Let $Ω$ denote the class of functions $f$ analytic in the open unit disc $Δ$, normalized by the condition $f(0)=f'(0)-1=0$ and satisfying the inequality \begin{equation*}
\left|zf'(z)-f(z)\right|<\frac{1}{2}\quad(z\inΔ). \end{equation*} The class $Ω$ was introduced recently by Peng and Zhong (Acta Math Sci {\bf37B(1)}:69--78, 2017). Also let $\mathcal{U}$ denote the class of functions $f$ analytic and normalized in $Δ$ and satisfying the condition \begin{equation*}
\left|\left(\frac{z}{f(z)}\right)^2f'(z)-1\right|<1\quad(z\inΔ). \end{equation*} In this article, we obtain some further results for the class $Ω$ including, an extremal function and more examples of $Ω$, inclusion relation between $Ω$ and $\mathcal{U}$, the radius of starlikeness, convexity and close--to--convexity and sufficient condition for function $f$ to be in $Ω$. Furthermore, along with the settlement of the coefficient problem and the Fekete--Szegö problem for the elements of $Ω$, the Toeplitz matrices for $Ω$ are also discussed in this article.
In this work, considering a general subclass of bi-univalent functions and using the Chebyshev polynomials, we obtain coefficient expansions for functions in this class.
In this paper, we define certain subclass of harmonic univalent function in the unit disc U = {z in C :|z|<1} by using q-differential operator. Also we obtain coefficient inequalities, growth and distortion theorems for this subclass.
In the present paper, we investigate connections between various subclasses of harmonic univalent functions by using a convolution operator involving the Pascal distribution series.
In this paper, we introduce the subclass $SHP^{-m}(α,β)$ using integral operator and give sufficient coefficient conditions for normalized harmonic univalent function in the subclass $SHP^{-m}(α,β)$.These conditions are also shown to be necessary when the coefficients are negative.
In this paper, we consider a general subclass of analytic and bi-univalent functions in the open unit disk in the complex plane. Making use of the Chebyshev polynomials, we obtain upper bound estimate for the second Hankel determinant for this function class.
The purpose of the present paper is to find the necessary and sufficient conditions for the subclasses of analytic functions associated with Pascal distribution to be in subclasses of spiral-like univalent functions and inclusion relations for such subclasses in the open unit disk $\mathbb{D}.$ Further, we consider the properties of integral operator related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.
In this paper we determine the disks $|z|<r\le1$ where for different classes of univalent functions, we have the property $${\rm Re}\left\{2\frac{zf'(z)}{f(z)}-\frac{z f''(z)}{f'(z)}\right\}>0\qquad (|z|<r).$$
In this work, we introduce two new subclasses S_{Σ_{m}}(α,λ) and S_{Σ_{m}}(\b{eta},λ) of Σ_{m} consisting of analytic and m-fold symmetric bi-univalent functions in the open unit disk U. Furthermore, for functions in each of the subclasses introduced in this paper, we obtain the coefficient bounds for |a_{m+1}| and |a_{2m+1}|.
In the present paper, we introduce a new subclass of harmonic functions in the unit disc U defined by using the generalized Mittag-Leffler type functions. Coefficient conditions, extreme points, distortion bounds, convex combination are studied.
In the present paper, we define a new general subclass of bi-univalent functions involving a differential operator in the open unit disk U. For this purpose, we use the Faber polynomial expansions. Several connections to some of the earlier known results are also pointed out.
Our objective in this paper is to introduce and investigate a newly-constructed subclass of normalized analytic and bi-univalent functions by means of the Chebyshev polynomials of the second kind. Upper bounds for the second and third Taylor-Maclaurin coefficients, and also Fekete-Szego inequalities of functions belonging to this subclass are founded. Several connections to some of the earlier known results are also pointed out.
The main object of the present paper is to, introduce the. class of meromorphic univalent functions Involving! hypergeomatrc function .We obtain~ some interesting geometric properties according to coefficient inequality , growth and distortion theorems , radii of starlikeness and convexity for the" functions in our subclass.
In this paper we introduce and investigate two new subclasses of the function class $Σ$ of bi-univalent functions of complex order defined in the open unit disk, which are associated with the Hohlov operator, and satisfying subordinate conditions. 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 these results are also pointed out.
In the present article, we define a new subclass of pseudo-type meromorphic bi-univalent functions class $Σ'$ of complex order $γ\in \mathbb{C}\backslash \{0\}$ and investigate the initial coefficient estimates $|b_0|, |b_1|$ and $|b_2|.$ Further we mention several new or known consequences of our result.
Our objective in this paper is to introduce and investigate comprehensive-constructed subclasses of normalized analytic and bi-univalent functions on the unit open disc. Bounds for the second and third Tayler-Maclaurin coefficients of functions belonging to this subclasses were investigated. Furthermore, some improvement and connections to some of the previous known results are also pointed out.
In this paper, new class of bi-univalent functions are introduced. Upper bound of the second Hankel determinant $|H_2(2)|$ of subclass of bi-univalant functions class $Σ$, which defined by subordination, investigated. Furthermore, some results concluded as a special case of our main results and corrected some previous researchers results which investigated by miscalculation.
In this paper, we introduce and investigate a novel class of analytic and univalent functions of negative coefficients in the open unit disk. For this function class, we obtain characterization and distortion theorems as well as the radii of close-to-convexity, starlikeness and convexity by using fractional calculus techniques.
In this paper our purpose is to find upper bound estimate for the second Hankel determinant $|a_{2}a_{4}-a_{3}^{2}|$ for functions defined by convolution belonging to the class $\mathcal{N}_σ^{μ,δ}(λ,t)$ by using Chebyshev polynomials.
In the present paper, a subclass of analytic and bi-univalent functions by means of (p; q)- Lucas polynomials is introduced. Certain coefficients bounds for functions belonging to this subclass are obtained. Furthermore, the Fekete-Szego problem for this subclass is solved.
In this work, we consider certain class of bi-univalent functions related with shell-like curves related to $κ-$Fibonacci numbers. Further, we obtain the estimates of initial Taylor-Maclaurin coefficients (second and third coefficients) and Fekete - Szegö inequalities. Also we discuss the special cases of the obtained results.
In this paper, a comprehensive subclass of bi-univalent functions class are introduced and investigated. Using the Faber polynomials, estimation of the coefficients $|a_n|$ and certain Fekete-Szegö inequality of Maclaurin expansion of functions in this subclass are concluded. Finally, some earlier results are pointed out and improved.
In the present paper, a new subclass of analytic and bi-univalent functions by means of Chebyshev polynomials is introduced. Certain coefficient bounds for functions belong to this subclass are obtained. Furthermore, the Fekete-Szego problem in this subclass is solved.
In this article, a subclass of univalent harmonic mapping is introduced by restricting its analytic part to lie in the class $\mathcal{S}^δ[α]$, $0\leq α< 1$, $-\infty < δ< \infty$ which has been introduced and studied by Kumar \cite{Kumar87} (see also \cite{Mishra95}, \cite{MishraChoudhury95}, \cite{MishraDas96}, \cite{MishraGochhayat06}). Coefficient estimations, growth and distortion properties, area theorem and covering estimates of functions in the newly defined class have been established. Furthermore, we also found bounds for the Bloch's constant for all functions in that family.
In this paper, we obtain the Fekete-Szegö problem for the $k$-th $(k\geq1)$ root transform of the analytic and normalized functions $f$ satisfying the condition \begin{equation*} 1+\frac{α-π}{2 \sin α}< {\rm Re}\left\{\frac{zf'(z)}{f(z)}\right\} < 1+\fracα{2\sin α} \quad (|z|<1), \end{equation*} where $π/2\leq α<π$. Afterwards, by the above two-sided inequality we introduce and investigate a certain subclass of analytic and bi-univalent functions in the disk $|z|<1$ and obtain upper bounds for the first few coefficients and Fekete-Szegö problem for functions belonging to this analytic and bi-u…
Inspired by the recent works of Srivastava et al. (2010), Frasin and Aouf (2011), and Caglar et al. (2013), we introduce and investigate in the present paper two new general subclasses of the class consisting of normalized analytic and bi-univalent functions in the open unit disk U. For functions belonging to these general subclasses introduced here, we obtain estimates on the Taylor-Maclaurin coefficients |a_2| and |a_3|. Several connections to some of the earlier known results are also pointed out. The results presented in this paper would generalize and improve those in related works of sev…
Let $\mathbb{D}:=\{z\in \mathbb{C}: |z|<1\}$ be the unit disk. For $0<α<1$, let $f_α(z)=z/(1-z^α)$ for $z \in \mathbb{D}$. We consider the class $\mathcal{F}$ of analytic functions $f_α$ which satisfy $\Re \left(1+zf"_α(z)/f'_α(z)\right) > β$ for $0<β<1$. In this paper, we determine the region of variability of $\log f'_α(z_0)$ for fixed $z_{0} \in \mathbb{D}$ when $f$ varies over the class ${\mathcal F}(λ):=\{f_α \in \mathcal{F}: f_α(0)=0, f'_α(0)=1 \, \mbox{and} \, f"_α(0)=2λ(1-β) \,\,\, \mbox{for} \,\, 0\leq λ\leq 1\}$.
In this paper, we introduce and investigate a new subclass of the function class $Σ$ of bi-univalent functions defined in the open unit disk, which are associated with the Sălăgean type $q-$ difference operator and satisfy some subordination conditions. Furthermore, we find estimates on the Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ for functions in the new subclass introduced here. Several (known or new) consequences of the results are also pointed out. Further we obtain Fekete-Szeg$\ddot{o}$ inequality for the new function class.
The purpose of the present paper is to establish connections between various subclasses of analytic univalent functions by applying certain convolution operator involving Touchard polynomials. To be more precise, we investigate such connections with the classes of analytic univalent functions with positive coefficients in the open unit disk U.
In this study, a subclass of an univalent function with negative coefficients which is defined by a new general Linear operator have been introduced. The sharp results for coefficients estimators, distortion and closure bounds, Hadamard product, and Neighborhood, and this paper deals with the utilizing of many of the results for classical hypergeometric function, where there can be generalized to m-hypergeometric functions. A subclasses of univalent functions are presented, and it has involving operator which generalizes many well-known. Denote A the class of functions f and we have other resu…
Making use of Chebyshev polynomials, we obtain upper bound estimate for the second Hankel determinant of a subclass $\mathcal{N}_{σ}^μ\left( λ,t\right) $ of bi-univalent function class $σ.$
Introducing a new method we give sharp estimates of the Hermitian Toepliz determinants of third order for the class $\mathcal{S}$ of functions univalent in the unit disc. The new approach is also illustrated on some subclasses of the class $\mathcal{S}$.
Sharp upper and lower bounds for the second and third order Hermitian-Toepilitz determinants are obtained for some generalized subclasses of starlike and convex functions. Applications of these results are also discussed for several widely known classes.
Estimates are obtained for the initial coefficients of a normalized analytic function $f$ in the unit disk $\mathbb{D}$ such that $f$ and the analytic extension of $f^{-1}$ to $\mathbb{D}$ belong to certain subclasses of univalent functions. The bounds obtained improve some existing known bounds.
The purpose of the present paper is to introduce a new subclasses of the function class of bi-univalent functions defined in the open unit disc. Furthermore, we obtain estimates on the coefficients $|a_{2}|$ and $|a_{3}|$ for functions of this class. Some results related to this work will be briefly indicated.
Let $\mathcal{V}_p(λ)$ be the collection of all functions $f$ defined in the unit disc $\ID$ having a simple pole at $z=p$ where $0<p<1$ and analytic in $\ID\setminus\{p\}$ with $f(0)=0=f'(0)-1$ and satisfying the differential inequality $|(z/f(z))^2 f'(z)-1|< λ$ for $z\in \ID$, $0<λ\leq 1$. Each $f\in\mathcal{V}_p(λ)$ has the following Taylor expansion: $$ f(z)=z+\sum_{n=2}^{\infty}a_n(f) z^n, \quad |z|<p. $$ In \cite{BF-3}, we conjectured that $$ |a_n(f)|\leq \frac{1-(λp^2)^n}{p^{n-1}(1-λp^2)}\quad \mbox{for}\quad n\geq3. $$ In the present article, we first obtain a representation formula fo…
In the present paper, we were mainly concerned with obtaining estimates for the general Taylor-Maclaurin coefficients for functions in a certain general subclass of analytic bi-univalent functions. For this purpose, we used the Faber polynomial expansions. Several connections to some of the earlier known results are also pointed out.
We consider a family of all analytic and univalent functions in the unit disk of the form $f(z)=z+a_2z^2+a_3z^3+\cdots$. The aim of this article is to investigate the bounds of the difference of moduli of initial successive coefficients, i.e. $\big | |a_{n+1}|-|a_n|\big |$ for $n=1,\,2$ and for some subclasses of analytic univalent functions. We found that all the estimations are sharp in nature by constructing some extremal functions.
Let $\mathcal{S}$ denote the family of all functions that are analytic and univalent in the unit disk $\mathbb{D}:=\{z: |z|<1\}$ and satisfy $f(0)=f^{\prime}(0)-1=0$. In the present paper, we consider certain subclasses of univalent functions associated with the exponential function, and obtain the sharp upper bounds on the initial coefficients and the difference of initial successive coefficients for functions belonging to these classes.
