In this paper, we consider the Ma-Minda classes of analytic functions $\mathcal{S}^{*}(ϕ):= \{f\in \mathcal{A} : ({zf'(z)}/{f(z)}) \prec ϕ(z) \}$ and $\mathcal{C}(ϕ):= \{f\in \mathcal{A} : (1+{zf''(z)}/{f'(z)}) \prec ϕ(z) \}$ defined on the unit disk $\mathbb{D}$ and show that the classes $\mathcal{S}^{*}(1+αz)$ and $\mathcal{C}(1+αz)$, $0<α\leq 1$ solve the problem of finding the sharp $\mathcal{S}^{*}(ϕ)$-radii and $\mathcal{C}(ϕ)$-radii for some normalized special functions, whenever $ϕ(-1)=1-α$. Radius of strongly starlikeness is also considered.
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.
We introduce three classes of analytic functions with fixed second coefficient which are defined using the class $\mathcal{P}$ of analytic functions with positive real part. The objective is to determine radii such that the three classes are contained in various subclasses of starlike functions. The radii estimated in the present investigation are better than the radii obtained earlier. Furthermore, connections with previous known results are shown.
Let $\mathcal{S}^*_{Ne}$ be the collection of all analytic functions $f(z)$ defined on the open unit disk $\mathbb{D}$ and satisfying the normalizations $f(0)=f'(0)-1=0$ such that the quantity $zf'(z)/f(z)$ assumes values from the range of the function $φ_{\scriptscriptstyle{Ne}}(z):=1+z-z^3/3\,,z\in\mathbb{D}$, which is the interior of the nephroid given by \begin{align*} \left((u-1)^2+v^2-\frac{4}{9}\right)^3-\frac{4 v^2}{3}=0. \end{align*} In this work, we find sharp $\mathcal{S}^*_{Ne}$-radii for several geometrically defined function classes introduced in the recent past. In particular, $…
In this paper our aim is to find the radii of $γ$-Spirallike of order $α$ and convex $γ$-Spirallike of order $α$ for three different kinds of normalizations of the function $N_ν(z)=az^2J_ν^{\prime\prime}(z)+bzJ_ν^{\prime}(z)+cJ_ν(z),$ where $J_ν(z)$ is the Bessel function of the first kind of order $ν.$ Moreover, the $\mathcal{S}^*\left(φ\right)-$radii and $\mathcal{C}\left(φ\right)-$radii of these normalized functions are investigated. The tables are created and visual verification with graphs are made by giving special values to the real numbers $a,$ $b$ and $c$ in the obtained results.
For $n\geq 4$ (even), the function $φ_{n\mathcal{L}}(z)=1+nz/(n+1)+z^n/(n+1)$ maps the unit disk $\mathbb{D}$ onto a domain bounded by an epicycloid with $n-1$ cusps. In this paper, the class $\mathcal{S}^*_{n\mathcal{L}} = \mathcal{S}^*(φ_{n\mathcal{L}})$ is studied and various inclusion relations are established with other subclasses of starlike functions. The bounds on initial coefficients is also computed. Various radii problems are also solved for the class $\mathcal{S}^*_{n\mathcal{L}}$.
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 introduce the class of analytic functions $$\mathcal{F}(ψ):= \left\{f\in \mathcal{A}: \left(\frac{zf'(z)}{f(z)}-1\right) \prec ψ(z),\; ψ(0)=0 \right\},$$ where $ψ$ is univalent and establish the growth theorem with some geometric conditions on $ψ$ and obtain the Koebe domain with some related sharp inequalities. Note that functions in this class may not be univalent. As an application, we obtain the growth theorem for the complete range of $α$ and $β$ for the functions in the classes $\mathcal{BS}(α):= \{f\in \mathcal{A} : ({zf'(z)}/{f(z)})-1 \prec {z}/{(1-αz^2)},\; α\in [0,1) \}$ and $\mat…
We consider three classes of functions defined using the class $\mathcal{P}$ of all analytic functions $p(z)=1+cz+\dotsb$ on the open unit disk having positive real part and study several radius problems for these classes. The first class consists of all normalized analytic functions $f$ with $f/g\in\mathcal{P}$ and $g/(zp)\in\mathcal{P}$ for some normalized analytic function $g$ and $p\in \mathcal{P}$. The second class is defined by replacing the condition $f/g\in\mathcal{P}$ by $|(f/g)-1|<1$ while the other class consists of normalized analytic functions $f$ with $f/(zp)\in\mathcal{P}$ for s…
In this investigation our main aim is to determine the radius of uniform convexity of the some normalized q-Bessel and Wright functions. Here we consider six different normalized forms of q-Bessel functions, while we apply three different kinds of normalizations of Wright function. Also, we have shown that the obtained radii are the smallest positive roots of some functional equations.
In this paper, we introduce a family of analytic functions given by $$ψ_{A,B}(z):= \dfrac{1}{A-B}\log{\dfrac{1+Az}{1+Bz}},$$ which maps univalently the unit disk onto either elliptical or strip domains, where either $A=-B=α$ or $A=αe^{iγ}$ and $B=αe^{-iγ}$ ($α\in(0,1]$ and $γ\in(0,π/2]$). We study a class of non-univalent analytic functions defined by \begin{equation*} \mathcal{F}[A,B]:=\left\{f\in\mathcal{A}:\left( \dfrac{zf'(z)}{f(z)}-1\right)\precψ_{A,B}(z)\right \}. \end{equation*} Further, we investigate various characteristic properties of $ψ_{A,B}(z)$ as well as functions in the class $…
In this paper we determine the upper bounds of the Hankel determinants of special type $H_{2}(3)(f)$ and $H_{2}(4)(f)$ for the class of univalent functions and for the class $\mathcal{U}$ defined by \[ \mathcal{U}=\left\{ f\in\mathcal{A} : \left|\left[\frac{z}{f(z)}\right]^2 f'(z)-1 \right|<1,\, z\in{\mathbb D} \right\}, \] where $\mathcal{A}$ is the class of functions analytic in the unit disk ${\mathbb D}$ and normalized such that $f(z)=z+a_2z^2+\cdots$.
The generalized Bohr radius $R_{p, q}(X), p, q\in[1, \infty)$ for a complex Banach space $X$ was introduced by Blasco in 2010. In this article, we determine the exact value of $R_{p, q}(\mathbb{C})$ for the cases (i) $p, q\in[1, 2]$, (ii) $p\in (2, \infty), q\in [1, 2]$ and (iii) $p, q\in [2, \infty)$. Moreover, we consider an $n$-variable version $R_{p, q}^n(X)$ of the quantity $R_{p, q}(X)$ and determine (i) $R_{p, q}^n(\mathcal{H})$ for an infinite dimensional complex Hilbert space $\mathcal{H}$, (ii) the precise asymptotic value of $R_{p, q}^n(X)$ as $n\to\infty$ for finite dimensional $X$…
