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Inequalities for Positive Series

Published online by Cambridge University Press:  31 October 2008

C. E. Walsh
C. E. Walsh
Affiliation:
34 Chelmsford Road, Ranelagh, Dublin
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Let f(x) ≡ (1 – x)b + b ab-1 x

ø (x) ≡ xc – c Βc-1 x

where b ≧ 1, c ≧ 1, 0 ≦ α ≦ 1, 0 ≦ β ≦ 1, and x is assumed to lie in the range (0, 1). By differentiation, or otherwise, it is easily shewn that f(x) and ø (x) have minima when x = 1 – α and when x = β, respectively. Hence

(1 – x)b + ab-1 x ≧ b ab-1 + (1 – b) ab

xc βc-1 x ≧ (1 – c)βc.

Type
Research Article
Information
Edinburgh Mathematical Notes , Volume 32 , January 1940 , pp. xxx - xxxii
Copyright
Copyright © Edinburgh Mathematical Society 1940

References

1 Copson, , Journal London Math. Society 32 (1927), 9–12CrossRefGoogle Scholar; 3(1928), 49–51. Elliott, Ibid., 1 (1926), 93–96; 4 (1929). 21–23.