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 Higher-order q-derivatives and their applications to subclasses of multivalent Janowski type q-starlike functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Bilal Khan ◽  
Zhi-Guo Liu ◽  
H. M. Srivastava ◽  
Serkan Araci ◽  
Nazar Khan ◽  
...  
Keyword(s):  
Starlike Functions ◽  
Higher Order ◽  
Sufficient Condition ◽  
Function Classes ◽  
Janowski Functions ◽  
Radius Problems ◽  
Coefficient Inequalities

AbstractIn the present investigation, with the help of certain higher-order q-derivatives, some new subclasses of multivalent q-starlike functions which are associated with the Janowski functions are defined. Then, certain interesting results, for example, radius problems and the results related to distortion, are derived. We also derive a sufficient condition and certain coefficient inequalities for our defined function classes. Some known consequences related to this subject are also highlighted. Finally, the well-demonstrated fact about the $(p,q)$ ( p , q ) -variations is also given in the concluding section.

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.

Some properties associated to a certain class of starlike functions

2019 ◽  
Vol 69 (6) ◽  
pp. 1329-1340 ◽  
Author(s):  
Vali Soltani Masih ◽  
Ali Ebadian ◽  
Sibel Yalçin
Keyword(s):  
Function Theory ◽  
Starlike Functions ◽  
Sharp Inequality ◽  
Open Unit ◽  
Geometric Function Theory ◽  
Open Unit Disk ◽  
Geometric Function ◽  
The Family ◽  
Logarithmic Coefficients ◽  
Dilogarithm Function

Abstract Let 𝓐 denote the family of analytic functions f with f(0) = f′(0) – 1 = 0, in the open unit disk Δ. We consider a class $$\begin{array}{} \displaystyle \mathcal{S}^{\ast}_{cs}(\alpha):=\left\{f\in\mathcal{A} : \left(\frac{zf'(z)}{f(z)}-1\right)\prec \frac{z}{1+\left(\alpha-1\right) z-\alpha z^2}, \,\, z\in \Delta\right\}, \end{array}$$ where 0 ≤ α ≤ 1/2, and ≺ is the subordination relation. The methods and techniques of geometric function theory are used to get characteristics of the functions in this class. Further, the sharp inequality for the logarithmic coefficients γn of f ∈ $\begin{array}{} \mathcal{S}^{\ast}_{cs} \end{array}$(α): $$\begin{array}{} \displaystyle \sum_{n=1}^{\infty}\left|\gamma_n\right|^2 \leq \frac{1}{4\left(1+\alpha\right)^2}\left(\frac{\pi^2}{6}-2 \mathrm{Li}_2\left(-\alpha\right)+ \mathrm{Li}_2\left(\alpha^2\right)\right), \end{array}$$ where Li2 denotes the dilogarithm function are investigated.

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 The Principle of Differential Subordination and Its Application to Analytic and p-Valent Functions Defined by a Generalized Fractional Differintegral Operator

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1083 ◽  
Author(s):  
Nak Eun Cho ◽  
Mohamed Kamal Aouf ◽  
Rekha Srivastava
Keyword(s):  
Fractional Derivative ◽  
Integral Operators ◽  
Differential Subordination ◽  
Open Unit ◽  
Open Unit Disk ◽  
The Family ◽  
Sandwich Type ◽  
Fractional Differintegral ◽  
Generalized Fractional Derivative ◽  
Fractional Differintegral Operator

A useful family of fractional derivative and integral operators plays a crucial role on the study of mathematics and applied science. In this paper, we introduce an operator defined on the family of analytic functions in the open unit disk by using the generalized fractional derivative and integral operator with convolution. For this operator, we study the subordination-preserving properties and their dual problems. Differential sandwich-type results for this operator are also investigated.

scholarly journals A Subclass of Analytic Functions Related tok-Uniformly Convex and Starlike Functions

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Saqib Hussain ◽  
Akhter Rasheed ◽  
Muhammad Asad Zaighum ◽  
Maslina Darus
Keyword(s):  
Convolution Operator ◽  
Starlike Functions ◽  
Distortion Theorem ◽  
Open Unit ◽  
Uniformly Convex ◽  
Open Unit Disc ◽  
Coefficient Inequalities ◽  
Uniformly Starlike Functions ◽  
Convex And Starlike Functions ◽  
Uniformly Starlike

We investigate some subclasses ofk-uniformly convex andk-uniformly starlike functions in open unit disc, which is generalization of class of convex and starlike functions. Some coefficient inequalities, a distortion theorem, the radii of close-to-convexity, and starlikeness and convexity for these classes of functions are studied. The behavior of these classes under a certain modified convolution operator is also discussed.

scholarly journals On the Fekete-Szegö problem

2000 ◽  
Vol 24 (9) ◽  
pp. 577-581 ◽  
Author(s):  
B. A. Frasin ◽  
Maslina Darus
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Upper Bound ◽  
Starlike Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Strongly Starlike Functions ◽  
Strongly Starlike

Letf(z)=z+a2z2+a3z3+⋯be an analytic function in the open unit disk. A sharp upper bound is obtained for|a3−μa22|by using the classes of strongly starlike functions of orderβand typeαwhenμ≥1.

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.

Multipliers of Fractional Cauchy Transforms and Smoothness Conditions

1998 ◽  
Vol 50 (3) ◽  
pp. 595-604 ◽  
Author(s):  
Donghan Luo ◽  
Thomas Macgregor
Keyword(s):  
Analytic Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Borel Measure ◽  
Open Unit ◽  
Open Unit Disk ◽  
Cauchy Transforms ◽  
The Family ◽  
Complex Borel Measure

AbstractThis paper studies conditions on an analytic function that imply it belongs to Mα, the set of multipliers of the family of functions given by where μ is a complex Borel measure on the unit circle and α > 0. There are two main theorems. The first asserts that if 0 < α < 1 and sup. The second asserts that if 0 < α < 1, ƒ ∈ H∞ and supt. The conditions in these theorems are shown to relate to a number of smoothness conditions on the unit circle for a function analytic in the open unit disk and continuous in its closure.

Sign in / Sign up

Export Citation Format

Share Document