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volume 7, issue 4, article 130, 2006.

Received 25 November, 2005;

accepted 06 October, 2006.

Communicated by:D. ¸Stef ˇanescu

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Journal of Inequalities in Pure and Applied Mathematics

GENERALIZATIONS OF THE KY FAN INEQUALITY

AI-JUN LI, XUE-MIN WANG AND CHAO-PING CHEN

Jiaozuo University

Jiaozuo City, Henan Province 454000, China

EMail:liaijun72@163.com EMail:wangxm881@163.com

College of Mathematics and Informatics Research Institute of Applied Mathematics Henan Polytechnic University

Jiaozuo City, Henan 454010, China EMail:chenchaoping@hpu.edu.cn

c

2000Victoria University ISSN (electronic): 1443-5756 346-05

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Generalizations of the Ky Fan Inequality

Ai-Jun Li, Xue-Min Wang and Chao-Ping Chen

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J. Ineq. Pure and Appl. Math. 7(4) Art. 130, 2006

http://jipam.vu.edu.au

Abstract

In this paper, we extend the Ky Fan inequality to several general integral forms, and obtain the monotonic properties of the function _L ^L^s^(a,b)

s(α−a,α−b) with α, a, b∈ (0,+∞)ands∈R.

2000 Mathematics Subject Classification:26A48, 26D20.

Key words: Generalized logarithmic mean, Monotonicity, Ky Fan inequality.

The authors were supported in part by the Science Foundation of the Project for Fostering Innovation Talents at Universities of Henan Province, China

1. Introduction

The following inequality proposed by Ky Fan was recorded in [1, p. 5] : If 0< x_i ≤ ¹₂ fori= 1,2, . . . , n, then

(1.1)

Qn i=1x_i Qn

i=1(1−x_i) _n¹

≤

Pn i=1x_i Pn

i=1(1−x_i), unlessx1 =x2 =· · ·=xn.

With the notation

(1.2) M_r(x) =







1 n

Pn i=1x^r_i¹_r

, r6= 0;

(Qn

i=1x_i)ⁿ¹ , r= 0,

whereM_r(x)denotes ther-order power mean ofx_i >0fori= 1,2, . . . , n, the inequality (1.1) can be written as

(1.3) M₀(x)

M₀(1−x) ≤ M₁(x) M₁(1−x).

In 1996, Zh. Wang, J. Chen and X. Li [12] found the necessary and sufficient condition for

(1.4) M_r(x)

Mr(1−x) ≤ M_s(x) Ms(1−x)

when r < s. Recently, Ch.-P. Chen proved that the function _L ^L^r^(a,b)

r(1−a,1−b) is strictly increasing for0< a < b≤ ¹₂ and strictly decreasing for ¹₂ ≤a < b <1,

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where r ∈ (−∞,∞) andL_r(a, b) is the generalized logarithmic mean of two positive numbersa, b, which is a special case of the extended meansE(r, s;x, y) defined by Stolarsky [10] in 1975. For more information about the extended means please refer to [4,6,8,11] and references therein.

Moreover, we have,

L_r(a, b) =a, a =b;

L_r(a, b) =

b^r+1−a^r+1 (r+ 1)(b−a)

¹_r

, a6=b, r6=−1,0;

L−1(a, b) = b−a

lnb−lna =L(a, b);

L₀(a, b) = 1 e

b^b a^a

_b−a¹

=I(a, b),

where L(a, b)and I(a, b) are respectively the logarithmic mean and the expo- nential mean of two positive numbersaandb. Whena6=b,L_r(a, b)is a strictly increasing function ofr. In particular,

r→−∞lim Lr(a, b) = min{a, b}, lim

r→+∞Lr(a, b) = max{a, b}, L₁(a, b) = A(a, b), L₋₂(a, b) = G(a, b),

whereA(a, b)andG(a, b)are the arithmetic and the geometric means, respectively. Fora6=b, the following well known inequality holds:

(1.5) G(a, b)< L(a, b)< I(a, b)< A(a, b).

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In this paper, motivated by inequality (1.4), we will extend the inequality (1.4) to general integral forms. Some monotonic properties of several related functions will be obtained.

Theorem 1.1. Let

f_α(s) =

Rb a x^sdx Rb

a(α−x)^sdx

!¹_s

= L_s(a, b) L_s(α−a, α−b),

s ∈(−∞,+∞)andαbe a positive number. Thenf_α(s)is a strictly increasing function for [a, b] ⊆ (0,^α₂], and is a strictly decreasing function for [a, b] ⊆ [^α₂, α).

Corollary 1.2. If[a, b]⊆(0,^α₂]andαis a positive number, then a

α−b < G(a, b)

G(α−a, α−b) < L(a, b) L(α−a, α−b)

< I(a, b)

I(α−a, α−b) < A(a, b)

A(α−a, α−b) < b α−a. (1.6)

If[a, b]⊆[^α₂, α), the inequalities (1.6) is reversed.

Corollary 1.3. Leth_α(s) = _R_b

ax^sdx Rα−a

α−b x^sdx

¹_s

,s∈(−∞,+∞)andαbe a positive number. Then h_α(s) is a strictly increasing function for[a, b] ⊆ (0,^α₂], or a strictly decreasing function for[a, b]⊆[^α₂, α).

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In [13], Feng Qi has proved that the function

r 7→

1 b−a

Rb a x^rdx

1 b+δ−a

Rb+δ a x^rdx

!¹_r

= L_r(a, b) L_r(a, b+δ)

is strictly decreasing withr ∈(−∞,+∞). Now, we will extend the conclusion in the following theorem.

Theorem 1.4. Let

f(s) =

1 b−a

Rb a x^sdx

1 d−c

Rd c x^sdx

!¹_s

= L_s(a, b) Ls(c, d),

s ∈ (−∞,+∞) anda, b, c, d be positive numbers. Then f(s) is a strictly in- creasing function forad < bc, or a strictly decreasing function forad > bc.

Corollary 1.5. Let

h(s) =

1 b−a

Rb a x^sdx

1 d−a

Rd a x^sdx

!¹_s

= L_s(a, b) Ls(a, d),

s ∈ (−∞,+∞) and a, b, d are positive numbers. Then h(s) is a strictly in- creasing function ford < b, or a strictly decreasing function ford > b.

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2. Proofs of Theorems

In order to prove Theorem1.1, we make use of the following elementary lemma which can be found in [3, p. 395].

Lemma 2.1 ([3, p. 395]). Let the second derivative ofφ(x)be continuous with x∈(−∞,∞)andφ(0) = 0. Define

(2.1) g(x) =





 φ(x)

x , x6= 0;

φ⁰(0), x= 0.

Then φ(x)is strictly convex (concave) if and only ifg(x)is strictly increasing (decreasing) withx∈(−∞,∞).

Remark 1. A general conclusion was given in [7, p. 18]: A functionφis convex on[a, b]if and only if ^φ(x)−φ(x_x−x ⁰⁾

0 is nondecreasing on[a, b]for every pointx₀ ∈ [a, b].

Proof of Theorem1.1. It is obvious that

f_α(s) =

Rb a x^sdx Rb

a(α−x)^sdx

!¹_s

=

b^s+1−a^s+1 (α−a)^s+1−(α−b)^s+1

¹_s

= L_s(a, b) Ls(α−a, α−b).

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Define fors∈(−∞,∞),

(2.2) ϕ(s) =









 ln

b^s+1−a^s+1 (α−a)^s+1−(α−b)^s+1

, s 6=−1;

ln

ln(b/a)

ln[(α−a)/(α−b)]

, s =−1.

Then

(2.3) lnf_α(s) =





 ϕ(s)

s , s6= 0;

ϕ⁰(0), s= 0.

In order to prove thatlnfα is strictly increasing (decreasing), it suffices to show that ϕ is strictly convex (concave) on (−∞,∞). A simple calculation reveals that

(2.4) ϕ(−1−s) =ϕ(−1 +s) +sln(α−a)(α−b)

ab ,

which implies that ϕ⁰⁰(−1−s) = ϕ⁰⁰(−1 +s), andϕ has the same convexity (concavity) on both(−∞,−1)and(−1,∞). Hence, it is sufficient to prove that ϕis strictly convex (concave) on(−1,∞).

A computation yields ϕ⁰(s) = b^s+1lnb−a^s+1lna

b^s+1−a^s+1 − (α−b)^s+1ln(α−b)−(α−a)^s+1ln(α−a) (α−b)^s+1−(α−a)^s+1 ,

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(s+ 1)²ϕ⁰⁰(s) (2.5)

= (s+ 1)²

"

−a^s+1b^s+1(ln ^a_b)²

(b^s+1−a^s+1)² +(α−a)^s+1(α−b)^s+1(ln ^α−b_α−a)² [(α−a)^s+1−(α−b)^s+1]²

#

=−(^a_b)^s+1[ln(^a_b)^s+1]²

[1−(^a_b)^s+1]² + (_α−a^α−b)^s+1[ln(^α−b_α−a)^s+1]² [1−(_α−a^α−b)^s+1]² . Define for0< t <1,

(2.6) ω(t) = t(lnt)²

(1−t)². Differentiation yields

(1−t)tlntω⁰(t)

ω(t) = (1 +t) lnt+ 2(1−t) (2.7)

=−

∞

X

n=2

n−1

n(n+ 1)(1−t)ⁿ⁺¹ <0, which implies thatω⁰(t)>0for0< t <1. It is easy to see that (2.8) 0<a

b s+1

<

α−b α−a

s+1

<1 for [a, b]⊆ 0,α

2 i

, s >−1,

(2.9) 0<

α−b α−a

s+1

<a b

s+1

<1 for [a, b]⊆hα 2, α

, s >−1,

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and thereforeϕ⁰⁰(s) >0for[a, b] ⊆(0,^α₂]ands >−1, ϕ⁰⁰(s) <0for[a, b]⊆ [^α₂, α)ands >−1. Thenϕis strictly convex (concave) on(−1,∞)for[a, b]⊆ (0,^α₂] ([a, b]⊆ [^α₂, α))respectively. By Lemma2.1 above, Theorem1.1holds.

Sincef_α(s)is a strictly increasing (decreasing) function for [a, b] ⊆ (0,^α₂] ([a, b] ⊆ [^α₂, α)), put s = −2,−1,0,1 respectively. The inequalities (1.6) are deduced.

Then, let(α−x) = t and apply it to the function ^R^b

ax^sdx Rb

a(α−x)^sdx

¹s

. We get Corollary1.3.

Proof of Theorem1.4. Using an analogous method of proof to that of Theorem 1.1, we get

f(s) =

1 b−a

Rb a x^sdx

1 d−c

Rd c x^sdx

!¹_s

=

"_b^s+1_−a^s+1

(s+1)(b−a) d^s+1−c^s+1 (s+1)(d−c)

#¹s

=

(d−c) (b−a)

(b^s+1−a^s+1) (d^s+1−c^s+1)

¹_s

= Ls(a, b) L_s(c, d). LetM = ^(d−c)_(b−a), and define fors∈(−∞,∞),

(2.10) ϕ(s) =









 ln

Mb^s+1−a^s+1 d^s+1−c^s+1

, s6=−1;

ln

Mln(b/a) ln(d/c)

, s=−1.

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Then

(2.11) lnf(s) =





 ϕ(s)

s , s6= 0;

ϕ⁰(0), s= 0,

andϕhas the same convexity (concavity) on both(−∞,−1)and(−1,∞).

A computation yields

(s+ 1)²ϕ⁰⁰(s) =−(^a_b)^s+1[ln(^a_b)^s+1]²

[1−(^a_b)^s+1]² +(^c_d)^s+1[ln(_d^c)^s+1]² [1−(^c_d)^s+1]² . Define for0< t <1,

(2.12) ω(t) = t(lnt)²

(1−t)².

Differentiation yieldsω⁰(t)>0for0< t <1. It is easy to see that (2.13) 0<

a b

s+1

<

c d

s+1

<1 for ad < bc, s >−1,

(2.14) 0<c d

s+1

<a b

s+1

<1 for ad > bc, s >−1,

and thereforeϕ⁰⁰(s) > 0forad < bc ands > −1, ϕ⁰⁰(s) < 0forad > bc and s > −1Thenϕ is strictly convex (concave) on(−1,∞)forad < bc (ad > bc) respectively. The proof is complete.

In Theorem1.4, let a = c. Then f(s)is a strictly increasing function for d < b, or a strictly decreasing function ford > b. Thus Corollary1.5holds.

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References

[1] E.F. BECKENBACH AND R. BELLMAN, Inequalities, Springer Verlag, 1961.

[2] CHAO-PING CHENANDFENG QI, An alternative proof of monotonicity for the extended mean values, Aust. J. Math. Anal. Appl., 1(2) (2004), Art.

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[4] E.B. LEACH AND M.C. SHOLANDER, Extended mean values, Amer.

Math. Monthly, 85 (1978), 84–90.

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[6] J.K. MERIKOWSKI, Extending means of two variables to several vari- ables, J. Ineq. Pure. Appl. Math., 5(3) (2004), Art. 65. [ONLINE:http:

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[7] D.S. MITRINOVI ´C, Analytic Inequalities, Springer-Verlag, Berlin, 1970.

[8] J. PE ˘CARI ´C AND V. ˘SIMI ´C, The Stolarsky-Tobey mean in n variables, Math. Inequal. Appl., 2 (1999), 325–341.

[9] F. QI, Logarithmic convexity of extended mean values, Proc. Amer. Math.

Soc., 130(6) (2002), 1787–1796 (electronic).

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[10] K.B. STOLARSKY, Generalizations of the logarithmic mean, Math. Mag., 48 (1975), 87–92.

[11] M.D. TOBEY, A two-parameter homogeneous mean value, Amer. Math.

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[12] ZH. WANG, J. CHENANDX. LI, A generalization of the Ky Fan inequal- ity, Univ. Beograd. Publ. Elektrotehn. Fak., 7 (1996), 9–17.

[13] CH.-P. CHEN AND F. QI, Monotonicity properties for generalized loga- rithmic means, Aust. J. Math. Anal. Appl., 1(2) (2004), Art. 2. [ONLINE:

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