PE ˇCARI ´C 1 ABDUSSALAMSCHOOL OFMATHEMATICALSCIENCES GC UNIVERSITY, LAHORE, PAKISTAN matloob_t@yahoo.com UNIVERSITYOFZAGREB FACULTYOFTEXTILETECHNOLOGY, CROATIA pecaric@mahazu.hazu.hr Received 14 July, 2008

CAUCHY’S MEANS OF LEVINSON TYPE

MATLOOB ANWAR AND J. PE ˇCARI ´C

1 ABDUSSALAMSCHOOL OFMATHEMATICALSCIENCES

GC UNIVERSITY, LAHORE, PAKISTAN

matloob_t@yahoo.com UNIVERSITYOFZAGREB

FACULTYOFTEXTILETECHNOLOGY, CROATIA

pecaric@mahazu.hazu.hr

Received 14 July, 2008; accepted 12 October, 2008 Communicated by S. Abramovich

ABSTRACT. In this paper we introduce Levinson means of Cauchy’s type. We show that these means are monotonic.

Key words and phrases: Convex function, Ky-Fan inequality, Weighted mean.

2000 Mathematics Subject Classification. Primary 26A51; Secondary 26A46, 26A48.

1. INTRODUCTION ANDPRELIMINARIES

Letx₁, x₂, . . . , x_nandp₁, p₂, . . . , p_nbe real numbers such thatx_i ∈[0,¹₂],p_i >0withP_n = Pn

i=1p_i. Let G_n and A_n be the weighted geometric mean and arithmetic mean respectively defined by

Gn =

n

Y

i=1

x^p_iⁱ

!_Pn¹

and An = 1 P_n

n

X

i=1

pixi =x.

In particular, consider the means

G⁰_n=

n

Y

i=1

(1−x_i)^pi

!_Pn¹

and A⁰_n = 1 P_n

n

X

i=1

p_i(1−x_i).

The well known Levinson inequality is the following ([1, 2] see also [6, p. 71]).

Theorem 1.1. Letf be a real valued 3-convex function on[0,2a]. Then for0< x_i < a,p_i >0 we have

The research of the second author was supported by the Croatian Ministry of Science, Education and Sports under the Research Grants 117-1170889-0888.

202-08

(1.1) 1 P_n

n

X

i=1

p_if(x_i)−f 1 P_n

n

X

i=1

p_ix_i

!

≤ 1 Pn

n

X

i=1

p_if(2a−x_i)−f 1 Pn

n

X

i=1

p_i(2a−x_i)

! .

In [4], the second author proved the following similar result.

Theorem 1.2. Letf be a real valued 3-convex function on[0,2a]andx_i (1≤i ≤n)npoints on[0, a]. Then

(1.2) 1 P_n

n

X

i=1

p_if(x_i)−f 1 P_n

n

X

i=1

p_ix_i

!

≤ 1 P_n

n

X

i=1

p_if(a+x_i)−f 1 P_n

n

X

i=1

p_i(a+x_i)

! .

Lemma 1.3. Letf be a log-convex function. If,x₁ ≤ y₁, x₂ ≤y₂, x₁ 6=x₂, y₁ 6=y₂, then the following inequality is valid:

(1.3)

f(x₂) f(x₁)

_x ¹

2−x1

≤

f(y₂) f(y₁)

_y ¹

2−y1

.

Lemma 1.4. Letf ∈C³(I)for some intervalI ⊆R, such thatf⁰⁰⁰is bounded andm = minf⁰⁰⁰ andM = maxf⁰⁰⁰.Consider the functionsφ₁, φ₂ defined as,

(1.4) φ₁(t) = M

6 t³−f(t),

(1.5) φ2(t) = f(t)− m

6t³, thenφ₁andφ₂ are3-convex functions.

Lemma 1.5. Let us define the function

(1.6) ϕ_s(x) =















x^s

s(s−1)(s−2), s6=0,1,2;

1

2logx, s=0;

−xlogx, s=1;

1

2x²logx, s=2.

Thenϕ⁰⁰⁰_s(x) =x^s−3,that isϕs(x)is3-convex forx >0.

LetI ⊆Rbe an interval and letF be some appropriately chosen vector space of real valued functions defined on I. LetΨbe a functional on F and letA : F → R be a linear operator, whereRis the vector of all real valued functions defined onI.

Suppose that for eachf ∈ F, there is aξ∈I such that

(1.7) Ψ(f) = A(f)(ξ).

J. Pe´cari´c, I. Peri´c and H. Srivastava in [5] proved the following important result forφ andA defined above.

Theorem 1.6. For everyf, g∈ F, there is aξ ∈I such that

(1.8) A(g)(ξ)Ψ(f) = A(f)(ξ)Ψ(g).

2. MAINRESULTS

Theorem 2.1. Let f ∈ C³(I). Then for xi > 0and pi > 0 there exist ξ ∈ I such that the following equality holds true,

(2.1) f 1 Pn

n

X

i=1

p_ix_i

!

− 1 Pn

n

X

i=1

p_if(x_i)

+ 1 P_n

n

X

i=1

p_if(2a−x_i)−f 1 P_n

n

X

i=1

p_i(2a−x_i)

!

= f⁰⁰⁰(ξ) 6



 1 P_n

n

X

i=1

p_ix_i

!3

− 1 P_n

n

X

i=1

p_ix³_i

+ 1 Pn

n

X

i=1

p_i(2a−x_i)³− 1 Pn

n

X

i=1

p_i(2a−x_i)

!3

.

Proof. Suppose that f⁰⁰⁰ is bounded, that is, minf⁰⁰⁰ = m, maxf⁰⁰⁰ = M. By applying the Levinsen inequality (1.1) to the functionsφ1andφ2defined in Lemma 1.4, we get the following inequalities,

(2.2) f 1 P_n

n

X

i=1

p_ix_i

!

− 1 P_n

n

X

i=1

p_if(x_i)

+ 1 P_n

n

X

i=1

p_if(2a−x_i)−f 1 P_n

n

X

i=1

p_i(2a−x_i)

!

≤ M 6



 1 P_n

n

X

i=1

p_ix_i

!3

− 1 P_n

n

X

i=1

p_ix³_i

+ 1 P_n

n

X

i=1

p_i(2a−x_i)³− 1 P_n

n

X

i=1

p_i(2a−x_i)

!3



and

(2.3) m 6



 1 P_n

n

X

i=1

p_ix_i

!3

− 1 P_n

n

X

i=1

p_ix³_i

+ 1 P_n

n

X

i=1

p_i(2a−x_i)³− 1 P_n

n

X

i=1

p_i(2a−x_i)

!3



≤f 1 P_n

n

X

i=1

pixi

!

− 1 P_n

n

X

i=1

pif(xi)

+ 1 Pn

n

X

i=1

p_if(2a−x_i)−f 1 Pn

n

X

i=1

p_i(2a−x_i)

! .

By combining both inequalities and using the fact that form ≤ ρ≤ M there existξ ∈I such thatf⁰⁰⁰(ξ) = ρ, we get (2.1). Moreover, iff⁰⁰⁰ is (for example) bounded from above we have that (2.2) is valid and again (2.1) is valid.

Of course (2.1) is obvious iff⁰⁰⁰is not bounded from above and below.

Theorem 2.2. Let f, g ∈ C³(I). Then forx_i > 0andp_i > 0, i = 1, . . . , nthere existξ ∈ I such that the following equality holds true,

(2.4) f⁰⁰⁰(ξ) g⁰⁰⁰(ξ) =

f

1 Pn

n

P

i=1

p_ix_i

−_P¹

n

n

P

i=1

p_if(x_i)+_P¹

n

n

P

i=1

p_if(2a−x_i)−f

1 Pn

n

P

i=1

p_i(2a−x_i)

g

1 Pn

n

P

i=1

pixi

−_P¹

n

n

P

i=1

pig(xi)+_P¹

n

n

P

i=1

pig(2a−xi)−g

1 Pn

n

P

i=1

pi(2a−xi) .

Proof. Consider the linear functionals Ψand A as in (1.7) for F = C³(I) and R the vector space of real valued functions such thatΨ(k) =A(k)(ξ)for some functionk. LetAbe defined as:

(2.5) A(f)(ξ) = f⁰⁰⁰(ξ) 6



 1 Pn

n

X

i=1

p_ix_i

!3

− 1 Pn

n

X

i=1

p_ix³_i

+ 1 P_n

n

X

i=1

pi(2a−xi)³− 1 P_n

n

X

i=1

pi(2a−xi)

!3

.

Also, consider the linear combinationk =c1f−c2g,wheref, g ∈C³(I)andc1, c2are defined by

c₁ = Ψ(g) (2.6)

=g 1 P_n

n

X

i=1

p_ix_i

!

− 1 P_n

n

X

i=1

p_ig(x_i)

+ 1 P_n

n

X

i=1

p_ig(2a−x_i)−g 1 P_n

n

X

i=1

p_i(2a−x_i)

! ,

c₂ = Ψ(f) (2.7)

=f 1 P_n

n

X

i=1

p_ix_i

!

− 1 P_n

n

X

i=1

p_if(x_i)

+ 1 P_n

n

X

i=1

p_if(2a−x_i)−f 1 P_n

n

X

i=1

p_i(2a−x_i)

! .

Obviously, we haveΨ(k) = 0. This implies that (as in Theorem 1.6):

(2.8) Ψ(g)A(f)(ξ) = Ψ(f)A(g)(ξ).

Now sinceΨ(g)6= 0andA(g)(ξ)6= 0we have from the last equation

(2.9) Ψ(f)

Ψ(g) = A(f)(ξ) A(g)(ξ).

After putting in the values we get (2.4).

Corollary 2.3. Let ^f_g000⁰⁰⁰ be invertible then (2.4) suggests new means. That is,

(2.10) ξ=

f⁰⁰⁰ g⁰⁰⁰

−1







 f

1 P_n

n

P

i=1

pixi

−_P¹

n

n

P

i=1

pif(xi)+_P¹

n

n

P

i=1

pif(2a−xi)−f

1 P_n

n

P

i=1

pi(2a−xi)

g

1 Pn

n

P

i=1

pixi

−_P¹

n

n

P

i=1

pig(xi)+_P¹

n

n

P

i=1

pig(2a−xi)−g

1 Pn

n

P

i=1

pi(2a−xi)









is a new mean.

Definition 2.1. Define the function

(2.11) ξ_s = 1

P_n

n

X

i=1

p_i

ϕ_s(2a−x_i)−ϕ_s(x_i)

−ϕ_s(2a−x) +ϕ_s(x),

whens 6= 0,1,2. s = 0,1,2are limiting cases defined by ξ₀ = 1

2ln

G^a_nA_n G_nA^a_n

,

where

G^a_n =

" _n Y

i=1

(2a−xi)^pi

#_Pn¹

and A^a_n = 1 P_n

n

X

i=1

pi(2a−xi),

ξ₁ = 1 P_n

n

X

i=1

p_i

x_ilnx_i−(2a−x_i) ln(2a−x_i)

+ (2a−x) ln(2a−x)−xlnx,

ξ₂ = 1 2

"

1 P_n

n

X

i=1

p_i

(2a−x_i)²ln(2a−x_i)−x²_i lnx_i

−(2a−x)²ln(2a−x) +x²lnx

# .

Then we define the new meansM_s,tas:

Definition 2.2. Let us denote:

(2.12) M_s,t =

ξ_s ξt

_s−t¹

fors 6=t 6= 0,1,2. We define these limiting cases as M_s,s = exp

"

η

1 Pn

Pn

i=1p_i((2a−x_i)^s−x^s_i)−(2a−x)¯ ^s+ ¯x^s − 3s²−6s+ 2 s(s−1)(s−2)

# ,

where η = 1

P_n

n

X

i=1

pi((2a−xi)^slog(2a−xi)−x^s_ilogxi)−(2a−x)¯ ^slog(2a−x) + ¯¯ x^slog ¯x fors 6= 0,1,2

M_0,0 = exp



 2

1 Pn

Pn

i=1p_i[(log(2a−x_i))²−(logx_i)²] 4ξ₀

− (log(2a−x))¯ ²−(log ¯x)²6ξ0

4ξ₀

,

M_1,1 = exp

" ₁

Pn

Pn

i=1p_i[(2a−x_i)(log(2a−x_i))² −x_i(logx_i)²] 2ξ₁

− (2a−x)(log(2a¯ −x))¯ ²−x(log ¯¯ x)² 2ξ₁

,

M_2,2 = exp

" ₁

Pn

Pn

i=1p_i[(2a−x_i)²(log(2a−x_i))²−x²_i(logx_i)²] 3ξ₂

− (2a−x)¯ ²(log(2a−x))¯ ²−x¯²(log ¯x)²

3ξ₂ −1

.

In our next result we prove that this new mean is monotonic.

Theorem 2.4. Letr≤s, t≤u, r 6=t, s 6=u, then the following inequality is valid:

(2.13) M_r,t ≤M_s,u.

Proof. Since ξ_s is log convex as proved in [3, Theorem 2.2], then applying Lemma 1.4 for

r≤s, t ≤u, r6=t, s6=uwe get our required result.

3. RELATEDRESULTS

Theorem 3.1. Letf ∈C³(I). Forx_i >0andp_i >0, i= 1, . . . , nthere existξ ∈I such that the following equality holds true,

(3.1) f 1 P_n

n

X

i=1

p_ix_i

!

− 1 P_n

n

X

i=1

p_if(x_i)

+ 1 P_n

n

X

i=1

p_if(a+x_i)−f 1 P_n

n

X

i=1

p_i(a+x_i)

!

= f⁰⁰⁰(ξ) 6



 1 P_n

n

X

i=1

p_ix_i

!3

− 1 P_n

n

X

i=1

p_ix³_i

+ 1 P_n

n

X

i=1

p_i(a+x_i)³− 1 P_n

n

X

i=1

p_i(a+x_i)

!3

.

Proof. Similar to proof of Theorem 2.1.

Theorem 3.2. Let f, g ∈ C³(I). Then forx_i > 0andp_i > 0, i = 1, . . . , nthere existξ ∈ I such that the following equality holds true,

(3.2) f⁰⁰⁰(ξ) g⁰⁰⁰(ξ) =

f

1 Pn

n

P

i=1

p_ix_i

−_P¹

n

n

P

i=1

p_if(x_i)+_P¹

n

n

P

i=1

p_if(a+x_i)−f

1 Pn

n

P

i=1

p_i(a+x_i)

g

1 Pn

n

P

i=1

p_ix_i

−_P¹

n

n

P

i=1

p_ig(x_i)+_P¹

n

n

P

i=1

p_ig(a+x_i)−g

1 Pn

n

P

i=1

p_i(a+x_i)

Proof. Similar to proof of Theorem 2.2.

Corollary 3.3. Let ^f_g000⁰⁰⁰ be invertible. Then (3.2) suggests new means. That is,

(3.3) ξ=

f⁰⁰⁰ g⁰⁰⁰

−1







 f

1 P_n

n

P

i=1

pixi

−_P¹

n

n

P

i=1

pif(xi)+_P¹

n

n

P

i=1

pif(a+xi)−f

1 P_n

n

P

i=1

pi(a+xi)

g

1 Pn

n

P

i=1

p_ix_i

−_P¹

n

n

P

i=1

p_ig(x_i)+_P¹

n

n

P

i=1

p_ig(a+x_i)−g

1 Pn

n

P

i=1

p_i(a+x_i)









is a new mean.

Definition 3.1. Define the function

(3.4) ξ¯_s = 1

P_n

n

X

i=1

p_i

ϕ_s(a+x_i)−ϕ_s(x_i)

−ϕ_s(a+x) +ϕ_s(x),

whens 6= 0,1,2. s = 0,1,2are limiting cases defined by ξ¯₀ = 1

2ln

G¯^a_nA_n G_nA¯^a_n

,

where

G¯^a_n =

n

Y

i=1

(a+x_i)^pi

!_Pn¹

, and A¯^a_n= 1 P_n

n

X

i=1

p_i(a+x_i),

ξ¯₁ = 1 P_n

n

X

i=1

p_i

x_ilnx_i−(a+x_i) ln(2a−x_i)

+ (a+x) ln(a+x)−xlnx,

ξ¯₂ = 1 2

"

1 Pn

n

X

i=1

p_i (a+x_i)²ln(a+x_i)−x²_i lnx_i

−(a+x)²ln(a+x) +x²lnx

# .

We now define new meansM_s,t as:

Definition 3.2. Let us denote:

(3.5) M_s,t =

ξ¯_s ξ¯_t

_s−t¹

fors 6=t 6= 0,1,2. We define these limiting cases as

M_s,s = exp η¯

1 Pn

Pn

i=1pi((a+xi)^s−x^s_i)−(a+ ¯x)^s+ ¯x^s − 3s²−6s+ 2 s(s−1)(s−2)

! ,

where

¯ η = 1

P_n

n

X

i=1

pi((a+xi)^slog(a+xi)−x^s_ilogxi)−(a+ ¯x)^slog(a+ ¯x) + ¯x^slog ¯x fors 6= 0,1,2

M_0,0 = exp



 2

1 Pn

Pn

i=1pi[(log(a+xi))²−(logxi)²]

4 ¯ξ₀

− (log(a+ ¯x))²−(log ¯x)²6 ¯ξ0

4 ¯ξ0

,

M_1,1 = exp

" ₁

Pn

Pn

i=1p_i((a+x_i)(log(a+x_i))²−x_i(logx_i)²) 2 ¯ξ1

− (a+ ¯x)(log(a+ ¯x))²−x(log ¯¯ x)² 2 ¯ξ₁

,

M_2,2 = exp

" ₁

Pn

Pn

i=1p_i((a+x_i)²(log(a+x_i))²−x²_i(logx_i)²) 3 ¯ξ₂

− (a+ ¯x)²(log(a+ ¯x))²−x¯²(log ¯x)²

3 ¯ξ₂ −1

.

In our next result we prove that this new mean is monotonic.

Theorem 3.4. Letr≤s, t≤u, r 6=t, s6=u, then the following inequality is valid:

(3.6) M_r,t ≤M_s,u.

Proof. Sinceξ¯_s is log convex as proved in [3, Theorem 2.5] (ξ¯_s = ρ_s), then applying Lemma 1.3 forr ≤s, t≤u, r6=t, s6=uwe get our required result.

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[3] M. ANWARANDJ.E. PE ˇCARI ´C, On logarithmic convexity for Ky-Fan inequality, J. Inequal. and Appl., (2008), Article ID 870950.

[4] J.E. PE ˇCARI ´C, An inequality for 3-convex functions, J. Math. Anal. Appl., 19 (1982), 213–218.

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