Download Multiplicative Inequalities of Carlson and Interpolation by Leo Larsson, Lech Maligranda, Josip Pečarić, Lars-Erik PDF
By Leo Larsson, Lech Maligranda, Josip Pečarić, Lars-Erik Persson
Gathering the entire effects at the specific sorts of inequalities, the assurance of this publication is exclusive between textbooks within the literature. The booklet makes a speciality of the historic improvement of the Carlson inequalities and their many generalizations and adaptations. in addition to just about all recognized effects relating those inequalities and all identified evidence suggestions, a few open questions compatible for extra learn are thought of. chapters are dedicated to clarifying the shut connection among interpolation idea 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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Extra resources for Multiplicative Inequalities of Carlson and Interpolation
Example text
17). 12 If m = 2 M + 1 is odd and if Xj = j — 1 in the above theorem, then 2M+1 = _ _ — - y (j - 1 ) = M . 2. 7 might not give the sharp constant. 1). 1 (with sharp constant). 1) if we put m = 2, Ai = 0, and A2 = 2. 7. +cmfc -)aP 0 > +1 j (ax>* +... 4 and note that the choice of Cj implies that m whereupon the desired result follows. 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 , .
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 +... + XNxaN = (XlXai +... + Xsxa°) + (Xs+ixa°+* +... + XNxa") XlXa> +... + Xsxa° Xs+1xa°+l +... (jV — S)""N>"(AI • • • Ajv) _77? ,N. l. Thus Since this can be done for any s, the conclusion follows upon taking ( r + l ) t h roots.
Bellman's basic trick was to split the integral on the left-hand side into a sum of two integrals. 1) with n2 replaced by 16. Proof by "Partition of Unity" 1. Since i y i = -1+2/ + i +y y ' it follows by the Schwarz inequality that if g is any positive function, then /»0O -I /»00 / 9{y) dy= Jo Jo + -I /»(X> •7——g{y) dy+ i + y io r——yg{y) dy i+ j ( f (irrt**),(f»vw*)1 °° 0 92{y)dy) "\ ^ / / f°° +[ / V0 Let S be any positive number, and put g(y) = f(y/S) tuting x for y/S, this yields 6 f(x)dx