By Richard Bellman (auth.), E. F. Beckenbach (eds.)
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Example text
As one easily sees, the following is true. • ,ßk ~ o real constants, Eßk = 1 and B p k (8) k are also price indices. Here B P is defined by We note that (8) with Bk = ßk follows fram (7) for B - 0. As one sees fram (7) or (8), the set of all price indices is a convex set C. The problem of determining this set C seems to be difficult. In order to solve this problem, the theorem of M. Krein and D. Milman [7] may be helpful. 28 Wolfgang Eichhorn I am indebted to H. Funke and J, Voeller for valuable ACKNCMLEDGMENT.
4. D. S. Mitrinovic, Analytic inequalities, Springer Verlag, Berlin, 1970. 35 ON TURÄN-TYPE INEQUALITIES Dieter K. Ross Department of Mathematics La Trabe University Bundoora, Victoria 3083 AUSTRALIA ABSTRACT. ny of the special functions yn which occur in mathematical physics satisfY an inequality of 2 the type yn+l- YnYn+ 2 ~ o, for n = 0,1,2, .... This paper deals primarily with a relationship between this inequality and a three-term recurrence relation which can be considered as the generator of the Yn· Only a few examples are given here, but many more can be derived just as easily.
242, 24J], [3, p. 66]), that (9) is an expression of the fact that the vectors 0 wj 1 w~ are the eigenvectors of C(A), 3. j 0,1, ... ,n-1 , with corresponding eigenvalues of f(wj). PROOF OF THEOREMS 1 AND 2 Since (j and since a 0 ,a1 , ... ,an-l = 0,1, ... ,n) , arereal numbers, we have (11) Proof of Theorem 1. For n odd, n = 2r + 1, (8) and (11) give 41 E. F. Beckenbach and Richard Bellman 42 det C(A) = f(w 0 ) (12) Since r 0 j=l lr(wj) 12 . n f(w 0 ) = f(O) = :E aj , j=O (13) and since ~=l lr(wj)l 2 ~ Proof of Theorem 2.