By Lizhen Ji

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**Additional info for Arithmetic Groups and Their Generalizations What, Why, and How**

**Example text**

D. thesis of A. Kamaly [39] and refer to the references given there. 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 .

Proof. P u t /»oo /-oo J' f2{x)dx and V= x2f2(x)dx. o Jo We may assume that both U and V are finite. Let a and /? be positive 47 48 Multiplicative Inequalities of Carlson Type and Interpolation numbers. )*)"- ( f ^Lpv^VM«x) IT 1 2 v¥ (aU + pV) With a = V and /3 = U, the rightmost expression is TrVt^V, which, upon squaring both sides, yields the desired inequality. Equality in the application of the Schwarz inequality is obtained precisely when for almost all x f(x) = A a + fix1 for some constant A or as claimed.

D The second result is yet another variation on Carlson's inequality. 2, and we therefore omit it. However, we state the corresponding lemma needed for the proof. 5 Suppose that po = 1 and pj > 0, j = 1 , . . , m. Put m , ( m *i = - + h \ 3 - - ) , where h > 0. ,m. 8 (Levin-Godunova, 1965) Let Xj be as in the above lemma. Then / oo \ (fe+l)(fc+2)/2 EH \fc=l m ^