Inequalities for Positive Series

Let f(x) ≡ (1 – x) b + b a b-1 x o (x) ≡ x c – c Β c-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 o ( x ) have minima when x = 1 – α and when x = β , respectively. Hence (1 – x) b + a b-1 x ≧ b a b-1 + (1 – b) a b x c β c-1 x ≧ (1 – c)β c .