Extension of the refined Gibbs inequality Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 6(24), No. 1, 2017, pp. 3-10

DOI: 10.15393/j3.art.2017.3750

3

UDC 517.52

VANDANJAV ADIYASUREN, TSERENDORJ BATBOLD

EXTENSION OF THE REFINED GIBBS' INEQUALITY

Abstract. In this note, we give an extension of the refined Gibbs' inequality containing arithmetic and geometric means. As an application, we obtain converse and refinement of the arithmetic-geometric mean inequality.

Key words: arithmetic-geometric mean inequality, Jensen's inequality, log-function, Gibbs' inequality

2010 Mathematical Subject Classification: 26D15, 94A15

1. Introduction. Let n — 2 and wi,...,wn be non-negative real numbers such that Y^n=1 Wj = 1. Let An and Gn denote the weighted arithmetic and geometric means of the positive real numbers xi,... ,xn, that is,

An = ^^ Wj Xj and Gn = JJ x

wj j

j=i j=i

The arithmetic-geometric mean inequality asserts that

A > G

■^n — ^ n*

For more details about the arithmetic-geometric mean inequality the reader is referred to [3]—[8].

It is interesting that some classical inequalities such as arithmetic-geometric mean inequality (see [9]), the Jensen inequality (see [10]), the Holder inequality (see [11]) play an important role in information sciences.

Let Pj ,Qj > 0 (j = 1..., n) and ¿n=i pj = j j. Then

n

0 < £ Pj log pj j=i Qj

(Petrozavodsk State University, 2017

with equality if and only if pj = qj (j = 1,...,n). This inequality is known in literature as the Gibbs' inequality (see [8, p. 382]). The Gibbs' inequality has many applications in information theory and also in mathematical statistics.

In 2004 Halliwell and Mercer [6] presented the following refinement of the Gibbs inequality:

Theorem 1. Let pj ,qj (j = 1,... ,n) be positive real numbers satisfying

En v yn m

j=i Pj = £ J=i qj ■ Then

y qj - pj)2 < ypj log Pj < yqj (? - Pj)2 (1) j=1 qj + Mj < j=Pj g qj < j= q? + m ()

where mj = min(pj, qj), Mj = max(p?, q?) (j = 1,..., n).

In 2014 H. Alzer [2] proved the following refinement of (1): Theorem 2. Let a, 3 G R. Then, inequalities

n

qj (qj - pj) „ ^ p log qj (qj- pj) (2) £ j+mj < h Pj log qj < h q?+m j> (2)

hold for positive real numbers pj, qj (j = 1,... ,n) withJ2 jj=]_ Pj = £ jj=i. qj if and only if a < 1/3 and 3 > 2/3.

In this note we give an extension of (2) containing arithmetic and geometric means. As an application, we obtain refinement of the left-hand inequality in (1) and we also give a converse of the arithmetic-geometric mean inequality.

2. Main results. In order to prove our main results, we need the following lemmas.

Lemma 1. (see [2]) (i) If 0 < x < 1, then

(x - 1)2 (x - 1)2

x - 1 ---^ < log x < x - 1 ---f- (3)

x + x1/3 " " x + 1 v 7

with equality if and only if x = 1. (ii) If x > 1, then

1 (x - 1)2 ! 1 (x - 1)2 m

x - 1 - ---- < log x < x - 1 - ---4rr. (4)

x + 1 x + xi/3

-r f / \ (^P aa) _ riii с

Lemma 2. Let fa(x) := -¡-^—+ logx, a > 0. Then fa

a(x2 + max{x2, а2 })

is a concave function on (0,

Proof. If x > а, then fa (x) = ^X-OX—+ log x, and consequently,

fa (x) = ^ < 0.

x(x — a)2

a(x2 + a2) yields

On the other hand, if 0 < x < а, then fa (x) = 9-+ log x, which

22

„,,, N a6 + 7a4x2 — 9a2x4 + x6 (a2 — x2)(a4 +8a2x2 — x4)

J (x) =--= — --—-- < 0

Ja (X) x2(x2 + a2)3 x2(x2 + a2)3 <

Therefore, the function fa (x) is concave for x > 0. □

Lemma 3. (see [1]) Let fa be as defined in Lemma 2, k E {2,..., n — 1}, and

k

k \ /J2 w,j x,j \ k

wj fA J j=1k- — ^ WW fAn (x,j )

'j=1 J v Ew,^ j=1

j=1

Sk := max

Then,

0 < S2 < S3 < ■ ■ ■ < Sn-1. First, we give an extension of (2) based on the corresponding result in

[2].

Theorem 3. Let a, ^ E R and mj = min(x2, A^, Mj = max(x2, An) (j = 1,..., n). Then inequalities

1 ^ wj xj (xj — An)2 ^ , , i n ^ 1 ^ wj xj (xj — An)2

An x2 + mj <log An — log Gn < An 2=i j^fMp"

(5)

hold if and only if a < 1/3 and ^ — 2/3. Proof. We follow the method of proof given in [2].

(Necessity) Since the sums on the left-hand side and on the right-hand side of (5) are increasing with respect to a and respectively, it suffices

to prove (5) for a = 1/3 and 3 = 2/3. Therefore, substituting xj/An instead of x in (3) and (4), then multiplying by Wj, and summing, we obtain

^L Y^ I Wjxj (xj - An)2

A I W" xj W"A" 2 . 2/3, ,1/3

1 v^ f zi Wj (xj - An)2

= A" ^ ^j xj - ^j--^ 1/3,2/3 1 <

" x3- <An \ xj + xj A" ,

t x, 1 / . Wj(x?- - A")2

< Z^ W" log 4" < T 2s W"xj - W"A"--jA-

A" A" xj + A"

1 V^ /„.. ™ ... 4 Wjxj (xj A")2

1 \ ^ I A Wj x, (x,

= ^ z^ I x" - a" "j j j

A" ^ \ " " " " x2 + 1/2 Mi/2

" xj < A„ \ xj + mj Mj

and

^L Y^ I A Wjxj (xj - A")2

4 Wj xj Wj A" 2 , i/2,,i/2 A"xj >A„ V x2 + mj Mj'

1 \—^ f a Wj (x, — An)2\ x—^ x,

= ( Wj xj - Wj A"-------- I < Wj log A- <

A" x. >A V x j +A" / x. >A A"

1 I A Wj (xj - A")2

< — I Wjxj - WjA" 1/^2/3

" x3- >An \ xj + xj A"

= ^L Y^ I - A Wjxj (xj - A")2

= A Wj xj Wj A" 2 , 1/3,,2/3

A"xj >A„ V xj + .

Further, utilizing inequalities m2/3MjL/3 < mi/2MjL/2 < mi/3Mj2/3, we get

1 \ ^ / w j x j (x j A") \ ^ x j A w"x" - w"a" - 2 i 2/^,1/J < Z^ w" log a" <

A"xj <aA x2 + Mj- / Xj < A A"

1 / _ A _ Wjxj (xj - A")2

< A Z^ Wj x" Wj A" 2 , 1/3,,2/3

A"xj <A„ V x2 + m/ M/

and

1 \ ^ ( A —jXj (xj An)2 \ ^ "v Xj

^ £ -X - - An - 2j 2/3 „1/3 < £ - lQg ~T <

1 I , wj xj (xj — An )2

<A- £ (- Xj - Wj An - -jj n)

An x>A \ j X2 + m1/3M2/3

x— >A„ \ Xj + mj Mj

Combining this together, we obtain

1 ^^ Wj Xj (Xj - An)2 ________1 ^^ Wj Xj (Xj — An)2

-2X+ j An;3 < log An - !og Gn < f£ 2 + 1/3 M2/3.

An j=1 X2 + m/ M/ An j=1 x2 + m/ Mj'

j j j=i j j

as desired.

(Sufficiency) Let s, t E R with 1 < t < s + 1. Set s +1 - t

xi =-, x2 = t, Xj = 1 (j = 3,..., n);

s1

Wi = ---7--7-, W2 = -----—, Wj =

s + 1 + (n - 2)t' s + 1 + (n - 2)t' j s + 1 + (n - 2)t

(j = 3,..., n). Now, the same computation as in the proof of Theorem 2 (see [2]), provides that a - 1/3 and p > 2/3. □

Remark. Putting Xj = —, Wj = pj /^ "=i Qj (j = 1,...,n), where

j

E"=i Pj = sn=i Qj; in (5), we get (2).

By using Theorem 3, we obtain the following consequence. Theorem 4. The inequality

An - Gn <V j (j (6)

X2 + mßj

holds for p > 2/3.

Proof. By the mean value theorem and Theorem 3, we have

An - Gn = exp(log An) - exp(log Gn) < An (log An - log Gn) <

< ^ Wj Xj (Xj - An)2

- j=i X2 + mf Mj^ ,

t

which completes the proof. □

Our next result refines sign of the first inequality in (5) for a = 0.

Theorem 5. If C = V Wj- A")2 , then

A" j= x2 + Mj ,

C < C + S2 < C + S3 <•••< C + S"_i < log(A") - log(G"). (7)

Equality occurs if and only if all x, are equal. Proof. By Lemma 3 we have

C < C + S2 < C + S3 <•••< C + S"_i.

Now, we have to prove the last inequality in (7). Let's choose an arbitrary xw G {ii,...,x„}, 1 < < < ••• < < n, with the corresponding weights Wj G {wl, ..., W"}, and let xin = {xL,..., x„} \ {x^,... }. Now, utilizing the first inequality in (5) with a = 0, we obtain

log(A") = log

' /y Wj) "j I >

ii_x ii_ I

jn in

E"_1

v - / j=1 Wlj

^ 1 Wjn xjn (xjn A") + A" xjn + max(xjn, A")

— 1 / — 1

EI— w,j I En=—i1 w,j A"

A" n—1 w x 2 n—1 w x 2

"jwrj + max I,A"

/ n — 1

f fr"-1 W x ^ j ^ -log xwr x|j

Ere_i

x . j = 1 Wlj

1 ^ Wj xj (xj - A" )2 1 "_1 Wjj x^j (x^j - A" )2 + A" j=1 x2 + max(x2 A") A" j=1 x|j + max(x|j , A")

2

n— 1

+ log(Gn) - £ log

XUj +

j=1

= log(Gn) + C +

n-1

£ wU; I \j=1

n-1

£ wUj \j=1

j=1 wUj XUj

wU; 1 fAn

En— 1 j = 1 xuj

En— 1

j = 1 wuj xuj

En— 1

j — 1 wUj n1

En — 1

.7 = 1 w

(xUj ) •

j = 1

Since , i = {1,...,k} are arbitrary, the last inequality in (7) holds. The theorem is proved. □

Putting xj = pj,wj = pj/En=1 qj (j = where ^n=1 p, =

n j=

= n=1 qj, in Theorem 5, we obtain the following refinement of the first inequality in (1).

Corollary 1. Let pj, qj (j = 1,..., n) be positive real numbers satisfying

En v yn m

,-=i Pj = E 7=i qj • Then

n f \2 n

£ qj (qj - Pj) = C < (7 + S2 < (7 + S3 <■■■< C + Sn—1 < V Pj log j=1 q2 + M j=1

Pj qj

(8)

where

Sk = max

1<Ui<U2<^"<Ufc <n

^ ^ ^E7 = 1 qUj

£pw I f1 1 k

_ \j = 1 / \^j = 1 qU

£ Pu, /1 (^ j VPuj

and Mj = max(p2, qj2) (j = 1,..., n).

j

Acknowledgment. The authors wish to express their thanks to Professor Mario KrniC for his valuable suggestions. This work was supported by the Asia Research Center at the National University of Mongolia and the Korea Foundation of Advanced Studies (Project No. 18, 2016-2017).

References

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Received October 16, 2016. In revised form, January 30, 2017. Accepted January 30, 2017. Published online April 5, 2017.

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