The geometric, the harmonic, and the quadratic mean values respectively are Gq(x

MATEMATIQKI VESNIK

57 (2005), 61–63

UDK 517.28 originalni nauqni rad research paper

ASYMPTOTIC PLANARITY OF DRESHER MEAN VALUES Momˇcilo 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 anda >0. Without loss of generality let the weights qi be normalized byq1+q2+· · ·+qn= 1. The geometric, the harmonic, and the quadratic mean values respectively are

Gq(x) = Yn

i=1

x^q_iⁱ, Aq(x) = Xn

i=1

qixi, Qq(x) = vu utXⁿ

i=1

qix²_i.

Note that σ²_q(x) = Q²_q(x)−A²_q(x) is a weighted variance of x, which satisfies σ²_q(x+a) =σ_q²(x). Dresher mean values [2] are a two-parameter family of means that increase with each parameter

Ds,t(x) =







 µP_n

i=1qix^s_i ÁP_n

j=1qjx^t_j

¶_1/(s−t)

, ifs6=t exp

µP_n

i=1qix^t_ilogxi

ÁP_n

j=1qjx^t_j

¶

, ifs=t.

Theorem. Dresher mean values for both casess6=tands=thave the unique asymptotic formulas

Ds,t(x) =Aq(x) +s+t−1

2a (Q²_q(x)−A²_q(x)) +o(Q²_q(x−a))

=Aq(x) + (s+t−1)(Qq(x)−Aq(x)) +o¡

Q²_q(x−a)¢

=Gq(x) + (s+t)(Aq(x)−Gq(x)) +o¡

Q²_q(x−a)¢

, x→a,

AMS Subject Classification: 26E60.

Keywords and phrases: Dresher mean values, asymptotic behavior.

62 M. Bjelica

and if a→ ∞, then

Ds,t(x+a) =a+Aq(x) +s+t−1

2a (Q²_q(x)−A²_q(x)) +o(1/a)

=Aq(x+a) + (s+t−1)(Qq(x+a)−Aq(x+a)) +o(1/a)

=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. Supposes6=t andh=x−a. Then x^s_i =a^s

µ 1 +hi

a

¶_s

=a^s µ

1 + s

ahi+s(s−1) 2a² h²_i +o

¶

, hi→0, Xn

i=1

qix^s_i =a^s µ

1 + s

aAq(h) +s(s−1)

2a² Q²_q(h) +o

¶ , whereo=o¡

h²_i¢

ando=o¡ Q²_q(h)¢

, respectively. Therefore logDs,t(x) = 1

s−t

· log

Xn

i=1

qix^s_i −log Xn

j=1

qjx^t_j

¸

= 1

s−t

·

loga^s+ log µ

1 + s

aAq(h) +s(s−1)

2a² Q²_q(h) +o

¶

−loga^t−log µ

1 + t

aAq(h) +t(t−1)

2a² Q²_q(h) +o

¶¸

= loga+ 1 s−t

·s

aAq(h) +s(s−1)

2a² Q²_q(h)− s² 2a²A²_q(h)

−t

aAq(h)−t(t−1)

2a² Q²_q(h) + t²

2a²A²_q(h) +o

¸

= loga+1

aAq(h)− 1

2a²A²_q(h) +s+t−1 2a²

¡Q²_q(h)−A²_q(h)¢ +o

= log

· a

µ 1 +1

aAq(h)

¶ µ

1 + s+t−1 2a²

¡Q²_q(h)−A²_q(h)¢¶¸

+o.

Since the obtained expression is well defined and continuous ats=t, for both cases s6=tands=t we have

Ds,t(x) =aexp µ1

aAq(h)− 1

2a²A²_q(h) +s+t−1 2a²

¡Q²_q(h)−A²_q(h)¢ +o

¶

=a µ

1 +1

aAq(h)− 1

2a²A²_q(h) +s+t−1 2a²

¡Q²_q(h)−A²_q(h)¢

+ 1

2a²A²_q(h) +o

¶

=a+Aq(h) +s+t−1 2a

¡Q²_q(h)−A²_q(h)¢ +o¡

Q²_q(h)−A²_q(h)¢ .

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 witha→ ∞, o =o(1/a²), ando =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→ ∞, whereM is any power mean value. Therefore

Q_q(x+a) +A_q(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 argumentx+aanda→ ∞. (The second formula also follows from the first one and from homogeneity of involved mean values.)

Conjecture. Let xbe a sequence of reals anda >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) +s+t−1

2a σ_q²(x) +o(σ_q²(x) 2a )

=Aq(a+x) + (s+t−1)(Qq(a+x)−Aq(a+x)) +o(σ²_q(x) 2a )

=Gq(a+x) + (s+t)(Aq(a+x)−Gq(a+x)) +o(σ²_q(x) 2a ), where either x→ 0or a→ ∞. Infinitesimals σ_q²(x)/2a, Qq(a+x)−Aq(a+x), andAq(a+x)−Gq(a+x)are equivalent.

Acknowledgement. The author thanks to the referee for suggestion to consider logDs,t in both cases, with aim to simplify algebra and obtain a unified development.

REFERENCES

[1] M. Bjelica,Asymptotic linearity of mean values, Mat. Vesnik51, 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:[email protected]