volume 1, issue 1, article 4, 2000.

*Received 4 November, 1999;*

*accepted 6 December, 1999.*

*Communicated by:**L. Debnath*

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**Journal of** **Inequalities in** **Pure and** **Applied** **Mathematics**

**GENERALIZED ABSTRACTED MEAN VALUES**

FENG QI

Department of Mathematics Jiaozuo Institute of Technology Jiaozuo City, Henan 454000

THE PEOPLE’S REPUBLIC OF CHINA
*EMail:*qifeng@jzit.edu.cn

2000c Victoria University ISSN (electronic): 1443-5756 013-99

**Generalized Abstracted Mean**
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**Abstract**

In this article, the author introduces the generalized abstracted mean values which extend the concepts of most means with two variables, and researches their basic properties and monotonicities.

*2000 Mathematics Subject Classification:*26A48, 26D15

*Key words: Generalized abstracted mean values, basic property, monotonicity.*

The author was supported in part by NSF of Henan Province, SF of the Education Committee of Henan Province (No. 1999110004), and Doctor Fund of Jiaozuo Insti- tute of Technology, China

**Contents**

1 Introduction. . . 3
2 Definitions and Basic Properties . . . 6
3 Monotonicities. . . 12
**References**

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**1.** **Introduction**

The simplest and classical means are the arithmetic mean, the geometric mean, and the harmonic mean. For a positive sequence a = (a1, . . . , an), they are defined respectively by

(1.1) A_{n}(a) = 1
n

n

X

i=1

a_{i}, G_{n}(a) = ^{n}
v
u
u
t

n

Y

i=1

a_{i}, H_{n}(a) = n

n

P

i=1

1
a_{i}
.

For a positive functionf defined on[x, y], the integral analogues of (1.1) are given by

A(f) = 1 y−x

Z y x

f(t)dt, G(f) = exp

1 y−x

Z y x

lnf(t)dt

, H(f) = y−x

Z y x

dt f(t)

. (1.2)

It is well-known that

(1.3) A_{n}(a)≥G_{n}(a)≥H_{n}(a), A(f)≥G(f)≥H(f)

are called the arithmetic mean–geometric mean–harmonic mean inequalities.

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These classical means have been generalized, extended and refined in many different directions. The study of various means has a rich literature, for details, please refer to [1,2], [4]–[8] and [19], especially to [9], and so on.

Some mean values also have applications in medicine [3,18].

Recently, the author [9] introduced the generalized weighted mean values
M_{p,f}(r, s;x, y)with two parametersrands, which are defined by

M_{p,f}(r, s;x, y) =
R^{y}

x p(u)f^{s}(u)du
Ry

x p(u)f^{r}(u)du

^{1/(s−r)}

, (r−s)(x−y)6= 0;

(1.4)

M_{p,f}(r, r;x, y) = exp
R^{y}

x p(u)f^{r}(u) lnf(u)du
Ry

x p(u)f^{r}(u)du

, x−y 6= 0;

(1.5)

M_{p,f}(r, s;x, x) = f(x),

where x, y, r, s ∈ R, p(u) 6≡ 0 is a nonnegative and integrable function and f(u)a positive and integrable function on the interval betweenxandy.

It was shown in [9,17] thatM_{p,f}(r, s;x, y)increases with bothrands and
has the same monotonicities asf in bothxandy. Sufficient conditions in order
that

M_{p}_{1}_{,f}(r, s;x, y)≥M_{p}_{2}_{,f}(r, s;x, y),
(1.6)

M_{p,f}_{1}(r, s;x, y)≥M_{p,f}_{2}(r, s;x, y)
(1.7)

were also given in [9].

It is clear thatM_{p,f}(r,0;x, y) =M^{[r]}(f;p;x, y). For the definition ofM^{[r]}(f;p;x, y),
please see [6].

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**Remark 1.1. As concrete applications of the monotonicities and properties of**
the generalized weighted mean values Mp,f(r, s;x, y), some monotonicity re-
sults and inequalities of the gamma and incomplete gamma functions are pre-
sented in [10].

Moreover, an inequality between the extended mean valuesE(r, s;x, y)and
the generalized weighted mean valuesM_{p,f}(r, s;x, y)for a convex functionfis
given in [14], which generalizes the well-known Hermite-Hadamard inequality.

The main purposes of this paper are to establish the definitions of the gen- eralized abstracted mean values, to research their basic properties, and to prove their monotonicities. In Section2, we introduce some definitions of mean val- ues and study their basic properties. In Section 3, the monotonicities of the generalized abstracted mean values, and the like, are proved.

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**2.** **Definitions and Basic Properties**

**Definition 2.1. Let**pbe a defined, positive and integrable function on[x, y]for
x, y ∈ R, f a real-valued and monotonic function on [α, β]. If g is a func-
tion valued on [α, β] and f ◦ g integrable on [x, y], the quasi-arithmetic non-
symmetrical mean ofg is defined by

(2.1) M_{f}(g;p;x, y) =f^{−1}
Ry

x p(t)f(g(t))dt Ry

x p(t)dt

!
,
wheref^{−1} is the inverse function off.

Forg(t) = t,f(t) = t^{r−1},p(t) = 1, the meanM_{f}(g;p;x, y)reduces to the
extended logarithmic means S_{r}(x, y); for p(t) = t^{r−1}, g(t) = f(t) = t, to the
one-parameter meanJ_{r}(x, y); forp(t) = f^{0}(t),g(t) = t, to the abstracted mean
M_{f}(x, y); for g(t) = t, p(t) =t^{r−1},f(t) = t^{s−r}, to the extended mean values
E(r, s;x, y); forf(t) = t^{r}, to the weighted mean of orderr of the functiong
with weight pon[x, y]. If we replacep(t)byp(t)f^{r}(t), f(t)byt^{s−r}, g(t)by
f(t)in (2.1), then we get the generalized weighted mean valuesM_{p,f}(r, s;x, y).

Hence, fromM_{f}(g;p;x, y)we can deduce most of the two variable means.

**Lemma 2.1 ([13]). Suppose that**f *and*g*are integrable, and*g *is non-negative,*
*on*[a, b], and that the ratiof(t)/g(t)*has finitely many removable discontinuity*
*points. Then there exists at least one point*θ∈(a, b)*such that*

(2.2)

Rb af(t)dt Rb

ag(t)dt = lim

t→θ

f(t) g(t).

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We call Lemma 2.1 the revised Cauchy’s mean value theorem in integral form.

*Proof. Since*f(t)/g(t)has finitely many removable discontinuity points, with-
out loss of generality, suppose it is continuous on [a, b]. Furthermore, using
g(t) ≥ 0, from the mean value theorem for integrals, there exists at least one
pointθ∈(a, b)satisfying

(2.3)

Z b a

f(t)dt= Z b

a

f(t) g(t)

g(t)dt= f(θ) g(θ)

Z b a

g(t)dt.

Lemma2.1follows.

* Theorem 2.2. The mean*M

_{f}(g;p;x, y)

*has the following properties:*

(2.4) α≤M_{f}(g;p;x, y)≤β,

Mf(g;p;x, y) = Mf(g;p;y, x),
*where*α= inf

t∈[x,y]g(t)*and*β = sup

t∈[x,y]

g(t).

*Proof. This follows from Lemma*2.1and standard arguments.

**Definition 2.2. For a sequence of positive numbers** a = (a_{1}, . . . , a_{n})and pos-
itive weights p = (p1, . . . , pn), the generalized weighted mean values of num-

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bersawith two parametersrandsis defined as

M_{n}(p;a;r, s) =

n

P

i=1

pia^{r}_{i}

n

P

i=1

p_{i}a^{s}_{i}

1/(r−s)

, r−s6= 0;

(2.5)

M_{n}(p;a;r, r) = exp

n

P

i=1

p_{i}a^{r}_{i} lna_{i}

n

P

i=1

p_{i}a^{r}_{i}

. (2.6)

Fors= 0we obtain the weighted meanMn^{[r]}(a;p)of orderrwhich is defined
in [2,5,6,7] and introduced above; fors = 0,r =−1, the weighted harmonic
mean; for s = 0, r = 0, the weighted geometric mean; and for s = 0, r = 1,
the weighted arithmetic mean.

The meanM_{n}(p;a;r, s)has some basic properties similar to those ofM_{p,f}(r, s;x, y),
for instance

* Theorem 2.3. The mean*M

_{n}(p;a;r, s)

*is a continuous function with respect to*(r, s)∈R

^{2}

*and has the following properties:*

(2.7)

m ≤M_{n}(p;a;r, s)≤M,
M_{n}(p;a;r, s) =M_{n}(p;a;s, r),

M_{n}^{s−r}(p;a;r, s) =M_{n}^{s−t}(p;a;t, s)·M_{n}^{t−r}(p;a;r, t),
*where*m= min

1≤i≤n{a_{i}},M = max

1≤i≤n{a_{i}}.

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*Proof. For an arbitrary sequence* b = (b_{1}, . . . , b_{n})and a positive sequencec =
(c1, . . . , cn), the following elementary inequalities [6, p. 204] are well-known

(2.8) min

1≤i≤n

nb_{i}
c_{i}

o

≤

n

P

i=1

b_{i}

n

P

i=1

ci

≤ max

1≤i≤n

nb_{i}
c_{i}

o .

This implies the inequality property.

The other properties follow from standard arguments.

**Definition 2.3. Let**f_{1}andf_{2} be real-valued functions such that the ratiof_{1}/f_{2}
is monotone on the closed interval [α, β]. Ifa = (a_{1}, . . . , a_{n})is a sequence of
real numbers from[α, β]andp = (p_{1}, . . . , p_{n})a sequence of positive numbers,
the generalized abstracted mean values of numbersa with respect to functions
f_{1} andf_{2}, with weightsp, is defined by

(2.9) M_{n}(p;a;f_{1}, f_{2}) =
f_{1}

f2

^{−1}

n

P

i=1

p_{i}f_{1}(a_{i})

n

P

i=1

p_{i}f_{2}(a_{i})

,

where(f_{1}/f_{2})^{−1}is the inverse function off_{1}/f_{2}.
The integral analogue of Definition2.3is given by

**Definition 2.4. Let** pbe a positive integrable function defined on[x, y], x, y ∈
R,f1 andf2real-valued functions and the ratiof1/f2 monotone on the interval

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[α, β]. In addition, let g be defined on [x, y] and valued on [α, β], andf_{i} ◦g
integrable on [x, y] for i = 1,2. The generalized abstracted mean values of
functiongwith respect to functionsf_{1} andf_{2}and with weightpis defined as
(2.10) M(p;g;f1, f2;x, y) =

f_{1}
f_{2}

−1 R^{y}

x p(t)f_{1}(g(t))dt
Ry

x p(t)f_{2}(g(t))dt

,
where(f_{1}/f_{2})^{−1}is the inverse function off_{1}/f_{2}.

**Remark 2.1. Set** f_{2} ≡ 1in Definition 2.4, then we can obtain Definition2.1
easily. Replacingf byf1/f2, p(t)byp(t)f2(g(t))in Definition2.1, we arrive
at Definition2.4directly. Analogously, formula (2.9) is equivalent toM_{f}(a;p),
see [6, p. 77]. Definition 2.1 and Definition 2.4 are equivalent to each other.

Similarly, so are Definition2.3and the quasi-arithmetic non-symmetrical mean
M_{f}(a;p)of numbersa= (a_{1}, . . . , a_{n})with weightsp= (p_{1}, . . . , p_{n}).

* Lemma 2.4. Suppose the ratio*f

_{1}/f

_{2}

*is monotonic on a given interval. Then*(2.11)

f1

f_{2}
−1

(x) = f2

f_{1}
−1

1 x

,
*where*(f_{1}/f_{2})^{−1}*is the inverse function of*f_{1}/f_{2}*.*

*Proof. This is a direct consequence of the definition of an inverse function.*

* Theorem 2.5. The means*M

_{n}(p;a;f

_{1}, f

_{2})

*and*M(p;g;f

_{1}, f

_{2};x, y)

*have the fol-*

*lowing properties:*

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*(i) Under the conditions of Definition2.3, we have*
(2.12) m≤M_{n}(p;a;f_{1}, f_{2})≤M,

M_{n}(p;a;f_{1}, f_{2}) =M_{n}(p;a;f_{2}, f_{1}),
*where*m = min

1≤i≤n{a_{i}},M = max

1≤i≤n{a_{i}};

*(ii) Under the conditions of Definition2.4, we have*

(2.13)

α ≤M(p;g;f_{1}, f_{2};x, y)≤β,
M(p;g;f_{1}, f_{2};x, y) =M(p;g;f_{1}, f_{2};y, x),
M(p;g;f1, f2;x, y) =M(p;g;f2, f1;x, y),
*where*α = inf

t∈[x,y]g(t)*and*β = sup

t∈[x,y]

g(t).

*Proof. These follow from inequality (2.8), Lemma*2.1, Lemma2.4, and stan-
dard arguments.

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**3.** **Monotonicities**

**Lemma 3.1 ([16]). Assume that the derivative of second order**f^{00}(t)*exists on*
R*. If*f(t)*is an increasing (or convex) function on*R*, then the arithmetic mean*
*of function*f(t),

(3.1) φ(r, s) =

1 s−r

Z s r

f(t)dt, r6=s,

f(r), r=s,

*is also increasing (or convex, respectively) with both*r*and*s*on*R*.*
*Proof. Direct calculation yields*

∂φ(r, s)

∂s = 1

(s−r)^{2}
h

(s−r)f(s)− Z s

r

f(t)dti , (3.2)

∂^{2}φ(r, s)

∂s^{2} = (s−r)^{2}f^{0}(s)−2(s−r)f(s) + 2Rs
r f(t)dt

(s−r)^{3} ≡ ϕ(r, s)

(s−r)^{3},
(3.3)

∂ϕ(r, s)

∂s = (s−r)^{2}f^{00}(s).

(3.4)

In the case off^{0}(t)≥ 0, we have∂φ(r, s)/∂s ≥0, thusφ(r, s)increases in
bothrands, sinceφ(r, s) = φ(s, r).

In the case of f^{00}(t) ≥ 0, ϕ(r, s) increases with s. Since ϕ(r, r) = 0, we
have∂^{2}φ(r, s)/∂s^{2} ≥ 0. Thereforeφ(r, s)is convex with respect to eitherror
s, sinceφ(r, s) =φ(s, r). This completes the proof.

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* Theorem 3.2. The mean*M

_{n}(p;a;r, s)

*of numbers*a= (a

_{1}, . . . , a

_{n})

*with weights*p= (p1, . . . , pn)

*and two parameters*r

*and*s

*is increasing in both*r

*and*s.

*Proof. Set*N_{n}= lnM_{n}, then we have

N_{n}(p;a;r, s) = 1
r−s

Z r s

n

P

i=1

p_{i}a^{t}_{i}lna_{i}

n

P

i=1

p_{i}a^{t}_{i}

dt, r−s6= 0;

(3.5)

Nn(p;a;r, r) =

n

P

i=1

p_{i}a^{r}_{i} lna_{i}

n

P

i=1

pia^{r}_{i}
.
(3.6)

By Cauchy’s inequality, direct calculation arrives at

(3.7)

n

P

i=1

pia^{t}_{i}lnai
n

P

i=1

p_{i}a^{t}_{i}

t

=

n

P

i=1

pia^{t}_{i}(lnai)^{2}

n

P

i=1

pia^{t}_{i}− n

P

i=1

pia^{t}_{i}lnai

2

^{n}
P

i=1

p_{i}a^{t}_{i}2 ≥0.

Combination of (3.7) with Lemma3.1yields the statement of Theorem3.2.

* Theorem 3.3. For a monotonic sequence of positive numbers* 0 < a

_{1}≤ a

_{2}≤

· · · *and positive weights*p= (p_{1}, p_{2}, . . .), ifm < n, then
(3.8) Mm(p;a;r, s)≤Mn(p;a;r, s).

*Equality holds if*a1 =a2 =· · ·*.*

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*Proof. For*r ≥s, inequality (3.8) reduces to

(3.9)

m

P

i=1

p_{i}a^{r}_{i}

m

P

i=1

p_{i}a^{s}_{i}

≤

n

P

i=1

p_{i}a^{r}_{i}

n

P

i=1

p_{i}a^{s}_{i}
.

Since 0 < a_{1} ≤ a_{2} ≤ · · ·, p_{i} > 0, i ≥ 1, the sequences

p_{i}a^{r}_{i} ^{∞}_{i=1} and
p_{i}a^{s}_{i} ^{∞}_{i=1} are positive and monotonic.

By mathematical induction and the elementary inequalities (2.8), we can easily obtain the inequality (3.9). The proof of Theorem3.3is completed.

* Lemma 3.4. If*A= (A

_{1}, . . . , A

_{n})

*and*B = (B

_{1}, . . . , B

_{n})

*are two nondecreas-*

*ing (or nonincreasing) sequences and*P = (P

_{1}, . . . , P

_{n})

*is a nonnegative se-*

*quence, then*

(3.10)

n

X

i=1

Pi n

X

i=1

PiAiBi ≥

n

X

i=1

PiAi n

X

i=1

PiBi,

*with equality if and only if at least one of the sequences*A*or*B *is constant.*

*If one of the sequences*A*or*B*is nonincreasing and the other nondecreasing,*
*then the inequality in (3.10) is reversed.*

The inequality (3.10) is known in the literature as Tchebycheff’s (or ˇCe- byšev’s) inequality in discrete form [7, p. 240].

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* Theorem 3.5. Let* p = (p

_{1}, . . . , p

_{n})

*and*q = (q

_{1}, . . . , q

_{n})

*be positive weights,*a = (a1, . . . , an)

*a sequence of positive numbers. If the sequences*(p1/q1, . . . , pn/qn)

*and*a

*are both nonincreasing or both nondecreasing, then*

(3.11) M_{n}(p;a;r, s)≥M_{n}(q;a;r, s).

*If one of the sequences of*(p1/q1, . . . , pn/qn)*or*a*is nonincreasing and the other*
*nondecreasing, the inequality (3.11) is reversed.*

*Proof. Substitution of* P = (q_{1}a^{s}_{1}, . . . , q_{n}a^{s}_{n}), A = (a^{r−s}_{1} , . . . , a^{r−s}_{n} ) and
B = (p1/q1, . . . , pn/qn)into inequality (3.10) and the standard arguments pro-
duce inequality (3.11). This completes the proof of Theorem3.5.

* Theorem 3.6. Let* p = (p

_{1}, . . . , p

_{n})

*be positive weights,*a = (a

_{1}, . . . , a

_{n})

*and*b = (b

_{1}, . . . , b

_{n})

*two sequences of positive numbers. If the sequences*(a

_{1}/b

_{1}, . . . , a

_{n}/b

_{n})

*and*b

*are both increasing or both decreasing, then*

(3.12) M_{n}(p;a;r, s)≥M_{n}(p;b;r, s)

*holds for* a_{i}/b_{i} ≥ 1, n ≥ i ≥ 1, and r, s ≥ 0 *or*r ≥ 0 ≥ s. The inequality
(3.12) is reversed fora_{i}/b_{i} ≤1,n ≥i≥1, andr, s≤0*or*s≥0≥r.

*If one of the sequences of*(a_{1}/b_{1}, . . . , a_{n}/b_{n}) *or*b *is nonincreasing and the*
*other nondecreasing, then inequality (3.12) is valid for* ai/bi ≥ 1, n ≥ i ≥ 1
*and* r, s ≥ 0 *or* s ≥ 0 ≥ r; the inequality (3.12) reverses for a_{i}/b_{i} ≤ 1,
n ≥i≥1, andr, s≥0*or*r≥0≥s,.

*Proof. The inequality (3.10) applied to*
(3.13) P_{i} =p_{i}b^{r}_{i}, A_{i} =a_{i}

b_{i}
r

, B_{i} =b^{s−r}_{i} , 1≤i≤n

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and the standard arguments yield Theorem3.6.

* Theorem 3.7. Suppose*p

*and*g

*are defined on*R

*. If*f

_{1}◦g

*has constant sign and*

*if*(f

_{1}/f

_{2})◦g

*is increasing (or decreasing, respectively), then*M(p;g;f

_{1}, f

_{2};x, y)

*have the inverse (or same) monotonicities as*f1/f2

*with both*x

*and*y.

*Proof. Without loss of generality, suppose*(f_{1}/f_{2})◦g increases. By straightfor-
ward computation and using Lemma2.1, we obtain

(3.14) d dy

R^{y}

x p(t)f_{1}(g(t))dt
Ry

x p(t)f_{2}(g(t))dt

= p(y)f_{1}(g(y))Ry

x p(t)f_{1}(g(t))dt
Ry

x p(t)f_{2}(g(t))dt2

R^{y}

x p(t)f_{2}(g(t))dt
Ry

x p(t)f_{1}(g(t))dt − f_{2}(g(y))
f_{1}(g(y))

≤0.

From Definition2.4and its suitable basic properties, Theorem3.7follows.

* Lemma 3.8. Let* G, H : [a, b] → R

*be integrable functions, both increasing*

*or both decreasing. Furthermore, let*Q : [a, b] → [0,+∞)

*be an integrable*

*function. Then*

(3.15) Z b

a

Q(u)G(u)du Z b

a

Q(u)H(u)du≤ Z b

a

Q(u)du Z b

a

Q(u)G(u)H(u)du.

*If one of the functions of*G*or*H *is nonincreasing and the other nondecreasing,*
*then the inequality (3.15) reverses.*

Inequality (3.15) is called Tchebycheff’s integral inequality, please refer to [1] and [4]–[7].

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**Remark 3.1. Using Tchebycheff’s integral inequality, some inequalities of the**
complete elliptic integrals are established in [15], many inequalities concerning
the probability function, the error function, and so on, are improved in [12].

* Theorem 3.9. Suppose* f

_{2}◦g

*has constant sign on*[x, y]. Wheng(t)

*increases*

*on*[x, y], ifp

_{1}/p

_{2}

*is increasing, we have*

(3.16) M(p_{1};g;f_{1}, f_{2};x, y)≥M(p_{2};g;f_{1}, f_{2};x, y);

*if*p_{1}/p_{2} *is decreasing, inequality (3.16) reverses.*

*When*g(t)*decreases on*[x, y], ifp_{1}/p_{2} *is increasing, then inequality (3.16)*
*is reversed; if*p_{1}/p_{2}*is decreasing, inequality (3.16) holds.*

*Proof. Substitution of*Q(t) = f2(g(t))p2(t),G(t) = (f1/f2)◦g(t)andH(t) =
p_{1}(t)/p_{2}(t) into Lemma 3.8 and the standard arguments produce inequality
(3.16). The proof of Theorem3.9is completed.

* Theorem 3.10. Suppose*f

_{2}◦g

_{2}

*does not change its sign on*[x, y].

*(i) When*f_{2}◦(g_{1}/g_{2})*and*(f_{1}/f_{2})◦g_{2}*are both increasing or both decreasing,*
*inequality*

(3.17) M(p;g_{1};f_{1}, f_{2};x, y)≥M(p;g_{2};f_{1}, f_{2};x, y)

*holds for*f_{1}/f_{2}*being increasing, or reverses for*f_{1}/f_{2}*being decreasing.*

*(ii) When one of the functions* f_{2} ◦(g_{1}/g_{2}) *or*(f_{1}/f_{2})◦g_{2} *is decreasing and*
*the other increasing, inequality (3.17) holds for*f_{1}/f_{2} *being decreasing,*
*or reverses for*f_{1}/f_{2}*being increasing.*

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*Proof. The inequality (3.15) applied to*Q(t) = p(t)(f_{2} ◦g_{2})(t), G(t) = f_{2} ◦
g_{1}

g_{2}

(t) and H(t) =
f_{1}

f_{2}

◦g_{2}(t), and standard arguments yield Theorem
3.10.

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**References**

[1] E.F. BECKENBACH AND*R. BELLMAN, Inequalities, Springer, Berlin,*
1983.

[2] P.S. BULLEN, D.S. MITRINOVI ´C AND P.M. VASI ´*C, Means and Their*
*Inequalities, D. Reidel Publ. Company, Dordrecht, 1988.*

*[3] Y. DING, Two classes of means and their applications, Mathematics in*
**Practice and Theory, 25(2) (1995), 16–20. (Chinese)**

[4] G.H. HARDY, J.E. LITTLEWOODAND*G. PÓLYA, Inequalities, 2nd edi-*
tion, Cambridge University Press, Cambridge, 1952.

*[5] J.-C. KUANG, Applied Inequalities (Changyong Budengshi), 2nd edition,*
Hunan Education Press, Changsha, China, 1993. (Chinese)

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

[7] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´CAND*A.M. FINK, Classical and New*
*Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.*

[8] J. PE ˇCARI ´C, F. QI, V. ŠIMI ´CANDS.-L. XU, Refinements and extensions
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*[9] F. QI, Generalized weighted mean values with two parameters, Proc. Roy.*

**Soc. London Ser. A, 454(1978) (1998), 2723–2732.**

[10] F. QI, Monotonicity results and inequalities for the gamma
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**2(7),** (1999), article 7. [ONLINE] Available online at
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*[11] F. QI, Studies on Problems in Topology and Geometry and on General-*
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[12] F. QI, L.-H. CUI AND S.-L. XU, Some inequalities constructed by
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**2(7)** (1999), article 11. [ONLINE] Available online at
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[15] F. QI AND Z. HUANG, Inequalities of the complete elliptic integrals,
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[16] F. QI, S.-L. XU AND L. DEBNATH, A new proof of monotonicity for
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[17] F. QI AND S.-Q. ZHANG, Note on monotonicity of generalized
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3259–3260.

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[18] M.-B. SUN, Inequalities for two-parameter mean of convex function,
* Mathematics in Practice and Theory, 27(3) (1997), 193–197. (Chinese)*
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