A sufficient condition for starlikeness of order α

We obtain a sufficient condition for starlikeness of order α, |f′(z)−λ(f(z)/z)+λ−1|<M=Mn(λ,α), where λ∈[0,1], α∈[0,1] and the function f(z)=z+an+1zn+1+⋯ is analytic in the unit disc U

Briot–Bouquet differential superordinations and sandwich theorems

AbstractBriot–Bouquet differential subordinations play a prominent role in the theory of differential subordinations. In this article we consider the dual problem of Briot–Bouquet differential superordinations. Let β and γ be complex numbers, and let Ω be any set in the complex plane C. The function p analytic in the unit disk U is said to be a solution of the Briot–Bouquet differential superordination ifΩ⊂{p(z)+zp′(z)βp(z)+γ|z∈U}. The authors determine properties of functions p satisfying this differential superordination and also some generalized versions of it.In addition, for sets Ω1 and Ω2 in the complex plane the authors determine properties of functions p satisfying a Briot–Bouquet sandwich of the formΩ1⊂{p(z)+zp′(z)βp(z)+γ|z∈U}⊂Ω2. Generalizations of this result are also considered

Convex subordination chains and injective mappings in Cn

AbstractIn this paper we continue the work related to convex subordination chains in C and Cn, and prove that if f(z)=z+∑k=2∞Ak(zk) is a holomorphic mapping on the Euclidean unit ball Bn in Cn such that ∑k=2∞k2‖Ak‖⩽1, a:[0,1]→[0,∞) is a function of class C2 on (0,1) and continuous on [0,1], such that a(1)=0, a(t)>0, ta′(t)>−1/2 for t∈(0,1), and if a(⋅) satisfies a differential equation on (0,1), then f(z,t)=a(t2)Df(tz)(tz)+f(tz) is a convex subordination chain over (0,1] and the mapping F(z)=a(‖z‖2)Df(z)(z)+f(z) is injective on Bn. We also present certain coefficient bounds which provide sufficient conditions for univalence, quasiregularity and starlikeness for the chain f(z,t). Finally we give some examples of convex subordination chains over (0,1]

Some properties of starlike functions with respect to symmetric-conjugate points

Let A be tile class of all analytic functions in the unit disk U such that f(0)=f′(0)−1=0. A function f∈A is called starlike with respect to 2n symmetric-conjugate points if Rezf′(z)/fn(z)>0 for z∈U, where fn(z)=12n∑k=0n−1[ω−kf(ωkz)+ωkf(ωkz˜)¯], ω=exp(2πi/n]. This class is denoted by Sn*, and was studied in [1]. A sufficient condition for starlikeness with respect to symmetric-conjugate points is obtained. In addition, images of some subclasses of Sn* under the integral operator I:A→A, I(f)=F where F(z)=c+1(g(z))c∫0zf(t)(g(t))c−1g′(t)dt,   c>0 and g∈A is given are determined

The Hardy classes for functions in the class MV[[alpha], k]

Suppose that f(z) = z + a2z2 + [middle dot][middle dot][middle dot] + anzn + [middle dot][middle dot][middle dot] is regular in the unit disc D with [f(z) f'(z)/z] [not equal to] 0 in D, and further let [alpha] [ges] 0 and k [ges] 2. If [is proportial to]2[pi]o | Re{(1 - [alpha])z[f'(z)/f(z)] + [alpha](1 + z[f"(z)/f'(z)])}| d[theta] [equal-or-less, slanted] k[pi] for z [epsilon] D, then f(z) is said to belong to the class MV[[alpha], k]. This class contains many of the special classes of regular and univalent functions. The authors determine the Hardy classes of which f(z), f'(z) and f"(z) belong and obtain growth estimates of an.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/22155/1/0000586.pd