SANDOR 4160Forteni Nr."79 R-Jud

Internat. J. Math. & Math. Sci.

VOL. 18 NO. 3 (1995) 621-623

621

RESEARCH NOTES

TWOINEQUALITIESFOR

MEANS

J. SANDOR 4160Forteni Nr."79

R-Jud. Harghita Romania

(Received September22, 1993 and in revisedform February22,

1994)

ABSTRACT. We provetwonewinequalitiesfor theidentric meanandameanrelatedtothe arithmetic andgeometricmeanoftwonumbers

KEY WORDS AND PHRASES. Identric mean,logarithmic mean,quadrature formulas.

1991AMSSUBJECT CLASSIFICATION CODES. 26D99,65D32.

1. INTRODUCTION.

Thelogarithmic andidentric means of twopositive numbers^aand b are definedby

’-’

fora

#

^b;

L(a, a)

^a

L L(a, b):

_logb-loga

and

I

I(a, b):

^!

(bb/a,)i/(b-’)

^fora

#

b;

I(o a)

a, respectively.

Let

m A(a, b): ,,t,

^and

^G G(a, b)"

denote the arithmetic andgeometricmeansofa and b, respectively. Manyinteresting results have beenproved for thesemeans, seee.g.

([1] [3], [5]

[10]). Letusintroduce themeanUdefinedby

The aimofthisnoteistoprove the following:

THEOREM. Fora

:/:

^{b onehas}

(U3G)

^/

< I <

U

(1.1)

2. PROOFOF

THE

THEOREM.

Forthefirstinequalityweapply the Newton quadratureformula

(see [4])

fa ⁺ -5--) + f(b)] ^(b-a)

₆₄₈

f(4)() (2.1)

where

E(a,b)

and

f:[a,b]--oR

^{has a continuous 4-th derivative on}

(a,b).

Let

f(x)

logx(x

> 0)

ⁱⁿ(2.1). Then

frO(x) >

0, andaftercertain transformationswe gettheleft side ofl.1.

622 ,J. SA,XIt4

Inorder toprovethe second inequalityof(1 !)dwide alltermsby ^a< ^{b and denotex.} > 1.

Then theinequalitytobeproved becomes

(4x’-’ +

10x

+ ^4)/(x + 1)9(x) > 9/e

⁽²²⁾

where g(x) x^x/I:--1 x> ^1.

Introduce the function

f. [1,

^{oe) ]definedby}

f(x)- (4x 2+lOx+4)/(x+l)g(x),

x> 1"

f(1)=lim f(x) =9/e

x--l Weshallprovethat

f

is strictlyincreasing,and thisproves(2 2) We have

ix-l)

and, aftersomeelementarycomputations,we candeduce

(x 1)g(x)f’(x) (4x +

^10x

+ ^4)(x +

^{1)logx-10x3 6x}

+

^6x

+

^{10 (2.3)}

Wenowshowthatthe rightsideof(2 3)isstrictly positive,orequivalently

L

< (8A + GO’)A/’(IOA +

^G

2) (2.4)

where

L L(x,

1) etc Since it is known that

L <

(2G+

A)/3

(See

[3])

we try to prove that

(2G + A)/3 < (SA + G2)A/(IOA G2).

Thisholdstrueiff14x³

20x2y +

^4xy +2y³

>

^{0, with}

x=A,y=G, ie,

(x-

y)(Tx 3xy-

y2) >

0 (2.5)

Wehave

7x 3xy

yZ [x ⁺

^y

(v/-11]-3)] [x

^y

(’/]-11]+3)]

^{>0 by 14}

^>

⁰

and 0

<

₁₄

<

^1. Thus

(2 5)

^isproved,concluding theproofof

(2.2)

^andof the theorem.

3o REMARKS.

(1)

Clearly, G

< U < A

(fora

- ^b).

^{Relation(1.1)}offers the improvement

G(.

(U3G)

^{1/4<: I}

<

U_X

<

U

< A (2.6) (2)

Itiswell-known that (seeeg.

[7]) A > I,

sofrom the fightinequalityin

(1.1)

wehave

912 <

^8A

+

^G

^(2.7)

Onthe otherhand, it isknown that[8] I

> (2A + G)/3,

whichaccordingto

A >

Gand(27)yields the following double-inequality:

4A

+

^{5G <:}

^{912 <}

^8A

+

^G

(2.8)

(3)

^Thetwosidesof(1

1)

imply

U

> A’G (2.9)

ACKNOWLEDGEMENTS. The authorwishes to thankthe referee for several useful remarks

two tNEQUAI.ITII’;S I:()1{ NEANS 623

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Rep.

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Math..⁴⁰(1990),26i-270.

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