f × ß r o 0 9 8 1 Y ( ä /| 2 3 8 1 Y X 7 0 3 1 /| Z g Š z Š * ñ D ~ â Z [ ¦ œ & Œ ¦ Y È 0* à ò ) 8 4 2 1 Z z g Z y Å Â [ § ] » ̧ ] ƒ Ð ¬ ¬ g s TM Z c* X â Z [ ñ ß s ò ä § ] ó g Å § ] z q Ñ ] Æ !* g } ~  1⁄4 7 – Ô Ü s Z y Æ * x * ò Z z g Z y Æ Ä Ð Z y Æ !* g } ~ É TM Z y Å Â [ § ] » ¿ c z s ® ð g ¶ Z v ¬ \ ä  [ â g Z Ñ w g ~ &+ TM § ] t t 1⁄4 – ì f × ß r o f TM H ì : L L Z â x ¦ Z z k Ð 3 _...

Let K denote the class of functions g(z) = z + a 2 z 2 + ··· which are regular and univalently convex in the unit disc E. In the present note, we prove that if f is regular in E, f (0) = 0, then for g ∈ K, f (z) + αzf (z) ≺ g(z) + αzg (z) in E implies that f (z) ≺ g(z) in E, where α > 0 is a real number and the symbol " ≺ " stands for subordination.

Z Š Ò ] Z s ò ~ Z E x z . y Æ j Z á Ð 3 z d$ Z z g Ñ b z q í Å g z Z e$ : Ü s 1 6, Z ã Z z g ¢ ì É ‰ & Ñ B 3 } é G E G ] z $ › Ò 3 ̈ é G G G ] Z z g Ñ z b z j Z Ù Â Z K ́ Z ' ] c* Z Ý Ð Ì i c* Š { c z s ƒ N Z z g  z » % œ/ I , ) 1 ( § ] ó g ~ Å @* g õ ~ Ì Z k n Å g z Z e$ ñ Š ì Š ~ œ ~ — ? çE ~ B â , œ ~ { ~ Æ ˆ 1 x Z y à § ] è I Z z g Z y Å Ñ z b z j Z Ù Ì % A$ ƒ ñ Z z g Z y ~ Ð ‰ ¥ â ] Å Ò ...

Let f (z) be an entire function and M(r) the maximum of I f (z) on z = r . We give some results on the density of the set of points at which f (z) I is small in comparison with M(r) ; although simple, these results seem not to have been noticed before . If E is a measurable set in the z-plane, we denote by DR (E) the ratio m(z c E, I z < R)/1rRZ and by D(E) and D(E) the upper and lower densitie...

Let E , E∗ be Hilbert spaces. Given a Hilbert space H of holomorphic functions in a domain Ω in C, consider the multiplier space MH(E , E∗). It is shown that for “nice enough” H, the following statements are equivalent for f ∈ MH(E , E∗): (1) There exists a g ∈ MH(E∗, E) such that g(z)f(z) = IE for all z ∈ Ω. (2) There exists a Hilbert space Ec and fc ∈ MH(Ec, E∗) such that F (z) := [ f(z) fc(z...

where we used the fact that (σ 2 z) = 1 . In general e−iφfn·fσ = 1 cos(φ)− in · nσ sin(φ) Now we consider a system composed of two subsystems H = H1 ⊗H2, each being a two-level system. To describe operators on the two subsystems we normally use the shorthanded notations A ≡ A ⊗ 1 2 and B ≡ 1 1 ⊗B2 and AB ≡ A ⊗B2 . a) What is (σσ) z z ? b) Consider the operator U = e− 1 iσ iσ φ⊗ z e− z : to whic...

We show that for α ∈ (0, 2], if f ∈ A with f ′(z) 6= 0, z ∈ E, satisfies the condition (1− α)f ′(z) + α ( 1 + zf ′′(z) f ′(z) ) ≺ F (z), then f is univalent in E, where F is the conformal mapping of the unit disk E with F (0) = 1 and F (E) = C \ { w ∈ C : < w = α, |= w| ≥ √ α(2− α) } . Our result extends the region of variability of the differential operator (1− α)f ′(z) + α ( 1 + zf ′′(z) f ′(...

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, a vertex labeling $f : V(G)rightarrow mathbb{Z}_2$ induces an edge labeling $ f^{+} : E(G)rightarrow mathbb{Z}_2$ defined by $f^{+}(xy) = f(x) + f(y)$, for each edge $ xyin E(G)$.  For each $i in mathbb{Z}_2$, let $ v_{f}(i)=|{u in V(G) : f(u) = i}|$ and $e_{f^+}(i)=|{xyin E(G) : f^{+}(xy) = i}|$. A vertex labeling $f$ of a graph $G...