Handbook of Means and Their Inequalities (Mathematics and by P.S. Bullen

By P.S. Bullen

There looks varieties of books on inequalities. at the one hand there are treatises that try to hide all or such a lot points of the topic, and the place an test is made to provide all ends up in their absolute best shape, including both a whole facts or a comic strip of the evidence including references to the place a whole facts are available. Such books, geared toward the pro natural and utilized mathematician, are infrequent. the 1st such, that introduced a few order to this untidy box, is the classical "Inequalities" of Hardy, Littlewood & P6lya, released in 1934. very important as this extraordinary paintings was once and nonetheless is, it made no test at completeness; quite it consisted of the full wisdom of 3 entrance rank mathematicians in a box during which each one had made basic contributions. large as this mixed wisdom was once there have been necessarily definite lacunre; a few vital effects, corresponding to Steffensen's inequality, weren't pointed out in any respect; the works of yes faculties of mathematicians have been passed over, and lots of very important principles weren't constructed, showing as workouts on the ends of chapters. The later booklet "Inequalities" by means of Beckenbach & Bellman, released in 1961, maintenance lots of those omissions. despite the fact that this final publication is way from a whole assurance of the sphere, both extensive or scope.

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N i=l The case of equality is easily considered. (ii) A very simple inductive proof for the equal weight case has been given by > 2 and that the result is Aczel, [Aczel1961a,b; Pecaric 1990c]. Assume that n known for all k, 2 1 f ( n n~ai ) < k < n. (n~ai) +n-1~f(ai)' 1 1 n ) 1 n- ) by the case n == 2, and the induction hypothesis. 1 Theorem 6(b), Remark (x). 1 (1) holds strictly if X # y' A # 0' 1 . With a slight change of notation we can rewrite (J) in this case as: f( 1- sx + sy) < (1- s)f(x) + sf(y), 0 < s < 1, with equality only if x == y.

N . -L((i-1)t(~- )+(n-i+1)t(~))>Lt( ~ ). n. n n . n+1 ~=1 ~=1 So n-1 . L ((i- 1)t(~-n 1 1 . 1 ) + (n- i + 1)t( ~ )) + -t(1) > n ~=1 n n+1 . L. t( n+1 ~ ) - t(1). ~=1 This on simplification gives the first inequality. A similar argument, starting with to the same inequality when REMARK (xii) 1 i f ( i + ) + (1 i )f ( i ) n+ 1 n+1 n+1 n+1 f is concave. leads D This result is a special case of a slightly more general result; [Kuang; Qi 2000a]. 122-123]. THEOREM 9 [HADAMARD-HERMITE INEQUALITY] Iff : [a, b] ~ 1R is convex and if a < c < d < b, tben c+d)< 1 1df

1 ) 1 fax (x-t)n f(n+l) (t) dt is the (n+l)-st Taylor remainder centred at a; also for some c, a

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