Radius of Limaçon starlikeness for Janowski starlike functions

Let [Formula: see text] be an analytic function defined on the open unit disc [Formula: see text], with [Formula: see text], satisfying the subordination [Formula: see text], where [Formula: see text]. The domain [Formula: see text] is bounded by a Limaçon and the function [Formula: see text] is called starlike function associated with Limaçon domain. For [Formula: see text], we find the smallest disc [Formula: see text] and the largest disc [Formula: see text], centered at [Formula: see text] such that the domain [Formula: see text] is contained in [Formula: see text] and contains [Formula: see text]. By using this result, we find the radius of Limaçon starlikeness for the class of starlike functions of order [Formula: see text] [Formula: see text] and the class of functions [Formula: see text] satisfying [Formula: see text], [Formula: see text]. We give extension of our results for Janowski starlike functions.

On Analytic Multivalent Function Associated with Cosine Hyperbolic Function

In this article, we set up some adequate conditions for analytic functions associated with cosine hyperbolic function. We determine conditions on α such that 1+α (ϰ^(2+P(j-1) ) g^' (ϰ))/(Pg^j (ϰ)),for each j=0,1,2,3 are subordinated by Janowski function, then (g(ϰ))/ϰ^P <cosℏ⁡(ϰ),(ϰ∈C). By choosing specific values of functions g we get some adequate conditions for multivalent starlike function related with cosine hyperbolic.

A Generalized Class of Close-to-Convex Functions

Let ℋαϕ(β) denote the class of functions f, analytic in the open unit disk ???? which satisfy the condition ( ( ) ) zf-′(z-)- zf-′′(z-) ℜ (1 − α) + α 1 + ′ > β, z ∈ ????, ϕ(z ) f (z ) where α, β are pre-assigned real numbers and ϕ(z) is a starlike function. The special cases of the class ℋαϕ(β) have been studied in literature by different authors. In 2007, Singh et al. [?] studied the class ℋαz(β) and they established that functions in ℋαz(β) are univalent for all real numbers α, β satisfying the condition α ≤ β < 1 and the result is sharp in the sense that constant β cannot be replaced by a real number smaller than α. Singh et al. [?] in 2005, proved that for 0 < α < 1 functions in class ℋαz(α) are univalent. In 1975, Al-Amiri and Reade [?] showed that functions in class ℋαz(0) are univalent for all α ≤ 0 and also for α = 1 in ????. In the present paper, we prove that members of the class ℋαϕ(β) are close-to-convex and hence univalent for real numbers α, β and for a starlike function ϕ satisfying the condition β + α − 1 < αℜ( ) zϕ′(z) ϕ(z)≤ β < 1.

On the Fourth Coefficient of the Inverse of a Starlike Function of Positive Order

We consider the inverse function $z=g(w)$ of a (normalized) starlike function $w=f(z)$ of order $\alpha$ on the unit disk of the complex plane with $0&lt;\alpha&lt;1.$ Krzy{\. z}, Libera and Z\l otkiewicz obtained sharp estimates of the second and the third coefficients of $g(w)$ in their 1979 paper. Prokhorov and Szynal gave sharp estimates of the fourth coefficient of $g(w)$ as a consequence of the solution to an extremal problem in 1981. We give a straightforward proof of the estimate of the fourth coefficient of $g(w)$ together with explicit forms of the extremal functions.

A subclass of uniformly convex functions and a corresponding subclass of starlike function with fixed coefficient associated with q-analogue of Ruscheweyh operator

Making use of Ruscheweyh q-differential operator, we define a new subclass of uniformly convex functions and corresponding subclass of starlike functions with negative coefficients. The main object of this paper is to obtain, coefficient estimates, closure theorems and extreme point for the functions belonging to this new class. The results are generalized to families with fixed finitely many coefficients.

ON SOME PROPERTIES OF A GENERALIZED CLASS OF CLOSE-TO-STARLIKE FUNCTIONS

In this paper, we consider a new class of close-to-starlike functions defined by the Carlson-Shaffer operator. Let denote the class of analytic univalent functions defined by then ifsatisfy the condition ,where and is a starlike function. Properties of the class such as the coefficient bounds, growth and distortion theorems and radius properties are investigated.

Extension Operators for Biholomorphic Mappings

Suppose that $D\subset \mathbb{C}$ is a simply connected subdomain containing the origin and $f(z_{1})$ is a normalized convex (resp., starlike) function on $D$. Let $$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{N}(D)=\bigg\{(z_{1},w_{1},\ldots ,w_{k})\in \mathbb{C}\times \mathbb{C}^{n_{1}}\times \cdots \times \mathbb{C}^{n_{k}}:\Vert w_{1}\Vert _{p_{1}}^{p_{1}}+\cdots +\Vert w_{k}\Vert _{p_{k}}^{p_{k}}<\frac{1}{\unicode[STIX]{x1D706}_{D}(z_{1})}\bigg\},\end{eqnarray}$$ where $p_{j}\geqslant 1$, $N=1+n_{1}+\cdots +n_{k}$, $w_{1}\in \mathbb{C}^{n_{1}},\ldots ,w_{k}\in \mathbb{C}^{n_{k}}$ and $\unicode[STIX]{x1D706}_{D}$ is the density of the hyperbolic metric on $D$. In this paper, we prove that $$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{N,1/p_{1},\ldots ,1/p_{k}}(f)(z_{1},w_{1},\ldots ,w_{k})=(f(z_{1}),(f^{\prime }(z_{1}))^{1/p_{1}}w_{1},\ldots ,(f^{\prime }(z_{1}))^{1/p_{k}}w_{k})\end{eqnarray}$$ is a normalized convex (resp., starlike) mapping on $\unicode[STIX]{x1D6FA}_{N}(D)$. If $D$ is the unit disk, then our result reduces to Gong and Liu via a new method. Moreover, we give a new operator for convex mapping construction on an unbounded domain in $\mathbb{C}^{2}$. Using a geometric approach, we prove that $\unicode[STIX]{x1D6F7}_{N,1/p_{1},\ldots ,1/p_{k}}(f)$ is a spiral-like mapping of type $\unicode[STIX]{x1D6FC}$ when $f$ is a spiral-like function of type $\unicode[STIX]{x1D6FC}$ on the unit disk.