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> 1 and 7r2o; —> 1 as u tends to infinity along the imaginary axis. Thus the nonstrict inequality (1.1) holds, and we can also conclude that for any e > 0,
Carlson's Inequalities
13
there is a sequence ak for which oo
x- -e)E^Efc2^-
5> \fc=l
oo
2
/
fc=l
fc=l
Hence the constant C = n2 is sharp. Finally, t o show t h a t we in fact have strict inequality in (1.1) for non-zero sequences, Carlson applies the continuous version of the inequality
(
/•oo
\ 4
-oo
/-oo
/ f(x)dx\ f c L f c (a:) >
f(x)
fc=0
where Lk denotes the fcth Laguerre polynomial and p is a real number < 1. W i t h this choice of / , (1.3) becomes
(5 > oo
fc=i
\
/
2 °°
°°
6fc2 6fc 2 a m, i.e. if the series are replaced by finite sums with m terms. If, however, we restrict attention to finite sums with a fixed number of terms, the constant 7r2 need no longer be (and is not) sharp. We have
18
Multiplicative
Inequalities of Carlson Type and
Interpolation
the following preliminary result by L. Larsson, Z. Pales and L.-E. Persson [52]. Proposition 1.1 Let a i , . . . , a m be non-negative numbers, not all zero. Then
(
m
\
Y,ak) fc=i
4
mm
oo is (1.1). Proof. Using the same notation and method as in Hardy's first proof above, we get
The integral can be evaluated exactly: fmdx_ 1 [0 / TT~^ = — F=^ arctan (\ — m). v J0 a + px2 v ^ Va ' Since S < T, if we put a = T and (3 — S, we will always have arctan (\ —m) < arctanm, l a
'
so that ( ^2 ak ) < arctan m(VST
(
+ VST) m
\ V2
m
y~j a2. 2 ^ fc2a2 J fc=i
fe=i
/
which, after squaring, yields the desired inequality. • Remark 1.8 It should be mentioned that the constant (2 arctan m) 2 in (1.8), although strictly smaller than n2 for each finite m, is not sharp. We consider the inequalities m \k=l
/
fc=l
m fe=l
Carlson's
Inequalities
19
and seek, for each m = 1,2,... the sharp constant Cm. Thus Proposition 1.1 says that Cm < (2arctanm) 2 . It is clear, however, that 7T 2
Ci = 1 < — = (2arctanl) 2 . 4 Moreover, numerical calculations show that (1+t*) 4 2 2 Q ^ ( l + a ) ( l + 4a )
C2 = s u p •
« 2.0311 < 4.9031 w (2arctan2) 2 . Note, also, that we can show that C2 < 4, as follows. By convexity (a + 6)4 = [(o + 6) 2 ] 2 < 4(a 2 + 6 2 ) 2 = 4(a2+&2)(a2+62) < 4 ( a 2 + 6 2 )(a 2 + 462). This gives a slightly better result than Proposition 1.1 for the case m = 2. Furthermore, consider Carlson's observation regarding the application of the Holder-Rogers inequality, but now to sums with m terms: f m
\
V.fc=l
/
4
/
/ m„
\fc=l
1
\ ^2
Til
771
/
fc=l
fc=l
\
If 1 < m < 10, it holds that m m
1 1
< 2arctanm, fc=i
so for such m, this method gives a better constant than Proposition 1.1. Problem 1 Find a formula for t^mj
m = 1,2,....
Carlson type inequalities for finite sums will be discussed further in Section 2.10 of Chapter 2.
Chapter 2
Some Extensions and Complements of Carlson's Inequalities In this chapter, we present some different variations of Carlson's inequalities, whose origins are spread in time from 1937 to 2005.
2.1
Gabriel
R. M. Gabriel [27] mentioned in a paper from 1937, that Hardy's method could be used to prove a more general version of Carlson's inequality. However, he chose to use a method similar to Carlson's original proof. We state Gabriel's result here. Theorem 2.1 (Gabriel, 1937) I f p > l a n d O < < J < p - l , then 2p
< C £ kp-l-5\ak\v £ fc=l
fcp-1+VI",
(2-1)
fe=i fc=i
where 4 C
/
1
1
B
-j28^ \2p-^2'2j^2)
\
2
P-2
•
(2
-2)
Remark 2.1 With p = 2 and S = 1, (2.1) reduces to Carlson's inequality (1.1). Remark 2.2 Note that this theorem allows the ak to be complex. Although this is merely a notational matter, it is an interesting observation, since most authors on the area at this time understand the ak to be real, sometimes without even mentioning this. 21
22
Multiplicative
Inequalities of Carlson Type and
Interpolation
Remark 2.3 Gabriel also proved that the inequality (2.1) fails for any choice of C when 6 = 0. In fact, it suffices to consider the sequence {afcj^j defined by a,k =
T.
a*; = 0,
fc = 1 , . . . ,m, k > m
and let m —> oo. 2.2
Levin
V. I. Levin [55] gave another variation of Carlson's inequality (1.1). Instead of using two factors on the right-hand side of the inequality, he allowed any odd number of factors. Theorem 2.2 (Levin, 1938) If ak > 0, k = 1,2,..., and m is a positive integer, then
(
oo
\ (m+l)(2m+l)
fe = l
2m
/
oo
j=0fc=l
unless all a/t are zero, where
c
= n i2^2™-1=(2m+i)2-+i n rm) 2m+l
2m
j=i
,„
j=o V J
>.
/
is the sharp constant. Remark 2.4 It is interesting to note that the sharp constant is an integer in every case covered by Theorem 2.2. Example 2.1 In the cases m = 1 and m = 2, we get the following sharp inequalities. oo
\ °
($ >
oo
\/c=l
oo
< 54 £ alj^kai £ fcVfc,
fc = l / / oo \ 1&
£
oo
afc
/C=l oo
< 300000 2 /
fc=l
fe=l oo
fc=l
af £ fca fc=l
oo
3 fe
oo
oo
£ fcVfe £ fc3af £
fc=l
fc=l
fc4a3.
A:=l
Thus there is an inequality with the speed of light as sharp constant!
Some Extensions
and Complements
of Carlson's Inequalities
23
As a step in the proof, Levin used the following interesting observation. Lemma 2.1 If m is a positive integer and po = 1, Pj > 0, j — 1,..., 2m, then 2m
/„
>
>. - 1 / m
/ 2fe
., - l / i m ( 2 m + l )
r(S(T)—) - ( n . )
Proof of Theorem 2.2. If Co,..., czm are any positive numbers, then, by writing
(
2m Zm
x >.
/2m
-l/(m+l)
v l/(m+l)
the Holder-Rogers inequality with exponents m +1
and m
implies that / oo
(It
\ (m+l)(2m+l)
7
^
0, the constant is shown to be sharp.
•
24 2.3
Multiplicative Inequalities of Carlson Type and Interpolation Caton
W . B . C a t o n [24] generalized t h e idea of looking at Carlson's inequality (1.1) as a limiting case of t h e Holder-Rogers inequality. He noted t h a t if Pi € (0, oo), i = l , 2 , 3 , are such t h a t
if t h e non-negative numbers ft, i = 1,2,3 satisfy /?l+/?2=/33, and if ai,ct2>0,
a i + 0 : 2 = 1,
t h e n Holder's inequality implies t h a t 00
/ 00
\ VPI
fe=l
\fc=l
/ 00
\ VP2
£( f c / 3 2 < 2 H
^0k 1, but tends to infinity as /?3P3 \ 1. C a t o n investigated under which conditions an inequality of t h e form (2.4), with a finite constant C, exists in t h e case /3 3 p 3 = 1. More precisely, he gave a necessary condition for inequality, and a lower bound for t h e constant C in (2.4). T h e main result may be stated as follows. T h e o r e m 2.3 ( C a t o n , 1 9 4 0 ) Let Ui = ciiPi and Tj = PiPi, i = 1,2. (2.4) holds with a\ = cr2 — a, and T\ > T 2 , t h e n necessarily
' (a - 1 - r 2 ) 1 / p i ( n + 1 x B (
where B(-,-)
_ J n - r2
a)1/P»
T2
n
{a - 1 ) ( T I - r 2 ) ' (ff - 1 ) ( T I - r 2 )
denotes t h e B e t a function.
i_\1/P3 p. Then a necessary and sufficient condition for the existence of a finite constant C such that the inequality (2.13) holds is that q>r-p.
(2-14)
If r-p °"
32
Multiplicative
Inequalities
of Carlson Type and
Interpolation
If p, q and r are such that „2^ fp(p + r) 4p(p + r)-r\ -m&X\^TT> ( r + l) /' 4
q
(2 15)
-
then the function / is convex. Our second lemma is another type of convexity result, in the guise of an interpolation inequality. This is what we need to prove Propostion 2.3. This was inspired by the Kjellberg Principle (see Proposition 3.2 of Chapter 3). Lemma 2.3 Let S(a,r) be as defined by (2.12). If 0 < 9 < 1 and 1, let 9=q-±± 2?
and
„=«fl. ' 2q
Then 0 = (1 - 77)(1 - g) + 7?(1 + 9) and 2 = {l-9)(l-q)
+ 0{l+q),
so again by Lemma 2.3, 5(0, l) 4
