Dictionary of Inequalities - download pdf or read online

By Peter S. Bullen

ISBN-10: 1482237628

ISBN-13: 9781482237627

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The left-hand side is always positive except when the domain is a disk, and then R = r. Comments (i) Definitions of inner and outer radius can be found in Isodiametric Inequality Comments (ii). (ii) See also: Gale’s Inequality. [EM, vol. 1, p. 420]. References Borel-Carath´ eodory Inequality and if A(r) = max|z|=r ℜf (z), then for r < R M∞ (r, f ) ≤ If f is analytic on {z; |z| ≤ R} R+r 2r A(R) + |f (0)|. R−r R−r (1) In particular if A(R) ≥ 0 then M∞ (r, f ) ≤ R+r A(R) + |f (0)| . R A(R) + f (0) (R − r)n+1 .

1 k n i=0 i (1+x)n > (1) (2) xi . (3) (c) If n > 1 and −1 < x < 1/(n − 1), x = 0, then, (1 + x)n < 1 + (d) [Karanicoloff] nx . 1 − (n − 1)x If 0 < q < p, 0 < m < n and 0 < x < 1 then, (1 − xp )m > (1 − xq )n . (4). If 0 < p < q and 0 < x < x0 , x0 being the unique root of (1 − xp )m = (1 − xq )n in ]0, 1[, then (∼4) also holds, while if x0 < x < 1 (4) holds. (e) If x + y = 1, 0 < x < 1 and if r, s > 1 then (1 − xr )s + (1 − y s )r > 1. (f ) [Bennett] If x ∈ R, m, n ∈ N with m, n > x then 1+ (g) ´] [Maleˇ sevic x m m 1− x n n < 1.

129], Titchmarsh [T86, pp. 174– References 176]. Bounded Variation Function Inequalities (a) If f, g are of bounded variation on [a, b] then so is f + g and V (f + g; a, b) ≤ V (f ; a, b) + V (g; a, b). (b) If in addition f (a) = g(a) = 0 then V (f g; a, b) ≤ V (f ; a, b)V (g; a, b). (c) If on [a, b], f is continuous, g of bounded variation, or f, g are both of bounded variation and g is continuous, then b a b f dg ≤ a |f (x)| dV (g; a, x) ≤ sup |f (x)| V (g; a, b). 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.

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Dictionary of Inequalities by Peter S. Bullen

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