By Percival Frost Joseph Wolstenholme

ISBN-10: 140217098X

ISBN-13: 9781402170980

This Elibron Classics e-book is a facsimile reprint of a 1863 variation by way of Macmillan and Co., Cambridge - London.

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**Additional resources for A Treatise on Solid Geometry**

**Example text**

However, for N > 2, the constant does not necessarily approach the sharp constant for the corresponding infinite series inequality when m —> oo. 10. ,N. fc=i If A i , . . , XN are any positive numbers, we write ak = (Aifc"1 + . . + XNkaN)'^ • (A^"1 + ... + XNkaN)^ak Some Extensions and Complements of Carlson's Inequalities 45 and apply the Holder-Rogers inequality with parameters and r + 1, r which yields m \ fc=i / ( r + 1 r / m \ Vfe=i / N i=i The first sum on the right-hand side can be estimated by the integral (Aix a i +...

13) holds is that q>r-p. (2-14) If r-p 0 is defined by „2 _ . / P ( P + Q MP + r)-r\ max 4(r+1) <7°- \^Tr' j. then we may choose C = ^ ) = <(iB(^,^)f, and this constant is sharp. In the case Q>Qo, we may choose C = C(p,q0,r). 5. If, in addition, 1 P<~2, then there are cases which give the best constant but which are not covered by earlier results. 6. We then get the inequality +i (£^<^<^))> gH)V for all non-zero sequences of non-negative numbers, and the constant is sharp. 11) with the sharp constant as announced.

3 Let q > 0. Then there exists a constant C such that S(0,l)4 \. The constant is sharp in the cases 0 < q < w | and q = 1. 5 for the case 0 < q < W | . For the remaining cases, an interpolation argument is used. 8 The sharp constant in the above result is unknown in the case w | < q < 1. By investigating the form of the sharp constant in known cases, one is tempted to guess that 30 Multiplicative Inequalities of Carlson Type and Interpolation is the sharp constant whenever 0 < q < 1.

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