Y. Polatoglu, M. Bolcal
The purpose of this paper is to give exact value of the radius of starlikeness and distortion theorem, Koebe domain for the class of Janowski's starlike functions of complex order. We note that the class of Janowski's starlike functions of complex order contain many interesting subclasses of univalent functions.
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For $-1\leq B<A\leq 1$, let $\mathcal{S}^*(A,B)$ denote the class of Janowski starlike functions which satisfy the subordination relation $zf'(z)/f(z)\prec (1+Az)/(1+Bz)$. In the present article, we determine the sharp of pre-Schwarzian norm for the functions in the class $\mathcal{S}^*(A,B)$.
Recently, authors [7] studied the logarithmic coefficient bounds for class of the Janowski type $(j,k)$-symmetric starlike functions $\mathcal{ST}_{[j,k]}(A,B)$ in ({\em Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.} (2022). DOI: 10.1007/s13398-022-01310-9). We pointed out that the proof of Theorem~3 in [7] is incorrect. In this article, we present the correct proof of Theorem~3. In addition, we also obtain some new results related to the logarithmic coefficient inequalities for the class $\mathcal{ST}_{[j,k]}(A,B)$.
Let A,B,D,E belong to [-1, 1] and let p(z) be an analytic function with fixed initial coefficient defined in the open unit disk. Conditions on A,B,D,and E are determined so that 1+αzp'(z) being subordinated to (1+Dz)/(1+Ez) implies that p(z) is subordinated to (1+Az)/(1+Bz) and other similar implications involving 1+αzp'(z)/p(z), αp2(z)+λzp'(z),αp(z)+(1-α)p2(z)+λzp'(z),and (1-α)p(z)+{alpha}(1+zp'(z)/p(z)). Also, sufficient conditions for Janowski starlikeness with fixed second coefficient are obtained.
The function $G_α(z)=1+ z/(1-αz^2)$, \, $0\leq α<1$, maps the open unit disc $\mathbb{D}$ onto the interior of a domain known as the Booth lemniscate. Associated with this function $G_α$ is the recently introduced class $\mathcal{BS}(α)$ consisting of normalized analytic functions $f$ on $\mathbb{D}$ satisfying the subordination $zf'(z)/f(z) \prec G_α(z)$. Of interest is its connection with known classes $\mathcal{M}$ of functions in the sense $g(z)=(1/r)f(rz)$ belongs to $\mathcal{BS}(α)$ for some $r$ in $(0,1)$ and all $f \in \mathcal{M}$. We find the largest radius $r$ for different classes…
Theory of differential subordination provides techniques to reduce differential subordination problems into verifying some simple algebraic condition called admissibility condition. We exploit the first order differential subordination theory to get several sufficient conditions for function satisfying several differential subordinations to be a Janowski function with positive real part. As applications, we obtain sufficient conditions for normalized analytic functions to be Janowski starlike functions.
We are interested in finding the sufficient conditions on $A$, $B$, $λ$, $b$ and $c$ which ensure that the generalized Bessel functions ${u}_λ:={u}_{λ,b,c}$ satisfies the subordination ${u}_λ(z) \prec (1+Az)/ (1+Bz)$. Also, conditions for which ${u}_λ(z)$ to be Janowski convex, and $z{u}'_λ(z)$ to be Janowski starlike in the unit disk $\mathbb{D}=\{z \in \mathbb{C}: |z|<1\}$ are obtained.
Sufficient conditions on $A$, $B$, $p$, $b$ and $c$ are determined that will ensure the generalized Bessel functions ${u}_{p,b,c}$ satisfies the subordination ${u}_{p,b,c}(z) \prec (1+Az)/ (1+Bz)$. In particular this gives conditions for $(-4κ/c)({u}_{p,b,c}(z)-1)$, $c \neq 0$ to be close-to-convex. Also, conditions for which ${u}_{p,b,c}(z)$ to be Janowski convex, and $z{u}_{p,b,c}(z)$ to be Janowski starlike in the unit disk $\mathbb{D}=\{z \in \mathbb{C}: |z|<1\}$ are obtained.
For $ -1 \leq B \leq 1$ and $A>B$, let $\mathcal{S}^*[A,B]$ denote the class of generalized Janowski starlike functions consisting of all normalized analytic functions $f$ defined by the subordination $z f'(z)/f(z) \prec (1+ A z)/(1+ B z)$ $(|z|<1)$. For $-1 \leq B \leq 1<A$, we investigate the inverse coefficient problem for functions in the class $\mathcal{S}^*[A,B]$ and its meromorphic counter part. Also, for $ -1 \leq B \leq 1 < A $, the sharp bounds for first five coefficients for inverse functions of generalized Janowski convex functions are determined. A simple and precise proof for inv…
Some sufficient conditions on certain constants which are involved in some first, second and third order differential subordinations associated with certain functions with positive real part like modified Sigmoid function, exponential function and Janowski function are obtained so that the analytic function p normalized by the condition p(0) = 1, is subordinate to Janowski function. The admissibility conditions for Janowski function are used as a tool in the proof of the results. As application, several sufficient conditions are also computed for Janowski starlikeness.
Let $h$ be a non-vanishing analytic function in the open unit disc with $h(0)=1$. Consider the class consisting of normalized analytic functions $f$ whose ratios $f(z)/g(z)$, $g(z)/z p(z)$, and $p(z)$ are each subordinate to $h$ for some analytic functions $g$ and $p$. The radius of starlikeness is obtained for this class when $h$ is chosen to be either $h(z)=\sqrt{1+z}$ or $h(z)=e^z$. Further $\mathcal{G}$-radius is also obtained for each of these two classes when $\mathcal{G}$ is a particular widely studied subclass of starlike functions. These include $\mathcal{G}$ consisting of the Janowsk…
Sharp Pre-Schwarzian Norm Bounds for Ma-Minda Starlike Classes
本文建立统一框架计算 Ma-Minda 星形类的预 Schwarz 范数，给出直接通用方法，推导出经典和新引入星形子类的显式范数公式。
Starlikeness Using Special Functions and Subordination
本文结合特殊函数与从属关系给出参数β的精确估计，并利用容许性建立高阶微分从属关系，进而证明多类解析函数的星形性。
The main purpose of the present paper is to ascertain the radii of starlikeness and convexity associated with lemniscate of Bernoulli and the Janowski function, $(1+Az)/(1+Bz)$ for $-1\leq B<A\leq 1,$ of normalized hyper-Bessel function.
The main purpose of the present paper is to determine the radii of starlikeness and convexity associated with lemniscate of Bernoulli and the Janowski function, $(1+Az)/(1+Bz)$ for $-1\leq B<A\leq 1,$ of normalized Wright functions. The key tools in the proof of our main results are the infinite product representation of Wright function and properties of real zeros of the Wright function and its derivative.
We investigate certain properties of tilted (oblique) domains, associated with the Janowski function $(1+Az)/(1+Bz),$ where $A,B\in\mathbb{C}$ with $A\neq B$ and $|B|\leq 1$. We find several bounds for these oblique domains. Further, we extend the various known argument, radius, and subordination results involving Janowski functions with complex parameters.
On the localization of the poles of the best Mobius approximations of f
本文研究单位圆盘上局部单叶函数的最佳Möbius逼近极点定位，利用Pommerenke线性不变阶建立尖锐几何界，并分析多类函数的极点行为，获得精确重数结果及新凸性条件。
Generalized Toeplitz determinants for Starlike Mappings in Several Complex Variables
本文研究了几复变量中星形映射的广义托普利茨行列式，推导了相关精确界并扩展到复巴拿赫空间和$\mathbb{C}^n$中的圆域。
Generalized Zalcman Conjecture for Starlike Mappings in Several Complex Variables
本文研究了几复变量中星形映射的推广Zalcman猜想，扩展了Ma提出的不等式至多复变量情形，并探讨了其在复Banach空间和Cⁿ中的应用。
Zalcman Conjecture for Starlike Mappings in Higher Dimensions
本文研究了高维星形映射中Zalcman猜想的验证，通过获得精确界确认了该猜想在n=3情况下的正确性。
Third-Order Toeplitz Determinant for a Subclass of Starlike Mappings in Higher Dimensions
本文建立了复Banach空间单位球和C^n中有界星形圆形域上一类星形映射的三阶Toeplitz行列式的精确界。
The Fekete-Szegö problem for starlike mappings of order $α$ and quasi-convex mappings of type $B$ in the unit balls of complex Banach spaces
该研究在复Banach空间单位球及ℂⁿ单位多圆盘上，针对两类全纯映射建立了广义Zalcman泛函与Fekete-Szegö不等式的精确估计，推广了经典结果并部分验证了高维广义Zalcman猜想。
Directional Coefficient Inequalities for Strongly Starlike Mappings in Complex Banach Spaces
该研究针对复Banach空间中满足标量剖面条件的α阶强星像映射，通过系数对应与提升定理，推导了方向系数相关不等式及Hankel行列式精确界，为极值问题提供了单变量对应关系。
This paper studies analytic functions $f$ defined on the open unit disk of the complex plane for which $f/g$ and $(1+z)g/z$ are both functions with positive real part for some analytic function $g$. We determine radius constants of these functions to belong to classes of strong starlike functions, starlike functions of order $α$, parabolic starlike functions, as well as to the classes of starlike functions associated with lemniscate of Bernoulli, cardioid, lune, reverse lemniscate, sine function, exponential function and a particular rational function. The results obtained are sharp.
Sharp Robertson-type coefficient estimates for radial starlike and quasi-convex of type B mappings of order α in complex Banach spaces
本文在复Banach空间中，利用控制概念建立α阶星型函数及α阶B型拟凸映射的系数精确估计，将Robertson1936年的相关结论推广到该空间框架。
Let $f$ and $g$ be analytic functions on the open unit disk of the complex plane with $f/g$ belonging to the class $\mathcal{P} $ of functions with positive real part consisting of functions $p$ with $p(0)=1$ and $\operatorname{Re} p(z)>0$ or to its subclass consisting of functions $p$ with $|p(z)-1|<1$. We obtain the sharp radius constants for the function $f$ to be starlike of order $α$, parabolic starlike, etc. when $g/k\in\mathcal{P}$ where $k$ denotes the Koebe function defined by $k(z)=z/(1-z)^2$.
On coefficient theorems for starlike and quasi-convex of type B mappings in complex Banach spaces
本文研究复Banach空间中特定形式的全纯映射，建立其为B型星形拟凸映射的充分系数条件，证明Bieberbach猜想对这类函数成立，并分析系数差异，将相关结论推广至复Banach空间场景。
The error function occurs widely in multiple areas of mathematics, mathematical physics and natural sciences. There has been no work in this area for the past four decades. In this article, we estimate the coefficient bounds with q-difference operator for certain classes of the spirallike starlike and convex error function associated with convolution product using subordination as well as quasi-subordination. Though this concept is an untrodden path in the field of complex function theory, it will prove to be an encouraging future study for researchers on error function.
Toeplitz determinants for starlike mappings of order α in several complex variables
该研究针对多复变量中α级星型映射，建立了涉及弗雷歇导数的两个托普利茨行列式的精确界，明确极值映射并验证了估计的精确性。
On sharp third Hankel determinant for certain starlike functions
本文推广了Rath等人的结果，给出了阶数$\alpha$在$[0,1/6]\cup\{1/2\}$范围内的星形函数三阶汉克尔行列式的锐利上界估计。
In this paper, we first obtain several sharp inequalities of homogeneous expansion for both the subclass of all normalized biholomorphic quasi-convex mappings of type B and order alpha and the subclass of all normalized biholomorphic almost starlike mappings of order alpha defined on the unit ball $B$ of a complex Banach space X. Then, with these sharp inequalities, we derive the sharp estimates of the third and fourth homogeneous expansions for the above mappings defined on the unit polydisk $D^n$ in C(n).
Let $h$ and $g$ be two analytic functions in the unit disc $Δ$ that $g(0)=1$. Also let $β$ be a complex number with ${\rm Re}\{β\}>-1/2$. A function $f$ is said to be log--harmonic mapping if it has the following representation \begin{equation*} f(z)=z |z|^{2β} h(z)\overline{g(z)}\quad (z\in Δ). \end{equation*} A log--harmonic mapping $f$ is said to be starlike log--harmonic mapping of order $α$, where $0\leq α<1$, if \begin{equation*} {\rm Re}\left\{\frac{zf_z -\overline{z}f_{\overline{z}}}{f}\right\}>α\quad(z\in Δ). \end{equation*} In this paper, by use of the subordination principle, we stu…
Given a domain $Ω$ in the complex plane $\mathbb{C}$ and a univalent function $q$ defined in an open unit disk $\mathbb{D}$ with nice boundary behaviour, Miller and Mocanu studied the class of admissible functions $Ψ(Ω,q)$ so that the differential subordination $ψ(p(z),zp(z),z^2p''(z);z)\prec h(z)$ implies $p(z)\prec q(z)$ where $p$ is an analytic function in $\mathbb{D}$ with $p(0)=1$, $ψ:\mathbb{C}^3\times \mathbb{D}\to\mathbb{C}$ and $Ω=h(\mathbb{D})$. This paper investigates the properties of this class for $q(z)=e^z$. As application, several sufficient conditions for normalized analytic f…
Assume that $Δ$ is the open unit disk in the complex plane and $\mathcal{A}$ is the class of normalized analytic functions in $Δ$. In this paper we introduce and study the class \begin{equation*} \mathcal{BS}(α):=\left\{f\in \mathcal{A}: \left(\frac{zf'(z)}{f(z)}-1\right)\prec \frac{z}{1-αz^2}, \, z\inΔ\right\}, \end{equation*} where $0\leqα\leq1$ and $\prec$ is the subordination relation. Some properties of this class like differential subordination, coefficients estimates and Fekete-Szegö inequality associated with the $k$-th root transform are considered.
Coefficient estimates for the class of q-starlike functions
本文利用卷积定义了 $q$-星象函数类，通过 Carathéodory-Toeplitz 定理和 Parseval 定理，分别推导了实数和复数 $q$ 下其 Taylor 系数及行列式的精确上界。
We will provide sufficient conditions for the shifted hypergeometric function $z_2F_1(a,b;c;z)$ to be a member of a specific subclass of starlike functions in terms of the complex parameters $a,b$ and $c.$ For example, we study starlikeness of order $α,$ $λ$-spirallikeness of order $α$ and strong starlikeness of order $α.$ In particular, those properties lead to univalence of the shifted hypergeometric functions on the unit disk.
Let $L_H$ denote the set of all normalized locally one-to-one and sense-preserving harmonic functions in the unit disc $Δ$. It is well-known that every complex-valued harmonic function in the unit disc $Δ$ can be uniquely represented as $f = h + \overline{g}$, where $h$ and $g$ are analytic in $Δ$. In particular the decomposition formula holds true for functions of the class $L_H$. For a fixed analytic function $h$, an interesting problem arises - to describe all functions $g$, such that $f$ belongs to $L_H$. The case when $f\in L_H$ and $h$ is the identity mapping was considered [2]. More gen…
This paper introduce the Bessel-Struve kernel functions $\mathcal{B}_ν$ defined on the unit disk in the complex plane. We studies the close-to-convexity of $\mathcal{B}_ν$ with respect to several starlike functions. Sufficient condition on $ν$ for which the function $z\mathcal{B}_ν$ is starlike is given.
Geometric properties of the Jackson and Hahn-Exton $q$-Bessel functions are studied. For each of them, three different normalizations are applied in such a way that the resulting functions are analytic in the unit disk of the complex plane. For each of the six functions we determine the radii of starlikeness and convexity precisely by using their Hadamard factorization. These are $q$-generalizations of some known results for Bessel functions of the first kind. The characterization of entire functions from the Laguerre-Pólya class via hyperbolic polynomials play an important role in this paper.…
Let $S^{*}(1-b)$ ($b \not= 0$ complex) denote the class of functions $f(z)=z+α_{2}z^{2}+...$ analytic in $D={z \mid | z | < 1}$ which satisfies,for $z=e^{iθ} \in D$, $(f(z)/z)\not= 0$ in D, and $Re \Biggr [ 1+ {1 \over b} \Biggr (z {{f^{'}(z)} \over f(z)}-1 \Biggl) \Biggl ] > 0$. The aim of this paper is to give the Koebe domain of the above mentioned class.
Let $p$ be an analytic function defined on the open unit disc $\mathbb{D}$ with $p(0)=1$ and $0< α\leq 1$. The conditions on complex valued functions $C$, $D$ and $E$ are obtained for $p$ to be subordinate to $((1+z)/(1-z))^α$ when $C(z) z^{2}p''(z)+D(z)zp'(z) + E(z)p(z)=0$. Sufficient conditions for confluent (Kummer) hypergeometric function and generalized and normalized Bessel function of the first kind of complex order to be subordinate to $((1+z)/(1-z))^α$ are obtained as applications. The conditions on $α$ and $β$ are derived for $p$ to be subordinate to $((1+z)/(1-z))^α$ when $1+βzp'(z)…
In this article we consider a family $\mathcal{C}(A, B)$ of analytic and locally univalent functions on the open unit disc $\ID=\{z :|z|<1\}$ in the complex plane that properly contains the well-known Janowski class of convex univalent functions. In this article, we determine the exact set of variability of $\log(f'(z_0))$ with fixed $z_0 \in \ID$ and $f"(0)$ whenever $f$ varies over the class $\mathcal{C}(A, B)$.
The classical Bohr inequality states that if $ f $ is an analytic function with the power series representation $ f(z)=\sum_{n=0}^{\infty}a_nz^n $ in the unit disk $ \mathbb{D}:=\{z\in\mathbb{C} : |z|<1\} $ such that $ |f(z)|\leq 1 $ for all $ z\in\mathbb{D} $, then \begin{equation*} \sum_{n=0}^{\infty}|a_n|r^n\leq 1\;\; \text{for}\;\; |z|=r\leq\frac{1}{3} \end{equation*} and the constant $ 1/3 $ cannot be improved. The constant $ r_0=1/3 $ is known as Bohr radius and the inequality $ \sum_{n=0}^{\infty}|a_n|r^n\leq 1 $ is known as Bohr inequality. Let $ \mathcal{H} $ be the class of complex-v…
Sharp Estimates for Hankel, Fekete-Szegö and Zalcman Functionals for $\mathcal{S}_\mathbb{B}^{*}(α)$ in Complex Banach Spaces
研究复Banach空间单位球上\(\alpha\)阶星形映射相关问题，利用弗雷歇导数和辅助引理建立二阶汉克尔行列式等泛函的精确上界，证明其精确性，还将相关经典结果扩展到复Banach空间。
We obtain sharp estimates for a generalized Zalcman coefficient functional with a complex parameter for the Hurwitz class and the Noshiro-Warschawski class of univalent functions as well as for the closed convex hulls of the convex and starlike functions by using an inequality from [6]. In particular, we generalize an inequality proved by Ma for starlike functions and answer a question from his paper [16]. Finally, we prove an asymptotic version of the generalized Zalcman conjecture for univalent functions and discuss various related or equivalent statements which may shed further light on the…
In this paper, by making use of a certain family of fractional derivative operators in the complex domain, we introduce and investigate a new subclass $\mathcal{P}_{τ,μ}(k,δ,γ)$ of analytic and univalent functions in the open unit disk $\mathbb{U}$. In particular, for functions in the class $\mathcal{P}_{τ,μ}(k,δ,γ)$, we derive sufficient coefficient inequalities, distortion theorems involving the above-mentioned fractional derivative operators, and the radii of starlikeness and convexity. In addition, some applications of functions in the class $\mathcal{P}_{τ,μ}(k,δ,γ)$ are also pointed out.
Sharp estimates on $β$ are determined so that an analytic function $p$ defined on the open unit disk in the complex plane normalized by $p(0)=1$ is subordinate to some well known starlike functions with positive real part whenever $1+βz p'(z), \,\,1+βz p'(z)/p(z), \,\,\mbox{or}\,\,1+βz p'(z)/p^{2}(z)$ is subordinate to $\sqrt{1+z}$. Our results provide sharp version of previously known results.
We describe the basic complex geometry and function theory of the {\em pentablock} $\mathcal{P}$, which is the bounded domain in $\mathbb{C}^3$ given by \[ \mathcal{P}= \{(a_{21}, \mathrm{tr} A, \det A): A= \begin{bmatrix} a_{ij}\end{bmatrix}_{i,j=1}^2 \in \mathbb{B}\} \] where $\mathbb{B}$ denotes the open unit ball in the space of $2\times 2$ complex matrices. We prove several characterizations of the domain. We describe its distinguished boundary and exhibit a $4$-parameter group of automorphisms of $\mathcal{P}$. We show that $\mathcal{P}$ is intimately connected with the problem of $μ$-sy…
In this paper, we first introduce the concept of biholomorphic convex mapping of order alpha on the unit ball B in a complex Hilbert space X. Next we provide some sufficient conditions that a locally biholomorphic mapping f is a biholomorphic convex mapping of order alpha and give an Alexander's theorem between the subclass of convex mappings and the subclass of starlike mappings on B in Hilbert space. We also obtain the order of starlikeness of biholomorphic convex mappings of order alpha on B in Hilbert spaces. Finally, we construct some concrete examples of biholomorphic convex mappings of …
In this paper, we introduce a class of complex-valued polyharmonic mappings, denoted by $HS_{p}(λ)$, and its subclass $HS_{p}^{0}(λ)$, where $λ\in [0,1]$ is a constant. These classes are natural generalizations of a class of mappings studied by Goodman in 1950's. We generalize the main results of Avci and Złotkiewicz from 1990's to the classes $HS_{p}(λ)$ and $HS_{p}^{0}(λ)$, showing that the mappings in $HS_{p}(λ)$ are univalent and sense preserving. We also prove that the mappings in $HS_{p}^{0}(λ)$ are starlike with respect to the origin, and characterize the extremal points of the above cl…
A $2p$-times continuously differentiable complex-valued function $f=u+iv$ in a simply connected domain $Ω\subseteq\mathbb{C}$ is \textit{p-harmonic} if $f$ satisfies the $p$-harmonic equation $Δ^pf=0.$ In this paper, we investigate the properties of $p$-harmonic mappings in the unit disk $|z|<1$. First, we discuss the convexity, the starlikeness and the region of variability of some classes of $p$-harmonic mappings. Then we prove the existence of Landau constant for the class of functions of the form $Df=zf_{z}-\barzf_{\barz}$, where $f$ is $p$-harmonic in $|z|<1$. Also, we discuss the region …
Let ${\mathcal H}$ denote the class of all normalized complex-valued harmonic functions $f=h+\bar{g}$ in the unit disk ${\mathbb D}$, and let $K=H+\bar{G}$ denote the harmonic Koebe function. Let $a_n,b_n, A_n, B_n$ denote the Maclaurin coefficients of $h,g,H,G$, and $${\mathcal F}=\{f=h+\bar{g}\in {\mathcal H}:\,|a_n|\leq A_n and |b_n|\leq B_n for n\geq 1}. $$ We show that the radius of univalence of the family ${\mathcal F}$ is $0.112903...$. We also show that this number is also the radius of the starlikeness of ${\mathcal F}$. Analogous results are proved for a subclass of the class of har…
Growth, Distortion, Pre-Schwarzian and Schwarzian norm estimates for Generalized Robertson class
本文研究广义罗伯特类函数的几何性质，推导其施瓦茨ian和预施瓦茨ian导数范数的精确界，并探讨函数的增长、扭曲及凹凸半径问题。
Hausdorff moment sequences and hypergeometric functions
本文完整刻画了系数为 Hausdorff 矩序列的超几何函数，并利用 Phragmèn-Lindelöf 型定理给出充分条件，进而得到平移超几何函数全星形的充要条件。
Alpha-convex functions in complex Banach spaces
本文研究复Banach空间单位球上形如F(z)=f(z)z的全纯映射，建立α-凸函数类的包含条件、刻画、对偶关系及汉克尔行列式界，将相关结论推广至复Banach空间。
Radius of convexity of certain classes of functions defined by convolution
研究由卷积定义的凹函数类与星形函数类的凸半径，利用卷积技巧得到凸半径的下界。
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