Journal of Inequalities in Pure and Applied Mathematics ON BEST EXTENSIONS OF HARDY-HILBERT’S INEQUALITY WITH TWO PARAMETERS volume 6, issue 3, article 81, 2005.
BICHENG YANG Department of Mathematics Guangdong Institute of Education Guangzhou, Guangdong 510303 People’s Republic of China
Received 23 February, 2005; accepted 17 June, 2005. Communicated by: W.S. Cheung
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2000 Victoria University ISSN (electronic): 1443-5756 055-05
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Abstract This paper deals with some extensions of Hardy-Hilbert’s inequality with the best constant factors by introducing two parameters λ and α and using the Beta function. The equivalent form and some reversions are considered. 2000 Mathematics Subject Classification: 26D15. Key words: Hardy-Hilbert’s inequality; Beta function; Hölder’s inequality.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Some Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 Some Best Extensions of (1.3) . . . . . . . . . . . . . . . . . . . . . . . . . . 26 References
On Best Extensions of Hardy-Hilbert’s Inequality with Two Parameters Bicheng Yang
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1.
Introduction
If an, bn ≥ 0 satisfy 0
B , [1 − θp (n)]np(1−φq )−1 apn , α α α n=1 h ip φp φq 1 1 where 0 < θp (n) = O nφp < 1, and the constant factor α B α , α is the best possible. Inequality (3.15) is equivalent to (3.8).
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Proof. Still setting " bn := n
pφp −1
∞ X
am α (m + nα )λ m=1
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#p−1 ,
J. Ineq. Pure and Appl. Math. 6(3) Art. 81, 2005
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by (3.8), one has (3.16)
0
1, p1 + 1q = 1, λ > 0, α > 2−min{p, q}, 1 (r = p, q), an, bn ≥ 0, satisfy 0
2 2 m +n 2 n=1 m=1
(∞ X n=1
1−
2 πn
apn n
) p1 ( ∞ )1 X bq q n
n=1
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and (3.24)
∞ X n=1
" n
p−1
∞ X
am 2 m + n2 m=1
#p >
∞ π p X
2
n=1
2 1− πn
apn , n
where the constant factors in the above inequalities are the best possible.
On Best Extensions of Hardy-Hilbert’s Inequality with Two Parameters Bicheng Yang
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4.
Some Best Extensions of (1.3)
Setting φr =
λα r
(r = p, q), by Theorems 3.1 – 3.4, one has
Corollary 4.1. If p > 1, satisfy 0
0, λα ≤ min{p, q}, an , bn ≥ 0,
0, 0 < + ≤ 1 (r = p, q) = Φ, 2 r it follows that both (4.7) and (4.8) do not possess reversions. Setting φr = 1 1 − r (λα − 2) + 1 (r = p, q), by Theorems 3.1 – 3.4, one has Corollary 4.3. If p > 1, p1 + 1q = 1, λ, α > 0, 2 − min{p, q} < λα ≤ 2, an , bn ≥ 0, satisfy ∞ ∞ X X 0< n1−λα apn < ∞ and 0 < n1−λα bqn < ∞, n=1
n=1
then one has the following equivalent inequalities: ∞ X ∞ X 1 p + λα − 2 q + λα − 2 am bn (4.13) < B , α + nα )λ (m α pα qα n=1 m=1 (∞ ) p1 ( ∞ ) 1q X X × n1−λα apn n1−λα bqn n=1
n=1
On Best Extensions of Hardy-Hilbert’s Inequality with Two Parameters Bicheng Yang
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and (4.14)
#p a m n(p−1)(λα−1) α + nα )λ (m n=1 m=1 p X ∞ 1 p + λα − 2 q + λα − 2 < B , n1−λα apn , α pα qα n=1
∞ X
"
∞ X
where the constant factors in the above inequalities are the best possible. If 0 < p < 1, one has (α/2, α) ∈ A and two equivalent reversions as: ) 1q (∞ ) p1 ( ∞ ∞ X ∞ X1 X X am b n 1 1 p bqn (4.15) > kα 1− an α + nα )2/α n (m k n n α n=1 m=1 n=1 n=1 and
Bicheng Yang
Title Page ∞ X
"
∞ X
#p
∞ X
am 1 1 p p > k 1 − an , α α + nα )2/α (m k n n α n=1 m=1 n=1 where kα = α1 B α1 , α1 , and the constant factors are the best possible. (4.16)
On Best Extensions of Hardy-Hilbert’s Inequality with Two Parameters
np−1
Proof. For 0 < p < 1, 1 (λα − 2) + 1 ≤ 1 φr = 1 − r
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(r = p, q),
we obtain λα = 2 and φr = 1. By (3.13), we find 0 < θp (n)
