In this paper we establish the coefficient bodies for a wide class of families of inverse functions. We also completely describe those functions that provide boundary points of that bodies in small dimensions. As an application we get sharp bounds for Fekete--Szegö functionals over some classes of functions defined by quasi-subordination as well as over classes of their inverses. As a biproduct we derive a formula for ordinary Bell polynomials that seems to be new.
We study the Fekete--Szegö problem on the open unit ball of a complex Banach space. Namely, the Fekete--Szegö inequalities are proved for the class of spirallike mappings relative to an arbitrary strongly accretive operator, and some of its subclasses. Next, we consider families of non-linear resolvents for holomorphically accretive mappings vanishing at the origin. We solve the Fekete--Szegö problem over these families.
In this paper, we obtain Fekete-Szegö inequality for the generalized bi-subordinate functions of complex order. The results, which are presented in this study, would generalize those in related works of several earlier authors.
In the present investigation, we derive Fekete-Szegö inequality for the class $\mathcal{S}^α_{\mathscr{L}_{g}}(ϕ)$, introduced here. In addition to that, certain applications of our results are also discussed.
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 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.
We extend our definition (in a recent paper \cite{KB}) of the coefficient determinants of analytic mappings of the unit disk to include many Fekete-Szeg$\ddot{o}$-type parameters, and compute the best possible bounds on certain specific determinants for the choice class of starlike functions.
The main purpose of this paper is to introduce a new comprehensive subclass of analytic close-to-convex functions and derive Fekete-Szegö inequalities for functions belonging to this new class by using a different way. Various known special cases of our results are also pointed out.
In this paper, two new subclasses of bi-univalent functions related to conic domains are defined by making use of symmetric $q$-differential operator. The initial bounds for Fekete-Szegö inequality for the functions $f$ in these classes are estimated.
In the present work, we propose to investigate the Fekete-Szegö inequalities certain classes of analytic and bi-univalent functions defined by subordination. The results in the bounds of the third coefficient which improve many known results concerning different classes of bi-univalent functions. Some interesting applications of the results presented here are also discussed.
In this paper we study an interpolation problem involving the Fekete--Szegö functional. It turns out that this problem links to the so-called filtration of infinitesimal generators. We introduce new filtration classes using the non-linear differential operator \[α\frac{f(z)}{z}+β\frac{zf'(z)}{f(z)}+(1-α-β)\left(1+\frac{zf''(z)}{f'(z)}\right)\] and establish certain properties of these classes. Sharp upper bounds of the modulus of the Fekete--Szegö functional over some filtration classes are found. We also present open problems for further study.
In this sequel to the recent work (see Azizi et al., 2015), we investigate a subclass of analytic and bi-univalent functions in the open unit disk. We obtain bounds for initial coefficients, the Fekete-Szegö inequality and the second Hankel determinant inequality for functions belonging to this subclass. We also discuss some new and known special cases, which can be deduced from our results.
We give coefficient estimates for a class of close-to-convex harmonic mappings, and discuss the Fekete-Szegő problem of it. We also introduce two classes of polyharmonic mappings $\mathcal{HS}_{p}$ and $\mathcal{HC}_{p}$, consider the starlikeness and convexity of them, and obtain coefficient estimates on them. Finally, we give a necessary condition for a mapping $F$ to be in the class $\mathcal{HC}_{p}$.
In this paper, we investigate a new subclass of analytic and m-fold symmetric bi-univalent functions satisfying subordination in the open unit disk U. We consider the Fekete-Szegö inequalities for this class. Also, we establish estimates for the coefficients for this subclas and several related classes are also considered and connections to earlier known results are made.
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…
In this note we present a remark on the paper "On the coefficient inequalities for a class of holomorphic mappings associated with spirallike mappings in several complex variables" by Y.~Lai and Q.~Xu \cite{LX} published recently in the journal {\it Results in Mathematics}. We show that one of the theorems in \cite{LX} concerning the finite-dimensional space $\mathbb{C}^n$ is a direct consequence of another one, so it does not need an independent proof. Moreover, we prove that a sharp norm estimate on the Fekete--Szegö functional over spirallike mappings in a general Banach space can be deduce…
In the present article, our goal is finding estimates on the Taylor-Maclaurin coefficients |a2| and |a3| for a new class of bi-univalent functions defined by means of the Horadam polynomials. Fekete-Szego inequalities of functions belonging to this subclass are also founded.
In the present investigation, certain subclasses of close-to-convex functions are investigated. In particular, we obtain an estimate for the Fekete-Szegö functional for functions belonging to the class, distortion, growth estimates and covering theorems.
In this paper, we mainly study the order of $q$-starlikeness of the well-known basic hypergeometric function. In addition, we obtain the Bieberbach-type problem for a generalized class of starlike functions. We also discuss the Fekete-szegö and the Hankel determinant problems for the same class of functions.
Estimation of linear and quadratic functionals over different classes of univalent functions is one of the classical problems in geometric function theory. In this paper we solve the problem over some classes of so-called non-linear resolvents, which arise as a fruitful tool in dynamic systems. Sharp estimates on early Taylor coefficients and the Fekete--Szegö functional are established.
In this paper we study class $\mathcal{S}^+$ of univalent functions $f$ such that $\frac{z}{f(z)}$ has real and positive coefficients. For such functions we give estimates of the Fekete-Szegő functional and sharp estimates of their initial coefficients and logarithmic coefficients. Also, we present necessary and sufficient conditions for $f\in \mathcal{S}^+$ to be starlike of order $1/2$.
In this note, we discuss the coefficient regions of analytic self-maps of the unit disk with a prescribed fixed point. As an application, we solve the Fekete-Szegő problem for normalized concave functions with a prescribed pole in the unit disk.
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, for every $q\in(0,1)$, we obtain the Herglotz representation theorem and discuss the Bieberbach type problem for the class of $q$-convex functions of order $α, 0\leα<1$. In addition, we discuss the Fekete-szegö problem and the Hankel determinant problem for the class of $q$-starlike functions, leading to couple of conjectures for the class of $q$-starlike functions of order $α, 0\leα<1$.
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.
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.
We consider a family of analytic and normalized functions that are related to the domains $\mathbb{H}(s)$, with a right branch of a hyperbolas $H(s)$ as a boundary. The hyperbola $H(s)$ is given by the relation $\frac{1}ρ=\left( 2\cos\fracφ{s}\right)^s\quad (0<s\le 1,\ |φ|<(πs)/2$). We mainly study a coefficient problem of the families of functions for which $zf'/f$ or $1+zf''/f'$ map the unit disk onto a subset of $\mathbb{H}(s)$. We find coefficients bounds, solve Fekete-Szegö problem and estimate the Hankel determinant.
In this paper, we introduce and investigate a new subclass of bi-prestarlike functions defined in the open unit disk, associated with Chebyshev Polynomials. Furthermore, we find estimates of first two coefficients of functions in these classes, making use of the Chebyshev polynomials. Also, we obtain the Fekete-Szegö inequalities for function in these classes. Several consequences of the results are also pointed out as corollaries.
Our present investigation is motivated essentially by the fact that, in Geometric Function Theory, one can find many interesting and fruitful usages of a wide variety of special functions and special polynomials. The main purpose of this article is to make use of the Horadam polynomials and the generating function, in order to introduce three new subclasses of the bi-univalent function class. For functions belonging to the defined classes, we then derive coefficient inequalities and the Fekete Szego inequalities. Some interesting observations of the results presented here are also discussed. W…
Ma-Minda class (of starlike functions) consists of all normalized analytic functions $f$ on the unit disk for which the image of $zf'(z)/f(z)$ is contained in the some starlike region in the right-half plane. We obtain the best possible bounds on the second and third coefficient for the inverse functions of functions in the Ma-Minda class. The bounds on the Fekete-Szegö functional and the second Hankel determinant of the inverse functions of the functions belonging to the Ma-Minda class are also determined. Further, the bounds on the first five coefficients of the inverse functions are investi…
Recently, in their pioneering work on the subject of bi-univalent functions, Srivastava et al. \cite{HMS-AKM-PG} actually revived the study of the coefficient problems involving bi-univalent functions. Inspired by the pioneering work of Srivastava et al. \cite{HMS-AKM-PG}, there has been triggering interest to study the coefficient problems for the different subclasses of bi-univalent functions. Motivated largely by Ali et al. \cite{Ali-Ravi-Ma-Mina-class}, Srivastava et al. \cite{HMS-AKM-PG} and Güney et al. \cite{HOG-GMS-JS-Fib-2018} in this paper, we consider certain classes of bi-univalent…
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 linear operator due to Hohlov \cite{6}, a new subclass $\mathcal{R}_{a,b}^{c}$ 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 t…
In this paper, we show that the Carathéodory function $φ_{\scriptscriptstyle {Ne}}(z)=1+z-z^3/3$ maps the open unit disk $\mathbb{D}$ onto the interior of the nephroid, a $2$-cusped kidney-shaped curve, \begin{align*} \left((u-1)^2+v^2-\frac{4}{9}\right)^3-\frac{4 v^2}{3}=0, \end{align*} and introduce new Ma-Minda type function classes $\mathcal{S}^*_{Ne}$ and $\mathcal{C}_{Ne}$ associated with it. Apart from studying the characteristic properties of the region bounded by this nephroid, the structural formulas, extremal functions, growth and distortion results, inclusion results, coefficient b…
For $0<λ\le 1$, let $\mathcal{U}(λ)$ be the class analytic functions $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk $\mathbb{D}$ satisfying $|f'(z)(z/f(z))^2-1|<λ$ and $\mathcal{U}:=\mathcal{U}(1)$. In the present article, we prove that the class $\mathcal{U}$ is contained in the closed convex hull of the class of starlike functions and using this fact, we solve some extremal problems such as integral mean problem and arc length problem for functions in $\mathcal{U}$. By means of the so-called theory of star functions, we also solve the integral mean problem for functions in $\mathcal{U…
