By Leo Larsson, Lech Maligranda, Josip Pečarić, Lars-Erik Persson
Amassing all of the effects at the certain types of inequalities, the assurance of this booklet is exclusive between textbooks within the literature. The booklet makes a speciality of the old improvement of the Carlson inequalities and their many generalizations and adaptations. in addition to just about all recognized effects bearing on those inequalities and all identified facts suggestions, a few open questions appropriate for extra study are thought of. chapters are dedicated to clarifying the shut connection among interpolation idea and this kind of inequality. different purposes also are incorporated, as well as a historic be aware on Fritz Carlson himself.
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Additional info for Multiplicative Inequalities of Carlson Type And Interpolation
Instead of using two factors on the right-hand side of the inequality, he allowed any odd number of factors. , and m is a positive integer, then ( oo \ (m+l)(2m+l) fe = l 2m / oo j=0fc=l unless all a/t are zero, where c = n i2^2™-1=(2m+i)2-+i n rm) 2m+l 2m j=i ,„ j=o V J >. / is the sharp constant. 2. 1 In the cases m = 1 and m = 2, we get the following sharp inequalities. oo \ ° ($ > oo \/c=l oo < 54 £ alj^kai £ fcVfc, fc = l / / oo \ 1& £ oo afc /C=l oo < 300000 2 / fc=l fe=l oo fc=l af £ fca fc=l oo 3 fe oo oo £ fcVfe £ fc3af £ fc=l fc=l fc4a3.
3. 2 of Chapter 3). 12). If 0 < 9 < 1 and <7i,Tj, i = 0,1 are real numbers, then S((l - 6)a0 + 0*i, (1 - 6)T0 + On) < Proof. Sivo^-'SitTuTi)". The Holder-Rogers inequality with exponents l-6> and 6 yields the desired inequality. • We can now prove our extensions of the Landau and Levin-Steckin results. 3. 5 if we put p = 1. V8 2 l + \ l = (l-9)(l-q) + l - \ l + v(^ + q), V— 1 Then, since J and « i 6=—I 7 i \ —. 3 and the case q = »/|, that 5(0,1) 4 < C-s(l - y 1 - z W l + y ^ , 2 < C - 5 ( l - g , 2 ) 1 - " 5 ( l + g,2)" S(l-q,2)1-eS0.
2=/33, and if ai,ct2>0, a i + 0 : 2 = 1, t h e n Holder's inequality implies t h a t 00 / 00 \ VPI fe=l \fc=l / 00 \ VP2 £( f c / 3 2 < 2 H ^0k
Multiplicative Inequalities of Carlson Type And Interpolation by Leo Larsson, Lech Maligranda, Josip Pečarić, Lars-Erik Persson