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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

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Generalized Abstracted Mean Values

Feng Qi

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J. Ineq. Pure and Appl. Math. 1(1) Art. 4, 2000

http://jipam.vu.edu.au

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

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_p₁_,f(r, s;x, y)≥M_p₂_,f(r, s;x, y), (1.6)

M_p,f₁(r, s;x, y)≥M_p,f₂(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 results 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 generalized abstracted mean values, to research their basic properties, and to prove their monotonicities. In Section2, we introduce some definitions of mean values 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. Letpbe 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 function 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) = f⁰(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 thatf andgare integrable, andg 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 pointθ∈(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. Sincef(t)/g(t)has finitely many removable discontinuity points, without 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 meanM_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), whereα= inf

t∈[x,y]g(t)andβ = sup

t∈[x,y]

g(t).

Proof. This follows from Lemma2.1and standard arguments.

Definition 2.2. For a sequence of positive numbers a = (a₁, . . . , a_n)and positive 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_ia^s_i







1/(r−s)

, r−s6= 0;

(2.5)

M_n(p;a;r, r) = exp







n

P

i=1

p_ia^r_i lna_i

n

P

i=1

p_ia^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 meanM_n(p;a;r, s)is a continuous function with respect to (r, s)∈R² 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), wherem= min

1≤i≤n{a_i},M = max

1≤i≤n{a_i}.

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Proof. For an arbitrary sequence b = (b₁, . . . , 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. Letf₁andf₂ be real-valued functions such that the ratiof₁/f₂ is monotone on the closed interval [α, β]. Ifa = (a₁, . . . , a_n)is a sequence of real numbers from[α, β]andp = (p₁, . . . , p_n)a sequence of positive numbers, the generalized abstracted mean values of numbersa with respect to functions f₁ andf₂, with weightsp, is defined by

(2.9) M_n(p;a;f₁, f₂) = f₁

f2

⁻¹







n

P

i=1

p_if₁(a_i)

n

P

i=1

p_if₂(a_i)





 ,

where(f₁/f₂)⁻¹is the inverse function off₁/f₂. 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 functionsf₁ andf₂and with weightpis defined as (2.10) M(p;g;f1, f2;x, y) =

f₁ f₂

−1 R^y

x p(t)f₁(g(t))dt Ry

x p(t)f₂(g(t))dt

, where(f₁/f₂)⁻¹is the inverse function off₁/f₂.

Remark 2.1. Set f₂ ≡ 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= (a₁, . . . , a_n)with weightsp= (p₁, . . . , p_n).

Lemma 2.4. Suppose the ratiof₁/f₂ is monotonic on a given interval. Then (2.11)

f1

f₂ −1

(x) = f2

f₁ −1

1 x

, where(f₁/f₂)⁻¹is the inverse function off₁/f₂.

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

Theorem 2.5. The meansM_n(p;a;f₁, f₂)andM(p;g;f₁, f₂;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;f₁, f₂)≤M,

M_n(p;a;f₁, f₂) =M_n(p;a;f₂, f₁), wherem = 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;f₁, f₂;x, y)≤β, M(p;g;f₁, f₂;x, y) =M(p;g;f₁, f₂;y, x), M(p;g;f1, f2;x, y) =M(p;g;f2, f1;x, y), whereα = inf

t∈[x,y]g(t)andβ = sup

t∈[x,y]

g(t).

Proof. These follow from inequality (2.8), Lemma2.1, Lemma2.4, and standard arguments.

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3. Monotonicities

Lemma 3.1 ([16]). Assume that the derivative of second orderf⁰⁰(t)exists on R. Iff(t)is an increasing (or convex) function onR, then the arithmetic mean of functionf(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 bothrandsonR. 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)

∂²φ(r, s)

∂s² = (s−r)²f⁰(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)²f⁰⁰(s).

(3.4)

In the case off⁰(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 f⁰⁰(t) ≥ 0, ϕ(r, s) increases with s. Since ϕ(r, r) = 0, we have∂²φ(r, s)/∂s² ≥ 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 meanM_n(p;a;r, s)of numbersa= (a₁, . . . , a_n)with weights p= (p1, . . . , pn)and two parametersrandsis increasing in bothrands.

Proof. SetN_n= lnM_n, then we have

N_n(p;a;r, s) = 1 r−s

Z r s

n

P

i=1

p_ia^t_ilna_i

n

P

i=1

p_ia^t_i

dt, r−s6= 0;

(3.5)

Nn(p;a;r, r) =

n

P

i=1

p_ia^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_ilnai n

P

i=1

p_ia^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_ilnai

2

ⁿ P

i=1

p_ia^t_i2 ≥0.

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

Theorem 3.3. For a monotonic sequence of positive numbers 0 < a₁ ≤ a₂ ≤

· · · and positive weightsp= (p₁, p₂, . . .), ifm < n, then (3.8) Mm(p;a;r, s)≤Mn(p;a;r, s).

Equality holds ifa1 =a2 =· · ·.

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Proof. Forr ≥s, inequality (3.8) reduces to

(3.9)

m

P

i=1

p_ia^r_i

m

P

i=1

p_ia^s_i

≤

n

P

i=1

p_ia^r_i

n

P

i=1

p_ia^s_i .

Since 0 < a₁ ≤ a₂ ≤ · · ·, p_i > 0, i ≥ 1, the sequences

p_ia^r_i ^∞_i=1 and p_ia^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. IfA= (A₁, . . . , A_n)andB = (B₁, . . . , B_n)are two nondecreas- ing (or nonincreasing) sequences and P = (P₁, . . . , 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 sequencesAorB is constant.

If one of the sequencesAorBis 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 = (p₁, . . . , p_n) andq = (q₁, . . . , q_n)be positive weights, a = (a1, . . . , an)a sequence of positive numbers. If the sequences(p1/q1, . . . , pn/qn) andaare 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)orais nonincreasing and the other nondecreasing, the inequality (3.11) is reversed.

Proof. Substitution of P = (q₁a^s₁, . . . , q_na^s_n), A = (a^r−s₁ , . . . , a^r−s_n ) and B = (p1/q1, . . . , pn/qn)into inequality (3.10) and the standard arguments produce inequality (3.11). This completes the proof of Theorem3.5.

Theorem 3.6. Let p = (p₁, . . . , p_n) be positive weights, a = (a₁, . . . , a_n) and b = (b₁, . . . , b_n) two sequences of positive numbers. If the sequences (a₁/b₁, . . . , a_n/b_n)andbare 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 orr ≥ 0 ≥ s. The inequality (3.12) is reversed fora_i/b_i ≤1,n ≥i≥1, andr, s≤0ors≥0≥r.

If one of the sequences of(a₁/b₁, . . . , a_n/b_n) orb 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≥0orr≥0≥s,.

Proof. The inequality (3.10) applied to (3.13) P_i =p_ib^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. Supposepandgare defined onR. Iff₁◦ghas constant sign and if(f₁/f₂)◦gis increasing (or decreasing, respectively), thenM(p;g;f₁, f₂;x, y) have the inverse (or same) monotonicities asf1/f2with bothxandy.

Proof. Without loss of generality, suppose(f₁/f₂)◦g increases. By straightfor- ward computation and using Lemma2.1, we obtain

(3.14) d dy

R^y

x p(t)f₂(g(t))dt

= p(y)f₁(g(y))Ry

x p(t)f₂(g(t))dt2

R^y

x p(t)f₂(g(t))dt Ry

x p(t)f₁(g(t))dt − f₂(g(y)) f₁(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 ofGorH 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 f₂ ◦g has constant sign on[x, y]. Wheng(t)increases on[x, y], ifp₁/p₂is increasing, we have

(3.16) M(p₁;g;f₁, f₂;x, y)≥M(p₂;g;f₁, f₂;x, y);

ifp₁/p₂ is decreasing, inequality (3.16) reverses.

Wheng(t)decreases on[x, y], ifp₁/p₂ is increasing, then inequality (3.16) is reversed; ifp₁/p₂is decreasing, inequality (3.16) holds.

Proof. Substitution ofQ(t) = f2(g(t))p2(t),G(t) = (f1/f2)◦g(t)andH(t) = p₁(t)/p₂(t) into Lemma 3.8 and the standard arguments produce inequality (3.16). The proof of Theorem3.9is completed.

Theorem 3.10. Supposef₂◦g₂ does not change its sign on[x, y].

(i) Whenf₂◦(g₁/g₂)and(f₁/f₂)◦g₂are both increasing or both decreasing, inequality

(3.17) M(p;g₁;f₁, f₂;x, y)≥M(p;g₂;f₁, f₂;x, y)

holds forf₁/f₂being increasing, or reverses forf₁/f₂being decreasing.

(ii) When one of the functions f₂ ◦(g₁/g₂) or(f₁/f₂)◦g₂ is decreasing and the other increasing, inequality (3.17) holds forf₁/f₂ being decreasing, or reverses forf₁/f₂being increasing.

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Proof. The inequality (3.15) applied toQ(t) = p(t)(f₂ ◦g₂)(t), G(t) = f₂ ◦ g₁

g₂

(t) and H(t) = f₁

f₂

◦g₂(t), and standard arguments yield Theorem 3.10.

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References

[1] E.F. BECKENBACH ANDR. 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. LITTLEWOODANDG. PÓLYA, Inequalities, 2nd edition, Cambridge University Press, Cambridge, 1952.

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

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