Useful Inequalities x August 10, 2021
[Pages:3]Useful Inequalities {x2 0} v0.40 ? August 13, 2023
binomial
Cauchy-Schwarz Minkowski Ho?lder Bernoulli
exponential
n
2
xiyi
i=1
n
x2i
i=1
n
yi2
i=1
n
|xi + yi|p
1 p
n
|xi|p
1 p
+
n
|yi|p
1 p
i=1
i=1
i=1
for p 1.
n
n
1/p n
1/q
|xiyi|
|xi|p
|yi|q
i=1
i=1
i=1
for p, q > 1,
1 p
+
1 q
=
1.
(1 + x)r 1 + rx for x -1, r R \ (0, 1). Reverse for r [0, 1].
(1 + x)r
1 1-rx
for
x
[-1,
1 r
),
r 0.
(1
+
x)r
1
+
x x+1
r
for x 0, r [-1, 0].
(1 + x)r 1 + (2r - 1)x for x [0, 1], r R \ (0, 1).
(1 + nx)n+1 (1 + (n + 1)x)n for x 0, n N.
(a + b)n an + nb(a + b)n-1 for a, b 0, n N.
ex
1
+
x n
n 1 + x;
1
+
x n
n ex
1
-
x2 n
for n 1, |x| n.
xn n!
+1
ex
1
+
x n
n+x/2 ;
ex
ex n n
for x, n > 0.
xy
+ yx
>
1;
xy
>
x x+y
;
ex
>
1
+
x y
y
>
xy
e x+y ;
x y
e
x-y x
for x, y > 0.
1 2-x
<
xx
< x2
- x + 1;
e2x
1+x 1-x
for x (0, 1).
x1/r(x - 1) rx(x1/r - 1) for x, r 1;
2-x
1
-
x 2
for x [0, 1].
xex
x
+
x2
+
x3 2
;
ex x + ex2 ;
ex + e-x 2ex2/2
for x R.
e-x
1
-
x 2
for x [0, 1.59];
ex 1 + x + x2
for x < 1.79.
1+
x p
p
1+
x q
q
for (i) x > 0, p > q > 0,
(ii) - p < -q < x < 0, (iii) - q > -p > x > 0. Reverse for:
(iv) q < 0 < p , -q > x > 0, (v) q < 0 < p , -p < x < 0.
binary entropy
Stirling means power means
Lehmer log mean Heinz Maclaurin-
Newton
logarithm
trigonometric hyperbolic square root
x 1+x
ln(1 + x)
x(6+x) 6+4x
x
for x > -1.
2 2+x
1
1+x+x2 /12
ln(1+x) x
1 x+1
2+x 2+2x
for x > -1.
ln(n) +
1 n+1
<
ln(n + 1)
<
ln(n) +
1 n
n i=1
1 i
ln(n) + 1
for n 1.
|ln x|
1 2
|x
-
1 x
|;
ln(x
+
y)
ln(x)
+
y x
;
1
ln x y(x y
- 1);
x, y > 0.
ln(1 + x)
x
-
x2 2
for x 0;
ln(1 + x) x - x2 for x -0.68.
x-
x3 2
x cos x
x cos x 1-x2 /3
x 3 cos x x - x3/6 x cos
x 3
sin x,
x cos x
x3 sinh2 x
x cos2 (x/2)
sin x (x cos x + 2x)/3
x2 sinh
x
,
max
2
,
2 -x2 2 +x2
sin x x
cos
x 2
11+
x2 3
tan x x
for x
0,
2
.
2 x+
1-2 x
<
1 x
<
x+1- x
-1
<
2 x-2 x
-1
for x 1.
x
x+1 2
-
(x-1)2 2
x
x+1 2
-
(x-1)2 8
for x [0, 1].
Jensen Chebyshev
rearrangement
max
{
nk kk
,
(n-k+1)k k!
}
n k
nk k!
en k
k;
n k
nn kk (n-k)n-k
.
nk 4k!
n k
for n k 0;
4n n
(1
-
1 8n
)
2n n
4n n
(1
-
1 9n
).
n1 n2
k1 k2
2
G
n1 +n2 k1 +k2
;
n n
G
for
tn k
tk
n k
for t 1.
G=
2nH() ,
2n(1-)
H(x) = - log2(xx(1-x)1-x).
d i=0
n i
min
nd + 1,
en d
d,
2n
for n d 1.
n i=0
n i
min
1- 1-2
n n
,
2nH () ,
2n e-2n
1 2
-
2
for
(0,
1 2
).
4x(1 - x) H(x) (4x(1 - x))1/ln 4;
H (x2 ) H (x)
1.618x
for
x
(0, 1).
1 - 5x2 H(1/2 - x) 1 - x2, for 0 < x 1/4.
e
n e
n
2n
n e
n e1/(12n+1)
n!
2n
n e
ne1/12n en
n e
n
min xi
n x- i 1
(
xi )1/n
1 n
xi
1 n
xi2
x2i xi
max xi
Mp Mq for p q, where Mp = i wi|xi|p 1/p, wi 0, i wi = 1. In the limit M0 = i |xi|wi , M- = mini{xi}, M = maxi{xi}.
i wi|xi|p i wi|xi|p-1
i wi|xi|q i wi|xi|q-1
for p q, wi 0.
xy
x+y 2
(xy)
1 4
x-y ln(x)-ln(y)
x+y 2
2
x+y 2
for x, y > 0.
xy
x1- y +x y1- 2
x+y 2
for x, y > 0, [0, 1].
Sk2 Sk-1Sk+1 and (Sk)1/k (Sk+1)1/(k+1)
Sk =
1 n
ai1 ai2 ? ? ? aik ,
k 1i1 ................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- maths genie free online gcse and a level maths revision
- logarithms university of utah
- logarithmic functions and log laws university of sydney
- what is a logarithm reed college
- propagation of errors—basic rules university of washington
- logarithms university of plymouth
- exponentials and logarithms 14f
- chapter8 logarithmsandexponentials log x
- worksheet logarithmic function department of mathematics
- unit 6 homework part 2 deer valley unified school district
Related searches
- 10 most useful languages
- 10 most useful command prompts
- x plane 10 aircraft downloads
- x plane 10 aircraft list
- x plane 10 default aircraft
- x plane 10 download
- x y 10 xy 5
- x plane 10 freeware aircraft
- icd 10 2021 changes
- useful windows 10 command prompts
- solving inequalities with x squared
- icd 10 2021 code for htn