Certain Sufficient Conditions for Starlike Functions

2017 ◽  
Vol 9 (1) ◽  
pp. 1-12
Author(s):  
Pardeep Kaur ◽  
Sukhwinder Singh Billing
Keyword(s):  
Sufficient Conditions ◽  
Starlike Functions

scholarly journals On sufficient conditions of meromorphic starlike functions

2014 ◽  
Vol 32 (2) ◽  
pp. 229
Author(s):  
Ali Muhammad
Keyword(s):  
Unit Disc ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Meromorphic Starlike Functions

In this paper, we investigate interesting properties and sufficient conditions for meromorphic starlike functions in the punctured unit disc.

scholarly journals Convex and starlike criteria

1999 ◽  
Vol 22 (1) ◽  
pp. 75-79 ◽  
Author(s):  
Herb Silverman
Keyword(s):  
Sufficient Conditions ◽  
Starlike Functions ◽  
Positive Order ◽  
Convex And Starlike Functions

We investigate an expression involving the quotient of the analytic representations of convex and starlike functions. Sufficient conditions are found for functions to be starlike of a positive order and convex.

scholarly journals Some Janowski Type Harmonic q-Starlike Functions Associated with Symmetrical Points

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 629 ◽  
Author(s):  
Muhammad Arif ◽  
Omar Barkub ◽  
Hari Srivastava ◽  
Saleem Abdullah ◽  
Sher Khan
Keyword(s):  
Differential Operator ◽  
Harmonic Functions ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Partial Sums ◽  
Circular Region ◽  
Necessary And Sufficient ◽  
New Criterion ◽  
Symmetrical Points ◽  
Convexity Conditions

The motive behind this article is to apply the notions of q-derivative by introducing some new families of harmonic functions associated with the symmetric circular region. We develop a new criterion for sense preserving and hence the univalency in terms of q-differential operator. The necessary and sufficient conditions are established for univalency for this newly defined class. We also discuss some other interesting properties such as distortion limits, convolution preserving, and convexity conditions. Further, by using sufficient inequality, we establish sharp bounds of the real parts of the ratios of harmonic functions to its sequences of partial sums. Some known consequences of the main results are also obtained by varying the parameters.

scholarly journals Certain Geometric Properties of Lommel and Hyper-Bessel Functions

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 240
Author(s):  
Saima Mushtaq ◽  
Mohsan Raza ◽  
Muhey Din
Keyword(s):  
Bessel Functions ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Geometric Properties ◽  
Lommel Functions

In this article, we are mainly interested in finding the sufficient conditions under which Lommel functions and hyper-Bessel functions are close-to-convex with respect to the certain starlike functions. Strongly starlikeness and convexity of Lommel functions and hyper-Bessel functions are also discussed. Some applications are also the part of our investigation.

scholarly journals On close-to-convex functions of complex order

1990 ◽  
Vol 13 (2) ◽  
pp. 321-330 ◽  
Author(s):  
H. S. Al-Amiri ◽  
Thotage S. Fernando
Keyword(s):  
Convex Functions ◽  
Nonnegative Integer ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Coefficient Bounds ◽  
Complex Order ◽  
Covering Theorems

The classS*(b)of starlike functions of complex orderbwas introduced and studied by M.K. Aouf and M.A. Nasr. The authors using the Ruscheweyh derivatives introduce the classK(b)of functions close-to-convex of complex orderb,b≠0and its generalization, the classesKn(b)wherenis a nonnegative integer. HereS*(b)⊂K(b)=K0(b). Sharp coefficient bounds are determined forKn(b)as well as several sufficient conditions for functions to belong toKn(b). The authors also obtain some distortion and covering theorems forKn(b)and determine the radius of the largest disk in which everyf∈Kn(b)belongs toKn(1). All results are sharp.

scholarly journals On meromorphically starlike functions of order $\alpha$ and type $\beta$, which satisfy Shah's differential equation

2018 ◽  
Vol 9 (2) ◽  
pp. 154-162
Author(s):  
Yu.S. Trukhan ◽  
O.M. Mulyava
Keyword(s):  
Differential Equation ◽  
Sufficient Conditions ◽  
Starlike Functions

According to M.L. Mogra, T.R. Reddy and O.P. Juneja an analytic in ${\mathbb D_0}=\{z: 0<|z|<1\}$ function $f(z)=\frac{1}{z}+\sum_{n=1}^{\infty}f_n z^{n}$ is said to be meromorphically starlike of order $\alpha\in [0,\,1)$ and type $\beta\in (0,\,1]$ if $|zf'(z)+f(z)|<\beta|zf'(z)+(2\alpha-1)f(z)|, \, z\in {\mathbb D_0}. $ Here we investigate conditions on complex parameters $\beta_0,\,\beta_1,\,\gamma_0,\,\gamma_1,\,\gamma_2$, under which the differential equation of S. Shah $z^2 w''+(\beta_0 z^2+\beta_1 z) w'+(\gamma_0 z^2+\gamma_1 z+\gamma_2)w=0$ has meromorphically starlike solutions of order $\alpha\in [0,\,1)$ and type $\beta\in (0,\,1]$. Beside the main case $n+\gamma_2\not=0, \, n\ge 1,$ cases $\gamma_2=-1$ and $\gamma_2=-2$ are considered. Also the possibility of the existence of the solutions of the form $f(z)=\frac{1}{z}+\sum_{n=1}^{m}f_n z^{n}, \, m\ge 2,$ is studied. In addition we call an analytic in ${\mathbb D_0}$ function $f(z)=\frac{1}{z}+\sum_{n=1}^{\infty}f_n z^{n}$ meromorphically convex of order $\alpha\in [0,1)$ and type $\beta\in (0,1]$ if $|zf''(z)+2f'(z)|<\beta|zf''(z)+2\alpha f'(z)|, \, z\in {\mathbb D_0}$ and investigate sufficient conditions on parameters $\beta_0,\,\beta_1,\,\gamma_0,$ $\gamma_1,\,\gamma_2$ under which the differential equation of S. Shah has meromorphically convex solutions of order $\alpha\in [0,\,1)$ and type $\beta\in (0,\,1]$. The same cases as for the meromorphically starlike solutions are considered. 

scholarly journals Extension of Nunokawa Lemma for Functions with Fixed Second Coefficient and Its Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mostafa Amani ◽  
Rasoul Aghalary ◽  
Ali Ebadian
Keyword(s):  
Analytic Functions ◽  
Univalent Functions ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Differential Subordination ◽  
Differential Inequalities ◽  
Initial Coefficient ◽  
Strongly Starlike Functions ◽  
Strongly Starlike

In this paper, we study some properties of analytic functions with fixed initial coefficients. The methodology of differential subordination is used for modification and improvements of several well-known results for subclasses of univalent functions by restricting the functions with fixed initial coefficients. Actually, by extending the Nunokawa lemma for fixed initial coefficient functions, we obtain some novel results on subclasses of univalent functions, such as differential inequalities for univalency or starlikeness of analytic functions. Also, we provide some new sufficient conditions for strongly starlike functions. The results of this paper extend and improve the previously known results by considering functions with fixed second coefficients.

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.

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.

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