scholarly journals Applications of Certain Conic Domains to a Subclass of q-Starlike Functions Associated with the Janowski Functions

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 574
Author(s):  
Bilal Khan ◽  
Hari Mohan Srivastava ◽  
Nazar Khan ◽  
Maslina Darus ◽  
Qazi Zahoor Ahmad ◽  
...  
Keyword(s):  
Lower Bounds ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Function Class ◽  
Partial Sums ◽  
Open Unit ◽  
Additional Parameter ◽  
Open Unit Disk ◽  
Janowski Functions ◽  
Distortion Theorems

In our present investigation, with the help of the basic (or q-) calculus, we first define a new domain which involves the Janowski function. We also define a new subclass of the class of q-starlike functions, which maps the open unit disk U, given by U= z:z∈C and z <1, onto this generalized conic type domain. We study here some such potentially useful results as, for example, the sufficient conditions, closure results, the Fekete-Szegö type inequalities and distortion theorems. We also obtain the lower bounds for the ratio of some functions which belong to this newly-defined function class and for the sequences of the partial sums. Our results are shown to be connected with several earlier works related to the field of our present investigation. Finally, in the concluding section, we have chosen to reiterate the well-demonstrated fact that any attempt to produce the rather straightforward (p,q)-variations of the results, which we have presented in this article, will be a rather trivial and inconsequential exercise, simply because the additional parameter p is obviously redundant.

scholarly journals Some general families of q-starlike functions associated with the Janowski functions

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2613-2626 ◽  
Author(s):  
H.M. Srivastava ◽  
Muhammad Tahir ◽  
Bilal Khan ◽  
Qazi Ahmad ◽  
Nazar Khan
Keyword(s):  
Sufficient Conditions ◽  
Starlike Functions ◽  
Open Unit ◽  
Additional Parameter ◽  
Open Unit Disk ◽  
Conic Domain ◽  
Janowski Functions ◽  
Distortion Theorems ◽  
The Relationship ◽  
Inclusion Results

By making use of the concept of basic (or q-) calculus, various families of q-extensions of starlike functions, which are associated with the Janowski functions in the open unit disk U, were introduced and studied from many different viewpoints and perspectives. In this paper, we first investigate the relationship between various known families of q-starlike functions which are associated with the Janowski functions. We then introduce and study a new subclass of q-starlike functions which involves the Janowski functions and is related with the conic domain. We also derive several properties of such families of q-starlike functions with negative coefficients including (for example) sufficient conditions, inclusion results and distortion theorems. In the last section on conclusion, we choose to point out the fact that the results for the q-analogues, which we consider in this article for 0 < q < 1, can easily (and possibly trivially) be translated into the corresponding results for the (p; q)-analogues (with 0 < q < p ?? 1) by applying some obvious parametric and argument variations, the additional parameter p being redundant.

scholarly journals Some General Classes of q-Starlike Functions Associated with the Janowski Functions

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 292 ◽  
Author(s):  
Hari Srivastava ◽  
Muhammad Tahir ◽  
Bilal Khan ◽  
Qazi Ahmad ◽  
Nazar Khan
Keyword(s):  
Unit Disk ◽  
Starlike Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Janowski Functions ◽  
Distortion Theorems ◽  
The Relationship

By making use of the concept of basic (or q-) calculus, various families of q-extensions of starlike functions of order α in the open unit disk U were introduced and studied from many different viewpoints and perspectives. In this paper, we first investigate the relationship between various known classes of q-starlike functions that are associated with the Janowski functions. We then introduce and study a new subclass of q-starlike functions that involves the Janowski functions. We also derive several properties of such families of q-starlike functions with negative coefficients including (for example) distortion theorems.

scholarly journals A Subclass of Janowski Starlike Functions Involving Mathieu-Type Series

Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 2
Author(s):  
Dong Liu ◽  
Serkan Araci ◽  
Bilal Khan
Keyword(s):  
Analytic Functions ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Function Class ◽  
Partial Sums ◽  
Invariant Functions ◽  
Type Series ◽  
Janowski Functions ◽  
Radius Problems ◽  
Symmetrical Points

To date, many interesting subclasses of analytic functions involving symmetrical points and other well celebrated domains have been investigated and studied. The aim of our present investigation is to make use of certain Janowski functions and a Mathieu-type series to define a new subclass of analytic (or invariant) functions. Our defined function class is symmetric under rotation. Some useful results like Fekete-Szegö functional, a number of sufficient conditions, radius problems, and results related to partial sums are derived.

Coefficient Estimates for Certain Subclasses of m-Fold Symmetric Bi-univalent Functions Associated with Pseudo-Starlike Functions

2021 ◽  
pp. 209-223
Author(s):  
Timilehin G. Shaba ◽  
Amol B. Patil
Keyword(s):  
Unit Disk ◽  
Univalent Function ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Function Class ◽  
Open Unit ◽  
Open Unit Disk ◽  
Coefficient Estimates

In the present investigation, we introduce the subclasses $\varLambda_{\Sigma}^{m}(\eta,\leftthreetimes,\phi)$ and $\varLambda_{\Sigma}^{m}(\eta,\leftthreetimes,\delta)$ of \textit{m}-fold symmetric bi-univalent function class $\Sigma_m$, which are associated with the pseudo-starlike functions and defined in the open unit disk $\mathbb{U}$. Moreover, we obtain estimates on the initial coefficients $|b_{m+1}|$ and $|b_{2m+1}|$ for the functions belong to these subclasses and identified correlations with some of the earlier known classes.

scholarly journals A Subclass of Multivalent Janowski Type q-Starlike Functions and Its Consequences

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1275
Author(s):  
Qiuxia Hu ◽  
Hari M. Srivastava ◽  
Bakhtiar Ahmad ◽  
Nazar Khan ◽  
Muhammad Ghaffar Khan ◽  
...  
Keyword(s):  
Analytic Functions ◽  
Convex Combination ◽  
Starlike Functions ◽  
Function Class ◽  
Additional Parameter ◽  
Growth Theorem ◽  
Function Classes ◽  
Geometric Function ◽  
Janowski Functions ◽  
Principle Of Subordination

In this article, by utilizing the theory of quantum (or q-) calculus, we define a new subclass of analytic and multivalent (or p-valent) functions class Ap, where class Ap is invariant (or symmetric) under rotations. The well-known class of Janowski functions are used with the help of the principle of subordination between analytic functions in order to define this subclass of analytic and p-valent functions. This function class generalizes various other subclasses of analytic functions, not only in classical Geometric Function Theory setting, but also some q-analogue of analytic multivalent function classes. We study and investigate some interesting properties such as sufficiency criteria, coefficient bounds, distortion problem, growth theorem, radii of starlikeness and convexity for this newly-defined class. Other properties such as those involving convex combination are also discussed for these functions. In the concluding part of the article, we have finally given the well-demonstrated fact that the results presented in this article can be obtained for the (p,q)-variations, by making some straightforward simplification and will be an inconsequential exercise simply because the additional parameter p is obviously unnecessary.

scholarly journals A Study of Some Families of Multivalent q-Starlike Functions Involving Higher-Order q-Derivatives

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1470
Author(s):  
Bilal Khan ◽  
Zhi-Guo Liu ◽  
Hari M. Srivastava ◽  
Nazar Khan ◽  
Maslina Darus ◽  
...  
Keyword(s):  
Starlike Functions ◽  
Higher Order ◽  
Open Unit ◽  
Sufficient Condition ◽  
Open Unit Disk ◽  
Function Classes ◽  
The Family ◽  
Radius Problems ◽  
Coefficient Inequalities ◽  
Distortion Theorems

In the present investigation, by using certain higher-order q-derivatives, the authors introduce and investigate several new subclasses of the family of multivalent q-starlike functions in the open unit disk. For each of these newly-defined function classes, several interesting properties and characteristics are systematically derived. These properties and characteristics include (for example) distortion theorems and radius problems. A number of coefficient inequalities and a sufficient condition for functions belonging to the subclasses studied here are also discussed. Relevant connections of the various results presented in this investigation with those in earlier works on this subject are also pointed out.

scholarly journals A Subclass of q-Starlike Functions Defined by Using a Symmetric q-Derivative Operator and Related with Generalized Symmetric Conic Domains

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 917
Author(s):  
Shahid Khan ◽  
Saqib Hussain ◽  
Muhammad Naeem ◽  
Maslina Darus ◽  
Akhter Rasheed
Keyword(s):  
Unit Disk ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Necessary And Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Derivative Operator ◽  
Structural Formulas ◽  
Necessary And Sufficient

In this paper, the concepts of symmetric q-calculus and conic regions are used to define a new domain Ωk,q,α˜, which generalizes the symmetric conic domains. By using the domain Ωk,q,α˜, we define a new subclass of analytic and q-starlike functions in the open unit disk U and establish some new results for functions of this class. We also investigate a number of useful properties and characteristics of this subclass, such as coefficients estimates, structural formulas, distortion inequalities, necessary and sufficient conditions, closure and subordination results. The proposed approach is also compared with some existing methods to show the reliability and effectiveness of the proposed methods.

scholarly journals Sufficient conditions for Carathéodory functions

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 1097-1106 ◽  
Author(s):  
Mamoru Nunokawa ◽  
Oh Kwon ◽  
Young Sim ◽  
Nak Cho
Keyword(s):  
Unit Disk ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Carathéodory Functions ◽  
Special Cases

In the present paper, we obtain several sufficient conditions for Carath?odory functions in the open unit disk U = {z ? C : ?z? < 1}. We also obtain sufficient conditions for p-valent or starlike functions. Moreover, we improve some results due to Nunokawa [Tsukuba J. Math. 13 (1989), 453-455] as some special cases of main results.

scholarly journals Sufficiency Criteria for q -Starlike Functions Associated with Cardioid

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Saira Zainab ◽  
Ayesha Shakeel ◽  
Muhammad Imran ◽  
Nazeer Muhammad ◽  
Hira Naz ◽  
...  
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Image Domain ◽  
Lemniscate Of Bernoulli ◽  
Differential Subordinations

This article deals with the q -differential subordinations for starlike functions associated with the lemniscate of Bernoulli and cardioid domain. The primary goal of this work is to find the conditions on γ for 1 + γ z ∂ q   h z / h n   z   ≺ 1 + z , where h z is analytic function and is subordinated by the function which is producing cardioid domain as its image domain while mapping the open unit disk. Along with this, certain sufficient conditions for q -starlikeness of analytic functions are determined.

scholarly journals Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator

2021 ◽  
Vol 19 (1) ◽  
pp. 329-337
Author(s):  
Huo Tang ◽  
Kaliappan Vijaya ◽  
Gangadharan Murugusundaramoorthy ◽  
Srikandan Sivasubramanian
Keyword(s):  
Analytic Function ◽  
Differential Operator ◽  
Lower Bounds ◽  
Analytic Functions ◽  
Function Class ◽  
Partial Sums ◽  
Generalized Function ◽  
Inclusion Relations ◽  
Alpha Beta

Abstract Let f k ( z ) = z + ∑ n = 2 k a n z n {f}_{k}\left(z)=z+{\sum }_{n=2}^{k}{a}_{n}{z}^{n} be the sequence of partial sums of the analytic function f ( z ) = z + ∑ n = 2 ∞ a n z n f\left(z)=z+{\sum }_{n=2}^{\infty }{a}_{n}{z}^{n} . In this paper, we determine sharp lower bounds for Re { f ( z ) / f k ( z ) } {\rm{Re}}\{f\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}\left(z)\} , Re { f k ( z ) / f ( z ) } {\rm{Re}}\{{f}_{k}\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}f\left(z)\} , Re { f ′ ( z ) / f k ′ ( z ) } {\rm{Re}}\{{f}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}^{^{\prime} }\left(z)\} and Re { f k ′ ( z ) / f ′ ( z ) } {\rm{Re}}\{{f}_{k}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}^{^{\prime} }\left(z)\} , where f ( z ) f\left(z) belongs to the subclass J p , q m ( μ , α , β ) {{\mathcal{J}}}_{p,q}^{m}\left(\mu ,\alpha ,\beta ) of analytic functions, defined by Sălăgean ( p , q ) \left(p,q) -differential operator. In addition, the inclusion relations involving N δ ( e ) {N}_{\delta }\left(e) of this generalized function class are considered.

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