For a function $f$ starlike of order $α$, $0\leqslant α<1$, a non-constant polynomial $Q$ of degree $n$ which is non-vanishing in the unit disc $\mathbb{D}$ and $β>0$, we consider the function $F:\mathbb{D}\to\mathbb{C}$ defined by $F(z)=f(z) (Q(z))^{β/n}$ and find the largest value of $r\in (0,1]$ such that $r^{-1} F(rz)$ lies in various known subclasses of starlike functions such as the class of starlike functions of order $λ$, the classes of starlike functions associated with the exponential function, cardioid, a rational function, nephroid domain and modified sigmoid function. Our radii results are sharp. We also discuss the correlation with known radii results as special cases.
For normalised analytic functions $f$ defined on the open unit disc $\mathbb{D}$ satisfying the condition $\sup_{z\in \mathbb{D}}(1-|z^2|) |f'(z)|\leq 1$, known as Bloch functions, we determine various starlikeness radii.
In this manuscript, we deal with three classes of quotient functions having fixed second coefficient described on open unit disk. The radius of strongly starlikeness, lemniscate starlikeness, lune starlikeness, parabolic starlikeness, sine starlikeness, exponential starlikeness and several other radius estimates for such classes are examined. Relevant relations of obtained radius estimates with the existing estimates are also discussed.
In this paper, it is aimed to determine the radii of starlikeness and convexity of the normalized generalized Struve functions for three different kinds of normalization and to find tight lower and upper bounds for the radius of starlikeness and convexity of these normalized Struve functions by making use of Euler-Rayleigh inequalities. The Laguerre-Pólya class of entire functions has a crucial role in constructing our main results.
For a normalised analytic function f defined on the open unit disk in the complex plane, we determine several sufficient conditions for starlikeness in terms of the quotients Q_{ST}:=zf'(z)/f(z), Q_{CV}:=1+zf"(z)/f'(z) and the Schwarzian derivative Q_{SD}:=z^2((f"(z)/f'(z))'-(f"(z)/f'(z))^2/2)$. These conditions were obtained by using the admissibility criteria of starlikeness in the theory of second order differential subordination.
We find the radius of starlikeness of order $α$, $0\leq α<1$, of normalized analytic functions $f$ on the unit disk satisfying either $\operatorname{Re}(f(z)/g(z))>0$ or $\left| (f(z)/g(z))-1\right|<1$ for some close-to-star function $g$ with $\operatorname{Re}(g(z)/(z+z^2/2))>0$ as well as of the class of close-to-star functions $f$ satisfying $\operatorname{Re}(f(z)/(z+z^2/2))>0$. Several other radii such as radius of univalence and parabolic starlikeness are shown to be the same as the radius of starlikeness of appropriate order.
The radii of starlikeness and convexity associated with lemniscate of Bernoulli and the Janowski function, $(1+Az)/(1+Bz)$ for $-1\leq B<A\leq 1$, have been determined for normalizations of $q$-Bessel function, Bessel function of first kind of order $ν$, Lommel function of first kind and Legendre polynomial of odd degree.
In this study, a normalized form of regular Coulomb wave function is considered. By using differential subordination method due to Miller and Mocanu, we determine some conditions on the parameters such that the normalized regular Coulomb wave function is lemniscate starlike and exponenetial starlike in the open unit disk, respectively.
In this paper we deal with the radii of starlikeness and convexity of the $q-$Mittag--Leffler function for three different kinds of normalization by making use of their Hadamard factorization in such a way that the resulting functions are analytic in the unit disk of the complex plane. By applying Euler-Rayleigh inequalities for the first positive zeros of these functions tight lower and upper bounds for the radii of starlikeness of these functions are obtained. The Laguerre-Pólya class of real entire functions plays a pivotal role in this investigation.
In this paper, we shall find the order of starlikeness and convexity for integral operators \begin{equation*} \mathbb{F}_{α_{j},β_{j},λ_{j},ζ}(z)=\left\{ ζ\int\limits_{0}^{z}t^{ζ-1}\prod_{j=1}^{n}\left( \frac{\mathbb{E} _{α_{j},β_{j}}(t)}{t}\right) ^{1/λ_{j}}dt\right\} ^{1/ζ}, \end{equation*} where the functions $\mathbb{E}_{α_{j},β_{j}}$ are the normalized Mittag-Leffler functions.
We consider normalized analytic function $f$ on the open unit disk for which either $\operatorname{Re} f(z)/g(z)>0$, $|f(z) /g(z) - 1|<1$ or $\operatorname{Re} (1-z^2) f(z) /z>0$ for some analytic function $g$ with $\operatorname{Re} (1-z^2) g(z) /z>0$. We have obtained the radii for these functions to belong to various subclasses of starlike functions. The subclasses considered include the classes of starlike functions of order $α$, lemniscate starlike functions and parabolic starlike functions.
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.
The asymptotic behaviour, with respect to the large order, of the radii of starlikeness of two types of normalised Bessel functions is considered. We derive complete asymptotic expansions for the radii of starlikeness and provide recurrence relations for the coefficients of these expansions. The proofs rely on the notion of Rayleigh sums and asymptotic inversion. The techniques employed in the paper could be useful to treat similar problems where inversion of asymptotic expansions is involved.
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…
Sufficient conditions are obtained on the parameters of Lommel function of the first kind, generalized Struve function of the first kind and the confluent hypergeometric function under which these special functions become exponential convex and exponential starlike in the open unit disk. The method of differential subordination is employed in proving the results. Few examples are also provided to illustrate the results obtained.
In this paper our aim is to determine the radii of univalence, starlikeness and convexity of the normalized regular Coulomb wave functions for two different kinds of normalization. The key tools in the proof of our main results are the Mittag-Leffler expansion for regular Coulomb wave functions, and properties of zeros of the regular Coulomb wave functions and their derivatives. Moreover, by using the technique of differential subordinations we present some conditions on the parameters of the regular Coulomb wave function in order to have a starlike normalized form. In addition, by using the E…
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.
For an analytic function $f$ on the unit disk $\mathbb{D}=\{z:|z|<1\}$ satisfying $f(0)=0=f'(0)-1,$ we obtain sufficient conditions so that $f$ satisfies $|(zf'(z)/f(z))^2-1|<1.$ The technique of differential subordination of first or second order is used. The admissibility conditions for lemniscate of Bernoulli are derived and employed in order to prove the main results.
The main purpose of this paper is to determine the radii of starlikeness and convexity of the generalized $\emph{k}-$Bessel functions for three different kinds of normalization by using their Hadamard factorization in such a way that the resulting functions are analytic in the unit disk of the complex plane. The characterization of entire functions from Laguerre-Pólya class plays an crucial role in this paper. Moreover, the interlacing properties of the zeros of $\emph{k}-$Bessel function and its derivative is also useful in the proof of the main results. By making use of the Euler-Rayleigh in…
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 $k(z)=z/(1-z)^2$.
This paper aims to pursue some classes of normalized analytic functions $f$ with fixed second coefficient defined on open unit disk, such that ${(1+z)^2f(z)}/{z}$ and ${(1+z)f(z)}/{z}$ are functions having positive real part. The radius of strongly starlikeness, the radius of lemniscate starlikeness, the radius of parabolic starlikeness and other starlikeness estimates are calculated for such functions. As well relevant connections of computed radii estimates with the existing one are also shown.
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…
In this paper our aim is to find the radii of starlikeness and convexity of the generalized Mittag-Leffler function for three different kinds of normalization by using their Hadamard factorization in such a way that the resulting functions are analytic in the unit disk of the complex plane. The characterization of entire functions from Laguerre-Pólya class and a result of H. Kumar and M.A. Pathan on the reality of the zeros of generalized Mittag-Leffler functions, which origins goes back to Dzhrbashyan, Ostrovskiĭ and Peresyolkova, play important roles in this paper. Moreover, the interlacing …
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.
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…
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…
A starlike univalent function $f$ is characterized by the function $zf'(z)/f(z)$; several subclasses of these functions were studied in the past by restricting the function $zf'(z)/f(z)$ to take values in a region $Ω$ on the right-half plane, or, equivalently, by requiring the function $zf'(z)/f(z)$ to be subordinate to the corresponding mapping of the unit disk $\mathbb{D}$ to the region $Ω$. The mappings $w_1(z):=z+\sqrt{1+z^2}, w_2(z):=\sqrt{1+z}$ and $w_3(z):=e^z$ maps the unit disk $\mathbb{D}$ to various regions in the right half plane. For normalized analytic functions $f$ satisfying th…
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.…
For a fixed $a \in \{1, 2, 3, \ldots\},$ the radius of starlikeness of positive order is obtained for each of the normalized analytic functions \begin{align*} \mathtt{f}_{a, ν}(z)&:= \bigg(2^{a ν-a+1} a^{-\frac{a(aν-a+1)}{2}} Γ(a ν+1) {}_a\mathtt{B}_{2a-1, a ν-a+1, 1}(a^{a/2} z)\bigg)^{\tfrac{1}{a ν-a+1}},\\ \mathtt{g}_{a, ν}(z)&:= 2^{a ν-a+1} a^{-\frac{a}{2}(aν-a+1)} Γ(a ν+1) z^{a-aν} {}_a\mathtt{B}_{2a-1, a ν-a+1, 1}(a^{a/2} z),\\ \mathtt{h}_{a, ν}(z)&:= 2^{a ν-a+1} a^{-\frac{a}{2}(aν-a+1)} Γ(a ν+1) z^{\frac{1}{2}(1+a-aν)} {}_a\mathtt{B}_{2a-1, a ν-a+1, 1}(a^{a/2} \sqrt{z}) \end{align*} in t…
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 $\mathcal{A}$ consists of analytic functions $f:\mathbb{D}\to\mathbb{C}$ satisfying $f(0)=f'(0)-1=0$. Let $\mathcal{S}^*_{Ne}$ be the recently introduced Ma-Minda type functions family associated with the $2$-cusped kidney-shaped {\it nephroid} curve $\left((u-1)^2+v^2-\frac{4}{9}\right)^3-\frac{4 v^2}{3}=0$ given by \begin{align*} \mathcal{S}^*_{Ne}:= \left\{f\in\mathcal{A}:\frac{zf'(z)}{f(z)}\precφ_{\scriptscriptstyle {Ne}}(z)=1+z-z^3/3\right\}. \end{align*} In this paper, we adopt a novel technique that uses the geometric properties of {\it hypergeometric functions} to determine sharp e…
For $α\geq 0$, $δ>0$, $β<1$ and $γ\geq 0$, the class $\mathcal{W}_β^δ(α,γ)$ consist of analytic and normalized functions $f$ along with the condition \begin{align*} {\rm Re\,} e^{iϕ}(\dfrac{}{}(1\!-\!α\!+\!2γ)\!({f}/{z})^δ+(α\!-\!3γ\!+\!γ[\dfrac{}{}(1-{1}/δ)({zf'}/{f})+ {1}/δ(1+{zf''}/{f'})]).\\ .\dfrac{}{}({f}/{z})^δ\!({zf'}/{f})-β)>0, \end{align*} where $ϕ\in\mathbb{R}$ and $|z|<1$, is taken into consideration. The class $\mathcal{S}^\ast_s(ζ)$ be the subclass of the univalent functions, defined by the analytic characterization ${\rm Re}{\,}({zf'}/{f})>ζ$, for $0\leq ζ< 1$, $0<δ\leq\frac{1}{…
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 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.
In the present paper, we study spirallikenss (including starlikeness) of the shifted hypergeometric function $f(z)=z_2F_1(a,b;c;z)$ with complex parameters $a,b,c,$ where $_2F_1(a,b;c;z)$ stands for the Gaussian hypergeometric function. First, we observe the asymptotic behaviour of $_2F_1(a,b;c;z)$ around the point $z=1$ to obtain necessary conditions for $f$ to be $λ$-spirallike for a given $λ$ with $- π/2< λ<π/2.$ We next give sufficient conditions for $f$ to be $λ$-spirallike. As special cases, we obtain sufficient conditions of strong starlikeness and examples of spirallike, but not starli…
We study nonlinear resolvents of holomorphic generators of one-parameter semigroups acting in the open unit disk. The class of nonlinear resolvents can be studied in the framework of geometric function theory because it consists of univalent functions. In this paper we establish distortion and covering results, find order of starlikeness and of strong starlikeness of resolvents. This provides that any resolvent admits quasiconformal extension to the complex plane $\C$. In addition, we obtain some characteristics of semigroups generated by these resolvents.
