A FEW NEW INEQUALITIES FOR THE MEAN

Zhao Changjian (Department of Mathematics, Shanghai University, Shanghai 200436,P.R. China;

Department of Mathematics, Binzhou Teachers College, Shandong 256604, P.R.China)

Mihaly Bencze (Str. Harmanului 6,RO-2212 Sacele, Jud. Brasov Romania) Zdravko F Starc (Zarka Zrenjanina 93, 26300 Vrsac, Yugoslavia)

1. INTRODUCTION AND LEMMAS

let a, b be positive real numbers.Let us denote the harmonic mean of a and b by H = _{\frac{1}{\frac{1/a+1/b}{2}}} = \frac{2ab}{a+b}, thegeometric mean by

G = \sqrt{ab}, the arithmetic mean by A = \frac{a+b}{2} and the quadratic mean by K = \sqrt{\frac{a^{2}+b^{2}}{2}}.

The following inequalities

\frac{2ab}{a + b} ≤ \sqrt{ab} ≤ \frac{a + b}{2} ≤ \sqrt{\frac{a^{2} + b^{2}}{2}}

that is

H ≤ G ≤ A ≤ K

are valid,and equalities occur for a = b.

Two new inequalities for the mean was given by Z.F.Starc in [1],these two new inequalities can be stated as follows

AK + AG ≥ 2G^{2}  (1)

and

AG + HK ≥ 2A^{2}  (2)

In the inqualities (1) and (2),equality occurs if and only if a = b.

In this paper we will give also a few new inequalities similar to (1) and (2)

We shall need the following well known results LEMMA^{[2]} 1 If x > 0, y > 0 and \frac{1}{p} + \frac{1}{q}, p > 1 then

x^{\frac{1}{p}}y^{\frac{1}{q}} ≤ \frac{x}{p} + \frac{y}{q}  (3)

equality holds if and only if x = y

LEMMA^{[3]} 2 If a_{1} ≤ ⋯ ≤ a_{n} and b_{1} ≤ ⋯ ≤ b_{n} or a_{1} ≥ ⋯ ≥ a_{n} and b_{1} ≥ ⋯ ≥ b_{n} then

(\frac{1}{n} n ; ∑ ; i=1 a_{i})(\frac{1}{n} n ; ∑ ; i=1 b_{i}) ≤ \frac{1}{n} n ; ∑ ; i=1 a_{i}b_{i}

equality holds if and only if a_{1} = a_{2} = ⋯ = a_{n} or b_{1} = b_{2} = · = b_{n}. 2 MAIN RESULTS Our main results are given in following RESULT 1

G^{2} + A^{2} − AG ≤ \frac{1}{2}(AK + HK)  (4)

equality holds if and only if a = b.

From (1) and (2), we can get inequality (4). RESULT 2

\frac{(1 + G)(1 + A)}{(1 + H)(1 + K)} ≥ (\frac{1 + \sqrt{GA}}{1 + \sqrt{HA}})^{2}  (5)

equality holds it and only if a = b.

Proof Because x + y ≥ 2x^{\frac{1}{2}}y^{\frac{1}{2}}, where x ≥ 0, y ≥ 0,equality holds and only if x = y.

We have 1 + x + y + xy ≥ 1 + 2x^{\frac{1}{2}}y^{\frac{1}{2}} + xy. Thus (1 + x)(1 + y) ≥ (1 + x^{\frac{1}{2}}y^{\frac{1}{2}})^{2}. Then from

(1 + G)(1 + A) ≥ (1 + G^{\frac{1}{2}}A^{\frac{1}{2}})^{2},

and

(1 + H)(1 + K) ≥ (1 + H^{\frac{1}{2}}K^{\frac{1}{2}})^{2},

we can get inequality (5).

RESULT 3

GAKH ≤ (\frac{G^{p}}{p} + \frac{A^{q}}{q})(\frac{K^{p}}{p} + \frac{H^{Q}}{q})  (6)

where \frac{1}{p} + \frac{1}{q} = 1 and p > 1, and equality holds if and only if a^{p} = b^{q}.

In Lemma 1,let x^{p} take the pace of x , y^{q} take the pace of y, we have

xy ≤ \frac{x^{p}}{p} + \frac{y^{q}}{q} ,

holds if and only if x^{p} = y^{q}.

Thus

GA ≤ \frac{G^{p}}{p} + \frac{A^{q}}{q} ,  (7)
KH ≤ \frac{K^{p}}{p} + \frac{H^{q}}{q} .  (8)

From (7) and (8), we easy get (6).

RESULT 4

H + G + A + K ≤ 2\sqrt{H^{2} + G^{2} + A^{2} + K^{2}}  (9)

equality holds if and only if a = b.

In Lemma 2 ,let a_{i} = b_{i} and n = 4,then

\frac{1}{4}( 4 ; ∑ ; i=1 a_{i})^{2} ≤ 4 ; ∑ ; i=1 a_{i}^{2}  (10)

where a_{1} ≤ a_{2} ≤ a_{3} ≤ a_{4} or a_{1} ≥ a_{2} ≥ a_{3} ≥ a_{4}.

On the orther hand

H ≤ G ≤ A ≤ K

Hence (9) is valid.

References

[1] Z.F.Starc. Two inequalities for the mean . Function 23(1999):153-154.

[2] Chang-jian Zhao and Mihaly Bencze, About Hilbert’s Integral Inequalities, Octogon Mathematical Magazine, 10(2002):9-12.

[3] D.S.Mitrinovic. Analytic inequalities,Springer-Verlag,1970.