Download Multiplicative Inequalities of Carlson Type And by Leo Larsson, Lech Maligranda, Josip Pečarić, Lars-Erik PDF
By Leo Larsson, Lech Maligranda, Josip Pečarić, Lars-Erik Persson
Amassing the entire effects at the certain types of inequalities, the insurance of this booklet is exclusive between textbooks within the literature. The booklet specializes in the ancient improvement of the Carlson inequalities and their many generalizations and diversifications. in addition to just about all identified effects pertaining to those inequalities and all identified facts strategies, a few open questions appropriate for extra learn are thought of. chapters are dedicated to clarifying the shut connection among interpolation concept and this kind of inequality. different functions also are incorporated, as well as a ancient be aware on Fritz Carlson himself.
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Additional info for Multiplicative Inequalities of Carlson Type And Interpolation
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1) with the constant n2 replaced by 47r2. 4 Carlson's Inequality for Finite Sums Suppose that m is a positive integer. 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. -E. Persson [52]. 1 Let a i , . . , a m be non-negative numbers, not all zero.
9. Let m m a = J2k S 'ark+1 and ^2ka2ark+1, T = k=l fe=l and let A be any positive number. TT . (AfcQl + jka*)^iak, we get by the Holder-Rogers inequality / m a E0 \r+\ / m A a, + A lfcaa <(D * \ r / _m_ ~ )~M E( ^(Xk^+x^k^yr) Afcai+A_lfca2 +1 \ K {XS + X^T). 21) can be estimated by an integral. ; A M B a 2 -ai A2 + m a 2 - a i is decreasing. 22), it follows that +1 (±aX <2\(*2 (-^—B( \t^[ J — ai a >~\, ( \{a2-oii)r r(ai+\-> ^ ^ - a\)r l + m a2_ai 2 «2- f —ai+r •S a2-<»l J 1 " 2 - i l . 20). 9 is not sharp for finite m.
1). 1). Here, however, we include a proof of the discrete version based on Hilbert's double sum inequality. 1), it still gives the same qualitative information. Proof by Hilbert 's Inequality. that fc=l 1=1 The well-known Hilbert inequality states p \k=l / \l=l / for all sequences {cik}k and {bi}i of non-negative numbers, with equality if and only if either all the a,k or all the 6; are zero. 7) that oo ak oo = J2 Y^ai k=l 1=1 SS' k+i 00 °° (k + l)akai oo oo *=i 1=1 /oo . K+ oo oo . + k=u=iK \ 3 / oo \ I 2„2 l <2* E°0 I>< fc=l / \(=1 provided not all Ofe are zero.