By Joe Diestel, Hans Jarchow, Andrew Tonge

We will be able to top comprehend many primary methods in research by means of learning and evaluating the summability of sequence in quite a few modes of convergence. this article presents the reader with uncomplicated wisdom of actual and sensible research, with an account of p-summing and similar operators. The account is panoramic, with particular expositions of the middle effects and hugely proper purposes to harmonic research, likelihood and degree conception, and operator idea. this can be the 1st time that the topic and its functions were awarded in such entire aspect in ebook shape. Graduate scholars and researchers in genuine, complicated and practical research, and chance conception will take advantage of this article.

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**Example text**

We state Gabriel's result here. 1). 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. 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 Levin V. I. 1). Instead of using two factors on the right-hand side of the inequality, he allowed any odd number of factors.

Levin and S. B. 8) then the inequality ( oo ft—1 \ 2P+2 oo / / 1 v p-q ft—1 oo / 1 x p+q ft—J. of Landau type holds. Here, we may choose C = C(p, q) = 4(2q)-2"B (J-, i ) ", and this constant is sharp. 10) Some Extensions and Complements of Carlson's Inequalities 27 Here, as usual, B(-, •) denotes the Beta-function. 8). In the next section, an extension of the above result is given. 5 here. 8) hold, then the function f(y) = Uyp~q + \yp+9) \ y>o is convex for any A > 0. Thus, for any positive integer k, it holds by Hadamard's inequality that 'HM-/'"** Moreover, we have + +a -I/P f>'" i»' )~ '«-hBih-h)Thus, if we write ak = [X(k~-j +-x[k-zy 1V-" + 1/ (»-r K*-i)' J iy+A1/ip+1) it follows by the Holder-Rogers inequality with exponents and P p +1 Multiplicative Inequalities of Carlson Type and Interpolation 28 that (£>*!

N. 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 +...