ON AN INEQUALITY OF FENG QI
TAMÁS F. MÓRI
Department of Probability Theory and Statistics
Loránd Eötvös University
Pázmány P. s. 1/C, H-1117 Budapest, Hungary
On an Inequality of Feng Qi
Tamás F. Móri
vol. 9, iss. 3, art. 87, 2008
EMail: moritamas@ludens.elte.hu
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Received:
17 January, 2008
Accepted:
04 August, 2008
Communicated by:
F. Qi
2000 AMS Sub. Class.:
26D15.
Key words:
Optimal inequality, Power sum, Subadditive, Superadditive, Power mean inequality.
Abstract:
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.
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Acknowledgement:
This research was supported by the Hungarian Scientific Research Fund (OTKA)
Grant K67961.
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Contents
1
Introduction
3
2
Converse Inequalities
6
3
Further Generalizations
8
On an Inequality of Feng Qi
4
Proofs
10
Tamás F. Móri
vol. 9, iss. 3, art. 87, 2008
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1.
Introduction
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 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.
On an Inequality of Feng Qi
Tamás F. Móri
vol. 9, iss. 3, art. 87, 2008
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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
wi xαi ≤ f
wi xi
(1.3)
C
i=1
i=1
On an Inequality of Feng Qi
is valid with
Tamás F. Móri
C=
(1.4)
w0α−1
inf x
−α
x>0
f (x),
where
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(
(1.5)
vol. 9, iss. 3, art. 87, 2008
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
−α
w0 xi minimizes x
Pn f (x) in (0, ∞).
−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 }∞
of positive weights such that
i=1 be an infinite sequenceP
∞
w0 := inf 1≤i<∞ wi > 0 when α ≥ 1, and w0 :=
i=1 wi < ∞ when α < 1.
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Then for an arbitrary nonnegative sequence {xi }∞
i=1 such that
following inequality holds.
!
∞
∞
X
X
C
wi xαi ≤ f
wi xi ,
i=1
P∞
i=1
wi xi < ∞ the
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].
On an Inequality of Feng Qi
Tamás F. Móri
vol. 9, iss. 3, art. 87, 2008
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2.
Converse Inequalities
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.
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
Tamás F. Móri
vol. 9, iss. 3, art. 87, 2008
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i=1
is valid with
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C = w0α−1 sup x−α f (x),
(2.3)
x>0
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where
(2.4)
On an Inequality of Feng Qi
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 P
x−α f (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 }∞
i=1 be an infinite sequence
P∞of positive weights such
that w0 := inf 1≤i<∞ wi > 0 when α > 1, and w0 := i=1 wiP
< ∞ when α < 1.
∞
Then for an arbitrary nonnegative sequence {xi }∞
such
that
i=1
i=1 wi xi < ∞ the
following inequality holds.
!
∞
∞
X
X
f
wi xi ≤ C
wi xαi ,
i=1
where C is defined in (2.3).
i=1
On an Inequality of Feng Qi
Tamás F. Móri
vol. 9, iss. 3, art. 87, 2008
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3.
Further Generalizations
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.
On an Inequality of Feng Qi
(3.1)
(3.2)
(3.3)
(3.4)
F1
F2
F3
F4
= {g
= {g
= {g
= {g
: g(x) + g(y) ≤ g(x + y), g(x)g(y) ≤ g(xy) for x, y ≥ 0} ,
: g is concave, g(x)g(y) ≤ g(xy) for x, y ≥ 0} ,
: g(x) + g(y) ≥ g(x + y), g(x)g(y) ≥ g(xy) for x, y ≥ 0} ,
: 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.
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Then
(3.6)
(3.7)
(3.8)
(3.9)
Tamás F. Móri
vol. 9, iss. 3, art. 87, 2008
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.
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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
On an Inequality of Feng Qi
g(wi )
f (x)
(3.11)
C = min
· inf
.
x>0 g(x)
1≤i≤n wi
Suppose g ∈ F2 . Then (3.10) holds with
g(w0 )
f (x)
(3.12)
C=
· inf
,
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 )
i=1
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i=1
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is valid with
(3.14)
Tamás F. Móri
vol. 9, iss. 3, art. 87, 2008
f (x)
g(wi )
· sup
.
1≤i≤n wi
x>0 g(x)
C = max
Suppose g ∈ F4 , and supx>0
(3.15)
where w0 = w1 + · · · + wn .
f (x)
g(x)
C=
< ∞. Then (3.13) holds with
g(w0 )
f (x)
· sup
,
w0
x>0 g(x)
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4.
Proofs
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
i=1
x>0
≥ inf x−α f (x)
x>0
i=1
n
X
On an Inequality of Feng Qi
Tamás F. Móri
(wi xi )α
vol. 9, iss. 3, art. 87, 2008
i=1
≥ w0α−1 inf x−α f (x) ·
n
X
x>0
wi xαi ,
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
≥ inf x
x>0
f (x) w0α
w0−1
n
X
α
wi xi
i=1
−α
f (x)
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i=1
w0α−1
n
X
i=1
wi xαi ,
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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 caseP
of α < 1 we need x1 = · · · = xn for equality in the α-power
mean inequality. Then ni=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.
On an Inequality of Feng Qi
Tamás F. Móri
vol. 9, iss. 3, art. 87, 2008
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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.
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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
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x>0
i=1
i=1
≤ sup x−α f (x)
n
X
x>0
≤
w0α−1
sup x
(wi xi )α
x>0
f (x) ·
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i=1
−α
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n
X
i=1
wi xαi .
If α > 1, we have to apply the α-power mean inequality again.
α
X
X
n
n
−α
(4.4)
f
wi xi ≤ sup x f (x)
wi xi
i=1
x>0
= sup x
x>0
i=1
−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).
On an Inequality of Feng Qi
Tamás F. Móri
vol. 9, iss. 3, art. 87, 2008
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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
On an Inequality of Feng Qi
Tamás F. Móri
vol. 9, iss. 3, art. 87, 2008
g(x)+g(y) = p1 (xα1 +y α1 ) ≥ p1 (xα2 +y α2 ) ≥ p1 (x+y)α2 ≥ p2 (x+y)α2 = g(x+y).
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If x ≤ 1 < y, then
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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)
·g
wi xi
f
wi xi ≥ inf
x>0 g(x)
i=1
i=1
n
f (x) X
≥ inf
g(wi xi )
x>0 g(x)
i=1
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n
f (x) X
≥ inf
g(wi )g(xi )
x>0 g(x)
i=1
n
g(wi ) X
f (x)
≥ inf
min
wi g(xi ).
x>0 g(x) 1≤i≤n wi
i=1
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 ),
x>0 g(x) w0
i=1
≥ inf
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.
On an Inequality of Feng Qi
Tamás F. Móri
vol. 9, iss. 3, art. 87, 2008
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References
[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.
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Tamás F. Móri
vol. 9, iss. 3, art. 87, 2008
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