Let $\mathcal{A}_n$ be the class of analytic functions $f(z)$ of the form $f(z)=z+\sum_{k=n+1}^\infty a_kz^k,n\in\mathbb{N}$ and let \begin{align*} Ω_n:=\left\{f\in\mathcal{A}_n:\left|zf'(z)-f(z)\right|<\frac{1}{2},\; z\in\mathbb{D}\right\}. \end{align*} We make use of differential subordination technique to obtain sufficient conditions for the class $Ω_n$, and then employ these conditions to construct functions which involve double integrals and members of $Ω_n$. We also consider a subclass $\widehatΩ_n\subsetΩ_n$ and obtain subordination results for members of $\widehatΩ_n$ besides a necessary and sufficient condition. Writing $Ω_1=Ω$, we obtain inclusion properties of $Ω$ with respect to functions defined on certain parabolic regions and as a consequence, establish a relation connecting the parabolic starlike class $\mathcal{S}_p$ and the uniformly starlike $UST$. Various radius problems for the class $Ω$ are considered and the sharpness of the radii estimates is obtained analytically besides graphical illustrations.
In this paper, two new subclasses of bi-univalent functions related to conic domains are defined by making use of symmetric $q$-differential operator. The initial bounds for Fekete-Szegö inequality for the functions $f$ in these classes are estimated.
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 the present investigation, we study the class of Sigmoid starlike functions, given by $\mathcal{S}^*_{SG}=\{f\in\mathcal{A}: {zf'(z)}/{f(z)}\prec 2/(1+e^{-z})\}$ in context of estimating the sharp radius constants associated with several known subclasses of starlike functions. Further, graphical validation for the sharpness of results is also provided.
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.
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 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.
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…
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.
By considering the polynomial function $ϕ_{car}(z)=1+z+z^2/2,$ we define the class $\Scar$ consisting of normalized analytic functions $f$ such that $zf'/f$ is subordinate to $ϕ_{car}$ in the unit disk. The inclusion relations and various radii constants associated with the class $\Scar$ and its connection with several well-known subclasses of starlike functions is established. As an application, the obtained results are applied to derive the properties of the partial sums and convolution.
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.
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…
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…
Radius constants for several classes of analytic functions on the unit disk are obtained. These include the radius of starlikeness of a positive order, radius of parabolic starlikeness, radius of Bernoulli lemniscate starlikeness, and radius of uniform convexity. In the main, the radius constants obtained are sharp. Conjectures on the non-sharp constants are given.
In 2011, Sokół (Comput. Math. Appl. 62, 611--619) introduced and studied the class $\mathcal{SK}(α)$ as a certain subclass of starlike functions, consists of all functions $f$ ($f(0)=0=f'(0)-1$) which satisfy in the following subordination relation: \begin{equation*} \frac{zf'(z)}{f(z)}\prec \frac{3}{3+(α-3)z-αz^2} \qquad |z|<1, \end{equation*} where $-3<α\leq1$. Also, he obtained some interesting results for the class $\mathcal{SK}(α)$. In this paper, some another properties of this class, including infimum of $\mathfrak{Re}\frac{f(z)}{z}$, order of strongly starlikeness, the sharp logarithmi…
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…
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…
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.
Let $f=h+\overline{g}$ be a normalized harmonic mapping in the unit disk $\ID$. In this paper, we obtain the sharp radius of univalence, fully starlikeness and fully convexity of the harmonic linear differential operators $D_f^ε=zf_{z}-ε\overline{z}f_{\overline{z}}~(|ε|=1)$ and $F_λ(z)=(1-λ)f+λD_f^ε~(0\leqλ\leq 1)$ when the coefficients of $h$ and $g$ satisfy harmonic Bieberbach coefficients conjecture conditions. Similar problems are also solved when the coefficients of $h$ and $g$ satisfy the corresponding necessary conditions of the harmonic convex function $f=h+\overline{g}$. All results a…
For an analytic function f(z)=z+\sum_{n=2}^\infty a_n z^n satisfying the inequality \sum_{n=2}^\infty n(n-1)|a_n|\leq β, sharp bound on $β$ is determined so that $f$ is either starlike or convex of order $α$. Several other coefficient inequalities related to certain subclasses are also investigated.
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.
This paper deals with some radius results and inclusion relations that are established for functions in a newly defined subclass of starlike functions associated with a petal shaped domain.
We introduce and study a class of starlike functions defined by \begin{equation*} \mathscr{S}^*_\wp:=\left\{f\in\mathcal{A}: \frac{zf'(z)}{f(z)}\prec 1+ze^z=:\wp(z)\right\}, \end{equation*} where $\wp$ maps the unit disk onto a cardioid domain. We find the radius of convexity of $\wp(z)$ and establish the inclusion relations between the class $ \mathscr{S}^*_\wp$ and some well-known classes. Further we derive sharp radius constants and coefficient related results for the class $ \mathscr{S}^*_\wp$.
We consider the class $\mathcal{S}^*(q_c)$ of normalized starlike functions $f$ analytic in the open unit disk $|z|<1$ that satisfying the inequality \begin{equation*} \left|\left(\frac{zf'(z)}{f(z)}\right)^2-1\right|<c \quad (0<c\leq1). \end{equation*} In this article, we present some subordination relations and these relations are then used to obtain some corollaries for some subclass of analytic functions.
In 1984, Clunie and Sheil-Small proved that a sense-preserving harmonic function whose analytic part is convex, is univalent and close-to-convex. In this paper, certain cases are discussed under which the conclusion of this result can be strengthened and extended to fully starlike and fully convex harmonic mappings. In addition, we investgate the properties of functions in the class $\mathcal{M}(α)$ $(|α|\leq 1)$ consisting of harmonic functions $f=h+\overline{g}$ with $g'(z)=αzh'(z)$, $\RE (1+{zh''(z)}/{h'(z)})>-{1}/{2} $ $ \mbox{for} |z|<1 $. The coefficient estimates, growth results, area t…
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…
In this note our aim is to determine the radius of starlikeness of the normalized Bessel functions of the first kind for three different kinds of normalization. The key tool in the proof of our main result is the Mittag-Leffler expansion for Bessel functions of the first kind and the fact that, according to Ismail and Muldoon [IM2], the smallest positive zeros of some Dini functions are less than the first positive zero of the Bessel function of the first kind.
In this paper, we consider a subclass of starlike functions associated with a vertical strip domain. Several results concerned with integral representations, convolutions, and coefficient inequalities for functions belonging to this class are obtained. Furthermore, we consider radius problems and inclusion relations involving certain classes of strongly starlike functions, parabolic starlike functions and other types of starlike functions. The results are essential improvements of the corresponding results obtained by Kargar et al., and the derivations are similar to those used earlier by Sun …
Let $ \mathcal{H}(\mathbb{D}) $ be the linear space of analytic functions on the unit disk $ \mathbb{D}=\{z\in\mathbb{C}: |z|<1\} $ and let $ \mathcal{B}=\{w\in \mathcal{H}(\mathbb{D}: |w(z)|<1)\} $. The classical Bohr's inequality states that if a power series $ f(z)=\sum_{n=0}^{\infty}a_nz^n $ converges in $ \mathbb{D} $ and $ |f(z)|<1 $ for $ z\in\mathbb{D} $, then \begin{equation*} \sum_{n=0}^{\infty}|a_n|r^n\leq 1\;\;\mbox{for}\;\; r\leq \frac{1}{3} \end{equation*} and the constant $ 1/3 $ is the best possible. The constant $ 1/3 $ is known as Bohr radius. A function $ f : \mathbb{D}\righ…
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.
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 $k(z)=z/(1-z)^2$.
Assume that $f$ lies in the class of starlike functions of order $α\in [0,1)$, that is, which are regular and univalent for $|z|<1$ and such that $${\rm Re} \left (\frac{zf'(z)}{f(z)} \right ) > α~\mbox{ for } |z|<1. $$ In this paper we show that for each $α\in [0,1)$, the following sharp inequality holds: $$ |f(re^{iθ})|^{-1} \int_{0}^{r}|f'(ue^{iθ})| du \leq \frac{Γ(\frac12)Γ(2-α)}{Γ(\frac32-α)} ~\mbox {for every $r<1$ and $θ$}. $$ This settles the conjecture of Hall (1980).
In this paper we investigate an interesting subclass $\mathcal{BS}(α)$ ($0\leq α<1$) of starlike functions in the unit disk $Δ$. The class $\mathcal{BS}(α)$ was introduced by Kargar et al. [R. Kargar, A. Ebadian and J. Sokół, {\it On Booth lemniscate and starlike functions}, Anal. Math. Phys. (2017) DOI: 10.1007/s13324-017-0187-3] which is strongly related to the Booth lemniscate. Some geometric properties of this class of analytic functions including, radius of starlikeness of order $γ$ ($0\leqγ<1$), the image of $f(\{z:|z|<r\})$ when $f\in \mathcal{BS}(α)$, an special example and estimate of…
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.
In this paper, we find the radius of the disk $Ω_{r}$ such that every starlike logharmonic mapping $f(z)$ of order $α,$ is starlike in $% |z|\leq r$ with respect to any point of $Ω_{r}.$ We also establish a relation between the set of starlike logharmonic mappings \ and the set of starlike logharmonic mappings of order alpha. Moreover, the radius of starlikeness and univalence for the set of close to starlike logharmonic mappings of order $α$ is determined.
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 the past several subclasses of starlike functions are defined involving real part and modulus of certain expressions of functions under study, combined by way of an inequality. In the similar fashion, we introduce a new class $\mathcal{S}^*_{q}(α)$, consisting of normalized analytic univalent functions $f$ in the open unit disk $\mathbb{D}$, satisfying $$\RE\left(\dfrac{z f'(z)}{f(z)}\right) \geq \left|1+\dfrac{z f''(z)}{f'(z)} -\dfrac{z f'(z)}{f(z)}-α\right| \quad (0 \leq α<1).$$ Evidently, $\mathcal{S}^*_{q}(α) \subset \mathcal{S}^*$, the class of starlike functions. We first establish $\…
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…
Let $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ be analytic in the unit disk with second coefficient $a_2$ satisfying $|a_2|=2b$, $0\leq b\leq1$. Sharp radius of Janowski starlikeness and other radius constants are obtained when $|a_n|\leq cn+d$ ($c,d\geq0$) or $|a_n|\leq c/n$ ($c>0$) for $n\geq3$.
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 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…
The main object of the present paper is to, introduce the. class of meromorphic univalent functions Involving! hypergeomatrc function .We obtain~ some interesting geometric properties according to coefficient inequality , growth and distortion theorems , radii of starlikeness and convexity for the" functions in our subclass.
In this paper, we prove various radius results and obtain sufficient conditions using the convolution for the Ma-Minda classes $\mathcal{S}^*(ψ)$ and $\mathcal{C}(ψ)$ of starlike and convex analytic functions. We also obtain the Bohr radius for the class $ S_{f}(ψ):= \{g(z)=\sum_{k=1}^{\infty}b_k z^k : g \prec f \}$ of subordinants, where $f\in \mathcal{S}^*(ψ).$ The results are improvements and generalizations of several well known results.
In this paper, we introduce a new subclass of harmonic functions $f=s+\overline{t}$ in the open unit disk $U =\left \{ z\in C:\left \vert z\right \vert <1\right \} $ satisfying ${\text{Re}}\left[ γs^{\prime }(z)+δzs^{\prime \prime }(z)+\left( \frac{δ-γ}{2}\right) z^{2}s^{\prime \prime \prime }\left( z\right) -λ\right] >\left \vert γt^{\prime }(z)+δzt^{\prime \prime }(z)+\left( \frac{δ-γ}{2}\right) z^{2}t ^{\prime \prime \prime }\left( z\right) \right \vert,$ where $0\leq λ<γ\leq δ, z\in U.$ We determine several properties of this class such as close-to-convexity, coefficient bounds, and growth…
The Bohr inequality, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r$, $0<r<1$, such that $\sum_{n=0}^\infty |a_n|r^n \leq 1$ holds whenever $|\sum_{n=0}^\infty a_nz^n|\leq 1$ in the unit disk $\mathbb{D}$ of the complex plane. The exact value of this largest radius, known as the \emph{Bohr radius}, has been established to be $1/3.$ This paper surveys recent advances and generalizations on the Bohr inequality. It discusses the Bohr radius for certain power series in $\mathbb{D},$ as well as for analytic functions from $\mathbb{D}$ into particular domains. Thes…
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…
