Volume 9 (2008), Issue 3, Article 87, 7 pp.
ON AN INEQUALITY OF FENG QI
TAMÁS F. MÓRI
D EPARTMENT OF P ROBABILITY T HEORY AND S TATISTICS
L ORÁND E ÖTVÖS U NIVERSITY
PÁZMÁNY P. S . 1/C, H-1117 B UDAPEST, H UNGARY
moritamas@ludens.elte.hu
Received 17 January, 2008; accepted 04 August, 2008
Communicated by F. Qi
A BSTRACT. Recently Feng Qi has presented a sharp inequality between the sum of squares and
the exponential of the sum of a nonnegative sequence. His result has been extended to more
general power sums by Huan-Nan Shi, and, independently, by Yu Miao, Li-Min Liu, and Feng
Qi. In this note we generalize those inequalitites by introducing weights and permitting more
general functions. Inequalities in the opposite direction are also presented.
Key words and phrases: Optimal inequality, Power sum, Subadditive, Superadditive, Power mean inequality.
2000 Mathematics Subject Classification. 26D15.
1. I NTRODUCTION
The following inequality is due to Feng Qi [2].
Let x1 , x2 , . . . , xn be arbitrary nonnegative numbers. Then
X
n
n
e2 X 2
(1.1)
x ≤ exp
xi .
4 i=1 i
i=1
Equality holds if and only if all but one of x1 , . . . , xn are 0, and the missing one is equal to 2.
Thus the constant e2 /4 is the best possible. Moreover, (1.1) is also valid for infinite sums.
In answer of an open question posed by Qi, Shi [3] extended (1.1) to more general power
sums on the left-hand side, proving that
X
n
n
eα X α
(1.2)
x ≤ exp
xi
αα i=1 i
i=1
for α ≥ 1, and n ≤ ∞.
After the present paper had been prepared, Yu Miao, Li-Min Liu, and Feng Qi also published
Shi’s result for integer values of α, see [1].
In papers [2] and [3], after taking the logarithm of both sides, the authors considered the
left-hand side expression as an n-variate function, and maximized it under the condition of
This research was supported by the Hungarian Scientific Research Fund (OTKA) Grant K67961.
022-08
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TAMÁS F. M ÓRI
x1 + · · · + xn fixed. To this end Qi applied differential calculus, while Shi used Schur convexity.
Both methods relied heavily on the properties of the log function.
On the other hand, [1] uses a probability theory argument, which also seems to utilize the
particular choice of functions in the inequality.
In the present note we present extensions of (1.2) by permitting arbitrary positive functions
on both sides and weights in the sums. Our method is simple and elementary.
Theorem 1.1. Let w1 , w2 , . . . , wn be positive weights, f a positive function defined on [0, ∞),
and let α > 0. Then for arbitrary nonnegative numbers x1 , x2 , . . . , xn the inequality
!
n
n
X
X
(1.3)
C
wi xαi ≤ f
wi xi
i=1
i=1
is valid with
C = w0α−1 inf x−α f (x),
(1.4)
x>0
where
(
(1.5)
w0 =
min{w1 , . . . , wn }
if α ≥ 1,
w1 + · · · + wn
if α < 1.
This inequality is sharp in the sense that C cannot be replaced by any greater constant.
Remark 1. The necessary and sufficient condition for equality in (1.3) is the following.
Case α > 1. There is exactly one xi differing from zero, for which wi = w0 and w0 xi
minimizes x−α fP
(x) in (0, ∞).
n
−α
Case α = 1.
f (x) in (0, ∞).
i=1 wi xi minimizes x
Case α < 1. x1 = · · · = xn , and w0 x1 minimizes x−α f (x) in (0, ∞).
Remark 2. Inequality (1.3) can be extended to infinite sums. Let f and α be as in Theorem 1.1,
and let {wi }∞
sequence of positive weights such that w0 := inf 1≤i<∞ wi > 0
i=1 be an infinite
P∞
when α ≥ 1, and w0 := P i=1 wi < ∞ when α < 1. Then for an arbitrary nonnegative
∞
sequence {xi }∞
i=1 such that
i=1 wi xi < ∞ the following inequality holds.
!
∞
∞
X
X
C
wi xαi ≤ f
wi xi ,
i=1
i=1
where C is defined in (1.4).
Remark 3. By setting α ≥ 1, f (x) = ex and w1 = w2 = · · · = 1 we get Theorems 1 and 2 of
[3]. In particular, taking α = 2 implies Theorems 1.1 and 1.2 of [2].
2. C ONVERSE I NEQUALITIES
Qi posed the problem of determining the optimal constant C for which
X
n
n
X
(2.1)
exp
xi ≤ C
xαi
i=1
i=1
holds for arbitrary nonnegative x1 , . . . , xn , with a given positive α. As Shi pointed out, such an
inequality is generally untenable, because the exponential function grows faster than any power
function. However, if the exponential function is replaced with a suitable one, the following
inequalities, analogous to those of Theorem 1.1, have sense.
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 87, 7 pp.
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A N I NEQUALITY
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Theorem 2.1. Let w1 , w2 , . . . , wn be positive weights, f a positive function defined on [0, ∞),
and let α > 0. Suppose supx>0 x−α f (x) < ∞. Then for arbitrary nonnegative numbers
x1 , x2 , . . . , xn the inequality
!
n
n
X
X
(2.2)
f
wi xi ≤ C
wi xαi
i=1
i=1
is valid with
C = w0α−1 sup x−α f (x),
(2.3)
x>0
where
(2.4)
w0 =
min{w1 , . . . , wn }
w1 + · · · + wn
if α ≤ 1,
if α > 1.
This inequality is sharp in the sense that C cannot be replaced by any smaller constant.
Remark 4. The necessary and sufficient condition for equality in (2.2) is the following.
Case α < 1. There is exactly one xi differing from zero, for which wi = w0 and w0 xi
maximizes x−α fP
(x) in (0, ∞).
n
−α
Case α = 1.
f (x) in (0, ∞).
i=1 wi xi maximizes x
Case α > 1. x1 = · · · = xn , and w0 x1 maximizes x−α f (x) in (0, ∞).
Remark 5. Inequality (2.2) also remains valid for infinite sums. Let f and α be as in Theorem
2.1, and let {wi }∞
sequence of positive weights such that w0 := inf 1≤i<∞ wi > 0
i=1 be an
Pinfinite
∞
when α > 1, and wP
0 :=
i=1 wi < ∞ when α < 1. Then for an arbitrary nonnegative sequence
∞
such
that
w
x
{xi }∞
i=1
i=1 i i < ∞ the following inequality holds.
!
∞
∞
X
X
f
wi xi ≤ C
wi xαi ,
i=1
i=1
where C is defined in (2.3).
3. F URTHER G ENERALIZATIONS
Inequalities (1.3) and (2.2) can be further generalized by replacing the power function with
more general functions. Unfortunately, the inequalities thus obtained are not necessarily sharp
anymore.
Let us introduce four classes of nonnegative power-like functions g : [0, ∞) → R that are
positive for positive x.
(3.1)
F1 = {g : g(x) + g(y) ≤ g(x + y), g(x)g(y) ≤ g(xy) for x, y ≥ 0} ,
(3.2)
F2 = {g : g is concave, g(x)g(y) ≤ g(xy) for x, y ≥ 0} ,
(3.3)
F3 = {g : g(x) + g(y) ≥ g(x + y), g(x)g(y) ≥ g(xy) for x, y ≥ 0} ,
(3.4)
F4 = {g : g is convex, g(x)g(y) ≥ g(xy) for x, y ≥ 0} .
Obviously, the power function g(x) = xα belongs to F1 and F4 if α ≥ 1, and to F2 and F3
if α ≤ 1. In fact, our classes are wider.
Theorem 3.1. Let p1 , p2 , α1 , α2 be positive parameters and
p1 xα1 , if 0 ≤ x ≤ 1,
(3.5)
g(x) =
p2 xα2 , if 1 < x.
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 87, 7 pp.
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4
TAMÁS F. M ÓRI
Then
(3.6)
(3.7)
(3.8)
(3.9)
p1 ≤ p2 ≤ 1,
p1 = p2 ≤ 1,
1 ≤ p2 ≤ p1 ,
1 ≤ p2 = p1 ,
1 ≤ α2 ≤ α1
α2 ≤ α1 ≤ 1
α1 ≤ α2 ≤ 1
1 ≤ α1 ≤ α2
⇒
⇒
⇒
⇒
g
g
g
g
∈ F1 ,
∈ F2 ,
∈ F3 ,
∈ F4 .
It would be of independent interest to characterize these four classes.
Our last theorem generalizes Theorems 1.1 and 2.1.
Theorem 3.2. Let w1 , w2 , . . . , wn be fixed positive weights, and x1 , x2 , . . . , xn arbitrary nonnegative numbers. Let f be a positive function defined on [0, ∞).
Suppose g ∈ F1 . Then
!
n
n
X
X
(3.10)
C
wi g(xi ) ≤ f
wi xi
i=1
i=1
is valid with
g(wi )
f (x)
· inf
.
x>0 g(x)
1≤i≤n wi
Suppose g ∈ F2 . Then (3.10) holds with
f (x)
g(w0 )
· inf
,
(3.12)
C=
w0 x>0 g(x)
where w0 = w1 + · · · + wn .
(x)
Suppose g ∈ F3 , and supx>0 fg(x)
< ∞. Then
!
n
n
X
X
(3.13)
f
wi xi ≤ C
wi g(xi )
C = min
(3.11)
i=1
i=1
is valid with
C = max
(3.14)
1≤i≤n
Suppose g ∈ F4 , and supx>0
f (x)
g(x)
< ∞. Then (3.13) holds with
C=
(3.15)
g(wi )
f (x)
· sup
.
wi
x>0 g(x)
g(w0 )
f (x)
· sup
,
w0
x>0 g(x)
where w0 = w1 + · · · + wn .
4. P ROOFS
Proof of Theorem 1.1. First, let α ≥ 1. Making use of the superadditive property of the α-power
function we obtain
X
X
α
n
n
−α
(4.1)
f
wi xi ≥ inf x f (x)
wi xi
x>0
i=1
i=1
≥ inf x−α f (x)
n
X
x>0
≥
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 87, 7 pp.
w0α−1
(wi xi )α
i=1
inf x
x>0
−α
f (x) ·
n
X
wi xαi ,
i=1
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A N I NEQUALITY
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5
which was to be proved.
Suppose (1.3) is valid for arbitrary nonnegative numbers xi with some constant C. Let xj = 0
for j 6= i, where i is chosen to satisfy wi = w0 . Then from (1.3) we obtain that Cw0 xαi ≤
f (w0 xi ) must hold for every xi > 0. Hence C ≤ w0α−1 inf x−α f (x).
The proof is similar for α < 1. By applying the α-power mean inequality we have
X
X
α
n
n
−α
(4.2)
f
wi xi ≥ inf x f (x)
wi xi
i=1
x>0
= inf x
i=1
−α
x>0
f (x) w0α
n
X
w0−1
α
wi xi
i=1
≥ inf x−α f (x) w0α−1
n
X
x>0
wi xαi ,
i=1
as required.
Again, if (1.3) is valid for arbitrary nonnegative numbers xi with some constant C, let x1 =
· · · = xn = x > 0. Then it follows that Cw0 xα ≤ f (w0 x) for every x > 0, implying
C ≤ w0α−1 inf x−α f (x).
Proof of Remark 1. Let α > 1. In the second inequality of (4.1) equality holds if and only if
there is at most one positive term in the sum. Since f is positive, for x1 = · · · = xn = 0
(1.3) holds true with strict inequality. Let xi be the only positive term in the sum, then the first
inequality fulfils with equality if and only if wi xi = arg min x−α f (x). The last inequality is
strict if wi > w0 .
Similarly, in the
of α < 1 we need x1 = · · · = xn for equality in the α-power mean
Pcase
n
inequality. Then i=1 wi xi = w0 x1 , and the first inequality of (4.2) is strict if w0 x1 does not
minimize x−α f (x).
Finally, the case of α = 1 is obvious.
Proof of Remark 2. The proof of (1.3) is valid for infinite sums, too, because both the superadditivity of power functions with exponent α ≥ 1, and the α-power mean inequality remain true
for an infinite number of terms.
Proof of Theorem 2.1. The proof of Theorem 1.1 can be repeated with obvious alterations. Let
α ≤ 1. Then, by the subadditivity of the α-power function we have
X
X
α
n
n
−α
(4.3)
f
wi xi ≤ sup x f (x)
wi xi
x>0
i=1
i=1
≤ sup x−α f (x)
n
X
x>0
≤
w0α−1
(wi xi )α
i=1
sup x
−α
f (x) ·
x>0
n
X
wi xαi .
i=1
If α > 1, we have to apply the α-power mean inequality again.
X
X
α
n
n
−α
(4.4)
f
wi xi ≤ sup x f (x)
wi xi
i=1
x>0
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 87, 7 pp.
i=1
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6
TAMÁS F. M ÓRI
= sup x
x>0
−α
f (x) w0α
w0−1
≤ sup x−α f (x) w0α−1
x>0
n
X
α
wi xi
i=1
n
X
wi xαi .
i=1
Suppose (2.2) is valid for arbitrary nonnegative numbers xi with some constant C. If α ≤ 1,
let xj = 0 for j 6= i, where i is chosen to satisfy wi = w0 , and let xi = x > 0. In the
complementary case let x1 = · · · = xn = x > 0. In both cases from (2.2) we obtain that
f (w0 x) ≤ Cw0 xα must hold for every x > 0. Hence C ≥ w0α−1 sup x−α f (x).
The proofs of Remarks 4 and 5, being straightforward adaptations of what we have done in
the proofs of Remarks 1 and 2, resp., are left to the reader.
Proof of Theorem 3.1. Throughout we will suppose that x ≤ y.
Proof of (3.6). First we show that g is superadditive. It obviously holds if x + y ≤ 1 or x > 1.
If y ≤ 1 < x + y, then
g(x) + g(y) = p1 (xα1 + y α1 ) ≤ p1 (xα2 + y α2 ) ≤ p1 (x + y)α2 ≤ p2 (x + y)α2 = g(x + y).
Finally, if x ≤ 1 < y, then
g(x) + g(y) = p1 xα1 + p2 y α2 ≤ p2 (xα2 + y α2 ) ≤ p2 (x + y)α2 = g(x + y).
Let us turn to supermultiplicativity. It is valid if y ≤ 1 or x > 1. Let x ≤ 1 < y, then
g(x)g(y) = p1 xα1 p2 y α2 ≤ p1 (xy)α1 , because p2 y α2 ≤ y α1 . On the other hand, g(x)g(y) ≤
p2 (xy)α2 , because p1 xα1 ≤ xα2 . Thus g(x)g(y) ≤ g(xy).
Proof of (3.7). g 0 (x) = p1 α1 xα1 −1 if 0 < x < 1, and g 0 (x) = p1 α2 xα2 −1 if x > 1. Thus g 0 (x)
is decreasing, hence g is concave. The proof of supermultiplicativity is the same as in the proof
of (3.6).
Proof of (3.8). It can be done along the lines of the proof of (3.6), but with all inequality
signs reversed. Let us begin with the subadditivity. It is obvious, if x + y ≤ 1 or x > 1. If
y ≤ 1 < x + y, then
g(x) + g(y) = p1 (xα1 + y α1 ) ≥ p1 (xα2 + y α2 ) ≥ p1 (x + y)α2 ≥ p2 (x + y)α2 = g(x + y).
If x ≤ 1 < y, then
g(x) + g(y) = p1 xα1 + p2 y α2 ≥ p2 (xα2 + y α2 ) ≥ p2 (x + y)α2 = g(x + y).
Concerning submultiplicativity, it obviously holds when y ≤ 1 or x > 1. Let x ≤ 1 < y.
Then g(x)g(y) = p1 xα1 p2 y α2 does not exceed p1 (xy)α1 on the one hand, and p2 (xy)α2 on the
other hand. Hence g(x)g(y) ≥ g(xy).
Proof of (3.9). This time g 0 (x) is increasing, thus g is convex. The submultiplicativity of g has
already been proved above.
Proof of Theorem 3.2. We proceed similarly to the proofs of Theorems 1.1 and 2.1.
Let g ∈ F1 . Then
X
X
n
n
f (x)
(4.5)
f
wi xi ≥ inf
·g
wi xi
x>0 g(x)
i=1
i=1
n
f (x) X
≥ inf
g(wi xi )
x>0 g(x)
i=1
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 87, 7 pp.
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A N I NEQUALITY
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n
f (x) X
≥ inf
g(wi )g(xi )
x>0 g(x)
i=1
n
f (x)
g(wi ) X
min
wi g(xi ).
x>0 g(x) 1≤i≤n wi
i=1
≥ inf
For the second inequality we applied the superadditivity of g, and for the third one the supermultiplicativity.
Let g ∈ F2 . Using concavity at first, then supermultiplicativity, we obtain that
X
X
n
n
f (x)
(4.6)
·g
wi xi
f
wi xi ≥ inf
x>0 g(x)
i=1
i=1
X
n
f (x)
1
= inf
·g
wi w0 xi
x>0 g(x)
w0 i=1
n
f (x) 1 X
≥ inf
·
wi g(w0 xi )
x>0 g(x) w0
i=1
n
f (x) 1 X
·
wi g(w0 )g(xi ),
≥ inf
x>0 g(x) w0
i=1
as required.
The proof of (3.13) in the cases of g ∈ F3 and g ∈ F4 can be performed analogously to (4.5)
and (4.6), resp., with every inequality sign reversed, and wherever inf or min appears they have
to be changed to sup and max, resp.
Unfortunately, nothing can be said about the condition of equality in the sub/supermultiplicative steps. This is why inequalities (3.10) and (3.13) are not sharp in general.
R EFERENCES
[1] Y. MIAO, L.-M. LIU AND F. QI, Refinements of inequalities between the sum of squares and the
exponential of sum of a nonnegative sequence, J. Inequal. Pure and Appl. Math., 9(2) (2008), Art.
53 [ONLINE: http://jipam.vu.edu.au/article.php?sid=985]
[2] F. QI, Inequalities between the sum of squares and the exponential of sums of a nonnegative sequence, J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 78 [ONLINE: http://jipam.vu.
edu.au/article.php?sid=895]
[3] H.-N. SHI, Solution of an open problem proposed by Feng Qi, RGMIA Research Report Collection,
10(4), Article 4, 2007.
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 87, 7 pp.
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