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>

COEFFICIENT ESTIMATES FOR CERTAIN SUBCLASSES OF ANALYTIC FUNCTIONS ASSOCIATED WITH HORADAM POLYNOMIAL

2020 ◽  
Vol 9 (12) ◽  
pp. 10091-10102
Author(s):  
D. Kavitha ◽  
K. Dhanalakshmi ◽  
N. Arulmozhi
Keyword(s):  
Present Article ◽  
Unit Disk ◽  
Univalent Function ◽  
Analytic Functions ◽  
Function Class ◽  
Open Unit ◽  
Open Unit Disk ◽  
The Novel ◽  
Coefficient Estimates

In this present article, we studied and examined the novel general subclasses of the function class $\Sigma$ of bi-univalent function defined in the open unit disk, which are associated with the Horadam polynomial. This study locates estimates on the Taylor - Maclaurin coefficients $|a_{2}|$ {\it and} $|a_{3}|$ in functions of the class which are considered. Additionally, Fekete-Szeg\"{o} inequality of functions belonging to this subclasses are also obtained.

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 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 of Certain Subclasses of Bi-Bazilevic Functions Associated with Chebyshev Polynomials and Mittag-Leffler Function

2020 ◽  
pp. 365-376
Author(s):  
Adeniyi Musibau Gbolagade ◽  
Ibrahim Tunji Awolere
Keyword(s):  
Unit Disk ◽  
Chebyshev Polynomials ◽  
Univalent Functions ◽  
Function Class ◽  
Open Unit ◽  
Open Unit Disk ◽  
Coefficient Estimates ◽  
Bazilevic Functions

In this present investigation, the authors introduced certain subclasses of the function class $ T^{\alpha}_{\theta}(\lambda, \beta, t)$ of bi-Bazilevic univalent functions defined in the open unit disk $U$, which are associated with Chebyshev polynomials and Mittag-Leffler function. We establish the Taylor Maclaurin coefficients $\left|a_{2}\right|$, $\left|a_{3}\right|$ and $\left|a_{4}\right|$ for functions in the new subclass introduced and the Fekete-Szego problem is solved.

scholarly journals New Specific and General Linearization Formulas of Some Classes of Jacobi Polynomials

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 74
Author(s):  
Waleed Mohamed Abd-Elhameed ◽  
Afnan Ali
Keyword(s):  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Algebraic Computation ◽  
Ultraspherical Polynomials ◽  
Current Article ◽  
Linearization Coefficients

The main purpose of the current article is to develop new specific and general linearization formulas of some classes of Jacobi polynomials. The basic idea behind the derivation of these formulas is based on reducing the linearization coefficients which are represented in terms of the Kampé de Fériet function for some particular choices of the involved parameters. In some cases, the required reduction is performed with the aid of some standard reduction formulas for certain hypergeometric functions of unit argument, while, in other cases, the reduction cannot be done via standard formulas, so we resort to certain symbolic algebraic computation, and specifically the algorithms of Zeilberger, Petkovsek, and van Hoeij. Some new linearization formulas of ultraspherical polynomials and third-and fourth-kinds Chebyshev polynomials are established.

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.

scholarly journals Second Hankel determinant for certain class of bi-univalent functions defined by Chebyshev polynomials

2019 ◽  
Vol 12 (02) ◽  
pp. 1950017
Author(s):  
H. Orhan ◽  
N. Magesh ◽  
V. K. Balaji
Keyword(s):  
Unit Disk ◽  
Upper Bound ◽  
Chebyshev Polynomials ◽  
Univalent Functions ◽  
Function Class ◽  
Open Unit ◽  
Open Unit Disk ◽  
Hankel Determinant ◽  
Second Hankel Determinant ◽  
Upper Bound Estimate

In this work, we obtain an upper bound estimate for the second Hankel determinant of a subclass [Formula: see text] of analytic bi-univalent function class [Formula: see text] which is associated with Chebyshev polynomials in the open unit disk.

Coefficient estimates for certain analytic functions satisfying subordinate condition

2017 ◽  
Author(s):  
Erhan Deniz ◽  
Yekta Gülsün ◽  
Nizami Mustafa
Keyword(s):  
Analytic Functions ◽  
Coefficient Estimates

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 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.

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