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]

Rectangular modulus, Birkhoff orthogonality and characterizations of inner product spaces

summary:Some characterizations of inner product spaces in terms of Birkhoff orthogonality are given. In this connection we define the rectangular modulus $\mu_{_X}$ of the normed space $X$. The values of the rectangular modulus at some noteworthy points are well-known constants of $X$. Characterizations (involving $\mu_{_X})$ of inner product spaces of dimension $\geq 2$, respectively $\geq 3$, are given and the behaviour of $\mu_{_X}$ is studied

Poezia română clasică : de la Dosoftei la Octavian Goga.

Includes bibliographical references