By Man Kam Kwong

Norm inequalities pertaining to (i) a functionality and of its derivatives and (ii) a chain and of its adjustments are studied. targeted simple proofs of simple inequalities are given. those are available to an individual with a history of complicated calculus and a rudimentary wisdom of the Lp and lp areas. The classical inequalities linked to the names of Landau, Hadamard, Hardy and Littlewood, Kolmogorov, Schoenberg and Caravetta, etc., are mentioned, in addition to their discrete analogues and weighted types. top constants and the life and nature of extremals are studied and lots of open questions raised. an intensive checklist of references is equipped, together with a few of the substantial Soviet literature in this subject.

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Suppose G(h;) < a for i = 1,2 and the pairs (A, C), (B, D) are not proportional. Then = t p (llh~ll~ + I[h~l[~) 2 = aP[AB + 2 ( A B C D ) 1/2 + CD] < aP(A + C)(B + D) = aPllhll~[Ih"ll~ = < a P [ ( A B ) 1/2 -[- (CD)]/2] 2 [Ih'll~ p. 2. 4. EQUIVALENT BOUNDED INTERVAL PROBLEMS FOR R Proof. 9) for i = 1, let ¢ > 0. T h e r e exists an f in k(p, R) - e < a(f). 1 there exists a g • C ~ ( R ) such t h a t a ( f ) < a(~) + ~. Hence k(p, M~) > G(g) > k(p, R) - 2e, where M~" = {y • W~([a,b]): y(a) = y(b) = 0} and the c o m p a c t interval [a,b] is chosen to contain the s u p p o r t of g.

Is K(2,1,p,p,p,R +) increasing for 2 _< p < oo? decreasing for 1 < p _< 2? We conjecture that the answer is yes. 7. Berdyshev [1971] found that K(2, 1, 1, 1, 1, R +) = v/5-/2. W h a t are the values of K ( n , k , i , l , R +) for 1 _< k < n, n = 3 , 4 , . . Problem 1. In the absence of knowledge of the exact value of K(n, k, p, q, r, J), find "good" upper and lower bounds. For work along these lines see the paper by Franco, Kaper, Kwong, and Zettl [1983]. 8. 25) for functions of more than one variable have also been studied.

F r o m a p r o p e r t y of integrals we see t h a t given e > 0 there is a number X near 1 such t h a t the Sobolev norm of h~, the restriction of h to [0, a], a e IX, 1] is so close to t h a t of h t h a t G(h~) > G ( h ) - e. Let a:N be the first zero such t h a t XN >_ X . by the continuity argument employed in Case 1 above we can establish the existence of points al C (xi, Xi+l), i -- 1, 2 , . . , N , such t h a t h ' ( a i ) / h ( a i ) = h'(O)/h(O). After scaling, the restriction of h on each [ai,ai+t], i = 0, 1 , .