scholarly journals Applications of Chebyshev polynomials on a Sakaguchi type class of analytic functions associated with quasi-subordination

2018 ◽  
Vol 22 (1) ◽  
pp. 49-56
Author(s):  
Emmanuel Jesuyon Dansu ◽  
Sunday Oluwafemi Olatunji
Keyword(s):  
Chebyshev Polynomials ◽  
Analytic Functions ◽  
Coefficient Estimates

We introduce a class of analytic functions which is defined in terms of a quasi-subordination. Coefficient estimates including the relevant classical Fekete–Szegö inequality of functions belonging to the aforementioned class are derived. Improved results for associated classes involving subordination and majorization are also discussed.

scholarly journals Coefficient estimates and Fekete-Szegö inequality for a class of analytic functions satisfying subordinate condition associated with Chebyshev polynomials

2019 ◽  
Vol 11 (2) ◽  
pp. 430-436
Author(s):  
Eszter Szatmari ◽  
Şahsene Altinkaya
Keyword(s):  
Chebyshev Polynomials ◽  
Analytic Functions ◽  
Coefficient Estimates

Abstract In this paper, we define a class of analytic functions, ℱ(ℋ, α, δ, µ), satisfying the following condition \left( {\alpha {{\left[ {{{{\rm{zf'}}({\rm{z}})} \over {{\rm{f}}(z)}}} \right]}^\delta } + (1 - \alpha ){{\left[ {{{{\rm{zf'}}\left( {\rm{z}} \right)} \over {{\rm{f}}(z)}}} \right]}^\mu }{{\left[ {1 + {{{\rm{zf''}}({\rm{z}})} \over {{\rm{f'}}({\rm{z}})}}} \right]}^{1 - \mu }}} \right)\,\, \prec \mathcal{H}({\rm{z}},{\rm{t}}), where α ∈ [0, 1], δ ∈ [1, 2] and µ ∈ [0, 1]. We give coefficient estimates and Fekete-Szegö inequality for this class.

scholarly journals A study of the Fekete-Szegö functional and coefficient estimates for subclasses of analytic functions satisfying a certain subordination condition and associated with the Gegenbauer polynomials

2021 ◽  
Vol 7 (2) ◽  
pp. 2568-2584
Author(s):  
H. M. Srivastava ◽  
◽  
Muhammet Kamalı ◽  
Anarkül Urdaletova ◽  
◽  
...  
Keyword(s):  
Chebyshev Polynomials ◽  
Analytic Functions ◽  
Function Class ◽  
Gegenbauer Polynomials ◽  
Coefficient Estimates ◽  
Ultraspherical Polynomials

<abstract><p>In this paper, we introduce and study a new subclass of normalized analytic functions, denoted by</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \mathcal F_{\left(\beta,\gamma\right)} \bigg(\alpha,\delta,\mu,H\big(z,C_{n}^{\left(\lambda \right)} \left(t\right)\big)\bigg), $\end{document} </tex-math></disp-formula></p> <p>satisfying the following subordination condition and associated with the Gegenbauer (or ultraspherical) polynomials $ C_{n}^{\left(\lambda\right)}(t) $ of order $ \lambda $ and degree $ n $ in $ t $:</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \alpha \left(\frac{zG^{'}\left(z\right)}{G\left(z\right)} \right)^{\delta}+\left(1-\alpha\right)\left(\frac{zG^{'} \left(z\right)}{G\left(z\right)}\right)^{\mu} \left(1+\frac{zG^{''}\left(z\right)}{G^{'} \left(z\right)} \right)^{1-\mu} \prec H\big(z,C_{n}^{\left(\lambda\right)} \left(t\right)\big), $\end{document} </tex-math></disp-formula></p> <p>where</p> <p><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ H\big(z,C_{n}^{\left(\lambda\right)}\left(t\right)\big) = \sum\limits_{n = 0}^{\infty} C_n^{(\lambda)}(t)\;z^n = \left(1-2tz+z^2\right)^{-\lambda}, $\end{document} </tex-math></disp-formula></p> <p><disp-formula> <label/> <tex-math id="FE4"> \begin{document}$ G\left(z\right) = \gamma \beta z^{2} f^{''} \left(z\right)+\left(\gamma-\beta \right)zf^{'} \left(z\right)+\left(1-\gamma+\beta\right)f\left(z\right), $\end{document} </tex-math></disp-formula></p> <p>$ 0\leqq \alpha \leqq 1, $ $ 1\leqq \delta \leqq 2, $ $ 0\leqq \mu \leqq 1, $ $ 0\leqq \beta \leqq \gamma \leqq 1 $, $ \lambda \geqq 0 $ and $ t\in \left(\frac{1}{\sqrt{2}}, 1\right] $. For functions in this function class, we first derive the estimates for the initial Taylor-Maclaurin coefficients $ \left|a_{2}\right| $ and $ \left|a_{3}\right| $ and then examine the Fekete-Szegö functional. Finally, the results obtained are applied to subclasses of normalized analytic functions satisfying the subordination condition and associated with the Legendre and Chebyshev polynomials. The basic or quantum (or $ q $-) calculus and its so-called trivially inconsequential $ (p, q) $-variations have also been considered as one of the concluding remarks.</p></abstract>

scholarly journals Coefficient estimates of certain subclasses of analytic functions associated with Hohlov operator

2019 ◽  
pp. 2150021
Author(s):  
P. Gochhayat ◽  
A. Prajapati ◽  
A. K. Sahoo
Keyword(s):  
Upper Bound ◽  
Analytic Functions ◽  
Gauss Hypergeometric Function ◽  
Open Unit ◽  
Sufficient Condition ◽  
Extremal Functions ◽  
Open Unit Disk ◽  
Hankel Determinant ◽  
Coefficient Estimates ◽  
Extremal Value

A typical quandary in geometric functions theory is to study a functional composed of amalgamations of the coefficients of the pristine function. Conventionally, there is a parameter over which the extremal value of the functional is needed. The present paper deals with consequential functional of this type. By making use of Hohlov operator, a new subclass [Formula: see text] of analytic functions defined in the open unit disk is introduced. For both real and complex parameter, the sharp bounds for the Fekete–Szegö problems are found. An attempt has also been taken to found the sharp upper bound to the second and third Hankel determinant for functions belonging to this class. All the extremal functions are express in term of Gauss hypergeometric function and convolution. Finally, the sufficient condition for functions to be in [Formula: see text] is derived. Relevant connections of the new results with well-known ones are pointed out.

scholarly journals Coefficient Estimates for a Subclass of Analytic Functions Associated with a Certain Leaf-Like Domain

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1334
Author(s):  
Bilal Khan ◽  
Hari M. Srivastava ◽  
Nazar Khan ◽  
Maslina Darus ◽  
Muhammad Tahir ◽  
...  
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Second Order ◽  
Open Unit ◽  
Open Unit Disk ◽  
Hankel Determinant ◽  
Coefficient Estimates ◽  
Principle Of Subordination

First, by making use of the concept of basic (or q-) calculus, as well as the principle of subordination between analytic functions, generalization Rq(h) of the class R(h) of analytic functions, which are associated with the leaf-like domain in the open unit disk U, is given. Then, the coefficient estimates, the Fekete–Szegö problem, and the second-order Hankel determinant H2(1) for functions belonging to this class Rq(h) are investigated. Furthermore, similar results are examined and presented for the functions zf(z) and f−1(z). For the validity of our results, relevant connections with those in earlier works are also pointed out.

Coefficient Estimates of Some Classes of Analytic Functions

1983 ◽  
Vol 26 (2) ◽  
pp. 202-208
Author(s):  
Nicolas Samaris
Keyword(s):  
Analytic Functions ◽  
Positive Real Part ◽  
Coefficient Estimates ◽  
Positive Real ◽  
Real Functions ◽  
Typically Real ◽  
Typically Real Functions

AbstractWe are concerned with coefficient estimates, and other similar problems, of the typically real functions and of the functions with positive real part. Following the stream of ideas in [1], new results and generalizations of others given in [1], [2] and [3] are obtained.

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 Faber Polynomial Coefficient Estimates for Bi-Univalent Functions Defined by Using Differential Subordination and a Certain Fractional Derivative Operator

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 172 ◽  
Author(s):  
Hari M. Srivastava ◽  
Ahmad Motamednezhad ◽  
Ebrahim Analouei Adegani
Keyword(s):  
Fractional Derivative ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Upper Bounds ◽  
Differential Subordination ◽  
Open Unit ◽  
Open Unit Disk ◽  
Coefficient Estimates ◽  
Derivative Operator ◽  
Faber Polynomial

In this article, we introduce a general family of analytic and bi-univalent functions in the open unit disk, which is defined by applying the principle of differential subordination between analytic functions and the Tremblay fractional derivative operator. The upper bounds for the general coefficients | a n | of functions in this subclass are found by using the Faber polynomial expansion. We have thereby generalized and improved some of the previously published results.

scholarly journals Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators, and certain subclasses of analytic functions

1987 ◽  
Vol 106 ◽  
pp. 1-28 ◽  
Author(s):  
H. M. Srivastava ◽  
Shigeyoshi Owa
Keyword(s):  
Fractional Calculus ◽  
Analytic Functions ◽  
Hypergeometric Functions ◽  
Linear Operators ◽  
Coefficient Estimates ◽  
Hadamard Products ◽  
Generalized Hypergeometric Functions ◽  
Hypergeometric Type ◽  
Distortion Theorems ◽  
Characterization Theorems

By using a certain linear operator defined by a Hadamard product or convolution, several interesting subclasses of analytic functions in the unit disk are introduced and studied systematically. The various results presented here include, for example, a number of coefficient estimates and distortion theorems for functions belonging to these subclasses, some interesting relationships between these subclasses, and a wide variety of characterization theorems involving a certain functional, some general functions of hypergeometric type, and operators of fractional calculus. Some of the coefficient estimates obtained here are fruitfully applied in the investigation of certain subclasses of analytic functions with fixed finitely many coefficients.

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