let $f$ be a locally univalent function on the unit disk $u$. we consider the normalized extensions of $f$ to the euclidean unit ball $b^nsubseteqmathbb{c}^n$ given by$$phi_{n,gamma}(f)(z)=left(f(z_1),(f'(z_1))^gammahat{z}right),$$ where $gammain[0,1/2]$, $z=(z_1,hat{z})in b^n$ and$$psi_{n,beta}(f)(z)=left(f(z_1),(frac{f(z_1)}{z_1})^betahat{z}right),$$in which $betain[0,1]$, $f(z_1)neq 0$ and $...

Let $f$ be a locally univalent function on the unit disk $U$. We consider the normalized extensions of $f$ to the Euclidean unit ball $B^nsubseteqmathbb{C}^n$ given by $$Phi_{n,gamma}(f)(z)=left(f(z_1),(f'(z_1))^gammahat{z}right),$$  where $gammain[0,1/2]$, $z=(z_1,hat{z})in B^n$ and $$Psi_{n,beta}(f)(z)=left(f(z_1),(frac{f(z_1)}{z_1})^betahat{z}right),$$ in which $betain[0,1]$, $f(z_1)neq 0$ a...

*Correspondence: [email protected] 1School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia Full list of author information is available at the end of the article Abstract The estimates for the second Hankel determinant a2a4 – a3 of the analytic function f (z) = z + a2z + a3z + · · · , for which either zf ′(z)/f (z) or 1 + zf ′′(z)/f ′(z) is subordinate to a certai...

A parabolic starlike function f of order ρ in the unit disk is characterized by the fact that the quantity z f ′(z)/ f (z) lies in a given parabolic region in the right half-plane. Denote the class of such functions by PS∗(ρ). This class is contained in the larger class of starlike functions of order ρ. Subordination results for PS∗(ρ) are established, which yield sharp growth, covering, and di...

The main object of this paper is to derive the sufficient conditions for the function z {pψq (z)} to be in the class of uniformly starlike and uniformly convex function associated with the parabolic region Re {ω} > |ω − 1| . Further, the hadamard product of the function which are analytic in the open unit disk with negative coefficients are also investigated. Finally, similar results using an i...

AbstractIn this paper we introduce the subclass K(μ, γ, η, α, β) of β-uniformly convex and β-uniformly starlike functions which are analytic and multivalent with negative coefficients defined by using fractional calculus operators. Characterization property exhibited by the functions in the class and the results of modified Hadamard product are discussed. Connections with the popular subclasses...

If the Lorentzian norm on a maximal surface in the 3-dimensional Lorentz-Minkowski space R 1 is positive and proper, then the surface is relative parabolic. As a consequence, entire maximal graphs with a closed set of isolated singularities are relative parabolic. Furthermore, maximal and minimal graphs over closed starlike domains in R 1 and R, respectively, are relative parabolic.

Abstract This paper studies analytic functions f defined on the open unit disk of complex plane for which / g and (1 + z ) are both with positive real part some function . We determine radius constants these to belong classes strong starlike functions, order ? , parabolic as well associated lemniscate Bernoulli, cardioid, lune, reverse lemniscate, sine function, exponential a particular rationa...

Let A be the class of analytic functions in the open unit disk U . For complex numbers a, b and c (c 6= 0,−1,−2, ....), let I c be the operator defined on A by (I c (f))(z) = z2F1(a, b; c; z) ∗ f(z) where 2F1(a, b; c; z) is the Gaussian hypergeometric function. The function f in A is said to be in the class k − SP c if I a,b c (f) is a k-parabolic starlike function. For this class the Fekete-Sz...

This article defines a new class of meromorphic parabolic starlike functions in the punctured unit disc D*={z∈C:0<|z|<1} that includes fixed second coefficients As,cdψ,τ,ν,η and q- hypergeometric functions. For function belonging to As,cdψ,τ,ν,η, some properties are obtained, including coefficient inequalities, closure theorems, radius convexity.