Foundations of Data Science

1 + x ≤ e x for all real x
(1 − x) n ≥ 1 − nx for 0 ≤ x ≤ 1
(x + y) 2 ≤ 2x 2 + 2y 2
Triangle Inequality
|x + y| ≤ |x| + |y|.
Cauchy-Schwartz Inequality
|x||y| ≥ x T y
Young’s Inequality For positive real numbers p and q where 1 + 1 p q
positive reals x and y,
xy ≤ 1 p xp + 1 q yq .
= 1 and
Hölder’s inequalityHölder’s inequality For positive real numbers p and q with 1 + p
1
= 1, q
(
n∑
n∑
) 1/p ( n∑
) 1/q
|x i y i | ≤ |x i | p |y i | q .
i=1
i=1
i=1
Jensen’s inequality For a convex function f,
f
( n∑
i=1
α i x i
)
≤
n∑
α i f (x i ),
i=1
Let g(x) = (1 − x) n − (1 − nx). We establish g(x) ≥ 0 for x in [0, 1] by taking
the derivative.
g ′ (x) = −n(1 − x) n−1 + n = n ( 1 − (1 − x) n−1) ≥ 0
for 0 ≤ x ≤ 1. Thus, g takes on its minimum for x in [0, 1] at x = 0 where g(0) = 0
proving the inequality.
(x + y) 2 ≤ 2x 2 + 2y 2
The inequality follows from (x + y) 2 + (x − y) 2 = 2x 2 + 2y 2 .
Lemma 12.1 For any nonnegative reals a 1 , a 2 , . . . , a n and any ρ ∈ [0, 1], ( ∑ n
i=1 a i) ρ
≤
∑ n
i=1 aρ i . 381