On Ozaki's condition for p-valency

Let f be an analytic function in a convex domain $D?\mathbb{C}$. A well-known theorem of Ozaki states that if f is analytic in D, and is given by $f(z)=z^p+{\sum }_{n=p+1}^{\infty }{a}_n{z}^n$ for $z\in D$, and

$\mathfrak{Re}\{{\mathrm{e}}^{\mathrm{i}\alpha }{f}^{(p)}(z)\}>0,(z\in D),$

for some real α, then f is at most p-valent in D. Ozaki's condition is a generalization of the well-known Noshiro–Warschawski univalence condition. The purpose of this paper is to provide some related sufficient conditions for functions analytic in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ to be p-valent in $\mathbb{D}$, and to give an improvement to Ozaki's sufficient condition for p-valence when $z\in \mathbb{D}$.     

Completely bounded norms of right module maps

It is well-known that if T is a $D_m-D_n$ bimodule map on the m × n complex matrices, then T is a Schur multiplier and $6T{6}_{\mathit{cb}}=6T6$. If n = 2 and T is merely assumed to be a right D₂-module map, then we show that $6T{6}_{\mathit{cb}}=6T6$. However, this property fails if m ? 2 and n ? 3. For m ? 2 and n = 3, 4 or n ? m² we give examples of maps T attaining the supremum$C(m\text{,}n)=\sup T{6}_{\mathit{cb}}:T\text{a right}{D}_n\text{-module map on}{M}_{m\text{,}n}\text{with}6T6\le 1\}\text{,}$we show that $C(m\text{,}{m}^2)=\sqrt{m}$ and succeed in finding sharp results for C(m, n) in certain other cases. As a consequence, if H is an infinite-dimensional Hilbert space and D is a masa in B(H), then there is a bounded right D-module map on K(H) which is not completely bounded.