].

3. A Set of Lemmas

Lemma 1 (see [13]). Let . If is convex univalent in , then

Lemma 2 (see [32]). If is an analytic function with positive real part in , then

The equality holds foror one of its relations. The equality holds for

or one of its relations. If , the equality holds if and only ifor one of rotation of it. The above upper bound is sharp as well, whenand

Lemma 3 (see [32]). If the function given by (8) belongs to the class of analytic functions and have positive real part in , then

The equality holds for

Lemma 4 (see [32]). If the function given by (8) belongs to the class of analytic functions and have positive real part in , then

Lemma 5. Let be fixed andthenwhere , , and given by (83), (84), (16), and (17).

Proof. From (39), we have the following equation:whereandBy using (15) and (75), we have the following equation:whereThe seriesandare convergent and convergent to 1, and , respectively.
Therefore (78) becomeswhereandwhich is our required result.

Remark 1. If we set , and let , in Lemma 5, we will arrive it that of a result given by Sim et al. [25].

Lemma 6. Let -, then

Proof. By Definition 9 and - if and only if is given by (39).
By using (82) and (86), we have the following equation:Now by using Lemma 1 on (87), we have the following equation:which is our required result.

4. Main Results

Theorem 2. An analytic function of the form (3) belongs to the class -, if the following condition is satisfied:where is given by (29) and

Proof. Suppose that (89) holds, then it is enough to show thatwhere is given by (56).
Now, we have the following equation:whereExpression in (93) is bounded above by 1 ifwhere and are defined by (90) and (91) respectively. Thus, we have completed the proof of our Theorem.
If in Theorem 1 we putthen the result was reduced to the following known one.

Corollary 1 (see [30]). An analytic function of the form (3) is in the class -, if it hold the condition

If in Theorem 1, we set

and let , then we get the following well-known consequence.

Corollary 2 (see [2]). An analytic function of the form (3) is in the class -, if it satisfies the condition

If we put

and let , in Theorem 2, then we have the following known result.

Corollary 3 (see [24]). An analytic and of the form (3) is in the class -, if it satisfies the condition

Theorem 3. If -, thenwhere , and , are defined by (16), (29) and (44).

Proof. Let - andwhich implies thatBy using (47) and (48) on (104), we have the following equation:By making use of the well-known Cauchy Product formula on Right hand side of (105), we get the following equation:On both sides of equation (106), comparing the coefficients of , we have the following equation:which implies thatBy using Lemma 6 together with (108), we have the following equation:We claim thatWe prove (110) by using the principle of the mathematical induction method.
For , from (109), we have the following equation:Again from (110), we have the following equation:Let the hypothesis be true for , then from (109), we have the following equation:From (110), we have the following equation:Using modulus properties we get the following equation:By the induction hypothesis, we have the following equation:Multiplying both sides of (116) by the following equation:we haveFurthermore,That is,Hence inequality is true for . Which completes the proof. □
If in Theorem 2, we setand let , we get the following known result.

Corollary 4 (see [30]). An analytic function of the form )(3) belongs to the class -, if it satisfies

If we set

and let , in Theorem 2, we have the following result.

Corollary 5 (see [3]). An analytic function of the form (3) belongs to the class -, if it satisfies

If we set

and let , also by taking , then , in Theorem 2, we have the following result.

Corollary 6 (see [1]). An analytic function of the form (3) is in the class , if it satisfies

Theorem 4. If the function given by (3) belongs to the class -, thenand

Proof. Let and defined by the following equation:where is a Schwarz function, such that and . It follows seasily thatIf -, thus from (55), we have the following equation:where is given in (15), thus from (131), we have the following equation:From (130), we have the following equation:After some simplification we have the following equation:Using (134) and (132), we have the following equation:From (47) and (48), we have the following equation:From (135) and (136), equating coefficients of and gives us the following equation:andBy using Lemma 3, on (137) and (138) we obtain the result asserted by Theorem 3.

Theorem 5. Let and be fixed and let , then for a complex number .where , and are given by (141), (142), (143), and (144), respectively.

Proof. Using (137) and (138), we havewhereandApply Lemma 2 on (140) we have.

Theorem 6. Let and let - of the form (3), then for a complex number where , given by (141), (142), (143), and (144).

Proof. From (140), we have the following equation:Apply Lemma 4 in conjunction with (146), we obtain the result asserted by Theorem 5.

5. Conclusion

We have successfully studied the uses of certain Ruscheweyh-type -differential operator to a new subclass of -starlike symmetric functions, which involving both the conic domains and the well-known celebrated Janowski functions in the open unit disk . We then investigated many properties for the newly defined functions class, including for example coefficients inequalities, the Fekete–Szegö Problems, and a sufficient condition. We have also emphasized certain known results of our key findings.

The interested readers should be advised not to be misled to believe that the so-called -Gamma function provides a “generalization” of the classical (Euler’s) Gamma function. Similar remarks will apply also to all of the usages of the so-called -Gamma function including (for example) the so-called -extensions of the Riemann–Liouville and other operators of fractional integral and fractional derivatives.

Data Availability

No data were used in the paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this manuscript and approved its final version.

Acknowledgments

The authors would like to thank in advance to the handling Editor and Reviewers for their suggestions and valuable comments on the manuscript under review.

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Copyright © 2022 Mohammad Faisal Khan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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