A starlike function $f$ is characterized by the quantity $zf'(z)/f(z)$ lying in the right half-plane. This paper deals with sharp bounds for certain symmetric Toeplitz determinants whose entries are the coefficients of the functions $f$ for which the quantity $zf'(z)/f(z)$ takes values in certain specific subset in the right half-plane. The results obtained include several new special cases and some known results.
For the standard Ma-Minda class $\mathcal{S}^{*}(ψ)$ of univalent starlike functions, we derive $\mathcal{S}^{*}(ψ)$-radii for some well-known special functions. In addition, we obtain the set of extremal functions for the classical problem $$\max_{f\in \mathcal{S}^{*}(ψ)}^{}\left|Φ\left(\log{(f(z)/z)}\right)\right| \quad \text{or} \quad \max_{f\in \mathcal{S}^{*}(ψ)}^{}\ReΦ\left(\log{(f(z)/z)}\right),$$ where $Φ$ is a non-constant entire function. Moreover, we prove certain results on convolution and radius estimates for the case when $ψ(\mathbb{D})$ is starlike.
Estimates on the initial coefficients are obtained for normalized analytic functions $f$ in the open unit disk with $f$ and its inverse $g=f^{-1}$ satisfying the conditions that $zf'(z)/f(z)$ and $zg'(z)/g(z)$ are both subordinate to a starlike univalent function whose range is symmetric with respect to the real axis. Several related classes of functions are also considered, and connections to earlier known results are made.
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…
We say that a class $\mathcal{F}$ consisting of analytic functions $f(z)=\sum_{n=0}^{\infty} a_{n}z^{n}$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}: |z|<1\}$ satisfies a Bohr phenomenon if there exists $r_{f} \in (0,1)$ such that $$ \sum_{n=1}^{\infty} |a_{n}z^{n}|\leq d(f(0),\partial f(\mathbb{D})) $$ for every function $f \in \mathcal{F}$ and $|z|=r\leq r_{f}$, where $d$ is the Euclidean distance. The largest radius $r_{f}$ is the Bohr radius for the class $\mathcal{F}$. In this paper, we establish the Bohr phenomenon for the classes consisting of Ma-Minda type starlike functions and Ma…
In this paper, we derive the sharp bounds of Toeplitz determinants for a class of holomorphic mappings on the bounded starlike circular domain $Ω$ in $\mathbb{C}^n$, which extend certain known bounds for various subclasses of normalized analytic univalent functions in the unit disk to higher dimensions.
In this study, we deal with the sharp bounds of certain Toeplitz determinants whose entries are the logarithmic coefficients of analytic univalent functions $f$ such that the quantity $z f'(z)/f(z)$ takes values in a specific domain lying in the right half plane. The established results provide the bounds for the classes of starlike and convex functions, as well as various of their subclasses.
This paper deals with a special type of Ma-Minda function introduced here with many fascinating facts and interesting applications. It is much akin in all aspects but differs by a condition from its Ma-Minda counterpart. Further, we consider the function:~$1-\log(1+z)$, a special Ma-Minda of the type introduced here, to define a subclass of starlike functions in a similar fashion as we do with Ma-Minda function and is studied for establishing inclusion and radius results. Apart from that, we also deal with the majorization and Bloch function norm problems for the same class. In addition, we ob…
We investigate harmonic mappings $f=h+\bar{g}$ defined in the unit disk, where $g$ and $h$ satisfy certain prescribed conditions and the analytic part $h$ belongs to the Ma and Minda class of starlike functions. Certain sharp radius results for univalency, close-to-convexity, fully starlikeness, fully convexity, and radius of strongly starlikeness are established. We also calculate the radius of uniformly starlikeness and convexity for these functions. Several results enhance the well-known radius results.
We introduce and study a new Ma-Minda subclass of starlike functions $\mathcal{S}^*_{\varrho},$ defined as $$\mathcal{S}^{*}_{\varrho}:=\left\{f\in\mathcal{A}:\frac{zf'(z)}{f(z)} \prec \cosh \sqrt{z}=:\varrho(z), z\in\mathbb{D} \right\},$$ associated with an analytic univalent function $\cosh \sqrt{z},$ where we choose the branch of the square root function so that $\cosh\sqrt{z}=1+z/2!+z^{2}/{4!}+\cdots.$ We establish certain inclusion relations for $\mathcal{S}^{*}_{\varrho}$ and deduce sharp $\mathcal{S}^{*}_{\varrho}-$radii for certain subclasses of analytic functions.
We prove a conjecture concerning the third Hankel determinant, proposed in ``Anal. Math. Phys., https://doi.org/10.1007/s13324-021-00483-7", which states that $|H_3(1)|\leq 1/9$ is sharp for the class $\mathcal{S}_{\wp}^{*}=\{zf'(z)/f(z) \prec φ(z):=1+ze^z\}$. In addition, we also establish bounds for sixth and seventh coefficient, and $|H_4(1)|$ for functions in $\mathcal{S}_{\wp}^{*}$. The general bounds for two and three-fold symmetric functions related to the Ma-Minda classes $\mathcal{S}^*(φ)$ of starlike functions are also obtained.
We investigate the non-univalent function's properties reminiscent of the theory of univalent starlike functions. Let the analytic function $ψ(z)=\sum_{i=1}^{\infty}A_i z^i$, $A_1\neq0$ be univalent in the unit disk. Non-univalent functions may be found in the class $\mathcal{F}(ψ)$ of analytic functions $f$ of the form $f(z)=z+\sum_{k=2}^{\infty}a_k z^k$ satisfying $({zf'(z)}/{f(z)}-1) \prec ψ(z)$. Such functions, like the Ma and Minda classes of starlike functions, also have nice geometric properties. For these functions, growth and distortion theorems have been established. Further, we obta…
In this paper, we generalize the Bohr-Rogosinski sum for the Ma-Minda classes of starlike and convex functions. Also the phenomenon is studied for the classes of starlike functions with respect to symmetric points and conjugate points along with their convex cases. Further, the connection between the derived results and the known ones are established with the suitable examples.
In this article, we establish the Bohr inequalities for the sense-preserving $K$-quasiconformal harmonic mappings defined in the unit disk $\mathbb{D}$ involving classes of Ma-Minda starlike and convex univalent functions, usually denoted by $\mathcal{S}^*(ψ)$ and $\mathcal{C}(ψ)$ respectively, and for $\log (f(z)/z)$ where $f$ belongs to the Ma-Minda classes or satisfies certain differential subordination. We also estimate Logarithmic coefficient's bounds for the functions in $\mathcal{C}(ψ)$ for the case $ψ(\mathbb{D})$ be convex.
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.
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…
In this paper, we determine the sharp estimates for Toeplitz determinants of a subclass of close-to-convex harmonic mappings. Moreover, we obtain an improved version of Bohr's inequalities for a subclass of close-to-convex harmonic mappings, whose analytic parts are Ma-Minda convex functions.
In Geometric function theory, occasionally attempts have been made to solve a particular problem for the Ma-Minda classes, $\mathcal{S}^*(ψ)$ and $\mathcal{C}(ψ)$ of univalent starlike and convex functions, respectively. Recently, a popular radius problem generally known as Bohr's phenomenon has been studied in various settings, however little is known about Rogosinski radius. In this article, for a fixed $f\in \mathcal{S}^*(ψ)$ or $\mathcal{C}(ψ),$ the class of analytic subordinants $S_{f}(ψ):= \{g : g\prec f \} $ is studied for the Bohr-Rogosinski phenomenon in a general setting. It's applic…
We deal with different kinds of generalizations of $\mathcal{S}^*(ψ)$, the class of Ma-Minda starlike functions, in addition to a majorization result of $\mathcal{C}(ψ),$ the class of Ma-Minda convex functions, which are enlisted as follows: 1. Let $h$ be an analytic function, $f$ be in $\mathcal{C}(ψ)$ and $h$ be majorized by $f$ in the unit disk $\mathbb{D},$ then for a given $ψ,$ we derive a general equation, which yields the radius constant $r_ψ$ such that $|h'(z)|\leq |f'(z)|$ in $|z|\leq r_ψ$. Consequently, obtain results associating $\mathcal{S}^*(ψ)$ and others. 2. We find the largest …
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 f…
