bluuebunny/crossref_metadata_2025_split huggingface dataset:
The difference between consecutive members of a difference set (𝑚𝑜𝑑 𝑣)
Authors: Spence E. 
Published: June 2010

Let

D

=
{

d
1

,

d
2

,
⋯
,

d
k

}

\mathcal {D} = \{ {d_1},{d_2}, \cdots ,{d_k}\}

be a difference set

(
mod

v
)

\pmod v

so that for any

d
≢
0

(
mod

v
)

d\not \equiv 0\pmod v

there are exactly

λ
\lambda

pairs

(

d
i

,

d
j

)
,

d
i

,

d
j

∈

D

({d_i},{d_j}),{d_i},{d_j}\in \mathcal {D}

such that

d
i

−

d
j

≡
d

(
mod

v
)

{d_i} - {d_j} \equiv d\pmod v

. Suppose further that

0
≤

d
1

>

d
2

>
⋯
>

d
k

>
v

0 \leq {d_1} > {d_2} > \cdots > {d_k} > v

and write

d

k
+
1

=
v
+

d
1

{d_{k + 1}} = v+ {d_1}

. The following two results are proved: (i)

Σ

i
=
1

k

(

d

i
+
1

−

d
i

)
2

=
O
(
(

v
2

/

k
)
log
⁡
k
)
,

\Sigma _{i = 1}^k{({d_{i + 1}} - {d_i})^2} = O(({v^2}/k)\log k),

(ii)

max

1
⩽
i
⩽
k

(

d

i
+
1

−

d
i

)
=
O
(
v

/

√
k
)
.

{\max _{1 \leqslant i \leqslant k}}({d_{i + 1}} - {d_i}) = O(v/\surd k).
Some properties of self-inversive polynomials
Authors: O’Hara P. J. , Rodriguez R. S. 
Published: June 2010

A complex polynomial is called self-inversive [5, p. 201] if its set of zeros (listing multiple zeros as many times as their multiplicity indicates) is symmetric with respect to the unit circle. We prove that if

P
P

is self-inversive and of degree

n
n

then

|

|

P
′

|

|

=

1
2

n

|

|

P

|

|

||P’|| = \tfrac {1}{2}n||P||

where

|

|

P
′

|

|

||P’||

and

|

|

P

|

|

||P||

denote the maximum modulus of

P
′

P’

and

P
P

, respectively, on the unit circle. This extends a theorem of P. Lax [4]. We also show that if

P
(
z
)
=

Σ

j
=
0

n

a
j

z
j

P(z) = \Sigma _{j = 0}^n{a_j}{z^j}

has all its zeros on

|

z

|

=
1

|z| = 1

then

2

Σ

j
=
0

n

|

a
j

|

2

≦

|

|

P

|

|

2

2\Sigma _{j = 0}^n|{a_j}{|^2} \leqq ||P|{|^2}

. Finally, as a consequence of this inequality, we show that when

P
P

has all its zeros on

|

z

|

=
1

|z| = 1

then

2

1

/

2

|

a

n

/

2

|

≦

|

|

P

|

|

{2^{1/2}}|{a_{n/2}}| \leqq ||P||

and

2

|

a
j

|

≦

|

|

P

|

|

2|{a_j}| \leqq ||P||

for

j
≠
n

/

2

j \ne n/2

. This answers in part a question presented in [3, p. 24].
Coefficient estimates for exponential series
Authors: Sledd W. T. 
Published: June 2010

Results are obtained which relate the size of the coefficients

a
n

{a_n}

of an exponential series

f
(
x
)
=

Σ

n
=
0

∞

a
n

ε

−

Λ
n

x

,
x
>
0
,
Re
⁡

Λ
n

>
0

f(x) = \Sigma _{n = 0}^\infty {a_n}{\varepsilon ^{ - {\Lambda _n}x}},x > 0,\operatorname {Re} {\Lambda _n} > 0

, to the function

f
f

. These results involve comparisons between weighted

l
p

{l^p}

sums of the sequence

(

a
n

)

({a_n})

and weighted

L
p

{L^p}

integrals of

f
f

on

[
0
,
∞
)

[0,\infty )

.
Representation theorems for unconditionally converging operators
Authors: Howard Joe , Melendez Kenneth 
Published: June 2010

Let

N
(
X
)

N(X)

be the set

{
F
ε
X

\{ F\varepsilon X

: there exists a weakly unconditionally converging series

Σ

x
n

\Sigma {x_n}

in

X
X

such that

F
=
σ
(
X
,

X
′

)
−

lim
n

Σ

i
=
1

n

J

x
i

}

F = \sigma (X,X’) - {\lim _n}\Sigma _{i = 1}^nJ{x_i}\}

. Representation theorems for the unconditionally converging operators (map weakly unconditionally converging series into unconditionally converging series) are developed by using the

σ
(

X
′

,
N
(
X
)
)

\sigma (X’,N(X))

topology of

X
′

X’

.
Coefficient differences and Hankel determinants of areally mean 𝑝-valent functions
Authors: Noonan James W. 
Published: June 2010

With

p
>
0

p > 0

, denote by

S
p

{S_p}

the class of functions analytic and areally mean

p
p

-valent in the open unit disc. If

f
∈

S
p

f \in {S_p}

, it is well known that

α
(
f
)
=

lim

r
→
1

(
1
−
r

)

2
p

M
(
r
,
f
)

\alpha (f) = {\lim _{r \to 1}}{(1 - r)^{2p}}M(r,f)

exists and is finite. If

q
≥
1

q \geq 1

is an integer, denote the

q
q

Hankel determinant of

f
f

by

H
q

(
n
,
f
)

{H_q}(n,f)

. In this paper the asymptotic behavior of

H
q

(
n
,
f
)

{H_q}(n,f)

, as

n
→
∞

n \to \infty

, is related to

α
(
f
)

\alpha (f)

. A typical result is: if

α
(
f
)
>
0

\alpha (f) > 0

, and if

p
>
q
−
3

/

4

p > q - 3/4

, then
\[

|

H
q

(
n
,
f
)

|

/

n

2
p
q
−

q
2

∼

|

Q
q

(
p
)

|

(
α
(
f
)

/

Γ
(
2
p
)

)
q

,

|{H_q}(n,f)|/{n^{2pq - {q^2}}} \sim |{Q_q}(p)|{(\alpha (f)/\Gamma (2p))^q},

\]
where

Q
q

{Q_q}

is a polynomial of degree at most

q
2

−
q

{q^2} - q

. In the course of the proof, asymptotic results are proved concerning certain coefficient differences, and in particular concerning

|

a
n

|

−

|

a

n
−
1

|

|{a_n}| - |{a_{n - 1}}|

.
Maximal sublattices of finite distributive lattices. II
Authors: Rival Ivan 
Published: June 2010

Let

L
L

be a lattice,

J
(
L
)
=
{
x
∈
L

|

x

J(L) = \{ x \in L|x

join-irreducible in

L
}

L\}

and

M
(
L
)
=
{
x
∈
L

|

x

M(L) = \{ x \in L|x

meet-irreducible in

L
}

L\}

. As is well known the sets

J
(
L
)

J(L)

and

M
(
L
)

M(L)

play a central role in the arithmetic of a lattice

L
L

of finite length and particularly, in the case that

L
L

is distributive. It is shown that the “quotient set”

Q
(
L
)
=
{
b

/

a

|

a
∈
J
(
L
)
,
b
∈
M
(
L
)
,
a
≦
b
}

Q(L) = \{ b/a|a \in J(L),b \in M(L),a \leqq b\}

plays a somewhat analogous role in the study of the sublattices of a lattice

L
L

of finite length. If

L
L

is a finite distributive lattice, its quotient set

Q
(
L
)

Q(L)

) in a natural way determines the lattice of all sublattices of

L
L

. By examining the connection between

J
(
K
)

J(K)

and

J
(
L
)

J(L)

, where

K
K

is a maximal proper sublattice of a finite distributive lattice

L
L

, the following is proven: every finite distributive lattice of order

n
≧
3

n \geqq 3

which contains a maximal proper sublattice of order

m
m

also contains sublattices of orders

n
−
m
,
2
(
n
−
m
)

n - m,2(n - m)

, and

3
(
n
−
m
)

3(n - m)

; and, every finite distributive lattice

L
L

contains a maximal proper sublattice

K
K

such that either

|

K

|

=

|

L

|

−
1

|K| = |L| - 1

or

|

K

|

≧
2
l
(
L
)

|K| \geqq 2l(L)

, where

l
(
L
)

l(L)

denotes the length of

L
L

.
Lattice points and Lie groups. III
Authors: Cahn Robert S. 
Published: June 2010

If a compact, simply connected, semisimple Lie group is considered as a Riemannian manifold with metric arising from the negative of the Killing form it is shown that its volume is
\[

(
4
π

)

dim
⁡
G

/

2

Γ
(
dim
⁡
G

/

2
+
1
)
(
1

/

|

w

|

)

∫

|

Λ

|

⩽
1

f

2

(
Λ
)
d
Λ

.

{(4\pi )^{\dim G/2}}\Gamma (\dim G/2 + 1)(1/|w|)\int _{|\Lambda | \leqslant 1} {{f^{{2_{(\Lambda )d\Lambda }}}}.}

\]
A biorthogonal expansion related to the zeros of an entire function
Authors: Benzinger Harold E. 
Published: June 2010

Let

D
(
λ
)
=

a
0

+

Σ

k
=
1

∞

a
k

e

λ

/

k

D(\lambda ) = {a_0} + \Sigma _{k = 1}^\infty {a_k}{e^{\lambda /k}}

, where the

a
k

{a_k}

’s are given complex constants, with

a
0

a
1

≠
0
,

Σ

k
=
1

∞

|

a
k

|

>
∞

{a_0}{a_1} \ne 0,\Sigma _{k = 1}^\infty |{a_k}| > \infty

. Let

{

λ
k

}
,
k
=
1
,
2
,
⋯

\{ {\lambda _k}\} ,k = 1,2, \cdots

, denote the sequence of distinct zeros of

D
(
λ
)

D(\lambda )

, labeled in order of increasing modulus, and with multiplicities

m
k

≥
1

{m_k} \geq 1

. Let

{

ϕ
l

(
x
)
}

\{ {\phi _l}(x)\}

denote the sequence of functions

{

x
j

exp
⁡
(

λ
k

x
)
:
j
=
0
,
⋯
,

m
k

−
1
;
k
=
1
,
2
,
⋯
;
0
>
x
≤
1
}

\{ {x^j}\exp ({\lambda _k}x):j = 0, \cdots ,{m_k} - 1;k = 1,2, \cdots ;0 > x \leq 1\}

. We show that for each

p
.1
≤
p
>
∞

p.1 \leq p > \infty

, there is a sequence

{

ψ
l

(
x
)
}

\{ {\psi _l}(x)\}

in

L
q

(
0
,
1
)

(
p
q
=
p
+
q
)

{L^q}(0,1)\;(pq = p + q)

such that

(

ϕ
l

,

ψ
m

)
=

δ

l
m

({\phi _l},{\psi _m}) = {\delta _{lm}}

. Then we show that

{

ϕ
l

}

\{ {\phi _l}\}

is complete in

L
p

(
0
,
1
)
,
1
≤
p
>
∞

{L^p}(0,1),1 \leq p > \infty

, and for

1
>
p
>
∞

1 > p > \infty

, we find a subspace of

L
p

(
0
,
1
)

{L^p}(0,1)

such that the biorthogonal expansion

f
=

Σ

k
=
1

∞

(
f
,

ψ
k

)

ϕ
k

f = \Sigma _{k = 1}^\infty (f,{\psi _k}){\phi _k}

is valid in the norm of

L
p

(
0
,
1
)

{L^p}(0,1)

.
New criteria for univalent functions
Authors: Ruscheweyh Stephan 
Published: June 2010

The classes

K
n

{K_n}

of functions

f
(
z
)

f(z)

regular in the unit disc

U

\mathfrak {U}

with

f
(
0
)
=
0
,

f
′

(
0
)
=
1

f(0) = 0,f’(0) = 1

satisfying
\[

Re
⁡
[

(

z
n

f

)

(
n
+
1
)

/

(

z

n
−
1

f

)

(
n
)

]
>
(
n
+
1
)

/

2

\operatorname {Re} [{({z^n}f)^{(n + 1)}}/{({z^{n - 1}}f)^{(n)}}] > (n + 1)/2

\]
in

U

\mathfrak {U}

are considered and

K

n
+
1

⊂

K
n

,
n
=
0
,
1
,
⋯

{K_{n + 1}} \subset {K_n},n = 0,1, \cdots

, is proved. Since

K
0

{K_0}

is the class of functions starlike of order 1/2 all functions in

K
n

{K_n}

are univalent. Some coefficient estimates are given and special elements of

K
n

{K_n}

are determined.
Fixed point theorems for mappings with a contractive iterate at a point
Authors: Matkowski Janusz 
Published: June 2010

Let (X,d) be a complete metric space,

T
:
X
→
X

T:X \to X

, and

α
:
[
0
,
∞

)
5

→
[
0
,
∞
)

\alpha :[0,\infty )^5 \to [0,\infty )

be nondecreasing with respect to each variable. Suppose that for the function

γ
(
t
)
=
α
(
t
,
t
,
t
,
2
t
,
2
t
)

\gamma (t) = \alpha (t,t,t,2t,2t)

, the sequence of iterates

γ
n

{\gamma ^n}

tends to 0 in

[
0
,
∞
)

[0,\infty )

and

lim

t
→
∞

(
t
−
γ
(
t
)
)
=
∞

{\lim _{t \to \infty }}(t - \gamma (t)) = \infty

. Furthermore, suppose that for each

x
∈
X

x \in X

there exists a positive integer

n
=
n
(
x
)

n = n(x)

such that for all

y
∈
X

y \in X

,
\[

d
(

T
n

x
,

T
n

y
)
⩽
α
(
d
(
x
,

T
n

x
)
,
d
(
x
,

T
n

y
)
,
d
(
x
,
y
)
,
d
(

T
n

x
,
y
)
,
d
(

T
n

y
,
y
)
)
.

d({T^n}x,{T^n}y) \leqslant \alpha (d(x,{T^n}x),d(x,{T^n}y),d(x,y),d({T^n}x,y),d({T^n}y,y)).

\]
Under these assumptions our main result states that T has a unique fixed point. This generalizes an earlier result of V. M. Sehgal and some recent results of L. Khazanchi and K. Iseki.
Noncrossed products of small exponent
Authors: Saltman David J. 
Published: June 2010

Using the generic division algebras, we construct new division algebras and prove the following. Theorem. Let

m
m

,

n
n

be positive integers,

m

|

n

m|n

, such that

m
m

is divisible by every prime which divides

n
n

. If

k
k

is any field, and there is a prime

p
p

such that

p
3

|

m

{p^3}|m

, then there is a noncrossed product division

k
k

-algebra of exponent

m
m

and degree

n
n

. If

k
k

is a global field of characteristic 0,

p
p

is an odd prime, and

k
k

does not contain a primitive

p
p

th root of 1, we need only assume

p
2

|

m

p^2|m

.
An extension of the Hausdorff-Toeplitz theorem on the numerical range
Authors: Au-Yeung Yik Hoi , Tsing Nam-Kiu 
Published: June 2010

Let

H

n

{\mathcal {H}_n}

be the set of all

n
×
n

n \times n

hermitian matrices and

U

n

{\mathcal {U}_n}

the set of all

n
×
n

n \times n

unitary matrices. For any

c
=
(

c
1

,
…
,

c
n

)
∈

R

n

c = ({c_1}, \ldots ,{c_n}) \in {{\mathbf {R}}^n}

and

A
1

{A_1}

,

A
2

{A_2}

,

A
3

∈

H

n

{A_3} \in {\mathcal {H}_n}

, let

W
(

A
1

,

A
2

,

A
3

)

W({A_1},{A_2},{A_3})

denote the set
\[

{
(

tr

[
c
]
U

A
1

U
∗

,

tr

[
c
]
U

A
2

U
∗

,

tr

[
c
]
U

A
3

U
∗

)
:
U
∈

U

n

}
,

\{ ({\operatorname {tr}}[c]U{A_1}{U^*},{\operatorname {tr}}[c]U{A_2}{U^*},{\operatorname {tr}}[c]U{A_3}{U^*}):U \in {\mathcal {U}_n}\} ,

\]
where

[
c
]

[c]

is the diagonal matrix with

c
1

,
…
,

c
n

{c_1}, \ldots ,{c_n}

as diagonal entries. In this present note, the authors prove that if

n
>
2

n > 2

, then

W
c

(

A
1

,

A
2

,

A
3

)

{W_c}({A_1},{A_2},{A_3})

is always convex. Equivalent statements of this result, in terms of definiteness and inclusion relations, are also given. These results extend the theorems of Hausdorff-Toeplitz, Finsler and Westwick on numerical ranges, respectively.
On a theorem of P. L. Uljanov
Authors: Stanojevic Vera B. 
Published: June 2010

It is shown that if

c

(
n
)

=
o

(
1
)

c\left ( n \right ) = o\left ( 1 \right )

,

|
n
|

→
∞

\left | n \right | \to \infty

, and

∑

|
n
|

>
∞

|

Δ
m

c

(
n
)

|

>
∞

{\sum _{\left | n \right |}}_{ > \infty }\left | {{\Delta ^m}c\left ( n \right )} \right | > \infty

, for some integer

m
⩾
1

m \geqslant 1

, then the series

∑

|
n
|

>
∞

c

(
n
)

e

int

{\sum _{\left | n \right |}}_{ > \infty }c\left ( n \right ){e^{\operatorname {int} }}

converges to some

f
∈

L
p

(

T

)

f \in {L^p}\left ( {\mathbf {T}} \right )

for any

0
>
p
>
1

/

m

0 > p > 1 / m

.
Elementary proof of a formula of Ramanujan
Authors: Lamphere Robert L. 
Published: June 2010

In this paper we use only elementary calculus to prove Ramanujan’s integral formula
\[

∫
0
∞

x

m
−
1

(
1
+
a
b
x
)
(
1
+
a

b
2

x
)
⋯
(
1
+
a

b
n

x
)
⋯

(
1
+
x
)
(
1
+
b
x
)
(
1
+

b
2

x
)
⋯
(
1
+

b
n

x
)
⋯

d
x
=

π

Sin

(
m
π
)

∏

k
=
1

∞

(
1
−

b

k
−
m

)
(
1
−
a

b
k

)

(
1
−

b
k

)
(
1
−
a

b

k
−
m

)

\int _0^\infty {{x^{m - 1}}} \frac {{(1 + abx)(1 + a{b^2}x) \cdots (1 + a{b^n}x) \cdots }}{{(1 + x)(1 + bx)(1 + {b^2}x) \cdots (1 + {b^n}x) \cdots }}dx = \frac {\pi }{{{\operatorname {Sin}}(m\pi )}}\prod \limits _{k = 1}^\infty {\frac {{(1 - {b^{k - m}})(1 - a{b^k})}}{{(1 - {b^k})(1 - a{b^{k - m}})}}}

\]
where

m
m

,

a
a

and

b
b

are positive with

b
>
1

b > 1

and

a
>

b

m
−
1

a > {b^{m - 1}}

.
Spectra of constructs of a system of operators
Authors: Carrillo Angel , Hernández Carlos 
Published: June 2010

This paper describes the spectrum and the upper and lower Fredholm spectra of

(
n
+
m
)

(n + m)

-tuples

(
F
(

A
1

)
,
⋯
,
F
(

A
n

)
,
G
(

B
1

)
,
⋯
,
G
(

B
m

)
)

(F({A_1}), \cdots ,F({A_n}),G({B_1}), \cdots ,G({B_m}))

of operators, where

(

A
i

)

({A_i})

and

(

B
j

)

({B_j})

are systems of operators in two Hilbert spaces

H

1

{\mathcal {H}_1}

and

H

2

{\mathcal {H}_2}

, and

F
F

and

G
G

are certain linear operators defined on

L

(

H

i

)

\mathcal {L}({\mathcal {H}_i})

. Using spectral mapping theorems the spectra of operators constructed by the action of a polynomial on a system

(
F
(

A
1

)
,
⋯
,
F
(

A
n

)
,
G
(

B
1

)
,
⋯
,
G
(

B
m

)
)

(F({A_1}), \cdots ,F({A_n}),G({B_1}), \cdots ,G({B_m}))

is obtained. In particular, the spectra of the elementary operator and tensor products of operators is determined.
Iterates of holomorphic self-maps of the unit ball in Hilbert space
Authors: Stachura Adam 
Published: June 2010

An example is given of a biholomorphic self-mapping

T
T

of the unit ball in infinite-dimensional Hilbert space satisfying

0
=
lim

inf
n

‖

T
n

(
0
)

‖

>
lim

sup
n

|

|

T
n

(
0
)

|

|

=
1

0 = \lim {\inf _n}\left \| {{T^n}\left ( 0 \right )} \right \| > \lim {\sup _n}||{T^n}(0)|| = 1

.
Strong limit theorems for orthogonal sequences in von Neumann algebras
Authors: Jajte R. 
Published: June 2010

Let

A
A

be a von Neumann algebra with a faithful normal state

ϕ
\phi

. It is shown that if a sequence

(

x
n

)

({x_n})

in

A
A

is orthogonal relative to

ϕ
\phi

and satisfies the condition
\[

∑

k

ϕ
(

|

x
k

|

2

)

(

log
⁡
k

k

)

2

>
∞
,

\sum \limits _{k} {\phi (|{x_k}{|^2}){{\left ( {\frac {{\log k}}{k}} \right )}^2} > \infty ,}

\]
then

n
1

∑

k
=
1

n

x
k

→
0

{}_n^1\sum \nolimits _{k = 1}^n {{x_k} \to 0}

almost uniformly in

A
A

. Some other results related to this theorem are also discussed.
Hermitian forms and the fibration of spheres
Authors: Binding Paul 
Published: June 2010

We identify the real

(
2
n
−
1
)

(2n - 1)

-dimensional sphere

S

2
n
−
1

{S^{2n - 1}}

with the unit sphere of

F

2

{{\mathbf {F}}^2}

, where

F

=

reals

{\mathbf {F}} = {\text {reals}}

, complexes or quaternions and

n
=
1
,
2

n = 1,2

or 4, respectively. It is shown how any Hermitian form over

F

2

{{\mathbf {F}}^2}

, restricted to

S

2
n
−
1

{S^{2n - 1}}

, is related to the (double covering for

n
=
1

n = 1

, Hopf for

n
=
2
,
4

n = 2,4

) fibration
\[

(

x
1

,

x
2

)
→
(

|

x
1

|

2

−

|

x
2

|

2

,
2

x
1

x
¯

2

)
:

S

2
n
−
1

→

S
n

.

({x_1},{x_2}) \to ({\left | {{x_1}} \right |^2} - {\left | {{x_2}} \right |^2},2{x_1}{\bar x_2}):{S^{2n - 1}} \to {S^n}.

\]
Convexity of the joint range of several Hermitian forms over the unit sphere of an arbitrary normed vector space

V
V

over

F

{\mathbf {F}}

, with

dim
⁡
V
>
2

\dim V > 2

, is deduced as a corollary.
Convex functions and Fourier coefficients
Authors: Wang Hann Tzong 
Published: June 2010

Let

f
f

be a continuous function defined on the interval

(
0
,
1
)

(0,1)

. For

n
=
1
,
2
,
…

n = 1,2, \ldots

and

0
>
s
>
t
>
1

0 > s > t > 1

, denote by

a
n

(
f
;
s
,
t
)
,

b
n

(
f
;
s
,
t
)

{a_n}(f;s,t),{b_n}(f;s,t)

the

n
n

th Fourier coefficients of

f

|

(
s
,
t
)

f|(s,t)

. It is shown that the following statements are equivalent: (i)

f
f

is strictly convex on

(
0
,
1
)

(0,1)

. (ii)

b
n

(
f
;
s
,
t
)
>
(
2

/

n
π
)
[
f
(
s
)
−
f
(
(
s
+
t
)
)

/

2
]

{b_n}(f;s,t) > (2/n\pi )[f(s) - f((s + t))/2]

for all

n
=
1
,
2
,
…

n = 1,2, \ldots

and whenever

0
>
s
>
t
>
1

0 > s > t > 1

. (iii)

b
n

(
f
;
s
,
t
)
>
(
2

/

n
π
)
[
f
(
(
s
+
t
)

/

2
)
−
f
(
t
)
]

{b_n}(f;s,t) > (2/n\pi )[f((s + t)/2) - f(t)]

for all

n
=
1
,
2
,
…

n = 1,2, \ldots

and whenever

0
>
s
>
t
>
1

0 > s > t > 1

. If, in addition,

f
f

is twice differentiable, then (i) and the following statement are also equivalent: (iv)

a
n

(
f
;
s
,
t
)
>
0

{a_n}(f;s,t) > 0

for all

n
=
1
,
2
,
…

n = 1,2, \ldots

and whenever

0
>
s
>
t
>
1

0 > s > t > 1

.
On unimodular rows
Authors: Roitman Moshe 
Published: June 2010

We prove here, among other results, that if

(

x
0

,
…
,

x
n

)

({x_0}, \ldots ,{x_n})

is a unimodular row over a commutative ring

A
A

,

n
⩾
2

n \geqslant 2

,

x
∈
A

x \in A

and
\[

x
≡

x
n

mod

J
(
A

x
0

+
⋯
+
A

x

n
−
2

)

x \equiv {x_n}\quad \mod J(A{x_0} + \cdots + A{x_{n - 2}})

\]
then

(

x
0

,
…
,

x

n
−
1

,

x
n

)

∼
E

(

x
0

,
…
,

x

n
−
1

,
x
)

({x_0}, \ldots ,{x_{n - 1}},{x_n}){ \sim _E}({x_0}, \ldots ,{x_{n - 1}},x)

.
A problem on the Bloch norm of functions in Doob’s class
Authors: Hwang J. S. 
Published: June 2010

Let

Δ
\Delta

denote the unit disc and

∂
Δ

\partial \Delta

denote the unit circle both in the complex plane. Define the Doob’s class

D
(
ρ
)

D(\rho )

,

0
>
ρ
>
2
π

0 > \rho > 2\pi

, as all holomorphic functions on

Δ
\Delta

satisfying (1)

f
(
0
)
=
0

f(0) = 0

, and (2) for some arc

Γ
f

⊆
∂
Δ

{\Gamma _f} \subseteq \partial \Delta

with arclength

ρ
\rho

, for all

p
∈
Γ

p \in \Gamma

,

lim
_

z
→
p

|

f
(
z
)

|

≥
1

{\underline {\lim } _{z \to p}}|f(z)| \geq 1

. Recently the author and Rung [6] proved a conjecture of Doob made in 1935 by showing that the norm
\[

|

|

f

|

|

=

sup

z
∈
Δ

(
1
−

|

z

|

2

)

|

f
′

(
z
)

|

≥

2
sin
⁡
θ
(
ρ
)

e
θ
(
ρ
)

,

0
≤
θ
(
ρ
)
≤
π
−
ρ

/

2.

||f|| = {\sup _{z \in \Delta }}(1 - |z{|^2})|f’(z)| \geq \frac {{2\sin \theta (\rho )}}{{e\theta (\rho )}},\quad 0 \leq \theta (\rho ) \leq \pi - \rho /2.

\]
We then conjecture that the result should be true if the arc

Γ
f

{\Gamma _f}

is replaced by a finite union of arcs whose total length is at least

ρ
\rho

. In this paper, we answer this problem. It turns out to be surprising that the answer depends on the connectivity of the union, namely, the answer is no for the disconnected case, but yes for the connected one.
Mixed Hadamard’s theorems
Authors: Furuta Takayuki 
Published: June 2010

An operator

T
T

means a bounded linear operator on a complex Hilbert space

H
H

. We give two types of mixed Hadamard’s theorems containing the terms

T
,

|
T
|

T,\left | T \right |

and

|

T
∗

|

\left | {{T^ * }} \right |

as extensions of Hadamard’s theorem and mixed Schwarz’s inequality

|

(
T
x
,
y
)

|

2

≤
(

|
T
|

x
,
x
)
(

|

T
∗

|

y
,
y
)

{\left | {(Tx,y)} \right |^2} \leq (\left | T \right |x,x)(\left | {{T^ * }} \right |y,y)

for any

T
T

and for any

x
x

and

y
y

in

H
H

. Also we scrutinize the cases when the equalities in these mixed Hadamard’s theorems hold.
Poisson semigroups and singular integrals
Authors: Dahlberg Björn E. J. 
Published: June 2010

Let

D
⊂

R

n

D \subset {{\mathbf {R}}^n}

be a Lipschitz domain and consider the bilinear form

∫
D

u

(

∂
v

/

∂
y

)

d
P

\int _D {u\left ( {\partial v/\partial y} \right )dP}

. We show that the form is bounded if

v
v

is harmonic with boundary values in

L
2

{L^2}

, if

u
u

is smooth with its nontangential maximal function in

L
2

{L^2}

and

∫
D

dist

{

P
,
∂
D

}

|

grad

u

|

2

d
P
>
∞

\int _D {{\text {dist}}\left \{ {P,\partial D} \right \}{{\left | {{\text {grad }}u} \right |}^2}dP > \infty }

.
Tangential limits of starlike univalent functions
Authors: Twomey J. B. 
Published: June 2010

Let

f
f

be starlike univalent in the unit disc, let

γ
>
1

\gamma > 1

and let

K
>
0

K > 0

. Then

f
(
z
)

f(z)

tends to a limit as

z
→

e

i
θ

z \to {e^{i\theta }}

inside

{
z
:

|

e

i
θ

−
z

|

≤
K

(
1
−

|
z
|

)

1

/

γ

}

\{ z:\left | {{e^{i\theta }} - z} \right | \leq K{(1 - \left | z \right |)^{1/\gamma }}\}

for every

θ
\theta

in

[
0
,
2
π
]

[0,2\pi ]

. This result is sharp.
Oscillation theorems for second order linear differential equations with damping
Authors: Yan Ju Rang 
Published: June 2010

In this paper, we present some criteria for the oscillation of the differential equation with damping
\[

(
r
(
t
)

x
′

(
t
)

)
′

+
p
(
t
)

x
′

(
t
)
+
q
(
t
)
x
(
t
)
=
0
,

t
∈
(

t
0

,
∞
)
,

(r(t)x’(t))’ + p(t)x’(t) + q(t)x(t) = 0,\quad t \in ({t_0},\infty ),

\]
where

p
(
t
)

p(t)

and

q
(
t
)

q(t)

are allowed to change sign on

[

t
0

,
∞
)

[{t_0},\infty )

, and

γ
(
t
)
>
0

\gamma (t) > 0

. One of our results is new even for the differential equations
\[

x
(
t
)
+
q
(
t
)
x
(
t
)
=
0
,

x(t) + q(t)x(t) = 0,

\]
and
\[

x
(
t
)
+
p
(
t
)

x
′

(
t
)
+
q
(
t
)
x
(
t
)
=
0.

x(t) + p(t)x’(t) + q(t)x(t) = 0.

\]
Marx-Strohhäcker differential subordination systems
Authors: Miller Sanford S. , Mocanu Petru T. 
Published: June 2010

Let

f
(
z
)
=
z
+

a
2

z
2

+
⋯

f(z) = z + {a_2}{z^2} + \cdots

be analytic in the unit disc

U
U

and let

k
(
z
)
=
z

/

(
1
−
z
)

k(z) = z/(1 - z)

. The classic Marx-Strohhäcker result, that a convex (univalent) function

f
f

is starlike of order

1
2

\frac {1}{2}

, can be written in terms of differential subordinations as
\[

z
f
(
z
)

/

f
′

(
z
)
≺
z
k
(
z
)

/

k
′

(
z
)
⇒
z

f
′

(
z
)

/

f
(
z
)
≺
z

k
′

(
z
)

/

k
(
z
)
.

zf(z)/f’(z) \prec zk(z)/k’(z) \Rightarrow zf’(z)/f(z) \prec zk’(z)/k(z).

\]
The authors determine general conditions on

k
k

for which this relation holds. They also determine a different set of general conditions on

k
k

for which
\[

z

f
′

(
z
)

/

f
(
z
)
≺
z

k
′

(
z
)

/

k
(
z
)
⇒
f
(
z
)

/

z
≺
k
(
z
)

/

z
.

zf’(z)/f(z) \prec zk’(z)/k(z) \Rightarrow f(z)/z \prec k(z)/z.

\]
Finally, differential subordinations with starlike superordinate functions are considered.