ISBN-10: 9812567089

ISBN-13: 9789812567086

Amassing the entire effects at the certain types of inequalities, the assurance of this ebook is exclusive between textbooks within the literature. The e-book makes a speciality of the old improvement of the Carlson inequalities and their many generalizations and diversifications. in addition to just about all recognized effects touching on those inequalities and all recognized facts ideas, a few open questions appropriate for additional examine are thought of. chapters are dedicated to clarifying the shut connection among interpolation thought and this sort of inequality. different functions also are integrated, as well as a historic notice on Fritz Carlson himself.

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Additional info for Multiplicative Inequalities of Carlson and Interpolation

Sample text

N. l. Thus Since this can be done for any s, the conclusion follows upon taking ( r + l ) t h roots. 10, the constant is not sharp when N > 2, even if we let m tend to infinity. 23). e. where the sums are replaced by integrals over bounded intervals. This will be discussed in Chapter 3. 1. 1 If / is a non-negative, measurable function on (0, oo), then a oo \ 4 f(x)dx) »oo oo f(x)dx X2f(x)dx.

5, for any values of the parameters p, q, and r for which g is 34 Multiplicative Inequalities of Carlson Type and Interpolation convex, we can apply Hadamard's inequality so that for any k it holds that 9 \ ~ 2) < 9 dy / ^ - It follows, then, from the Holder-Rogers inequality with exponents r + 1 and r + 1, that 00 S(0,l)=5>fc k=l 00 r P+Q\ -£4>-\r+k*-\ +l k=l P-Q *<*-;r4H) p+g\ r + 1 ak P q ~ Uk-±V** < 7+T r+1 fc \S(p-q,r + l) + -S(p + q,r + l) /•OO / < \S(p-q,r g{y) dy + l) + jS(p + q,r + l) Jo In the integral above, we can subsitute X1/qt1/2q x), which yields f00 / \ „ « 1 D+aV* , for y, then put £ = x/(l A00 « + (r-,) .

Fc=i If A i , . . , XN are any positive numbers, we write ak = (Aifc"1 + . . + XNkaN)'^ • (A^"1 + ... + XNkaN)^ak Some Extensions and Complements of Carlson's Inequalities 45 and apply the Holder-Rogers inequality with parameters and r + 1, r which yields m \ fc=i / ( r + 1 r / m \ Vfe=i / N i=i The first sum on the right-hand side can be estimated by the integral (Aix a i +... + XNxaN) 1= Jo r dx. 26). Moreover, put Di = s(\i • • • \3)i and D2 = (N - s)(A s + 1 • • • XN)^. By the arithmetic-geometric mean inequality, it holds that X1xai +...