# Download An elementary treatise on elliptic functions by Arthur Cayley PDF By Arthur Cayley

This quantity is made out of electronic photos from the Cornell college Library historic arithmetic Monographs assortment.

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3. Jessen's and Related Inequalities 35 holds f o r some )~ > 1 in cases (i) and (i t) or )~ ~ (0, 1) in cases (ii) and (iit). 3) may be determined as follows. Set lz -- ( ~ ( M ) - ~ ( m ) ) / ( M - m). 3). I f lZ ~ O, let x = s be the unique solution in [m, M] o f the equation /z4~(x) - q~t(x) {q~(m) + / z ( x - m)} = O, then )~ = lZ/~t(s cases (i) and (ii). 3). Moreover, we have m < s < M in the (b) Let all the hypotheses o f (a) hold except that now dp is concave on I with qb" (x) <<,O, with equality f o r at most isolated points of I.

As, bl >/... >1 bs and ql . . . s-1, t=l ~q,a,=~q,b,. t=l t=l If g is convex on I and f is g-convex dominated on I, then the following inequality holds: ~q,(f(b,)t--I f(a,)) <~~ i=1 (g (b,) - g (ai)). 12) Chapter 1. Inequalities Involving Convex Functions 52 PROOF. Consider the functional J(f) = Lql(f(b,)- f(a,)), f ~ F(I). 1. 12). D The following theorem given in  deals with an improvement of the JensenSteffensen inequality. 5. Let x and p be two n-tuples of real numbers such that x~ ~ I, 1 <~ i <~n, and I is an interval from R and Pn > O.

Ii) 0<. Pk <. P n f o r a l l k = 1,2 . . . n - 1. PROOF. (i) =:~ (ii) is obviously the Jensen-Steffensen inequality. (ii) :=~(i) considers the functional ~ptf(xt)-J ( f ) = --~n 1---1 f E /-1 ptXl , f ~ F(I). 2. 11). 1. 6. Let x be a nonincreasing n-tuple of real numbers, x~ E I, 1 <~ i <<,n, p real n-tuple, and there exists x j, j ~ (1, 2 , . . , n), such that k 1 p~(x~ - x j ) <. 13) n Y ~ Pl (Xl -- Xj ) ~ 0 for every k such that xk <~x. 5. 11) holds. 1 given above. 5 Hadamard's Inequalities In 1893, J.