On a New Generalization of Martins' Inequality

On a New Generalization of Martins' Inequality

This is the Published version of the following publication

Qi, Feng (2002) On a New Generalization of Martins' Inequality. Research report collection, 5 (3).

The publisher’s official version can be found at

Note that access to this version may require subscription.

Downloaded from VU Research Repository https://vuir.vu.edu.au/17739/

ON A NEW GENERALIZATION OF MARTINS’ INEQUALITY

FENG QI

Abstract. Letn, m∈Nand{a_i}^n+m_i=1 be an increasing, logarithmically con- cave, positive, and nonconstant sequence such that the sequence

ia_i+1 a_i − 1 ^n+m−1_i=1 is increasing. Then the following inequality between ratios of the power means and of the geometic means holds:

1 n

n

X

i=1

a^r_i 1

n+m

n+m

X

i=1

a^r_i

!1/r

<

n√ an!

n+mp an+m!,

where r is a positive number,ai! denotes the sequence factorial defined by Qn

i=1ai. The upper bound is the best possible.

1. Introduction It is well-known that the following inequality

n

n+ 1 < 1 n

n

X

i=1

i^r 1

n+ 1

n+1

X

i=1

i^r

!^1/r

<

√n

n!

n+1p

(n+ 1)! (1)

holds for r > 0 andn ∈ N. We call the left-hand side of this inequality Alzer’s inequality [1], and the right-hand side Martins’s inequality [7].

Alzer’s inequality has invoked the interest of several mathematicians, we refer the reader to [5, 9,16,18] and the references therein.

Recently, F. Qi and L. Debnath in [15] proved that: Letn, m∈N and{ai}^∞_i=1 be an increasing sequence of positive real numbers satisfying

(k+ 2)a^r_k+2−(k+ 1)a^r_k+1 (k+ 1)a^r_k+1−ka^r_k ≥

ak+2

ak+1

r

(2) for a given positive real numberrandk∈N. Then

an

an+m

≤

(1/n)Pn i=1a^r_i (1/(n+m))Pn+m

i=1 a^r_i 1/r

. (3)

The lower bound of (3) is the best possible.

2000Mathematics Subject Classification. Primary 26D15.

Key words and phrases. Martins’s inequality, Alzer’s inequality, K¨onig’s inequality, increasing sequence, logarithmically concave, ratio, power mean, geometric mean.

The author was supported in part by NSF (#10001016) of China, SF for the Prominent Youth of Henan Province (#0112000200), SF of Henan Innovation Talents at Universities, NSF of Henan Province (#004051800), SF for Pure Research of Natural Science of the Education Department of Henan Province (#1999110004), Doctor Fund of Jiaozuo Institute of Technology, China.

This paper was typeset usingAMS-L^ATEX.

1

In [12,13,14,19,20,21], F. Qi and others proved the following inequalities and other more general results:

n+k+ 1 n+m+k+ 1 <

n+k

Y

i=k+1

i

!^1/n, _n+m+k Y

i=k+1

i

!^1/(n+m)

≤

r n+k

n+m+k, (4) a(n+k+ 1) +b

a(n+m+k+ 1) +b <

hQn+k

i=k+1(ai+b)i¹_n hQn+m+k

i=k+1 (ai+b)i_n+m¹

≤ s

a(n+k) +b

a(n+m+k) +b, (5) wheren, m∈N,kis a nonnegative integer,aa positive constant, andba nonegative constant. The equalities in (4) and (5) is valid forn= 1 andm= 1.

In [17], the following monotonicity results for the gamma function were obtained:

The function

[Γ(x+y+ 1)/Γ(y+ 1)]^1/x

x+y+ 1 (6)

is decreasing inx≥1 for fixed y≥0. Then, for positive real numbersxandy, we have

x+y+ 1

x+y+ 2 ≤ [Γ(x+y+ 1)/Γ(y+ 1)]^1/x

[Γ(x+y+ 2)/Γ(y+ 1)]^1/(x+1). (7) In [11,15], F. Qi proved that: Letnandmbe natural numbers,ka nonnegative integer. Then

n+k

n+m+k < 1 n

n+k

X

i=k+1

i^r 1

n+m

n+m+k

X

i=k+1

i^r

!^1/r

, (8)

whereris a given positive real number. The lower bound is the best possible.

In [4,18], some more general results for the lower bound of ratio of power means

1 n

Pn

i=1a^r_{i 1}

n+m

Pn+m i=1 a^r_i^1/r

for positive sequence{a_i}_i∈Nwere obtained.

An open problem in [10,11] asked for the validity of the following inequality:

1 n

n+k

X

i=k+1

i^r 1

n+m

n+m+k

X

i=k+1

i^r

!^1/r

<

pn

(n+k)!/k!

n+mp

(n+m+k)!/k!, (9) wherer >0,n, m∈N,k∈Z⁺.

Let{ai}i∈N be a positive sequence. If a_i+1a_i−1 ≥a²_i fori≥2, we call {ai}i∈N

a logarithmically convex sequence; if a_i+1a_i−1 ≤ a²_i for i ≥ 2, we call {a_i}_i∈N a logarithmically concave sequence. See [8, p. 284].

In [3], the open problem mentioned above was solved and generalized affirma- tively: Let {a_i}^n+m_i=1 be an increasing, logarithmically concave, positive, and non- constant sequence satisfying (a`+1/a`)^{`</sup> ≥ (a`/a`−1)<sup>`−1} for any positive integer

` >1, then _n¹Pn

i=1a^r_{i 1}

n+m

Pn+m i=1 a^r_i1/r

< √ⁿ

an!_n+mp

an+m!, where ris a posi- tive number,n, m∈N, and ai! denotes the sequence factorialQn

i=1ai. The upper bound is best possible.

The purpose of this paper is to give a new generalization of inequality (9) as follows.

Theorem 1. Letn, m∈Nand{ai}^n+m_i=1 be an increasing, logarithmically concave, positive, and nonconstant sequence such that the sequence

ia_i+1

a_i −1 ^n+m−1_i=1 is

ON A NEW GENERALIZATION OF MARTINS’ INEQUALITY 3

increasing. Then the following inequality between ratios of the power means and of the geometic means holds:

1 n

n

X

i=1

a^r_i 1

n+m

n+m

X

i=1

a^r_i

!^1/r

<

√n

a_n!

n+mp

a_n+m!, (10)

where r is a positive number and ai! denotes the sequence factorial Qn

i=1ai. The upper bound is the best possible.

As an easy consequence of Theorem1 by taking {ai}^n+m_i=1 ={(i+k+b)^α}^n+m_i=1 for a positive constantα, we have

Corollary 1. Letαbe a positive real number,ka nonnegative integer andba real number such that k+b >0, andm, n∈N. If the sequence

i

1 + 1

i+k+b ^α

−1

^n+m−1

i=1

(11) is increasing, then for any real number r >0, we have

1 n

Pn+k

i=k+1[(i+b)^α]^r

1 n+m

Pn+m+k

i=k+1 [(i+b)^α]^r

!^1/r

<

n

q Qn+k

i=k+1(i+b)^α

n+mq

Qn+m+k

i=k+1 (i+b)^α

. (12)

The upper bound is the best possible.

Remark 1. By lettingα= 1 andb= 0 in (12), we recover inequality (9).

Takingα= 2 in Corollary1 leads to the following

Corollary 2. Letkbe a nonnegative integer, andba real number such thatk+b≥

1

2, andm, n∈N. Then, for any real number r >0, we have

1 n

Pn+k

i=k+1[(i+b)²]^r

1 n+m

Pn+m+k

i=k+1 [(i+b)²]^r

!^1/r

<

n

q Qn+k

i=k+1(i+b)²

n+mq

Qn+m+k i=k+1 (i+b)²

. (13)

The upper bound is the best possible.

Considering {ai}i∈N= e^iα

i∈N in Theorem1 and standard argument gives us the following

Corollary 3. Letm, n∈N. If the constant 0< α <1such that inequality e^(1+x)α−e^xα

x^α−1−(1 +x)^α−1 ≥αxe^(1+x)α (14) holds with xon[1,∞), then, for any real numberr >0, we have

1 n

Pn i=1e^iαr

1 n+m

Pn+m i=1 e^iαr

!1/r

<exp 1

n

n

X

i=1

i^α− 1 n+m

n+m

X

i=1

i^α

. (15)

The upper bound is the best possible.

2. Lemmas

To prove our main results, the following lemmas are neccessary.

Lemma 1. Let n, m ∈ N, and {ai}^n+m+1_i=1 a nonconstant positive sequence such that the sequence

iai+1

a_i −1 ^n+m_i=1 is increasing, then the sequence (√i

ai! ai+1

)^n+m

i=1

(16) is decreasing. As a simple consequence, we have the following

√n

a_n!

n+mp

an+m! > a_n+1 an+m+1

, (17)

wherea_n! denotes the sequence factorial defined byQn i=1a_i.

Proof. For 1≤i≤n+m−1, the monotonicity of the sequence (16) is equivalent to the following

√i

ai! a_i+1 ≥

i+1p a_i+1!

a_i+2 , (18)

⇐⇒

i

Y

k=1

ak

a_i+1

!1/i

≥

i+1

Y

k=1

ak

a_i+2

!1/(i+1)

,

⇐⇒ 1

i

i

X

k=1

ln ak

ai+1

≥ 1 i+ 1

i+1

X

k=1

ln ak

ai+2

,

⇐⇒ i

i+ 1

i+1

X

k=1

ln a_k ai+2

≤

i

X

k=1

ln a_k ai+1

. (19)

Since lnxis concave on (0,∞), by definition of concaveness, it follows that, for 1≤k≤i,

k

i+ 1lna_k+1 ai+2

+i−k+ 1 i+ 1 ln a_k

ai+2

≤ln k

i+ 1 ·a_k+1 ai+2

+i−k+ 1 i+ 1 · a_k

ai+2

!

(20)

= ln

kak+1+ (i−k+ 1)ak

(i+ 1)ai+2

.

Since the sequence ia_i+1

ai −1 ^n+m_i=1 is increasing, we have, for 1≤i≤n+m−1 and 1≤k≤i, the following

(i+ 1)a_i+2 ai+1

−(i+ 1)≥ ia_i+1 ai

−i,

⇐⇒ (i+ 1)a_i+2

ai+1

−(i+ 1)≥ ka_k+1 ak

−k,

⇐⇒ ka_k+1+ (i−k+ 1)a_k ak

≤ (i+ 1)a_i+2 ai+1

,

⇐⇒ ka_k+1+ (i−k+ 1)a_k (i+ 1)ai+2

≤ a_k ai+1

.

ON A NEW GENERALIZATION OF MARTINS’ INEQUALITY 5

Combining the last line above with (20) yields k

i+ 1lnak+1

a_i+2 +i−k+ 1 i+ 1 ln ak

a_i+2 ≤ln ak

a_i+1. (21)

Summing up on both sides of (21) with k from 1 to i and simplifying reveals inequality (19). The monotonicity follows.

Since {ai}^n+m+1_i=1 is a nonconstant positive sequence, there exists at least one number 1 ≤i₀ ≤ n+m−1 such that a_i0 6= a_i0+1. The function lnx is strictly concave on (0,∞). Then, for anyisuch thati₀≤i≤n+m−1, we have

i0

i+ 1lnai₀+1

a_i+2 +i−i0+ 1 i+ 1 ln ai₀

a_i+2

<ln i0

i+ 1 ·ai₀+1

a_i+2 +i−i0+ 1 i+ 1 · ai₀

a_i+2

!

= ln

i₀a_i0+1+ (i−i₀+ 1)a_i0 (i+ 1)ai+2

≤ln ai₀

a_i+1,

(22)

notice that the last line follows from the sequence ia_i+1

ai −1 ^n+m_i=1 being increasing.

Therefore, for anyisuch thati0≤i≤n+m−1, inequality (18) is strict. Inequality (17) is proved.

The proof is complete.

Lemma 2. Let n >1 be a positive integer and{ai}ⁿ_i=1 an increasing nonconstant positive sequence such that

iai+1

a_i −1 ⁿ⁻¹_i=1 is increasing. Then the sequence ( ai

(a_i!)^1/i )ⁿ

i=1

(23) is increasing, and, for any positive integer `satisfying 1≤` < n,

a`

a_n < (a`!)<sup>1/`</sup>

(an!)^1/n

, (24)

wherean! denotes the sequence factorial Qn i=1ai.

Proof. For 1≤`≤n−1, the monotonicity of the sequence (23) is equivalent to a<sub>`</sub>

(a`!)<sup>1/`</sup> ≤ a<sub>`+1</sub> (a`+1!)^1/(`+1)

,

⇐⇒

`

Y

j=1

aj

a_`

!¹_`

≥

`+1

Y

j=1

aj

a_`+1

!_`+1¹ ,

⇐⇒ 1

`

`−1

X

j=1

lnaj

a`

≥ 1

`+ 1

`

X

j=1

ln aj

a`+1

,

⇐⇒

`−1

X

j=1

lna_j a`

≥ `

`+ 1

`

X

j=1

ln a_j a`+1

. (25)

Since lnxis concave on (0,∞), by definition of concaveness, it follows that, for 1≤j ≤`,

j

`+ 1lnaj+1

a`+1

+`−j+ 1

`+ 1 ln aj

a`+1

≤ln j

`+ 1· aj+1

a`+1

+`−j+ 1

`+ 1 · aj

a`+1

(26)

= ln

jaj+1+ (`−j+ 1)aj

(`+ 1)a<sub>`+1</sub>

.

Straightforward computation gives us

`

X

j=1

j

`+ 1lnaj+1

a`+1

+`−j+ 1

`+ 1 ln aj

a`+1

= `

`+ 1

`

X

j=1

ln a_j a`+1

+

`

X

j=1

j

`+ 1lna<sub>j+1</sub> a`+1

−

`

X

j=1

j−1

`+ 1ln a<sub>j</sub> a`+1

= `

`+ 1

`

X

j=1

ln aj

a_`+1 +

`+1

X

j=2

j−1

`+ 1ln aj

a_`+1

−

`

X

j=1

j−1

`+ 1ln aj

a_`+1

= `

`+ 1

`

X

j=1

ln a_j a`+1

.

(27)

From combining of (25), (26) and (27), it suffices to prove for 1≤j≤` jaj+1+ (`−j+ 1)aj

(`+ 1)a`+1

≤ aj

a`

,

⇐⇒ jaj+1+ (`−j+ 1)aj

a_j ≤ (`+ 1)a`+1

a_` ,

⇐⇒ jaj+1

aj

+`−j+ 1≤ (`+ 1)a`+1

a`

,

⇐⇒ (`+ 1)ha<sub>`+1</sub> a`

−1i

≥jha_j+1 aj

−1i

. (28)

Since the sequences{ai}ⁿ_i=1and ia_i+1

a_i −1 ⁿ⁻¹_i=1 are increasing, the inequality (28) holds.

Moreover, the sequence{ai}ⁿ_i=1is nonconstant positive, then there exists at least one number 1 ≤ i1 ≤ n−1 such that ai₁ 6= ai₁+1. The function lnx is strictly concave on (0,∞). Then, for any`such thati1< `≤n−1, we have

i1

`+ 1lnai₁+1

a<sub>`+1</sub> +`−i1+ 1

`+ 1 ln ai₁

a_`+1

<ln i₁

`+ 1 ·a<sub>i1+1</sub> a`+1

+`−i₁+ 1

`+ 1 · a<sub>i1</sub> a`+1

= ln

i1ai₁+1+ (`−i1+ 1)ai₁

(`+ 1)a<sub>`+1</sub>

≤lnai₁

a_` .

(29)

ON A NEW GENERALIZATION OF MARTINS’ INEQUALITY 7

Therefore, for any`such thati1+ 1≤` < n, inequality (25) is strict, and a`

a`+1

< (a`!)<sup>1/`</sup>

(a`+1!)<sup>1/(`+1)</sup>

, (30)

and then inequality (24) is strict. The proof is complete.

Remark 2. Some problems similar to Lemma1and Lemma2were discussed in [19]

by the author and B.-N. Guo.

The methods proving Lemma1 and Lemma2had been used in [18] and others.

Lemma 3 (K¨onig’s inequality [2, p. 149] and [7,22]). Let {ai}ⁿ_i=1 and{bi}ⁿ_i=1 be decreasing nonnegativen-tuples such that

k

Y

i=1

bi≤

k

Y

i=1

ai, 1≤k≤n, (31)

then, for r >0, we have

k

X

i=1

b^r_i ≤

k

X

i=1

a^r_i, 1≤k≤n. (32)

Remark 3. Lemma 3 is a well-known result due to K¨onig used to give a proof of Weyl’s inequality (cf. Corollary 1.b.8 of [6, p. 24]).

3. Proofs of Theorem1

Inequality (10) holds for n = 1 by the power mean inequality and its case of equality.

Forn≥2, inequality (10) is equivalent to 1

n

n

X

i=1

a^r_i 1

n+ 1

n+1

X

i=1

a^r_i

!^1/r

<

√n

a_n!

n+1p

a_n+1!, (33)

which is equivalent to 1 n

n

X

i=1

a_i

√n

an! r

< 1 n+ 1

n+1

X

i=1

a_i

n+1p an+1!

!^r

. (34)

Set

b_jn+1=b_jn+2=· · ·=b_jn+n= a_n+1−j

n+1p

a_n+1!, 0≤j≤n; (35) c_j(n+1)+1=c_j(n+1)+2=· · ·=cj(n+1)+(n+1)= an−j

√n

an!, 0≤j ≤n−1. (36) Direct calculation yields

n(n+1)

X

i=1

b^r_i =

n

X

j=0 n

X

k=1

b^r_jn+k

=n

n

X

j=0

a_n+1−j

n+1p a_n+1!

!r

=n

n+1

X

i=1

a_i

n+1p a_n+1!

!^r

(37)

and

n(n+1)

X

i=1

c^r_i = (n+ 1)

n

X

i=1

a_i

√n

a_n! ^r

. (38)

Since{ai}ⁿ⁺¹_i=1 is increasing, the sequence{bi}ⁿ⁽ⁿ⁺¹⁾_i=1 and{ci}ⁿ⁽ⁿ⁺¹⁾_i=1 are decreas- ing. Therefore, by Lemma 3, to obtain inequality (34), it is sufficient to prove inequality

bm!≥cm! (39)

for 1≤m≤n(n+ 1).

It is easy to see thatb_n(n+1)! =c_n(n+1)! = 1. Thus, inequality (39) is equivalent to

n(n+1)

Y

i=m

bi≤

n(n+1)

Y

i=m

ci (40)

for 2≤m≤n(n+ 1).

For 0≤`≤nand 0≤j ≤n−2, we have 2≤(n−`)n+ (n−j) = (n−`)(n+ 1) + (`−j)≤n(n+ 1). Then

n(n+1)

Y

i=(n−`)n+(n−j)

bi= (a`+1)<sup>j+1</sup>(a`!)ⁿ (an+1!)^`n+j+1n+1

; (41)

n(n+1)

Y

i=(n−`)(n+1)+(`−j)

ci= (a`)<sup>n−`+j+2</sup>(a`−1!)ⁿ⁺¹

(a_n!)<sup>`n+j+1n</sup> , ` > j; (42)

n(n+1)

Y

i=(n−`)(n+1)+(`−j)

ci=

n(n+1)

Y

i=(n−`−1)(n+1)+(n+1+`−j)

ci

= (a`+1)<sup>j−`+1</sup>(a`!)<sup>n+1</sup> (an!)<sup>`n+j+1n</sup>

, `≤j;

(43)

wherea0= 1.

The last term in (43) is bigger than the right term in (42), so, without loss of generality, we can assumej < `. Therefore, from formulae (41) and (42), inequality (40) is reduced to

(a`+1)<sup>j+1</sup>(a`!)ⁿ(an+1!)^`−j−1n+1

(an+1!)<sup>`</sup> ≤ (a`)<sup>n−`+j+2</sup>(a`−1!)ⁿ⁺¹

(an!)^`(an!)^j+1n , (44) that is

(a`+1)<sup>j+1</sup>(an+1!)<sup>`−j−1n+1</sup>

(a_{`</sub>!)(a<sub>`})^{j−`+1</sup> ≤ (an+1)<sup>`}(an!)^{−`n</sup> (an!)<sup>j−`+1n}

, (45)

this is further equivalent to

(a`+1)<sup>j+1</sup>(an+1!)<sup>`−j−1n+1</sup> a`!(a`)^{j−`+1</sup>(an!)<sup>`−j−1n}

≤(an+1)^`

(an!)^n` , (46)

which can be rearranged as a`+1

a`

·

√n

an!

n+1p an+1!

!^j+1_`

≤

√`

a`! a`

· an+1

n+1p

an+1!, j+ 1≤`≤n. (47)

ON A NEW GENERALIZATION OF MARTINS’ INEQUALITY 9

Utilizing Lemma 2 and the logarithmical concaveness of the sequence {ai}ⁿ⁺¹_i=1 yields

√n

a_n!

n+1p

an+1! > a_n an+1

≥ a<sub>`</sub> a`+1

. (48)

Since ^j+1<sub>`</sub> ≤1 and <sup>a`+1</sup>_a

` · ⁿ

√an!

n+1√

a_n+1! >1 by (48), thus, to obtain (47), it suffices to prove

a`+1ⁿ

pan!< an+1^`

pa`!, (49) this follows from Lemma1.

Since the sequence{a_i}^n+m_i=1 is nonconstant, the inequality (10) is strict.

By the L’Hospital rule, easy calculation produces

r→∞lim 1

n

n

X

i=1

a^r_i 1

n+m

n+m

X

i=1

a^r_i ^1/r

=

√n

an!

n+mp

an+m!, (50)

thus, the upper bound is the best possible. The proof is complete.

Remark 4. Recently, some new inequalities for the ratios of the mean values of functions were established in [23].

References

[1] H. Alzer, On an inequality of H. Minc and L. Sathre, J. Math. Anal. Appl. 179(1993), 396–402. 1

[2] P. S. Bullen, A Dictionary of Inequalities, Pitman Monographs and Surveys in Pure and Applied Mathematics97, Addison Wesley Longman Limited, 1998. 7

[3] T. H. Chan, P. Gao and F. Qi,On a generalization of Martins’ inequality, Monatsh. Math.

(2002), accepted. RGMIA Res. Rep. Coll.4(2001), no. 1, Art. 12, 93–101. Available online athttp://rgmia.vu.edu.au/v4n1.html. 2

[4] Ch.-P. Chen, F. Qi, P. Cerone, and S. S. Dragomir, Monotonicity of sequences involving convex and concave functions, RGMIA Res. Rep. Coll.5 (2002), no. 1, Art. 1. Available online athttp://rgmia.vu.edu.au/v5n1.html. 2

[5] B.-N. Guo and F. Qi,An algebraic inequality, II, RGMIA Res. Rep. Coll.4(2001), no. 1, Art. 8, 55–61. Available online athttp://rgmia.vu.edu.au/v4n1.html. 1

[6] H. K¨onig,Eigenvalue Distribution of Compact Operators, Birkha¨user, Basel, 1986. 7 [7] J. S. Martins,Arithmetic and geometric means, an application to Lorentz sequence spaces,

Math. Nachr.139(1988), 281–288. 1,7

[8] J. Peˇcari´c, F. Proschan, and Y. L. Tong,Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering187, Academic Press, 1992. 2 [9] F. Qi,An algebraic inequality, J. Inequal. Pure and Appl. Math.2(2001), no. 1, Art. 13.

Available online athttp://jipam.vu.edu.au/v2n1/006_00.html. RGMIA Res. Rep. Coll.2 (1999), no. 1, Art. 8, 81–83. Available online athttp://rgmia.vu.edu.au/v2n1.html. 1 [10] F. Qi, Generalizations of Alzer’s and Kuang’s inequality, Tamkang J. Math. 31 (2000),

no. 3, 223–227. RGMIA Res. Rep. Coll.2 (1999), no. 6, Art. 12, 891–895. Available online athttp://rgmia.vu.edu.au/v2n6.html. 2

[11] F. Qi, Generalization of H. Alzer’s inequality, J. Math. Anal. Appl. 240 (1999), no. 1, 294–297. 2

[12] F. Qi,Inequalities and monotonicity of the ratio for the geometric means of a positive arith- metic sequence with arbitrary difference, Tamkang. J. Math.34(2003), no. 3, in press. 2 [13] F. Qi,Inequalities and monotonicity of the ratio for the geometric means of a positive arith-

metic sequence with unit difference, submitted. 2

[14] F. Qi,Inequalities and monotonicity of sequences involving ⁿp

(n+k)!/k! , RGMIA Res. Rep.

Coll.2 (1999), no. 5, Art. 8, 685–692. Available online athttp://rgmia.vu.edu.au/v2n5.

html. 2

[15] F. Qi and L. Debnath, On a new generalization of Alzer’s inequality, Internat. J. Math.

Math. Sci.23(2000), no. 12, 815–818. 1,2

[16] F. Qi and B.-N. Guo, An inequality between ratio of the extended logarithmic means and ratio of the exponential means, Taiwanese J. Math.7(2003), no. 2, in press. 1

[17] F. Qi and B.-N. Guo,Inequalities and monotonicity for the ratio of gamma functions, Tai- wanese J. Math.7(2003), no. 2, in press. 2

[18] F. Qi and B.-N. Guo, Monotonicity of sequences involving convex function and sequence, RGMIA Res. Rep. Coll.3(2000), no. 2, Art. 14, 321–329. Available online athttp://rgmia.

vu.edu.au/v3n2.html. 1,2,7

[19] F. Qi and B.-N. Guo,Monotonicity of sequences involving geometric means of positive se- quences with logarithmical convexity, RGMIA Res. Rep. Coll.5(2002), no. 3, Art. 10. Avail- able online athttp://rgmia.vu.edu.au/v5n3.html. 2,7

[20] F. Qi and B.-N. Guo,Some inequalities involving the geometric mean of natural numbers and the ratio of gamma functions, RGMIA Res. Rep. Coll.4 (2001), no. 1, Art. 6, 41–48.

Available online athttp://rgmia.vu.edu.au/v4n1.html. 2

[21] F. Qi and Q.-M. Luo, Generalization of H. Minc and J. Sathre’s inequality, Tamkang J.

Math.31(2000), no. 2, 145–148. RGMIA Res. Rep. Coll.2(1999), no. 6, Art. 14, 909–912.

Available online athttp://rgmia.vu.edu.au/v2n6.html. 2

[22] J. A. Sampaio Martins, Inequalities of Rado-Popoviciu type, In: Marques de S´a, Eduardo (ed.) et al. Mathematical studies. Homage to Professor Doctor Lu´ıs de Albuquerque. Coimbra:

Universidade de Coimbra, Faculdade de Ciˆencias e Tecnologia, Departamento de Matem´atica, (1994), 169–175. 7

[23] N. Towghi and F. Qi,An inequality for the ratios of the arithmetic means of functions with a positive parameter, RGMIA Res. Rep. Coll.4(2001), no. 2, Art. 15, 305–309. Available online athttp://rgmia.vu.edu.au/v4n2.html. 9

(F. Qi)Department of Applied Mathematics and Informatics, Jiaozuo Institute of Technology, #142, Mid-Jiefang Road, Jiaozuo City, Henan 454000, China

E-mail address:qifeng@jzit.edu.cn URL:http://rgmia.vu.edu.au/qi.html