MATEMATIQKI VESNIK
UDK 517.28
originalniresearch
nauqni
rad
paper
57 (2005), 61–63
ASYMPTOTIC PLANARITY OF DRESHER MEAN VALUES
Momčilo Bjelica
Abstract. A family of Dresher mean values is asymptotically planar with respect to its two
parameters. An asymptotic formula presenting this property holds if: (a) all variables converge
to the same value; and, equivalently, because of means homogeneity, (b) for variables with same
additive increment converging to infinity.
Suppose x = (x1 , x2 , . . . , xn ), a = (a, a, . . . , a), and q = (q1 , q2 , . . . , qn ) are
sequences of nonnegative reals and a > 0. Without loss of generality let the weights
qi be normalized by q1 + q2 + · · · + qn = 1. The geometric, the harmonic, and the
quadratic mean values respectively are
v
u n
n
n
Y
X
uX
qi x2i .
Gq (x) =
xqi i , Aq (x) =
qi xi , Qq (x) = t
i=1
i=1
i=1
Note that σq2 (x) = Q2q (x) − A2q (x) is a weighted variance of x, which satisfies
σq2 (x + a) = σq2 (x). Dresher mean values [2] are a two-parameter family of means
that increase with each parameter
¶1/(s−t)
Á
µ
Pn
Pn

t
s

,
if s 6= t

j=1 qj xj
i=1 qi xi
Ds,t (x) =
¶
Á
µ

Pn

t
 exp Pn qi xt log xi
i
j=1 qj xj , if s = t.
i=1
Theorem. Dresher mean values for both cases s 6= t and s = t have the unique
asymptotic formulas
s+t−1 2
(Qq (x) − A2q (x)) + o(Q2q (x − a))
2a
¡
¢
= Aq (x) + (s + t − 1)(Qq (x) − Aq (x)) + o Q2q (x − a)
¡
¢
= Gq (x) + (s + t)(Aq (x) − Gq (x)) + o Q2q (x − a) , x → a,
Ds,t (x) = Aq (x) +
AMS Subject Classification: 26E60.
Keywords and phrases: Dresher mean values, asymptotic behavior.
62
M. Bjelica
and if a → ∞, then
s+t−1 2
(Qq (x) − A2q (x)) + o(1/a)
2a
= Aq (x + a) + (s + t − 1)(Qq (x + a) − Aq (x + a)) + o(1/a)
Ds,t (x + a) = a + Aq (x) +
= Gq (x + a) + (s + t)(Aq (x + a) − Gq (x + a)) + o(1/a).
Asymptotic planarity implies Hoehn and Niven property for Dresher mean values
Ds,t (x + a) − a → Aq (x),
a → ∞.
Proof. Suppose s 6= t and h = x − a. Then
µ
¶s
µ
¶
hi
s
s(s − 1) 2
xsi = as 1 +
= a s 1 + hi +
+
o
, hi → 0,
h
i
a
a
2a2
µ
¶
n
X
s
s(s − 1) 2
s
s
qi xi = a 1 + Aq (h) +
Qq (h) + o ,
a
2a2
i=1
¡
¢
¡ ¢
where o = o h2i and o = o Q2q (h) , respectively. Therefore
· X
¸
n
n
X
1
log Ds,t (x) =
log
qi xsi − log
qj xtj
s−t
i=1
j=1
·
µ
¶
1
s
s(s − 1) 2
s
=
log a + log 1 + Aq (h) +
Qq (h) + o
s−t
a
2a2
µ
¶¸
t
t(t − 1) 2
− log at − log 1 + Aq (h) +
(h)
+
o
Q
q
a
2a2
·
1
s
s(s − 1) 2
s2 2
= log a +
Aq (h) +
Q
(h)
−
A (h)
q
s−t a
2a2
2a2 q
¸
t
t(t − 1) 2
t2 2
− Aq (h) −
Q
(h)
+
A (h) + o
q
a
2a2
2a2 q
¢
1
1
s+t−1¡ 2
= log a + Aq (h) − 2 A2q (h) +
Qq (h) − A2q (h) + o
2
2a¶ µ
2a
¶¸
· µa
¢
1
s+t−1¡ 2
2
+ o.
= log a 1 + Aq (h)
1+
Qq (h) − Aq (h)
a
2a2
Since the obtained expression is well defined and continuous at s = t, for both cases
s 6= t and s = t we have
µ
¶
¢
1
s+t−1¡ 2
1
2
Ds,t (x) = a exp
(h)
−
A
(h)
+
o
Aq (h) − 2 A2q (h) +
Q
q
q
a
2a
2a2
µ
¢
1
1
s+t−1¡ 2
= a 1 + Aq (h) − 2 A2q (h) +
Qq (h) − A2q (h)
2
a
2a
2a
¶
1 2
+ 2 Aq (h) + o
2a
¢
¡
¢
s+t−1¡ 2
= a + Aq (h) +
Qq (h) − A2q (h) + o Q2q (h) − A2q (h) .
2a
Asymptotic planarity of Dresher mean values
63
This gives the first line of the first formula. The third line follows from asymptotic
linearity of power mean values [1], particularly
(Qq (x) − Aq (x))/(Aq (x) − Gq (x)) → 1,
x → a.
(1)
In the second formula the first line follows from the above proof with a → ∞,
o = o(1/a2 ), and o = o(1/a) in the last unspecified appearance of o. Hoehn and
Niven property [2], which is a consequence of asymptotic linearity property [1],
states
Mq (x + a) − Aq (x + a) → 0, a → ∞,
where M is any power mean value. Therefore
Qq (x + a) + Aq (x + a)/2a → 1,
a → ∞,
what implies the second line. The third line follows from the asymptotic linearity
formula at infinity, i.e. (1) for the argument x+a and a → ∞. (The second formula
also follows from the first one and from homogeneity of involved mean values.)
Conjecture. Let x be a sequence of reals and a > 0. The unified asymptotic
formula for Dresher mean values holds for convergent variables, as well as for an
additive infinitely increasing parameter
Ds,t (a + x) = a + Aq (x) +
σq2 (x)
s+t−1 2
σq (x) + o(
)
2a
2a
σq2 (x)
)
2a
2
σq (x)
= Gq (a + x) + (s + t)(Aq (a + x) − Gq (a + x)) + o(
),
2a
= Aq (a + x) + (s + t − 1)(Qq (a + x) − Aq (a + x)) + o(
where either x → 0 or a → ∞. Infinitesimals σq2 (x)/2a, Qq (a + x) − Aq (a + x),
and Aq (a + x) − Gq (a + x) are equivalent.
Acknowledgement. The author thanks to the referee for suggestion to
consider log Ds,t in both cases, with aim to simplify algebra and obtain a unified
development.
REFERENCES
[1] M. Bjelica, Asymptotic linearity of mean values, Mat. Vesnik 51, 1–2 (1999), 15–19.
[2] J. L. Brenner, B. C. Carlson, Homogeneous mean values: weights and asymptotics, J. Math.
Anal. Appl. 123 (1987), 265–280.
(received 31.12.2002, in revised form 26.04.2005)
University of Novi Sad, “Mihajlo Pupin”, Zrenjanin 23000, Serbia & Montenegro
E-mail: bjelica@tf.zr.ac.yu