By P. S. Bullen, D. S. Mitrinović, P. M. Vasić (auth.)
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Extra info for Means and Their Inequalities
It is not difficult to see that equality occurs in all the inequalities (10) for (10) for B see Remark (4). ~ iff it occurs in all the corresponding inequalities ~ and~. n If d n = cnx , nLN, n > 0, then ~ and 2. are a-logarithmically concave (convex) together so we can immediately state the following generalisation. Corollary 8. If and ~ bare logarithmically convex (concave) sequences then for every x > 0, Y > 0 so i8 the sequence nt Theorem 9. (a) If (b) If a ~ > ~(x,y) N. and bare logarithmically concave so is 0, ß > 0, a + ß • , and ~ (13) ~*~.
E is logarithmically convex iff c n2 -< c n+l c n_ 1 which for positive sequences i8 equivalent to ( 10) 15 INTRODUCTION c < s -c s+1 r, s == 1, 2, ••• ( 11) in the case of logarithmic concavity inequalities (10) and (11) are reversed. The results of this section show that the logarithmic concavity property is conserved under the operation of convolution. Not all the results are needed in the sequel but are included for the sake of completeness. Theorem 7. If ~ and ~ are logarithmically convex (concave) so is c Proof (1) Convex Case.
If ~ and ware two positive n-tples then (1) with equality iff a 1 = ••• Inequality (1) will be referred to as a n GA, or as the GA-inequality, for short. Corollary 2. If ~ and ware two positive n-tples then H (a;w) < G (a;w) < A (a;w) n-- - with equality iff Proof. 1. 4#17. 2. Some Preliminary Results. Before proving Theorem 1 we will consider some results that help to simplify the later work. We first consider 1(9)1 that is GA in its simplest form, n Lemma 3. If a > 0 and b > 0 then Iäb < with equality iff Remark (1).