An extension of Stolarsky means

Abstract

In this article we give an extension of the well known Sto- larsky means to the multi-variable case in a simple and applicable way. Some basic inequalities concerning this matter are also established with applications in Analysis and Probability Theory.

Full text

Novi Sad J. Math.
Vol. 38, No. 3, 2008, 81-89
AN EXTENSION OF STOLARSKY MEANS
Slavko Simi´c
1
Abstract. In this article we give an extension of the well known Sto-
larsky means to the multi-variable case in a simple and applicable way.
Some basic inequalities concerning this matter are also established with
applications in Analysis and Probability Theory.
AMS Mathematics Subject Classification (2000): 26A51, 60E15
Key words and phrases: Logarithmic convexity, extended mean values,
generalized power means, shifted Stolarsky means
1. Introduction
1.1.
There is a huge number of papers (cf. [2], [3], [6], [7], [8]) investigating
properties of the so-called Stolarsky (or extended) two-parametric mean value,
defined for positive values of x, y by the following
E
r,s
(x, y) =



















³
r(x
s
−y
s
)
s(x
r
−y
r
)
´
1/(s−r)
, rs(r − s) 6= 0
exp
³
−
1
s
+
x
s
log x−y
s
log y
x
s
−y
s
´
, r = s 6= 0
³
x
s
−y
s
s(log x−log y)
´
1/s
, s 6= 0, r = 0
√
xy, r = s = 0,
x, x = y > 0.
In this form it is introduced by Keneth Stolarsky in [1].
Most of the classical two variable means are special cases of the class E.
For example, E
1,2
=
x+y
2
is the arithmetic mean, E
0,0
=
√
xy is the geometric
mean, E
0,1
=
x−y
log x−log y
is the logarithmic mean, E
1,1
= (x
x
/y
y
)
1
x−y
/e is the
identric mean, etc. More generally, the r-th power mean
³
x
r
+y
r
2
´
1/r
is equal to
E
r,2r
.
Recently, several papers have been produced trying to define an extension
of the class E to n, n > 2 variables. Unfortunately, this is done in a highly
artificial mode (cf. [4], [5], [9]), without a practical background. Here is an
1
Mathematical Institute SANU, Kneza Mihaila 36, 11000 Belgrade, Serbia, e-mail:
ssimic@turing.mi.sanu.ac.yu

82 S. Simi´c
illustration of this point; recently J. Merikowski ([9]) has proposed the following
generalization of the Stolarsky mean E
r,s
to several variables
E
r,s
(X) :=
h
L(X
s
)
L(X
r
)
i
1
s−r
, r 6= s,
where X = (x
1
, ··· , x
n
) is an n-tuple of positive numbers and
L(X
s
) := (n − 1)!
Z
E
n−1
n
Y
i=1
x
su
i
i
du
1
···du
n−1
.
The symbol E
n−1
stands for the Euclidean simplex which is defined by
E
n−1
:= {(u
1
, ··· , u
n−1
) : u
i
≥ 0, 1 ≤ i ≤ n − 1; u
1
+ ··· + u
n−1
≤ 1}.
In this paper we give another attempt to generalize Stolarsky means to the
multi-variable case in a simple and applicable way. The proposed task can be
accomplished by founding a ”weighted” variant of the class E, wherefrom the
mentioned generalization follows naturally.
In the sequel we shall need notions of the weighted geometric mean G =
G(p, q; x, y) and weighted r-th power mean S
r
= S
r
(p, q; x, y), defined by
G := x
p
y
q
; S
r
:= (px
r
+ qy
r
)
1/r
,
where
p, q, x, y ∈ R
+
; p + q = 1; r ∈ R/{0}.
Note that (S
r
)
r
> (G)
r
for x 6= y, r 6= 0 and lim
r→0
S
r
= G.
1.2.
We introduce here a class W of weighted two parameters means which
includes the Stolarsky class E as a particular case. Namely, for p, q, x, y ∈
R
+
, p + q = 1, rs(r − s)(x − y) 6= 0, we define
W = W
r,s
(p, q; x, y) :=
³
r
2
s
2
(S
s
)
s
− (G)
s
(S
r
)
r
− (G)
r
´
1
s−r
=
³
r
2
s
2
px
s
+ qy
s
− x
ps
y
q s
px
r
+ qy
r
− x
pr
y
qr
´
1
s−r
.
Various identities concerning the means W can be established; some of them
are the following
W
r,s
(p, q; x, y) = W
s,r
(p, q; x, y)
W
r,s
(p, q; x, y) = W
r,s
(q, p; y, x); W
r,s
(p, q; y, x) = xyW
r,s
(p, q; x
−1
, y
−1
);
W
ar,as
(p, q; x, y) = (W
r,s
(p, q; x
a
, y
a
))
1/a
, a 6= 0.
Note that
W
2r,2s
(1/2, 1/2; x, y) =
³
r
2
s
2
x
2s
+ y
2s
− 2(
√
xy)
2s
x
2r
+ y
2r
− 2(
√
xy)
2r
´
1/2(s−r)

An extension of Stolarsky means 83
=
³
r
2
s
2
(x
s
− y
s
)
2
(x
r
− y
r
)
2
´
1/2(s−r)
= E(r, s; x, y).
Hence E ⊂ W .
The weighted means from the class W can be extended continuously to the
domain
D = {(r, s, x, y)|r, s ∈ R, x, y ∈ R
+
}.
This extension is given by
W
r,s
(p, q; x, y) =

























³
r
2
s
2
px
s
+qy
s
−x
ps
y
qs
px
r
+qy
r
−x
pr
y
qr
´
1/(s−r)
, rs(r − s)(x −y) 6= 0
³
2
px
s
+q y
s
−x
ps
y
qs
pqs
2
log
2
(x/y )
´
1/s
, s(x − y) 6= 0, r = 0
exp
³
−2
s
+
px
s
log x+qy
s
log y
px
s
+qy
s
−x
ps
y
qs
−
(p log x+q log y)x
ps
y
qs
px
s
+qy
s
−x
ps
y
qs
´
, s(x − y) 6= 0, r = s
x
(p+1)/3
y
(q+1) /3
, x 6= y, r = s = 0
x, x = y.
1.3.
A natural generalization to the multi-variable case gives
W
r,s
(p; x) =

















³
r
2
(
P
p
i
x
s
i
−(
Q
x
p
i
i
)
s
)
s
2
(
P
p
i
x
r
i
−(
Q
x
p
i
i
)
r
)
´
1/(s−r)
, rs(s − r) 6= 0;
³
2
s
2
P
p
i
x
s
i
−(
Q
x
p
i
i
)
s
P
p
i
log
2
x
i
−(
P
p
i
log x
i
)
2
´
1/s
, r = 0, s 6= 0;
exp
³
−2
s
+
P
p
i
x
s
i
log x
i
−(
P
p
i
log x
i
)(
Q
x
p
i
i
)
s
P
p
i
x
s
i
−(
Q
x
p
i
i
)
s
´
, r = s 6= 0;
exp
³
P
p
i
log
3
x
i
−(
P
p
i
log x
i
)
3
3(
P
p
i
log
2
x
i
−(
P
p
i
log x
i
)
2
)
´
, r = s = 0.
where x = (x
1
, x
2
, ··· , x
n
) ∈ R
n
+
, n ≥ 2, p is an arbitrary positive weight
sequence associated with x and W
r,s
(p; x
0
) = a for x
0
= (a, a, ··· , a).
We also write
P
(·),
Q
(·) instead of
P
n
1
(·),
Q
n
1
(·).
The above formulae are obtained by an appropriate limit process, imply-
ing continuity. For example W
s,s
(p, x) = lim
r→s
W
r,s
(p, x) and W
0,0
(p, x) =
lim
s→0
W
0,s
(p, x).
2. Results and applications
Our main result is contained in the following
Proposition 1. The means W
r,s
(p, x) are monotone increasing in both vari-
ables r and s.
Passing to the continuous variable case, we get the following definition of
the class
¯
W
r,s
(p, x).

84 S. Simi´c
Assuming that all integrals exist
¯
W
r,s
(p, x) =























³
r
2
(
R
p(t)x
s
(t)dt−exp(s
R
p(t) log x(t)dt))
s
2
(
R
p(t)x
r
(t)dt−exp(r
R
p(t) log x(t)dt)
´
1/(s−r)
, rs(s − r) 6= 0;
³
2
s
2
R
p(t)x
s
(t)dt−exp(s
R
p(t) log x(t)dt)
R
p(t) log
2
x(t)dt−(
R
p(t) log x(t)dt)
2
´
1/s
, r = 0, s 6= 0;
exp
³
−2
s
+
R
p(t)x
s
(t) log x(t)dt
R
p(t)x
s
(t)dt−exp(s
R
p(t) log x(t)dt)
−
(
R
p(t) log x(t)dt) exp(s
R
p(t) log x(t)dt)
R
p(t)x
s
(t)dt−exp(s
R
p(t) log x(t)dt)
´
, r = s 6= 0;
exp
³
R
p(t) log
3
x(t)dt−(
R
p(t) log x(t)dt)
3
3(
R
p(t) log
2
x(t)dt−(
R
p(t) log x(t)dt)
2
)
´
, r = s = 0
where x(t) is a positive integrable function and p(t) is a non-negative function
with
R
p(t)dt = 1.
¿From our former considerations a very applicable assertion follows
Proposition 2.
¯
W
r,s
(p, x) is monotone increasing in either r or s.
As an illustration we give the following
Proposition 3. The function w(s) defined by
w ( s) :=



³
12
(πs )
2
(Γ(1 + s) − e
−γs
)
´
1/s
, s 6= 0;
exp(−γ −
4ξ(3)
π
2
), s = 0,
is monotone increasing for s ∈ (−1, ∞).
In particular, for s ∈ (−1, 1) we have
Γ(1 − s)e
−γs
+ Γ(1 + s)e
γs
−
πs
sin(πs)
≤ 1 −
(πs)
4
144
,
where Γ(·), ξ(·), γ stands for the Gamma function, Zeta function and the Euler’s
constant, respectively.
Applications in Probability Theory
For a random variable X and an arbitrary distribution with support on
(−∞, +∞), it is well known that
Ee
X
≥ e
EX
.
Denoting the central moment of order k by µ
k
= µ
k
(X) := E(X − EX)
k
,
we improve the above inequality to the following
Proposition 4. For an arbitrary probability law with support on R, we have
Ee
X
≥ (1 + (µ
2
/2) exp (µ
3
/3µ
2
))e
EX
.

An extension of Stolarsky means 85
Proposition 5. We also have that
³
Ee
sX
− e
sEX
s
2
σ
2
X
/2
´
1/s
is monotone increasing in s.
Especially interesting is studying of the shifted Stolarsky means E
∗
, defined
by
E
∗
r,s
(x, y) := lim
p→0
+
W
r,s
(p, q; x, y).
Their analytic continuation to the whole (r, s) plane is given by
E
∗
r,s
(x, y) =



















³
r
2
(x
s
−y
s
(1+s log(x/y)))
s
2
(x
r
−y
r
(1+r log(x/y)))
´
1/(s−r)
, rs(r − s)(x − y) 6= 0;
³
2
s
2
x
s
−y
s
(1+s log(x/y))
log
2
(x/y )
´
1/s
, s(x − y) 6= 0, r = 0;
exp
³
−2
s
+
(x
s
−y
s
) log x−sy
s
log y log(x/y)
x
s
−y
s
(1+s log(x/y ))
´
, s(x − y) 6= 0, r = s;
x
1/3
y
2/3
, r = s = 0;
x, x = y.
Main results concerning the means E
∗
are the following
Proposition 6. Means E
∗
r,s
(x, y) are monotone increasing in either r or s for
each fixed x, y ∈ R
+
.
Proposition 7. Means E
∗
r,s
(x, y) are monotone increasing in either x or y for
each r, s ∈ R.
The well known result of Feng Qi ([11]) states that the means E
r,s
(x, y) are
logarithmically concave for each fixed x, y > 0 and r, s ∈ [0, +∞); also, they are
logarithmically convex for r, s ∈ (−∞, 0].
According to this, we propose the following
3. Open question
Is there any compact interval I, I ⊂ R such that the means E
∗
r,s
(x, y) are
logarithmically convex (concave) for r, s ∈ I and each x, y ∈ R
+
?
A partial answer to this problem is given in the next
Proposition 8. On any interval I which includes zero and r, s ∈ I,
(i) E
∗
r,s
(x, y) are not logarithmically convex (concave);
(ii) W
r,s
(p, q; x, y) are logarithmically convex (concave) if and only if p =
q = 1/2.

86 S. Simi´c
4. Proofs
We prove first a global theorem concerning log-convexity of the Jensen’s
functional with a parameter, which can be very usable (cf [10]).
Theorem 1. Let f
s
(x) be a twice continuously differentiable function in x with
a parameter s. If f
00
s
(x) is log-convex in s for s ∈ I := (a, b); x ∈ J := (c, d),
then the form
Φ
f
(w, x; s) = Φ(s) :=
X
w
i
f
s
(x
i
) − f
s
(
X
w
i
x
i
),
is log-convex in s for s ∈ I, x
i
∈ J, i = 1, 2, ···, where w = {w
i
} is any positive
weight sequence.
At the beginning we need some preliminary lemmas.
Lemma 1. A positive function f is log-convex on I if and only if the relation
f(s)u
2
+ 2f (
s + t
2
)uw + f(t)w
2
≥ 0,
holds for each real u, w and s, t ∈ I.
This assertion is nothing more than the discriminant test for the nonnega-
tivity of second-order polynomials.
Another well known assertions are the following (cf [12], p. 74, 97-98),
Lemma 2 (Jensen’s inequality). If g(x) is twice continuously differentiable and
g
00
(x) ≥ 0 on J, then g(x) is convex on J and the inequality
X
w
i
g(x
i
) − g(
X
w
i
x
i
) ≥ 0
holds for each x
i
∈ J, i = 1, 2, ··· and any positive weight sequence {w
i
},
P
w
i
= 1.
Lemma 3. For a convex f , the expression
f(s) − f(r)
s − r
is increasing in both variables.
Proof of Theorem 1.
Consider the function F (x) defined as
F (x) = F (u, v, s, t; x) := u
2
f
s
(x) + 2uvf
s+t
2
(x) + v
2
f
t
(x),
where u, v ∈ R; s, t ∈ I are real parameters independent of the variable x ∈ J.
Since
F
00
(x) = u
2
f
00
s
(x) + 2uvf
00
s+t
2
(x) + v
2
f
00
t
(x),

An extension of Stolarsky means 87
and by the assumption f
00
s
(x) is log-convex in s, it follows from Lemma 1 that
F
00
(x) ≥ 0, x ∈ J.
Therefore, by Lemma 2 we get
X
w
i
F (x
i
) − F (
X
w
i
x
i
) ≥ 0, x
i
∈ J,
which is equivalent to
u
2
Φ(s) + 2uvΦ(
s + t
2
) + v
2
Φ(t) ≥ 0.
According to Lemma 1 again, this is possible only if Φ(s) is log-convex and
proof is done. 2
Proof of Proposition 1.
Define the auxiliary function g
s
(x) by
g
s
(x) :=
(
(e
sx
− sx − 1)/s
2
, s 6= 0;
x
2
/2, s = 0.
Since
g
0
s
(x) =
(
(e
sx
− 1)/s, s 6= 0;
x, s = 0,
and
g
00
s
(x) = e
sx
, s ∈ R,
we see that g
s
(x) is twice continuously differentiable and that g
00
s
(x) is a log-
convex function for each real s, x.
Applying Theorem 1, we conclude that the form
Φ
g
(w , x; s) = Φ(s) :=
(
(
P
w
i
e
sx
i
− e
s
P
w
i
x
i
)/s
2
, s 6= 0;
(
P
w
i
x
2
i
− (
P
w
i
x
i
)
2
)/2, s = 0,
is log-convex in s.
By Lemma 3, with f(s) = log Φ(s), we find out that
log Φ(s) − log Φ(r)
s − r
= log
³
Φ(s)
Φ(r)
´
1
s−r
,
is monotone increasing either in s or r. Therefore, by changing variable x
i
→
log x
i
, we finally obtain the proof of Proposition 1. 2
Proof of Proposition 2. The assertion of Proposition 2 follows from Propo-
sition 1 by the standard argument (cf [12], pp. 131-134). Details are left to the
reader. 2
Proof of Proposition 3. The proof follows putting f ( t) = t, p(t) = e
−t
, t ∈
(0, +∞) and applying Proposition 2. 2
Proof of Proposition 4. By Proposition 2, we get
W
0,1
(p, e
x
) ≥ W
0,0
(p, e
x
),

88 S. Simi´c
i. e.,
Ee
X
− e
EX
µ
2
/2
≥ exp(
EX
3
− (EX)
3
3µ
2
).
Using the identity EX
3
−(EX)
3
= µ
3
+ 3µ
2
EX, we obtain the proof of Propo-
sition 4. 2
Proof of Proposition 5. This assertion is a straightforward consequence of
the fact that W
0,s
(p, e
x
) is monotone increasing in s. 2
Proof of Proposition 6 Direct consequence of Proposition 1. 2
Proof of Proposition 7 This is left as an easy exercise to the readers. 2
Proof of Proposition 8 We prove only the part (ii). The proof of (i) goes
along the same lines.
Suppose that 0 ∈ (a, b ) := I and that E
r,s
(p, q; x, y) are log-convex (concave)
for r, s ∈ I and any fixed x, y ∈ R
+
. Then there should be an s, s > 0 such that
F
s
(p, q; x, y) := W
0,s
(p, q; x, y)W
0,−s
(p, q; x, y) − (W
0,0
(p, q; x, y))
2
is of constant sign for each x, y > 0.
Substituting (x/y)
s
:= e
w
, w ∈ R, after some calculations we get that the
above is equivalent to the assertion that F (p, q; w) is of constant sign, where
F (p, q; w) := pe
w
+ q − e
pw
− e
2
3
(1+p)w
(pe
−w
+ q − e
−pw
).
Developing in power series in w, we get
F (p, q; w) =
1
1620
pq(1 + p)(2 − p)(1 − 2p)w
5
+ O(w
6
).
Therefore, F (p, q; w) can be of constant sign for each w ∈ R only if p =
1/2(= q).
Suppose now that I is of the form I := [0, a) or I := (−a, 0]. Then there
should be an s, s 6= 0, s ∈ I such that
W
0,0
(p, q; x, y)W
0,2s
(p, q; x, y) − (W
0,s
(p, q; x, y))
2
is of constant sign for each x, y ∈ R
+
.
Proceeding as above, this is equivalent to the assertion that G(p, q; w) is of
constant sign with
G(p, q; w ) := p
3
q
3
w
6
e
2
3
(p+1)w
(pe
2w
+ q − e
2pw
) − (pe
w
+ q − e
pw
)
4
.
But,
G(p, q; w ) =
2
405
p
4
q
4
(1 + p)(1 + q)(q − p)w
11
+ O(w
12
).
Hence we conclude that G(p, q; w) can be of constant sign for a sufficiently
small w, w ∈ R only if p = q = 1/2. Combining this with the Feng Qi theorem,
the assertion from Proposition 8 follows. 2

An extension of Stolarsky means 89
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Received by the editors September 16, 2008