bluuebunny/crossref_metadata_2025_split huggingface dataset:
Some inequalities related to Hölder’s inequality
Authors: Neugebauer C. J. 
Published: June 2010

In this paper we generalize the Jodeit-Jones-Moser inequalities relating

0
⩽
x
−

(

∫
0
x

f

)

p

,

if

‖
f
‖

q

⩽
1

0 \leqslant x - {\left ( {\int _0^xf} \right )^p},{\text {if}}{\left \| f \right \|_q} \leqslant 1

.
Π-regular variation
Authors: Geluk J. L. 
Published: June 2010

A function

U
:

R
+

→

R
+

U:{R^ + } \to {R^ + }

is said to be

Π
\Pi

-regularly varying with exponent

α
\alpha

if

U
(
x
)

x

−
α

U(x){x^{ - \alpha }}

is nondecreasing and there exists a positive function

L
L

such that
\[

U
(
λ
x
)

/

λ
α

−
U
(
x
)

x
α

L
(
x
)

→
log
⁡
λ

(
x
→
∞
)

for

λ

> 0

.

\frac {{U(\lambda x)/{\lambda ^\alpha } - U(x)}} {{{x^\alpha }L(x)}} \to \log \lambda \quad (x \to \infty ){\text {for}}\lambda {\text { > 0}}{\text {.}}

\]
Suppose
\[

U
^

(
t
)
=

∫
0
∞

e

−
t
x

d
U
(
x
)

exists for

t
>
0.

\hat {U}(t)=\int _0^\infty {e^{ - tx}}dU(x){\text {exists for }}t > 0.

\]
We prove that

U
U

is

Π
\Pi

-regularly varying iff

U
U

is

Π
\Pi

-regularly varying.
An Abel-Tauber theorem for partitions
Authors: Geluk J. L. 
Published: June 2010

Suppose

Λ
=
{

λ
1

,

λ
2

,
…
}

\Lambda = \{ {\lambda _1},{\lambda _2}, \ldots \}

is a given set of real numbers such that

0
>

λ
1

>

λ
2

>
…
.

Let

n
(
u
)
=

∑

λ
k

⩽
u

1

0 > {\lambda _1} > {\lambda _2} > \ldots .{\text {Let }}n(u) = {\sum _{{\lambda _k} \leqslant u}}1

and

P
(
u
)

P(u)

the number of solutions of

n
1

λ
1

+

n
2

λ
2

+
…
⩽
u

{n_1}{\lambda _1} + {n_2}{\lambda _2} + \ldots \leqslant u

in integers

n
i

⩾
0

{n_i} \geqslant 0

. An Abel-Tauber theorem concerning

n
(
u
)

n(u)

and log

P
(
u
)

P(u)

is proved for the case where

n
(
t
x
)

/

n
(
t
)
→
1
(
t
→
∞
)

n(tx)/n(t) \to 1(t \to \infty )

for

x
>
0

x > 0

.
A generalized Bessel inequality
Authors: Prestini Elena 
Published: June 2010

To any pair of diadic intervals

[
I
,
ω
]

[I,\omega ]

with / Q

I
⊆
[
0
,
1
]

I \subseteq [0,1]

and

ω
=
[

N
ω

,

N
ω

+

|

I

|

−
1

)

\omega = [{N_\omega },{N_\omega } + |I{|^{ - 1}})

we associate the function

u

[
I
,
ω
]

(
x
)
=

|

I

|

−
1

/

2

e

i

N
w

x

χ
I

(
x
)

{u_{[I,\omega ]}}(x) = |I{|^{ - 1/2}}{e^{i{N_w}x}}{\chi _I}(x)

. In this paper we will give a condition under which a collection

B
B

of such pairs satisfies the inequality

B
B

for any

a
b

ab

in

f
f

.
Distance between normal operators
Authors: Sunder V. S. 
Published: June 2010

Lidskii and Wielandt have proved independently that if

A
A

and

B
B

are selfadjoint operators on an

n
n

-dimensional space

H
H

, with eigenvalues

{

α
k

}

k
=
1

n

\{ {\alpha _k}\} _{k = 1}^n

and

{

β
k

}

k
=
1

n

\{ {\beta _k}\} _{k = 1}^n

respectively (counting multiplicity), then,
\[

‖

A
−
B

‖

⩾

min

σ
∈

S
n

‖

diag

(

α
k

−

β

σ
(
k
)

)

‖

\left \| {A - B} \right \| \geqslant \min \limits _{\sigma \in {S_n}} \left \| {{\text {diag}}\left ( {{\alpha _k} - {\beta _{\sigma (k)}}} \right )} \right \|

\]
for any unitarily invariant norm on

L
(
H
)

L(H)

. In this note an example is given to show that this result is no longer true if

A
A

and

B
B

are only required to be normal (even unitary). It is also shown that the above inequality holds in the operator norm, if

A
A

is selfadjoint and

B
B

is skew-self-adjoint.
Mean growth and smoothness of analytic functions
Authors: Matheson A. 
Published: June 2010

Let

G
α

{G_\alpha }

denote the class of functions

f
(
z
)

f(z)

analytic in the unit disk such that
\[

∫
0
1

(
1
−
r
)

−
α

M
∞

(

f
′

,
r
)

d
r
>
∞
,

\int _0^1 {{{(1 - r)}^{ - \alpha }}{M_\infty }(f’,r)} dr > \infty ,

\]
for some

α
(
0
>
α
>
1
)

\alpha (0 > \alpha > 1)

. A characterization of

G
α

{G_\alpha }

is given in terms of moduli of continuity and an application is given to Riesz factorization of functions in

G
α

{G_\alpha }

.
Recoverability of some classes of analytic functions from their boundary values
Authors: Zayed Ahmed I. 
Published: June 2010

The technique devised by D. J. Patil to recover the functions of the Hardy space

H
p

(
1
⩽
p
⩽
∞
)

{H^p}(1 \leqslant p \leqslant \infty )

from the restrictions of their boundary values to a set of positive measure on the unit circle was modified by S. E. Zarantonello in order to extend the result to

H
p

(
0
>
p
>
1
)

{H^p}(0 > p > 1)

. In this paper, we show that Zarantonello’s technique can be slightly modified to extend the result to a larger class of analytic functions in the unit disc. In particular, if

f
(
z
)

f(z)

is analytic in the unit disc and satisfies
\[

lim

r
→
1

(
1
−
r

)
β

log
⁡
M
(
r
,
f
)
=
0

for some

β
⩾

1,

\lim \limits _{r \to 1} {(1 - r)^\beta }\log M(r,f) = 0\;{\text {for some }}\beta \geqslant {\text {1,}}

\]
then

f
(
z
)

f(z)

can be recovered from the restriction of its boundary value to an open arc.
Systems of differential equations subject to mild integral conditions
Authors: Trench William F. 
Published: June 2010

It is shown that solutions of a system

x
′

=
f
(
t
,
x
)

x’ = f(t,x)

approach constant vectors as

t
→
∞

t \to \infty

, under assumptions which do not require that

|

|

f
(
t
,
x
)

|

|

⩽
w
(
t
,

|

|

x

|

|

)

|| {f(t,x)} || \leqslant w(t,|| x ||)

, where

w
w

is nondecreasing in

|

|

x

|

|

|| x ||

, and which permit some or all of the integral smallness conditions on

f
f

to be stated in terms of ordinary—rather than absolute—convergence. Estimates of the order of convergence are given.
The structure of the automorphism group of a free group on two generators
Authors: Đoković Dragomir Ž. 
Published: June 2010

Let

F
2

=
Z
∗
Z

{F_2} = Z * Z

be a free group of rank two. We show that

Aut
⁡

F
2

{\operatorname {Aut} F_2}

can be built up from cyclic groups by using only the free products and semidirect products. Explicitly we have

Aut
⁡

F
2

=
(
(
Z
∗
Z
)
⋊
(

Z
3

∗

Z
3

)
)
⋊
(

Z
4

⋊

Z
2

)

\operatorname {Aut} {F_2} = ((Z * Z) \rtimes ({Z_3} * {Z_3})) \rtimes ({Z_4} \rtimes {Z_2})

. As a corollary we obtain a simple presentation of

Aut
⁡

F
2

\operatorname {Aut} {F_2}

.
Coefficients and integral means of some classes of analytic functions
Authors: Sheil-Small T. 
Published: June 2010

The sharp coefficient bounds for the classes

V
k

{V_k}

of functions of bounded boundary rotation are obtained by a short and elementary argument. Elementary methods are also applied for the coefficients of related classes characterised by a generalised Kaplan condition. The result

(
1
+
x
z

)
α

(
1
−
z

)

−
β

≪

(
1
+
z

)
α

(
1
−
z

)

−
β

{(1 + xz)^\alpha }{(1 - z)^{ - \beta }} \ll {(1 + z)^\alpha }{(1 - z)^{ - \beta }}

(

|
x
|

=
1
,
α
⩾
1
,
β
⩾
1
)

(\left | x \right | = 1,\alpha \geqslant 1,\beta \geqslant 1)

is proved simply. It is further shown that the functions

(
1
+
z

)
α

(
1
−
z

)

−
β

{(1 + z)^\alpha }{(1 - z)^{ - \beta }}

are extremal for the

p
p

th means (

p
p

an arbitrary real) of all Kaplan classes

K
(
α
,
β
)

K(\alpha ,\beta )

.
A note on polynomial operator approximation
Authors: Smith R. R. , Ward J. D. 
Published: June 2010

An example is given of an operator

T
T

contained in a block-diagonal algebra of operators

A

\mathcal {A}

, an ideal

J
⊂

A

J \subset \mathcal {A}

and an infinite set of polynomials

P

\mathcal {P}

for which there is a

K
∈
J

K \in J

satisfying

‖

p
(
T
+
K
)

‖

A

=

‖

p
(
T
+
K
)

‖

A

/

J

{\left \| {p(T + K)} \right \|_{\mathcal {A}}} = {\left \| {p(T + K)} \right \|_{\mathcal {A}/J}}

for any finite subset of

P

\mathcal {P}

but for which there is no

K
∈
J

K \in J

satisfying

‖

p
(
T
+
K
)

‖

A

=

‖

p
(
T
+
K
)

‖

A

/

J

{\left \| {p(T + K)} \right \|_{\mathcal {A}}} = {\left \| {p(T + K)} \right \|_{\mathcal {A}/J}}

for all

p
∈

P

p \in \mathcal {P}

. This sheds some light on a well-known question of C. Olsen.
On a singular elliptic equation
Authors: Ni Wei Ming 
Published: June 2010

In this paper, we study the singular elliptic equation

L
u
+
K
(
x
)

u
p

=
0

Lu + K(x){u^p} = 0

, where

L
L

is a uniformly elliptic operator of divergence form,

p
>
1

p > 1

and

K
(
x
)

K(x)

has a singularity at the origin. We prove that this equation does not possess any positive (local) solution in any punctured neighborhood of the origin if there exist two constants

C
1

{C_1}

,

C
2

{C_2}

such that

C
1

|

x

|

σ

⩾
K
(
x
)
⩾

C
2

|

x

|

σ

{C_1}|x{|^\sigma } \geqslant K(x) \geqslant {C_2}|x{|^\sigma }

near the origin for some

σ
⩽
−
2

\sigma \leqslant - 2

(with no other condition on the gradient of

K
K

). In fact, an integral condition is derived.
An inequality for derivatives of polynomials whose zeros are in a half-plane
Authors: Abi-Khuzam Faruk F. 
Published: June 2010

Let

Q
Q

be a real polynomial of degree

N
N

all of whose zeros lie in the half-plane

Re
⁡
z
⩽
0

\operatorname {Re} z \leqslant 0

. Let

M
(
r
,
Q
)

M(r,Q)

be the maximum of

|

Q
(
z
)

|

on

|
z
|

=
r

\left | {Q(z)} \right |{\text {on}}\left | z \right | = r

and

n
(
r
,
0
)

n(r,0)

the counting function of the zeros of

Q
Q

. It is shown that the inequality

M
(
r
,

Q
′

)
⩽

(
2
r

)

−
1

{

N
+
n
(
r
,
0
)

}

M
(
r
,
Q
)

M(r,Q’) \leqslant {(2r)^{ - 1}}\left \{ {N + n(r,0)} \right \}M(r,Q)

holds for

r
>
0

r > 0

. It is also shown that Bernstein’s inequality characterizes polynomials.
Two Tauberian theorems for nonconvolution Volterra integral operators
Authors: Gripenberg Gustaf 
Published: June 2010

Two sets of sufficient conditions on the kernel

k
(
t
,
s
)

k(t,s)

are given so that one can prove that if

x
x

is a bounded function such that
\[

lim

t
→
∞

τ
→
0

|

x
(
t
+
τ
)
−
x
(
t
)

|

=
0

and

lim

t
→
∞

∫
0
t

k
(
t
,
s
)
x
(
s
)
d
s

\lim \limits _{\begin {array}{*{20}{c}} {t \to \infty } \\ {\tau \to 0} \\ \end {array} } \left | {x(t + \tau ) - x(t)} \right | = 0\quad {\text {and}}\quad \lim \limits _{t \to \infty } \int _0^t {k(t,s)x(s)ds}

\]
exists, then

lim

t
→
∞

x
(
t
)

{\lim _{t \to \infty }}x(t)

exists.
Countable models of stable theories
Authors: Pillay Anand 
Published: June 2010

The notion of a normal theory such a theory

T
,

I
(

ℵ
0

,
T
)
=
1

or

⩾

ℵ
0

T,\;I({\aleph _0},T) = 1{\text { or }} \geqslant {\aleph _0}

. theorem that for superstable

T
,

I
(

ℵ
0

,
T
)
=
1

or

⩾

ℵ
0

T,\;I({\aleph _0},T) = 1{\text { or }} \geqslant {\aleph _0}

stronger than stability but incomparable is introduced, and it is proved that for We also include a short proof of Lachlan’s (The property of normality is to superstability.)
Almost sure stability of partial sums of uniformly bounded random variables
Authors: Hill Theodore P. 
Published: June 2010

Suppose

a
1

{a_1}

,

a
2

,
…

{a_2}, \ldots

is a sequence of real numbers with

a
n

→
∞

{a_n} \to \infty

. If

(

X
1

+
⋯
+

X
n

)

/

a
n

=
α

({X_1} + \cdots + {X_n})/{a_n} = \alpha

a.s. for every sequence of independent nonnegative uniformly bounded random variables

X
1

{X_1}

,

X
2

,
…

{X_2}, \ldots

satisfying some hypothesis condition A, then for every (arbitrarily-dependent) sequence of nonnegative uniformly bounded random variables

Y
1

{Y_1}

,

Y
2

,
…
,

{Y_2}, \ldots ,

,

lim

sup

(

Y
1

+
⋯
+

Y
n

)

/

a
n

=
α

\lim {\text { sup}}({Y_1} + \cdots + {Y_n})/{a_n} = \alpha

a.s. on the set where the conditional distributions (given the past) satisfy precisely the same condition A. If, in addition,

Σ
∞

a
n

−
2

>
∞

\Sigma ^\infty a_n^{ - 2} > \infty

, then the assumption of nonnegativity may be dropped.
Analyticity preserving properties of resolvents for degenerate diffusion operators in one dimension
Authors: Tsuchiya Masaaki 
Published: June 2010

Let

L
=
a
(
x
)
(

d
2

/

d

x
2

)
+
b
(
x
)
(
d

/

d
x
)
+
c
(
x
)

L = a(x)({d^2}/d{x^2}) + b(x)(d/dx) + c(x)

be a diffusion operator on a compact interval

I
=
[

r
0

,

r
1

]

I = [{r_0},{r_1}]

investigated by S. N. Ethier. Here, assume that

a
a

,

b
b

and

c
c

are real analytic functions on

I
I

,

a
(
x
)
>
0

a(x) > 0

for

x
∈
(

r
0

,

r
1

)

x \in ({r_0},{r_1})

,

a
(

r
i

)
=
0
⩽

(
−
1

)
i

b
(

r
i

)
(
i
=
0
,
1
)

a({r_i}) = 0 \leqslant {( - 1)^i}b({r_i})(i = 0,1)

, and both

r
i

(
i
=
0
,
1
)

{r_i}(i = 0,1)

are simple zeros of

a
(
x
)

a(x)

. It is shown that the resolvent

{

G
λ

}

\left \{ {{G_\lambda }} \right \}

for

L
L

has the analyticity preserving property for sufficiently large

λ
\lambda

, so that the equation

(
L
−
λ
)
u
=
f

(L - \lambda )u = f

is solvable in the space of real analytic functions on

I
I

. Some examples are given to show that the condition on

L
L

is best possible.
Two results concerning cardinal functions on compact spaces
Authors: Juhász I. , Szentmiklóssy Z. 
Published: June 2010

We show that for

X
X

compact

T
2

:

(
i
)

d

(
X
)

⩽
s

(
X
)

⋅

F
^

(
X
)

{T_2}:\left ( i \right )d\left ( X \right ) \leqslant s\left ( X \right ) \cdot \hat F\left ( X \right )

; (ii) if the pair

(

κ
,

F
^

(
X
)

)

\left ( {\kappa ,\hat F\left ( X \right )} \right )

is a caliber of

X
X

then

π

(
X
)

>
κ

\pi \left ( X \right ) > \kappa

. These strengthen results of Šapirovskii from [3 and 5], respectively. Moreover, (i) settles a problem raised in [2] implying that there are no compact

T
2

{T_2}

κ
\kappa

-examples for any singular cardinal

κ
\kappa

.
Primitive obstructions in the cohomology of loopspaces
Authors: Williams Frank 
Published: June 2010

Let

X
X

and

X
′

X’

be

H
H

-spaces. If

f
:
Ω
X
→
Ω

X
′

f:\Omega X \to \Omega X’

is an

H
H

-map then the obstruction to

f
f

being a homotopy-commutative map is a subset

{

c
2

(
f
)

}

⊂

[

Ω
X
Λ
Ω
X
;

Ω
2

X
′

]

\left \{ {{c_2}(f)} \right \} \subset \left [ {\Omega X\Lambda \Omega X;{\Omega ^2}X’} \right ]

. In this paper we prove:

I
f
[
f
]

If[f]

is in the image of the composition
\[

[

P

k
+
m

Ω
X
;

X
′

]

→

[

Σ
Ω
X
;

X
′

]

→

≈

[

Ω
X
;
Ω

X
′

]

,

\left [ P_{k + m} \Omega X;X’ \right ] \rightarrow \left [ {\Sigma \Omega X;X’} \right ] \stackrel {\approx }{\rightarrow } \left [ {\Omega X;\Omega X’} \right ],

\]
then

{

c
2

(
f
)

}

\left \{ {{c_2}(f)} \right \}

is in the image of the composition
\[

[

P
k

Ω
X
Λ

P
m

Ω
X
;

X
′

]

→

[
Σ
Ω
X
Λ
Σ
Ω
X
;

X
′

]

→

≈

[
Ω
X
Λ
Ω
X
;

Ω
2

X
′

]

.

\left [ P_k \Omega X\Lambda {P_m}\Omega X;X’ \right ] \rightarrow \left [ \Sigma \Omega X\Lambda \Sigma \Omega X;X’ \right ] \stackrel {\approx }{\rightarrow } \left [ \Omega X\Lambda \Omega X;{\Omega ^2}X’ \right ].

\]
Consequently if

α
∈

H
n

(
Ω
X
;

Z
p

)

\alpha \in {H^n}(\Omega X;{Z_p})

is an

A
3

{A_3}

-class in the sense of Stasheff then each element of

{

c
2

(
f
)

}

\left \{ {{c_2}(f)} \right \}

is of the form

∑

c
′

i

⊗

c
i

\sum {{{c’}_i}} \otimes {c_i}

where the

c
i

{c_i}

are primitive.
A simple evaluation of Askey and Wilson’s 𝑞-beta integral
Authors: Rahman Mizan 
Published: June 2010

By using the well-known sum of

2

ϕ
1

(

a
,
b
;
c
;
c

/

a
b

)

_2{\phi _1}\left ( {a,b;c;c/ab} \right )

and Sears’ identity for the sum of two nonterminating balanced

3

ϕ
2

_3{\phi _2}

series, a simple evaluation is given for Askey and Wilson’s

q
q

-beta type integral
\[

∫

−
1

1

h
(
x
;
1
)
h
(
x
;
−
1
)
h
(
x
;

q

)
h
(
x
:
−

q
)

h
(
x
;
a
)
h
(
x
;
b
)
h
(
x
;
c
)
h
(
x
;
d
)

d
x

1
−

x
2

,

\int _{ - 1}^1 {\frac {{h(x;1)h(x; - 1)h(x;\sqrt q )h(x: - \sqrt {q)} }}{{h(x;a)h(x;b)h(x;c)h(x;d)}}} \frac {{dx}}{{\sqrt {1 - {x^2}} }},

\]
where

max

(

|
q
|

,

|
a
|

,

|
b
|

,

|
c
|

,

|
d
|

>
1

)

\max \left ( {\left | q \right |,\left | a \right |,\left | b \right |,\left | c \right |,\left | d \right | > 1} \right )

.
A canonical trace class approximant
Authors: Legg D. A. , Ward J. D. 
Published: June 2010

Let

H
H

be a finite-dimensional Hilbert space,

B

(
H
)

B\left ( H \right )

the space of bounded linear operators on

H
H

, and

C
C

a convex subset of

B

(
H
)

B\left ( H \right )

. If

A
A

is a fixed operator in

B

(
H
)

B\left ( H \right )

, then

A
A

has a unique best approximant from

C
C

in the

C
P

{C_P}

norm for

1
>
p
>
∞

1 > p > \infty

. However, in the

C
1

{C_1}

(trace) norm,

A
A

may have many best approximants from

C
C

. In this paper, it is shown that the best

C
p

{C_p}

approximants to

A
A

converge to a select trace class approximant

A
1

{A_1}

as

p
→
1

p \to 1

. Furthermore,

A
1

{A_1}

is the unique trace class approximant minimizing

∑

i
=
1

n

S
i

(

A
−
B

)

ln
⁡

S
i

(

A
−
B

)

\sum \nolimits _{i = 1}^n {{S_i}\left ( {A - B} \right )\operatorname {ln }{S_i}\left ( {A - B} \right )}

over all trace class approximants

B
B

. The numbers

S
i

(
T
)

{S_i}\left ( T \right )

are the eigenvalues of the positive part

|
T
|

\left | T \right |

of

T
T

.
On successive coefficients of univalent functions
Authors: Hu Ke 
Published: June 2010

Let

f
(
z
)
∈
S

f(z) \in S

, that is,

f
(
z
)

f(z)

is analytic and univalent in the unit disk

|
z
|

>
1

\left | z \right | > 1

, normalized by

f
(
0
)
=

f
′

(
0
)
−
1
=
0

f(0) = f’(0) - 1 = 0

. Let

p
p

be real and
\[

{
f
(
z
)

/

z

}
p

=
1
+

∑

n
=
1

∞

D
n

(
p
)

z
n

.

{\{ f(z)/z\} ^p} = 1 + \sum \limits _{n = 1}^\infty {{D_n}(p){z^n}} .

\]
Lucas proved that
\[

|

D
n

(
p
)

|

−

|

D

n
+
1

(
p
)

|

|

≤
A

n

(
t
(
p
)
−
1
)

/

2

log

3

/

2

n
,

n
=
2
,
3
,
…
,

\left | {{D_n}(p)} \right | - \left | {{D_{n + 1}}(p)} \right |\left | { \leq A{n^{(t(p) - 1)/2}}{{\log }^{3/2}}n,\quad n = 2,3, \ldots ,} \right .

\]
for some absolute constant

A
A

and

t
(
p
)
=

(
2

p

−
1

)
2

t(p) = {(2\sqrt p - 1)^2}

. In this paper we improve

t
(
p
)

t(p)

as follows:
\[

T
(
p
)
=

4
p
−
1

2
p
+
t
(
p
)

t
(
p
)
.

T(p) = \frac {{4p - 1}}{{2p + t(p)}}t(p).

\]
On the stability of almost convex functions
Authors: Parnami J. C. , Vasudeva H. L. 
Published: June 2010

Let

R

{\mathbf {R}}

denote the set of real numbers and

I
I

an open interval of

R

{\mathbf {R}}

. A function

f
:
I
→
R

f:I \to R

is said to be almost

δ
\delta

-convex iff

f
(
t
x
+
(
1
−
t
)
y
)
≤
t
f
(
x
)
+
(
1
−
t
)
f
(
y
)
+
δ

f(tx + (1 - t)y) \leq tf(x) + (1 - t)f(y) + \delta

holds for all

(
x
,
y
)
∈
I
×
I
∖
N

(x,y) \in I \times I\backslash N

, where

N
⊂
I
×
I

N \subset I \times I

is of measure zero, each

t
∈
[
0
,
1
]

t \in [0,1]

and some

δ
≥
0

\delta \geq 0

. It is proved that such a function is uniformly close to a convex function almost everywhere.
The best bound in the 𝐿𝑙𝑜𝑔𝐿 inequality of Hardy and Littlewood and its martingale counterpart
Authors: Gilat David 
Published: June 2010

The smallest positive constant

c
c

for which the Hardy and Littlewood [8]

L
log
⁡
L

L\log L

inequality (1)
\[

M
(
f
)
≡
∫

d
x

x

∫
0
x

|

f

|

≤
c
(
1
+
∫

|

f

|

log

+

|

f

|

)

≡
c
(
1
+
L
(
f
)
)

M(f) \equiv \int \frac {{dx}}{x}\int _0^x {|f| \leq c(1 + \int |f|{{\log }^ + }|f|)} \quad \equiv c(1 + L(f))

\]
is valid is proved to be the unique positive solution

c
0

{c_0}

of the equation

e

−
c

=

(
c
−
1

)
2

{e^{ - c}} = {(c - 1)^2}

. This settles a question raised by Dubins and Gilat (1978) [6] and, again, more recently, by D. Cox (1984) [3]. Numerically,

c
0

≐
1.478

{c_0} \doteq 1.478

. This should be compared with

c
=
e

(
e
−
1

)

−
1

≐
1.582

c = e{(e - 1)^{ - 1}} \doteq 1.582

, obtained by Doob (1953) [5] in the context of martingale theory and, since then, widely used in the probability literature. Curiously enough, Doob’s coefficient is the best upper bound, but for a slightly different inequality. If only the plus sign is removed from

log
+

⁡

|
f
|

\log ^ + \left | f \right |

in (1), then

c
c

must be at least

e

(
e
−
1

)

−
1

e{(e - 1)^{ - 1}}

for (1), so modified, to be valid. The original inequality (1) is a normalized form of the two-parameter inequality (2)
\[

M
(
f
)
≤
c
L
(
f
)
+
d
,

for all integrable

f
.

M(f) \leq cL(f) + d,\quad {\text {for all integrable}}\quad f.

\]
The set of all ordered pairs

(
c
,
d
)

(c,d)

for which (2) holds is identified as
\[

{
(
c
,
d
)
:
c
>
1
,
d
≥
1
+

e

−
c

/

(
c
−
1
)
}
.

\{ (c,d):c > 1,d \geq 1 + {e^{ - c}}/(c - 1)\} .

\]
Furthermore, for each point on the lower boundary of this set, there is a unique

f
f

(up to null sets) which attains equality in (2).
Character values of finite groups as eigenvalues of nonnegative integer matrices
Authors: Chillag David 
Published: June 2010

Let

C
1

{C_1}

,

C
2

,
…

{C_2}, \ldots

,

C
k

{C_k}

be the conjugacy classes of the finite group

G
G

and choose

x
i

∈

C
i

{x_i} \in {C_i}

, for

i
=
1

i = 1

,

2
,
…

2, \ldots

,

k
k

. For every complex character

θ
\theta

of

G
G

there is a

k
×
k

k \times k

matrix

M
(
θ
)

M(\theta )

whose entries are nonnegative integers such that

X

−
1

M
(
θ
)
X
=

diag

(
θ
(

x
1

)
,
θ
(

x
2

)
,
…
,
θ
(

x
k

)
)

{X^{ - 1}}M(\theta )X = {\text {diag}}(\theta ({x_1}),\theta ({x_2}), \ldots ,\theta ({x_k}))

where

X
X

is the character table matrix of

G
G

. Some consequences are shown.
The singular cohomology of the inverse limit of a Postnikov tower is representable
Authors: Dydak Jerzy , Geoghegan Ross 
Published: June 2010

Let

X
1

←

X
2

←
⋯

{X_1} \leftarrow {X_2} \leftarrow \cdots

be an inverse sequence of spaces and maps satisfying (i) each

X
n

{X_n}

has the homotopy type of a CW complex, (ii) each

f
n

{f_n}

is a Hurewicz fibration, and (iii) the connectivity of the fiber of

f
n

{f_n}

goes to

∞
\infty

with

n
n

. Let

X
^

\hat X

be the inverse limit of the sequence. It is shown that the natural homomorphism

H
ˇ

k

(

X
^

,
G
)
→

H
k

(

X
^

,
G
)

\check {H}^k(\hat {X},G) \to H^k(\hat {X}, G)

(from Čech cohomology to singular cohomology, with ordinary coefficient module

G
G

) is an isomorphism for all

k
k

. It follows that

lim

→
n

[

X
n

,
K
(
G
,
k
)
]
≅
[

X
^

,
K
(
G
,
k
)
]

{\lim _{ \to n}}[{X_n},K(G,k)] \cong [\hat X,K(G,k)]

for any Eilenberg- Mac Lane space

K
(
G
,
k
)

K(G,k)

. It is also shown that, except in trivial cases,

X
X

does not have the homotopy type of a CW complex.
A remark on singular Calderón-Zygmund theory
Authors: Christ Michael , Stein Elias M. 
Published: June 2010

It is shown that in

R

n

{{\mathbf {R}}^n}

the operator
\[

H
f

(
x
)

=
p
v

∫

−
∞

+
∞

f

(

x
1

−
t
,
…

x
n

−

t
n

)

t

−
1

d
t

Hf\left ( x \right ) = pv\int _{ - \infty }^{ + \infty } {f\left ( {{x_1} - t, \ldots {x_n} - {t^n}} \right ){t^{ - 1}}dt}

\]
maps

L

(

log
⁡
L

)

L\left ( {\log L} \right )

to weak

L
1

{L^1}

locally. A slight variant of the Calderón-Zygmund procedure provides a new approach to the previously known

L
p

{L^p}

boundedness of

H
,
1
>
p
>
∞

H,1 > p > \infty

. Relatively sharp bounds are obtained as

p
→

1
+

p \to {1^ + }

, and extrapolation produces the result for

L

(

log
⁡
L

)

L\left ( {\log L} \right )

.
Perturbations near zero of the leading coefficient of solutions to a nonlinear differential equation
Authors: Komkov Vadim 
Published: June 2010

The equation
\[

d

d
t

(

a

(
t
)

⋅
ψ

(
x
)

d
x

d
t

)

+
f

(
x
)

⋅

d
x

d
t

+
c

(
t
)

x
=
g

(
t
)

\frac {d}{{dt}}\left ( {a\left ( t \right ) \cdot \psi \left ( x \right )\frac {{dx}}{{dt}}} \right ) + f\left ( x \right ) \cdot \frac {{dx}}{{dt}} + c\left ( t \right )x = g\left ( t \right )

\]
that generalizes the more commonly studied selfadjoint second order equation
\[

d

d
t

(

a

(
t
)

d
x

d
t

)

+
c

(
t
)

x
=
g

(
t
)

\frac {d}{{dt}}\left ( {a\left ( t \right )\frac {{dx}}{{dt}}} \right ) + c\left ( t \right )x = g\left ( t \right )

\]
occurs in celestial dynamics, in the study of gyroscopic systems, and in other problems of nonlinear mechanics. Using techniques of nonstandard analysis we derive some properties of the "duck"-type cycles for the solutions of this equation.