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Iii) The proof of (3) is quite complex. (iv) There is a left-hand side for the inequality in (d) but the constant is not known; see Bennett, [Be]. (v) For other inequalities due to Bennett see: Abel’s Inequalities Related Results (b), Copson’s Inequality Extensions, Hardy’s Inequality Extensions (c), Hilbert’s Inequalities Extensions (b), Littlewood’s Conjectures, Zeta Function Inequalities. References Bennett [Be, pp. 47–56]; Alzer [20, 22, 23], Bennett [57]. Benson’s Inequalities (a) Given three functions u(x), P (u, x) > 0, G(u, x) write p(x) = P u(x), x , g(x) = G u(x), x .

Com/new+%26+forthcoming+titles+(default)/book/ 978-3-7643-2737-8 General Inequalities 7: Proceedings of Seventh International Conference on General Inequalities, Oberwolfach. org/details/TheTheoryOfFunctions Univerzitet u Beogradu Publikacije Elektrotehniˇckog Fakulteta. org/ © 2015 by Taylor & Francis Group, LLC ✐ ✐ ✐ ✐ ✐ ✐ “K23178” — 2015/5/15 — ✐ ✐ 1 Abel–Arithmetic Abel’s Inequalities (a) If w, a are n-tuples and a is monotonic then n i=1 wi ai ≤ ≤ max {|Wi |} |a1 | + 2|an | . 1≤i≤n (b) Let a be an n-tuple with Ai > 0, 1 ≤ i ≤ n, then n An ≤ exp A1 i=2 ai Ai−1 .

A≤x≤b (d) If on [a, b], f is continuous, g of bounded variation and if F (x) = a ≤ x ≤ b then x a f dg, x V (F ; a, x) ≤ a |f | |dg|, a ≤ x ≤ b. © 2015 by Taylor & Francis Group, LLC ✐ ✐ ✐ ✐ ✐ ✐ “K23178” — 2015/5/15 — ✐ ✐ 2 Backward–Bushell 33 (e) If f is of bounded variation on [0, 1] then 1 0 f− 1 n n f i=1 i n ≤ V (f ; 0, 1) , n n = 1, 2, . . Comments (i) Results (a) and (b) have been extended by A. M. Russell to functions of bounded higher order variation. The integrals above are RiemannStieltjes integrals.

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