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 $1+βzp'(z)/p^{n}(z)$ with $n=1,2$ is subordinate to $1+4z/3+2z^{2}/3=:φ_{CAR}(z)$. Similar problems were investigated for $\RE p(z)>0$ when the functions $p(z)+βzp'(z)/p^{n}(z)$ with $n=0,2$ is subordinate to $φ_{CAR}(z)$. The condition on $β$ is determined for $p$ to be subordinate to $((1+z)/(1-z))^α$ when $p(z)+βzp'(z)/p^{n}(z)$ with $n=0,1,2$ is subordinate to $((1+z)/(1-z))^α$.
For functions $f(z)=z^p+a_{n+1}z^{p+1}+...$ defined on the open unit disk, the condition $\Re (f'(z)/z^{p-1})>0$ is sufficient for close-to-convexity of $f$. By making use of this result, several sufficient conditions for close-to-convexity are investigated and relevant connections with previously known results are indicated.
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.
There are many results for sufficient conditions of functions f(z) which are analytic in the open unit disc U to be starlike and convex in U. The object of the present paper is to derive some interesting sufficient conditions for f(z) to be starlike of order α and convex of order α in U concerned with Jack's lemma. Some examples for our results are also considered with the help of Mathematica 5.2.
For analytic functions f(z) in the open unit disk U with f(0)=f'(0)-1=0, P. T. Mocanu (Mathematica (Cluj), 11(34) (1969)) have considered Mocanu functions. The object of the present paper is to discuss some sufficient problems for f(z) to be starlike of order α.
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.
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.
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.
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…
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.
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.…
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.
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.
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.
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…
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 $α\geq 0$, $β<1$ and $γ\geq 0$, the class $\mathcal{W}_β(α,γ)$ satisfies the condition \begin{align*} {\rm Re\,} \left( e^{iϕ}\left((1-α+2γ)f/z+(α-2γ)f'+ γzf''-β\right)\frac{}{}\right)>0, \quad ϕ\in {\mathbb{R}},{\,}z\in {\mathbb{D}}; \end{align*} is taken into consideration. The Pascu class of $ξ$-convex functions of order $σ$ $(M(σ,{\,}ξ))$, having analytic characterization \begin{align*} {\rm Re\,}\frac{ξz(zf'(z))'+(1-ξ)zf'(z)}{ξzf'(z)+(1-ξ)f(z)}>σ,\quad 0\leq σ< 1,\quad z\in {\mathbb{D}}, \end{align*} unifies starlike and convex functions class of order $σ$.The admissible and sufficien…
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}{…
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.
For analytic functions f(z) in the open unit disk U with f(0)=f'(0)-1=0, S. S. Miller and P. T. Mocanu (Integral Transform. Spec. Funct. 19(2008)) have considered some sufficient problems for starlikeness. The object of the present paper is to discuss some sufficient problems for f(z) to be in some classes of order α.
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.
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.
In this paper our aim is to investigate the radii of $η-$uniformly convexity, $α-$convexity, $η-$parabolic starlikeness and strong starlikeness of order $ρ$ 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. The basic tool of this study is the Mittag-Leffler function in series. Also we have shown that the obtained radii are the smallest positive roots of some functional equations.
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.
Some geometric properties of certain class of Le Roy type functions
Indian Institute of Technology Hyderabad（印度理工学院海德拉巴分校）；Shiv Nadar University（希夫纳达尔大学）；Dr. Shyama Prasad Mukherjee University（S.P.穆尔杰博士大学）；Indian Institute of Technology Madras（印度理工学院马德拉斯分校）
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…
Applying the subordination principle for analytic functions in the open unit disk U, I. H. Kim and N. E. Cho (Comput. Math. Appl. 59(2010), 2067-2073) considered some sufficient conditions for Carathéodory functions. The purpose of this paper is to discuss some sufficient conditions for the class of strongly Carathéodory functions in U.
In this paper we shall find some sufficient conditions for a uniqueness polynomial to be a strong uniqueness polynomial, as this type of problem was never investigated by researchers earlier.We also exhibit some examples to substantiate our theorems.
For analytic functions f(z) in the open unit disk U with f(0)=f'(0)-1=0, a class STC(μ) is defined. The object of the present paper is to discuss some sufficient problems for f(z) to be strongly close-to-convex of order μ in U.
