CAUCHY’S MEANS OF LEVINSON TYPE

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Cauchy’s Means of Levinson Type M. Anwar and J. Peˇcari´c vol. 9, iss. 4, art. 120, 2008

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CAUCHY’S MEANS OF LEVINSON TYPE

MATLOOB ANWAR J. PE ˇCARI ´C

1- Abdus Salam School of Mathematical Sciences University Of Zagreb GC University, Lahore, Faculty Of Textile Technology

Pakistan Croatia

EMail:matloob_t@yahoo.com EMail:pecaric@mahazu.hazu.hr

Received: 14 July, 2008

Accepted: 12 October, 2008 Communicated by: S. Abramovich

2000 AMS Sub. Class.: Primary 26A51; Secondary 26A46, 26A48.

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

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

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

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1. Introduction and Preliminaries

Letx₁, x₂, . . . , x_n andp₁, p₂, . . . , p_n be real numbers such that x_i ∈ [0,¹₂], p_i > 0 withP_n=Pn

i=1p_i. LetG_nandA_nbe the weighted geometric mean and arithmetic mean respectively defined by

G_n =

n

Y

i=1

x^p_iⁱ

!_Pn¹

and A_n = 1 P_n

n

X

i=1

p_ix_i =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. Let f be a real valued 3-convex function on [0,2a]. Then for 0 <

x_i < a,p_i >0we have

(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 P_n

n

X

i=1

p_if(2a−x_i)−f 1 P_n

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]and x_i (1≤ i ≤ n)npoints on[0, a]. Then

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(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 interval I ⊆ R, such thatf⁰⁰⁰ is bounded and m= minf⁰⁰⁰andM = maxf⁰⁰⁰.Consider the functionsφ₁, φ₂ defined as,

φ₁(t) = M

6 t³−f(t), (1.4)

φ2(t) = f(t)− m 6t³, (1.5)

thenφ₁andφ₂are3-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.

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LetI ⊆Rbe an interval and letF be some appropriately chosen vector space of real valued functions defined onI. LetΨbe a functional onF 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. Pećarić, I. Perić and H. Srivastava in [5] proved the following important result for φandAdefined above.

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

(1.8) A(g)(ξ)Ψ(f) = A(f)(ξ)Ψ(g).

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2. Main Results

Theorem 2.1. Letf ∈ C³(I). Then forx_i > 0and p_i > 0there exist ξ ∈ I such that the following equality holds true,

(2.1) 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

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 P_n

n

X

i=1

pi(2a−xi)³− 1 P_n

n

X

i=1

pi(2a−xi)

!3

.

Proof. Suppose thatf⁰⁰⁰ is bounded, that is,minf⁰⁰⁰ =m,maxf⁰⁰⁰ =M. By applying the Levinsen inequality (1.1) to the functionsφ₁ andφ₂ defined in Lemma1.4, 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

n

X

i=1

p_i(2a−x_i)

!

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≤ M 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

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

pi(2a−xi)³− 1 P_n

n

X

i=1

pi(2a−xi)

!3



≤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

pif(2a−xi)−f 1 P_n

n

X

i=1

pi(2a−xi)

! .

By combining both inequalities and using the fact that form ≤ ρ ≤ M there exist ξ ∈I such thatf⁰⁰⁰(ξ) = ρ, we get (2.1). Moreover, iff⁰⁰⁰is (for example) bounded from above we have that (2.2) is valid and again (2.1) is valid.

Of course (2.1) is obvious iff⁰⁰⁰ is not bounded from above and below.

Theorem 2.2. Letf, g ∈ C³(I). Then for x_i > 0andp_i > 0, i = 1, . . . , nthere

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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

P

i=1

p_if(x_i)+_P¹

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

p_ix_i

−_P¹

n

P

i=1

p_ig(x_i)+_P¹

n

P

i=1

p_ig(2a−x_i)−g

1 Pn

n

P

i=1

p_i(2a−x_i) .

Proof. Consider the linear functionalsΨandAas in (1.7) forF =C³(I)andRthe vector space of real valued functions such thatΨ(k) = A(k)(ξ)for some function k. LetAbe defined as:

(2.5) A(f)(ξ) = 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

.

Also, consider the linear combinationk =c1f −c2g,wheref, g ∈C³(I)andc1, c2

are defined by

c₁ = Ψ(g) =g 1 P_n

n

X

i=1

p_ix_i

!

− 1 P_n

n

X

i=1

p_ig(x_i) (2.6)

+ 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)

! ,

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c₂ = Ψ(f) =f 1 P_n

n

X

i=1

p_ix_i

!

− 1 P_n

n

X

i=1

p_if(x_i) (2.7)

+ 1 P_n

n

X

i=1

n

X

i=1

p_i(2a−x_i)

! .

Obviously, we haveΨ(k) = 0. This implies that (as in Theorem1.6):

(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 Pn

n

P

i=1

pixi

− ¹

Pn n

P

i=1

pif(xi)+_P¹

n n

P

i=1

pif(2a−x_i)−f

1 Pn

n

P

i=1

pi(2a−x_i)

g

1 P_n

n

P

i=1

pixi

−_P¹

n n

P

i=1

pig(xi)+_P¹

n n

P

i=1

pig(2a−xi)−g

1 P_n

n

P

i=1

pi(2a−xi)







is a new mean.

Definition 2.4. 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),

whens6= 0,1,2. s= 0,1,2are limiting cases defined by ξ₀ = 1

2ln

G^a_nA_n G_nA^a_n

,

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where

G^a_n=

" _n Y

i=1

(2a−x_i)^pⁱ

#_Pn¹

and A^a_n= 1 P_n

n

X

i=1

p_i(2a−x_i),

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 2.5. Let us denote:

(2.12) M_s,t =

ξ_s ξ_t

_s−t¹

fors6=t6= 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 Pn

n

X

i=1

p_i((2a−x_i)^slog(2a−x_i)−x^s_i logx_i)−(2a−x)¯ ^slog(2a−x) + ¯¯ x^slog ¯x

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fors6= 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

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

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 2.6. 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. Sinceξ_s is log convex as proved in [3, Theorem 2.2], then applying Lemma 1.4forr≤s, t ≤u, r6=t, s6=uwe get our required result.

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3. Related Results

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 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

p_i(a+x_i)³− 1 P_n

n

X

i=1

p_i(a+x_i)

!3

.

Proof. Similar to proof of Theorem2.1.

Theorem 3.2. Letf, g ∈ C³(I). Then for x_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

P

i=1

p_if(x_i)+_P¹

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

P

i=1

p_ig(x_i)+_P¹

n

P

i=1

p_ig(a+x_i)−g

1 Pn

n

P

i=1

p_i(a+x_i)

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Proof. Similar to proof of Theorem2.2.

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

pixi

− ¹

Pn n

P

i=1

pig(xi)+_P¹

n n

P

i=1

pig(a+xi)−g

1 Pn

n

P

i=1

pi(a+xi)







is a new mean.

Definition 3.4. 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),

whens6= 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 P_n

n

X

i=1

p_i (a+x_i)²ln(a+x_i)−x²_i lnx_i

−(a+x)²ln(a+x) +x²lnx

# .

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We now define new meansM_s,tas:

Definition 3.5. Let us denote:

(3.5) M_s,t =

ξ¯_s ξ¯_t

s−t¹

fors6=t6= 0,1,2. We define these limiting cases as

M_s,s = exp η¯

1 Pn

Pn

i=1p_i((a+x_i)^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 fors6= 0,1,2

M_0,0 = exp



 2

1 Pn

Pn

i=1p_i[(log(a+x_i))²−(logx_i)²]

4 ¯ξ₀

− (log(a+ ¯x))²−(log ¯x)²6 ¯ξ₀ 4 ¯ξ₀

,

M_1,1 = exp

" ₁

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

,

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M_2,2 = exp

" ₁

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 3.6. 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¯_sis log convex as proved in [3, Theorem 2.5] (ξ¯_s =ρ_s), then applying Lemma1.3forr ≤s, t≤u, r6=t, s6=uwe get our required result.

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References

[1] S. LAWRENCEANDD. SEGALMAN, A generalization of two inequalities in- volving means, Proc. Amer. Math. Soc., 35(1) (1972), 96–100.

[2] N. LEVINSON, Generalization of an inequality of Ky Fan, J. Math. Anal. Appl., 8 (1964), 133–134.

[3] M. ANWARANDJ.E. PE ˇCARI ´C, On logarithmic convexity for Ky-Fan inequal- ity, 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.

[5] J.E. PE ˇCARI ´C, I. PERI ´C AND H. SRIVASTAVA A family of the Cauchy type mean-value theorems, J. Math. Anal. Appl., 306 (2005), 730–739.

[6] J.E. PE ˇCARI ´C, F. PROSCHAN AND Y.C. TONG, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, New York, 1992.