INVARIANCE IN THE CLASS OF WEIGHTED
LEHMER MEANS
IULIA COSTIN
GHEORGHE TOADER
Department of Computer Science
Technical University of Cluj,
Romania
Department of Mathematics
Technical University of Cluj,
Romania
EMail: iulia.costin@cs.utcluj.ro
EMail: gheorghe.toader@math.utcluj.ro
Weighted Lehmer Means
Iulia Costin and
Gheorghe Toader
vol. 9, iss. 2, art. 54, 2008
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Received:
15 July, 2007
Accepted:
26 March, 2008
Communicated by:
L. Losonczi
2000 AMS Sub. Class.:
26E60.
Key words:
Lehmer means, Generalized means, Invariant means, Complementary means,
Double sequences.
Abstract:
We study invariance in the class of weighted Lehmer means. Thus we look at
triples of weighted Lehmer means with the property that one is invariant with
respect to the other two.
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Contents
1
Means
3
2
Invariant Means
5
3
Series Expansion of Means
7
4
Cp,λ −Complementary of Means
8
Weighted Lehmer Means
Iulia Costin and
Gheorghe Toader
vol. 9, iss. 2, art. 54, 2008
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1.
Means
The abstract definitions of means are usually given as:
Definition 1.1. A mean is a function M : R2+ → R+ , with the property
min(a, b) ≤ M (a, b) ≤ max(a, b),
∀a, b > 0.
A mean M is called symmetric if
Weighted Lehmer Means
Iulia Costin and
∀a, b > 0.
M (a, b) = M (b, a),
Gheorghe Toader
vol. 9, iss. 2, art. 54, 2008
In [12] the following definition was given:
Definition 1.2. The function M is called a generalized mean if it has the property
M (a, a) = a,
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∀a > 0.
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A generalized mean is called in [10] a pre-mean, which seems more adequate.
Of course, each mean is reflexive, thus it is a generalized mean.
In what follows, we use the weighted Lehmer means Cp;λ defined by
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λ · ap + (1 − λ) · bp
Cp;λ (a, b) =
,
λ · ap−1 + (1 − λ) · bp−1
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with λ ∈ [0, 1] fixed. Important special cases are the weighted arithmetic mean and
the weighted harmonic mean, given respectively by
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Aλ = C1;λ
and
Hλ = C0;λ .
For λ = 1/2 we get the symmetric means denoted by Cp , A and H. Note that the
geometric mean can also be obtained, but the weighted geometric mean cannot:
C1/2 = G
but
C1/2;λ 6= Gλ
for
λ 6= 1/2.
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For λ = 0 and λ = 1 we have
Cp;0 = Π2
respectively Cp;1 = Π1 ,
∀p ∈ R,
where Π1 and Π2 are the first and the second projections, defined respectively by
Π1 (a, b) = a,
Π2 (a, b) = b,
∀a, b ≥ 0.
If λ ∈
/ [0, 1] the functions Cp;λ are generalized means only.
Weighted Lehmer Means
Iulia Costin and
Gheorghe Toader
vol. 9, iss. 2, art. 54, 2008
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2.
Invariant Means
Given three means P, Q and R , their compound
P (Q, R)(a, b) = P (Q(a, b), R(a, b)),
∀a, b > 0,
defines also a mean P (Q, R).
Definition 2.1. A mean P is called (Q, R)−invariant if it verifies
Weighted Lehmer Means
Iulia Costin and
Gheorghe Toader
P (Q, R) = P.
vol. 9, iss. 2, art. 54, 2008
Remark 1. Using the property of (A, G) −invariance of the mean
"Z
#−1
π/2
dθ
π
√
M (a, b) = ·
,
2
a2 cos2 θ + b2 sin2 θ
0
Gauss showed that this mean gives the limit of the arithmetic-geometric double sequence. As was proved in [1], this property is generally valid: the mean P which is
(Q, R)−invariant gives the limit of the double sequence of Gauss type defined with
the means Q and R :
an+1 = Q(an , bn ),
bn+1 = R(an , bn ),
n ≥ 0.
Moreover, the validity of this property for generalized means is proved in [14] (if the
limit L exists and P (L, L) is defined).
Remark 2. In this paper, we are interested in the problem of invariance in a family M of means. It consists of determining all the triples of means (P, Q, R) from
M such that P is (Q, R)−invariant. This problem was considered for the first time
for the class of quasi-arithmetic means by Sutô in [11] and many years later by J.
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Matkowski in [8]. It was called the problem of Matkowski-Sutô and was completely
solved in [4]. The invariance problem was also solved for the class of weighted
quasi-arithmetic means in [6], for the class of Greek means in [13] and for the class
of Gini-Beckenbach means in [9]. In this paper we are interested in the problem
of invariance in the class of weighted Lehmer means. We use the method of series expansion of means, as in [13]. The other papers mentioned before have used
functional equations methods.
Weighted Lehmer Means
Iulia Costin and
Gheorghe Toader
vol. 9, iss. 2, art. 54, 2008
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3.
Series Expansion of Means
For the study of some problems related to a mean M, in [7] the power series expansions of the normalized function M (1, 1 − x) is used. For some means it is very
difficult, or even impossible to determine all the coefficients. In these cases, a recurrence relation for the coefficients is very useful. Such a formula is presented in [5]
as Euler’s formula.
Theorem 3.1. If the function f has the Taylor series
f (x) =
∞
X
Weighted Lehmer Means
Iulia Costin and
Gheorghe Toader
vol. 9, iss. 2, art. 54, 2008
an · x n ,
n−0
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p is a real number and
p
[f (x)] =
∞
X
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bn · x n ,
n−0
then we have the recurrence relation
n
X
[k(p + 1) − n] · ak · bn−k = 0,
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n ≥ 0.
k=0
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Using it in [3], the series expansion of the weighted Lehmer mean is given by:
Cp;λ (1, 1 − x)
= 1 − (1 − λ) x + λ (1 − λ) (p − 1) x2 − λ (1 − λ) (p − 1) [2λ (p − 1) − p]
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x3
2
x4
+ λ (1 − λ) (p − 1) 6λ2 (p − 1)2 − 6λp (p − 1) + p (p + 1) ·
+ ··· .
6
4.
Cp,λ −Complementary of Means
If the mean P is (Q, R)−invariant, the mean R is called complementary to Q
with respect to P (or P −complementary to Q). If a given mean Q has a unique
P −complementary mean R, we denote it by R = QP .
Some obvious general examples are given in the following
Proposition 4.1. For every mean M we have
M M = M,
Weighted Lehmer Means
Iulia Costin and
ΠM
1 = Π2 ,
M Π2 = Π2 .
Gheorghe Toader
vol. 9, iss. 2, art. 54, 2008
If M is a symmetric mean we have also
ΠM
2 = Π1 .
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We shall call these results trivial cases of complementariness.
Denote the Cp;λ −complementary of the mean M by M C(p;λ) , or by M C(p) if λ =
1/2. Using Euler’s formula, we can establish the following.
Theorem 4.2. If the mean M has the series expansion
M (1, 1 − x) = 1 +
∞
X
an x n ,
n=0
then the first terms of the series expansion of M C(p;λ) , for λ 6= 0, 1, are
M C(p;λ) (1, 1 − x)
λ
1 − λ + λa1
x−
=1−
[(p − 1) a1 (a1 + 2 (1 − λ)) + a2 (1 − λ)] · x2
1−λ
(1 − λ)2
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λ
3
2
3 a1 (p − 1) 2λ p − λ (p + 2) − 4λ (p − 1) + 3p − 2
2 (1 − λ)
+ a21 (p − 1) 2λ2 (1 − 3p) + λ (3p + 2) + 3p − 4 + a31 (p − 1) (2λp + p − 2)
+ 4a2 (p − 1) (1 − λ)2 + 4a1 a2 (p − 1) (1 − λ) +2a3 (1 − λ)2 · x3 + · · · .
−
C(p;λ)
Corollary 4.3. The first terms of the series expansion of Cr;µ
are
C(p;λ)
Cr;µ
(1, 1
− x)
1 − 2λ + λµ
=1−
x
1−µ
2
λ (1 − µ) +
p
(1
−
2λ
+
µ)
+
µr
(λ
−
1)
−
1
+
2λ
−
λµ
x
(1 − λ)2
λ (1 − µ) 2
+
p 2λ3 + 2λµ2 − 6λ2 µ − λµ + 5λ2
(1 − λ)3
+ µ2 + µ − 5λ + 1 + 4pr λµ2 + λµ − λ2 µ −µ2
+ r2 2λµ − 4λµ2 − λ2 µ − µ + 2µ2
2 2
2
2
3
2
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vol. 9, iss. 2, art. 54, 2008
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2
+ p 2λ µ + 12λ µ − 6λµ − 2λ − 9λ + µ − λµ + 7λ − µ − 1
+ r 5λ2 µ − 4λ2 µ2 + 4λµ2 − 6λµ + µ
+ 2λ2 µ2 + 4λ2 − 6λ2 µ +2λµ − 2λ] x3 + · · · .
Weighted Lehmer Means
Iulia Costin and
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Using them we can prove the following main result.
Corollary 4.4. We have
Cp;λ (Cr;µ , Cu;ν ) = Cp;λ
if we are in one of the following non-trivial cases:
i)
C1;λ (C1;(2λ−1)/λ , Cu;1 ) = C1;λ ;
ii)
C0;λ (C0;(2λ−1)/λ , Cu;1 ) = C0;λ ;
iii)
C0 (Cr;µ , C−r;1−µ ) = C0 ;
iv)
C1/2 (Cr;µ , C1−r;1−µ ) = C1/2 ;
v)
C1 (Cr;µ , C2−r;1−µ ) = C1 ;
vi)
C0;λ (C0;(3λ−1)/2λ , C0;1/2 ) = C0;λ ;
vii)
C1;λ (C1;(3λ−1)/2λ , C1 ) = C1;λ ;
viii)
C0,1/3 (Cr;0 , C0 ) = C0;1/3 ;
ix)
C1,1/3 (Cr;0 , C1 ) = C1;1/3 ;
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x)
C2,1/4 (C1;−1/2 , C1 ) = C2,1/4 ;
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xi)
C−1,1/4 (C0;−1/2 , C0 ) = C−1,1/4 ;
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xii)
C0;λ (C0 , C0;λ/(2−2λ) ) = C0;λ ;
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xiii)
C1;λ (C1 , C1;λ/(2−2λ) ) = C1;λ ;
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xiv)
C−1;3/4 (C0 , C0;3/2 ) = C−1;3/4 ;
xv)
C2;3/4 (C1 , C1;3/2 ) = C2;3/4 .
Proof. We consider the equivalent condition
C(p;λ)
Cr;µ
= Cu;ν which gives
C(p;λ)
Cr;µ
(1, 1 − x) = Cu;ν (1, 1 − x).
Equating the coefficients of xk , k = 1, 2, . . . , 5, we get the following table of solutions with corresponding conclusions:
Weighted Lehmer Means
Iulia Costin and
Gheorghe Toader
vol. 9, iss. 2, art. 54, 2008
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Case
λ
µ
ν
p
r
u
C(p;λ)
= Cu;ν
Case
Π(2)
Cr;µ
= Π2
Trivial
Cr;µ
1
0
µ
0
p
r
u
2
λ
1
0
p
r
u
3
1
2
0
1
p
r
u
Π2
4
λ
2λ−1
λ
1
1
1
u
A 2λ−1 = Π1
5
λ
9
1
2
1
2
1
2
1
2
10
λ
6
7
8
11
12
13
14
λ
1
3
1
3
1
4
1
4
15
16
λ
17
λ
18
19
3
4
3
4
2λ−1
λ
C(p;λ)
Π1
C(p)
1
0
A(λ)
0
0
u
H(λ)
H 2λ−1 = Π1
1
2
3λ−1
2λ
H
Cr;µ
= C−r;1−µ
G
Cr;µ
= C1−r;1−µ
A
Cr;µ = C2−r;1−µ
C(p)
Cp = Cp
H(λ)
1
r
r
r
−r
1−r
2−r
1
2
1
2
p
p
p
0
0
0
3λ−1
2λ
1
2
1
0
1
2
1
2
1
2
1
2
λ
2(1−λ)
λ
2(1−λ)
3
2
3
2
0
r
0
Π2
1
r
1
Π2
2
1
1
A−1/2
−1
0
0
0
1
−1
2
- 12
1
2
1
2
1
2
1
2
Trivial
i)
ii)
Weighted Lehmer Means
Iulia Costin and
Gheorghe Toader
λ
1−µ
1−µ
1−µ
− 12
= Π1
Trivial
λ
µ
µ
µ
0
= Π2
1
2
H 3λ−1 = H
iii)
iv)
v)
vol. 9, iss. 2, art. 54, 2008
Trivial
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vi)
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vii)
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2λ
1
1
A(λ)
A 3λ−1 = A
2λ
H(1/3)
=H
viii)
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A(1/3)
=A
ix)
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=A
x)
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0
0
H−1/2 = H
HH(λ) = H λ
xi)
xii)
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1
1
AA(λ) = A
xiii)
0
1
0
1
C(2;1/4)
C(2;1/4)
2(1−λ)
λ
2(1−λ)
HC(−1;3/4) = H3/2
AC(2;3/4) = A3/2
xiv)
xv)
Remark 3. Equating the coefficients of x1 , x2 , ..., xn , we have a system of n equations with six unknowns (the parameters of the means). For n = 2, 3, 4, solving the
system, we get relations among the parameters such as:
ν=
λ (1 − µ)
,
1−λ
u=
λµr − µr + pµ − 2λp + p
,
1 − 2λ + λµ
r=
Z
,
λ−1
where
Z 2 µ (µ − 1) + 2pµZ (λ − λµ + µ − 1) + λ2 p − 2λ2 µ2 p − λ2 p2 + 2λ3 p2 − 2λ3 p
+ 3λ2 µ2 p2 − λµ3 p2 − λ3 µp2 + λ3 µp + λµ3 p + 4λ2 µp + 4λµp2
Weighted Lehmer Means
Iulia Costin and
Gheorghe Toader
vol. 9, iss. 2, art. 54, 2008
− 5λ2 µp2 − 2λµp − 2λµ2 p + µ2 p − µp2 = 0.
For n = 5 we obtained the table of solutions given in the previous corollary. For
n = 6, however, the system could not even be solved using Maple. As a result, we
are not certain that we have obtained all the solutions for the problem of invariance.
Remark 4. The cases i)-ii), vi)-vii), xii)-xiii) and xiv)-xv), involve C1;λ = Aλ and
C0;λ = Hλ . There are, however, no similar cases for C1/2;λ . Instead we have the
following results for Gλ :
G(λ)
G 2λ−1 = Π1 ,
λ
G(1/3)
Π2
= G,
G(λ)
G 3λ−1 = G,
G
G(λ)
2λ
=G
λ
2(1−λ)
,
but these are not Lehmer means.
Remark 5. It is easy to see that not all of the generalized means that appear in the
above results are means. In such a case, the result given in Remark 1 can be negative.
For example, in the case xv), if we consider
an+1 = C1 (an , bn ),
bn+1 = C1;3/2 (an , bn ),
n ≥ 0,
for a0 = 10 and b0 = 1, we get a2 = a0 and b2 = b0 , thus the sequences are
divergent. Also, in the case xii), if we take λ = 4/5, the double sequence
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an+1 = C0 (an , bn ),
bn+1 = C0;2 (an , bn ),
n ≥ 0,
has the limit zero for a0 = 10 and b0 = 1, which is different from C0;4/5 (10, 1). This
is because C0;4/5 is not defined in (0, 0) , thus the proof of the Invariance Principle in
[14] does not work.
Corollary 4.5. For means we have
Cp;λ (Cr;µ , Cu;ν ) = Cp;λ
Weighted Lehmer Means
Iulia Costin and
if we are in one of the following non-trivial cases:
Gheorghe Toader
vol. 9, iss. 2, art. 54, 2008
i)
C1;λ (C1;(2λ−1)/λ , Cu;1 ) = C1;λ ,
λ ∈ [1/2, 1];
ii)
C0;λ (C0;(2λ−1)/λ , Cu;1 ) = C0;λ ,
λ ∈ [1/2, 1];
iii)
C0 (Cr;µ , C−r;1−µ ) = C0 ;
iv)
C1/2 (Cr;µ , C1−r;1−µ ) = C1/2 ;
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v)
C1 (Cr;µ , C2−r;1−µ ) = C1 ;
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vi)
C0;λ (C0;(3λ−1)/2λ , C0;1/2 ) = C0;λ , λ ∈ [1/3, 1];
vii)
C1;λ (C1;(3λ−1)/2λ , C1 ) = C1;λ ,
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λ ∈ [1/3, 1];
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viii)
C0,1/3 (Cr;0 , C0 ) = C0;1/3 ;
ix)
C1,1/3 (Cr;0 , C1 ) = C1;1/3 ;
x)
C0;λ (C0 , C0;λ/(2−2λ) ) = C0;λ ,
λ ∈ [0, 2/3];
xi)
C1;λ (C1 , C1;λ/(2−2λ) ) = C1;λ ,
λ ∈ [0, 2/3].
Close
Remark 6. Each of the above results allows us to define a double sequence of Gauss
type with known limit.
Corollary 4.6. For symmetric means, we have
Cp (Cr , Cu ) = Cp
if and only if we are in the following non-trivial cases:
i)
C0 (Cr , C−r ) = C0 ;
ii)
C1/2 (Cr , C1−r ) = C1/2 ;
iii)
C1 (Cr , C2−r ) = C1 .
Weighted Lehmer Means
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Gheorghe Toader
vol. 9, iss. 2, art. 54, 2008
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References
[1] J.M. BORWEIN AND P.B. BORWEIN, Pi and the AGM - a Study in Analytic Number Theory and Computational Complexity, John Wiley & Sons, New
York, 1986.
[2] P.S. BULLEN, Handbook of Means and their Inequalities, Kluwer Academic
Publishers, Dordrecht/ Boston/ London, 2003.
[3] I. COSTIN AND G. TOADER, A weighted Gini mean, Proceedings of the
International Symposium Specialization, Integration and Development, Section Quantitative Economics, Babeş-Bolyai University Cluj-Napoca, Romania,
137–142, 2003.
[4] Z. DARÓCZY AND Zs. PÁLES, Gauss-composition of means and the solution
of the solution of the Matkowski-Sutô problem, Publ. Math. Debrecen, 61(1-2)
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[5] H.W. GOULD, Coefficient identities for powers of Taylor and Dirichlet series,
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[6] J. JARCZYK AND J. MATKOWSKI, Invariance in the class of weighted quasiarithmetic means, Ann. Polon. Math., 88(1) (2006), 39–51.
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Gheorghe Toader
vol. 9, iss. 2, art. 54, 2008
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[7] D.H. LEHMER, On the compounding of certain means, J. Math. Anal. Appl.,
36 (1971), 183–200.
[8] J. MATKOWSKI, Invariant and complementary quasi-arithmetic means, Aequationes Math., 57 (1999), 87–107.
Close
[9] J. MATKOWSKI, On invariant generalized Beckenbach-Gini means, Functional Equations - Results and Advances (Z. Daróczy and Zs. Páles, eds.), Advances in Mathematics, Vol. 3, Kluwer Acad. Publ., Dordrecht, 2002, 219-230.
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[11] O. SUTÔ, Studies on some functional equations I, Tôhoku Math. J., 6 (1914),
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vol. 9, iss. 2, art. 54, 2008
[12] G. TOADER, Integral generalized means, Math. Inequal. Appl., 5(3) (2002),
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[13] G. TOADER AND S. TOADER, Greek Means and the Arithmetic-Geometric
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