AN ELEMENTARY INEQUALITY
SEYMOUR HABER
United States Department of Commerce National Bureau of Standards
Washington, D.C. 20234 (Received November 8, 1978)
ABSTRACT. An elementary inequality is proved in this note.
KEY WORDS AND PHRASES. Inequy, Compound
interest.AMS (MOS) SUBJECT CLASSIFICATION
(1970)CODES.
26DT5.I. INTRODUCTION AND RESULTS.
The following theorems present a series of closely related inequalities. The form that was found originally is given first.
THEOREM i. If a, b _> 0 and n is a positive integer, then
n
an-I
an-2b2
bn
(a +
b na
+
b+
+’’’+ > (I i)n+
I 2PROOF. Let Ix] be, as usual, the integer part of x, and let the [n/2
],
symbol be defined by
0
Enl2]
o
x0
+
x1
+...+ X[n/2
] x0+
xI
+...+ X[n/2]_
Iif n is odd
+
1 x[n/2] if n is even.
Assuming, without loss of generality, that a > b, we divide through by an
in (I.I), and set x b/a, obtaining
x2 n n
1+
x+ +...+
x>
(1 + x)
n+l 2 O-<x<- 1 (1.2)
Now
En/2]
(i
+ x)n * ()
(xi+ xn-f)
o and
n
xn-i
i
+
x+
x2+...+
x <xi+
0 so that we may rewrite (1.2) as
2-
()) (xi+
xn-i)
>- 0 0 < x -< I (1.3)n n-I 2 n-2
We now note that I
+
x > x+
x > x+
x >...> x[n/2] n-[n/2]+
x This isan example of the rearrangements inequality ([I ], p.261), since we can write k n-k k+I/2 -I/2 n-k-I/2
XI/2
X +X X X +X
and
k+l n-k-I k+I/2 1/2 n-k-I/2 -1/2
X
+
X X X+
X Xk+I/2 n-k-i/2
and for 0 < k _< [n/2] I, x and x and xi/2 It follows that if we set
are in the same order as x-1/2
n-i
xi+x
ai
xI
+
xn-i 2if i < n/2
if n is even and i n/2
> >
then a 0 -> a
I
a[n/2
]. If we also set(n)
bi n+l
2n i
i I, 2, In/2]
-> b >-...>
b[
and since (i.i) (1.3) are equalities it is clear that bI 2 n/2]
when a b, the sum of the b. is zero. (1.3) is then an immediate consequence of the following obvious lemma, which is also related to the rearrangements inequality.
LEMMA. If k is a positive integer and a
I > a2 >...> a
k >_ 0, and
bI > b2 >...> b
k and bI
+
b2
+...+
bk 0, then
alb
I+ a2b
2+...+ akb
k > 0.COROLLARIES.
n
)n-I
x -I > n(x-l)
n x+l.n-I
x-I < n(x-l)
(--)
x >_ 1 and n 0,1,2, (1.4)
0 _< x _< I and n 0,1,2, (1.5)
Inequality (1.4) is a sharpening of Hardy, Littlewood, and Polya’s (2.15.3) for part of the latter’s range, while (1.5) is complementary to (2.15.3) for another part of its range. These are immediate consequences of (1.2), which is valid for
all x _> 0 by Theorem i. (The case n 0 is not a consequence of Theorem I. but
is obvious; similarly (1.2) holds for n 0 if we interpret I
+
x+
x2+...+
xn as 1 for that case.)Setting x l+t in (1.4) and (1.5) gives us the inequalities n-I
(l+t)n >
l+nt(l+, )
t > 0 and n 0,1,2, (1.6) andn-I
(l+t)n <
l+nt(l+ )
-I _< t < 0 and n 0,1,2 (1.7) Putting -t in place of t gives an alternative form:(l-t)
n-I
<
l-nt(l-)
0 < t -< and n 0,1,2, (1.8) Replacing t by I/t in (1.6) gives usn-i (t+l)n > tn
+
n(t+)
t > 0 and n 0,1,2, (1.9)This is better than (1.6) for t > 1 though not as good for t < (for n > 2).
Similarly, from (1.8) we obtain
(t_l)n
< tn I n-In(t-)
t > and n 0,1,2, (I.I0)(1.6) is a sharpening of the observation that compound interest beats simple interest: a period rate t, compounded for n periods, is better than simple interest at the rate t(l+t/2)n-I
(1.6) (i.i0) were obtained from (1.4) or (1.5) by invertible bilinear changes of variable; similarly each of (1.4) and (1.5) may be obtained from the other by replacing x by I/x. Thus (1.4)-(1.10) are equivalent, and equivalent to Theorem I.
Another bilinear transformation- setting x (l+t)/(l-t) in (1.4) gives us
(l+t)n (l-t)n > 2nt 0 < t < and n 0,1,2, (I.II) which is obvious’. This provides a quick alternate proof of Theorem i., but one which does not show the donnection with the rearrangements inequality.
The form of inequalities (1.4) (i. ii) suggests consideration of non-integral n. The inequalities are equivalent in this precise sense; for any given n, one of these inequalities holds for its stated range of values of the other variable
(x or t) if and only if all the other inequalities hold, for that n, for their stated ranges of the other variable. Thus for each n we may choose which of (1.4)
(I.II) to study, at our convenience.
n n
For 0 < n < I (and 0 < t < I) the binomial expansion of (l+t) (l-t) is
and each of the coefficients
() ()
is positive. It follows that I.holds for n between 0 and I.
For n between I and 2, all of
(), (),
are negative. So for I < n < 2 (and in fact I < n < 2), (I.II) holds with the inequality reversed.For non-integral n > 2 we look at (1.6). Set
i
+ nt(l)
n-If(t,n)
(l+t)n
f(0,n) I. To show that (1.6) holds, it is sufficient to show that
- (l+t)n+l (I +
(I + )
I
is < 0 for t > 0, n > 2. Setting s n-2, the quantity in brackets is
( -) (
+)
t s.
This is certainly negative when 1 st/2 is negative; when the latter is non- negative, the whole quantity is
Summing up we have
st
< (z---) (e z_<o
THEOREM 2. Each of the inequalities (1.4) (I.II) holds also for non-lntegral n, when 0 < n < I and when n > 2: for I < n < 2 each of (1.4) (i.ii) holds with the direction of the inequality reversed.
REFERENCES
i. Hardy, G.H., J.E. Littlewood and G. Polya, Inequalities, Cambridge University Press (1959).
Special Issue on
Time-Dependent Billiards
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This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi accelera- tion (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.
This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
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