scholarly journals PALEY'S INEQUALITY FOR THE JACOBI EXPANSIONS

2001 ◽  
Vol 33 (4) ◽  
pp. 483-491 ◽  
Author(s):  
YUICHI KANJIN ◽  
KUNIO SATO
Keyword(s):  
Analytic Function ◽  
Unit Disc ◽  
Jacobi Expansions

Let F(z) = [sum ]∞n=0anzn be an analytic function in the unit disc satisfying[formula here]Then [formula here], which is familiar as Paley's inequality. In this paper, an analogue of this inequality with respect to the Jacobi expansions is established.

scholarly journals Class of Analytic Function Related with Uniformly Convex and Janowski’s Functions

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Akhter Rasheed ◽  
Saqib Hussain ◽  
Muhammad Asad Zaighum ◽  
Maslina Darus
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Unit Disc ◽  
Extreme Points ◽  
Distortion Theorem ◽  
Open Unit ◽  
Uniformly Convex ◽  
Coefficient Estimates ◽  
Open Unit Disc

In this paper, we introduce a new subclass of analytic functions in open unit disc. We obtain coefficient estimates, extreme points, and distortion theorem. We also derived the radii of close-to-convexity and starlikeness for this class.

scholarly journals Norm of a Volterra Integral Operator on Some Analytic Function Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Hao Li ◽  
Songxiao Li
Keyword(s):  
Analytic Function ◽  
Integral Operator ◽  
Unit Disc ◽  
Function Spaces ◽  
Volterra Integral Operator ◽  
Analytic Function Spaces

Let f be an analytic function in the unit disc 𝔻. The Volterra integral operator If is defined as follows: If(h)(z)=∫0zf(w)h'(w)dw,h∈H(𝔻),z∈𝔻. In this paper, we compute the norm of If on some analytic function spaces.

scholarly journals Hankel determinant for certain class of analytic function defined by generalized derivative operator

2012 ◽  
Vol 43 (3) ◽  
pp. 445-453
Author(s):  
Ma'moun Harayzeh Al-Abbadi ◽  
Maslina Darus
Keyword(s):  
Analytic Function ◽  
Upper Bound ◽  
Unit Disc ◽  
Univalent Functions ◽  
Open Unit ◽  
Hankel Determinant ◽  
Open Unit Disc ◽  
Derivative Operator ◽  
Generalized Derivative ◽  
Generalised Derivatives

The authors in \cite{mam1} have recently introduced a new generalised derivatives operator $ \mu_{\lambda _1 ,\lambda _2 }^{n,m},$ which generalised many well-known operators studied earlier by many different authors. By making use of the generalised derivative operator $\mu_{\lambda_1 ,\lambda _2 }^{n,m}$, the authors derive the class of function denoted by $ \mathcal{H}_{\lambda _1 ,\lambda _2 }^{n,m}$, which contain normalised analytic univalent functions $f$ defined on the open unit disc $U=\left\{{z\,\in\mathbb{C}:\,\left| z \right|\,<\,1} \right\}$ and satisfy \begin{equation*}{\mathop{\rm Re}\nolimits} \left( {\mu _{\lambda _1 ,\lambda _2 }^{n,m} f(z)} \right)^\prime > 0,\,\,\,\,\,\,\,\,\,(z \in U).\end{equation*}This paper focuses on attaining sharp upper bound for the functional $\left| {a_2 a_4 - a_3^2 } \right|$ for functions $f(z)=z+ \sum\limits_{k = 2}^\infty {a_k \,z^k }$ belonging to the class $\mathcal{H}_{\lambda _1 ,\lambda _2 }^{n,m}$.

Radius of Limaçon starlikeness for Janowski starlike functions

2021 ◽  
Author(s):  
R. Kanaga ◽  
V. Ravichandran
Keyword(s):  
Analytic Function ◽  
Unit Disc ◽  
Starlike Functions ◽  
Starlike Function ◽  
Open Unit ◽  
Open Unit Disc ◽  
Class Of Functions

Let [Formula: see text] be an analytic function defined on the open unit disc [Formula: see text], with [Formula: see text], satisfying the subordination [Formula: see text], where [Formula: see text]. The domain [Formula: see text] is bounded by a Limaçon and the function [Formula: see text] is called starlike function associated with Limaçon domain. For [Formula: see text], we find the smallest disc [Formula: see text] and the largest disc [Formula: see text], centered at [Formula: see text] such that the domain [Formula: see text] is contained in [Formula: see text] and contains [Formula: see text]. By using this result, we find the radius of Limaçon starlikeness for the class of starlike functions of order [Formula: see text] [Formula: see text] and the class of functions [Formula: see text] satisfying [Formula: see text], [Formula: see text]. We give extension of our results for Janowski starlike functions.

scholarly journals Subclasses of Bi-Univalent Functions Defined by Frasin Differential Operator

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 783 ◽  
Author(s):  
Ibtisam Aldawish ◽  
Tariq Al-Hawary ◽  
B. A. Frasin
Keyword(s):  
Analytic Function ◽  
Differential Operator ◽  
Unit Disc ◽  
Univalent Functions ◽  
Function Class ◽  
Open Unit ◽  
Open Unit Disk ◽  
Open Unit Disc ◽  
Class A ◽  
Class Of Functions

Let Ω denote the class of functions f ( z ) = z + a 2 z 2 + a 3 z 3 + ⋯ belonging to the normalized analytic function class A in the open unit disk U = z : z < 1 , which are bi-univalent in U , that is, both the function f and its inverse f − 1 are univalent in U . In this paper, we introduce and investigate two new subclasses of the function class Ω of bi-univalent functions defined in the open unit disc U , which are associated with a new differential operator of analytic functions involving binomial series. Furthermore, we find estimates on the Taylor–Maclaurin coefficients | a 2 | and | a 3 | for functions in these new subclasses. Several (known or new) consequences of the results are also pointed out.

Some results of Lindelöf type involving the segmental behaviour of holomorphic functions

1993 ◽  
Vol 114 (1) ◽  
pp. 57-65 ◽  
Author(s):  
Francisca Bravo ◽  
Daniel Girela
Keyword(s):  
Analytic Function ◽  
Unit Disc ◽  
Slow Growth ◽  
Holomorphic Functions ◽  
Maximum Modulus ◽  
Classical Theorem ◽  
Asymptotic Value

AbstractA classical theorem of Lindelöf asserts that if ƒ is a function analytic and bounded in the unit disc δ which has the asymptotic value L at a point ξ ε ∂ δ then it has the non-tangential limit L at ξ. This result does not remain true for functions f analytic in δ whose maximum modulus grows to infinity arbitrarily slowly. However, the second author has recently obtained some results of Lindelöf type valid for these functions. In this paper we obtain new results of this kind. We prove that if f is an analytic function of slow growth in δ and ξ ε ∂ δ, then certain restrictions on the growth of ƒ′ along a segment which ends at ξ do imply that ƒ has a non-tangential limit at ξ.

scholarly journals On a subclass of analytic functions defined by Ruscheweyh operator

2012 ◽  
Vol 21 (1) ◽  
pp. 49-56
Author(s):  
ABDUL RAHMAN SALMAN JUMA ◽  
◽  
LUMINITA-IOANA COTIRLA ◽  
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Unit Disc ◽  
Inclusion Relation ◽  
Distortion Bounds ◽  
Ruscheweyh Derivative ◽  
Coefficient Condition ◽  
Necessary And Sufficient

By using the Ruscheweyh derivative, we have introduced a subclass of analytic functions with negative coefficients in the unit disc. Some properties of analytic function as necessary and sufficient coefficient condition for this class are provided. Distortion bounds, inclusion relation and various properties are also determined.

scholarly journals A CONVOLUTION APPROACH TO CERTAIN SUBCLASSES OF STARLIKE FUNCTIONS

1992 ◽  
Vol 23 (4) ◽  
pp. 311-320
Author(s):  
T . RAM REDDY ◽  
O. P. JUNEJA ◽  
K. SATHYANARAYANA
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Unit Disc ◽  
Starlike Functions ◽  
Integral Transforms ◽  
Open Unit ◽  
Sufficient Condition ◽  
Open Unit Disc ◽  
Containment Property ◽  
Convolution Approach

The class $R_\gamma(A,B)$ for $-1\le B < A\le 1$ and $\gamma> (A- 1)/(1- B)$ consisting of normalised analytic functions in the open unit disc is defined with the help of Convolution technique. It consists of univalent starlike functions for $\gamma\ge 0$. We establish containment property, integral transforms and a sufficient condition for an analytic function to be in $R\gamma(A,B)$. Using the concept of dual spaces we find a convolution condition for a function in this class.

scholarly journals Powers of p–valent functions

1978 ◽  
Vol 25 (1) ◽  
pp. 66-70 ◽  
Author(s):  
J. W. Noonan
Keyword(s):  
Analytic Function ◽  
Unit Disc ◽  
Recent Literature

AbstractIf f is areally mean p–valent in the unit disc, if λ>0 and if fλ is defined as a single-valued analytic function on the unit disc with finitely many arcs removed, several results in the recent literature suggest that fλ might be areally mean pλ–valent. The purpose of this note is to determine the valence of fλ when f is areally mean p–valent, and also to characterize those functions for which fλ is pλ–valent for all λ>0. Analogous results are obtained for functions which are either s–dimensionally mean p–valent or logarithmically mean p–valent.

Analytic Continuation of Power Series by Regular Generalized Weighted Means

1983 ◽  
Vol 26 (4) ◽  
pp. 454-463
Author(s):  
Bruce L. R. Shawyer ◽  
Ludwig Tomm
Keyword(s):  
Analytic Function ◽  
Power Series ◽  
Analytic Continuation ◽  
Unit Disc ◽  
Meromorphic Functions ◽  
Geometric Series ◽  
Summability Methods ◽  
Weighted Means ◽  
Simply Connected Region ◽  
Simply Connected

AbstractThe behaviour of summability transforms of power series outside their circles of convergence has been studied by many authors. In the case of the geometric series Luh [6] and Tomm [10] showed that there exist regular methods A which provide an analytic continuation into any given simply connected region G that contains the unit disc but not the point 1. Moreover, the Atransforms of the geometric series may be required to converge to any chosen analytic function on prescribed regions outside the unit circle. In this paper, these results are extended to power series representing other meromorphic functions. It is also shown that the summability methods involved may be chosen to be generalized weighted means previously introduced by Faulstich [1].

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