On Steffensen Pairs

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FENG QI AND BAI-NI GUO

Abstract. In this article, by mathematical induction and properties of the generalized weighted mean values, some general Steffensen pairs are established.

1. Introduction

Let f and g be integrable functions on [a, b] such thatf is decreasing and 0 6 g(x) 61 for

x∈[a, b]. Then

Z b

b−λ

f(x)dx6 Z b

a

f(x)g(x)dx6 Z a+λ

a

f(x)dx, (1)

whereλ=R_abg(x)dx.

The inequality (1) is called Steffensen’s inequality in [3, 8].

In [1], its discrete analogue of the inequality (1) was proved: Let{xi}ni=1be a decreasing finite

sequence of nonnegative real numbers, {yi}ni=1 be a finite sequence of real numbers such that

06yi61 for 16i6n. Letk1, k2∈ {1,2, . . . , n} be such thatk26

n

P

i=1

yi6k1. Then

n

X

i=n−k2+1

xi6 n

X

i=1 xiyi6

k1

X

i=1

xi. (2)

As a direct consequence of inequality (2), we obtain: Let {xi}ni=1 be a finite sequence of

non-negative real numbers such that

n

P

i=1

xi 6 A and n

P

i=1 x2

i > B2, where A and B are positive real

numbers. Let k ∈ {1,2, . . . , n} be such that k > _BA. Then there are k numbers among {xi}ni=1

whose sum is not less thanB.

From above results, the so-called Steffensen pair was defined in [2] by Dr. H. Gauchman as follows:

Date: July 13, 2000.

1991Mathematics Subject Classification. Primary 26D15.

Key words and phrases. Steffensen inequality, Steffensen pair, generalized weighted mean values, mathematical induction, absolutely monotonic function.

The authors were supported in part by NSF of Henan Province (no. 004051800), SF for Pure Research of the Education Committee of Henan Province (no. 1999110004), and Doctor Fund of Jiaozuo Institute of Technology, The People’s Republic of China.

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Definition 1. Letϕ: [c,∞)→[0,∞) andτ : (0,∞)→(0,∞) be two strictly increasing functions,

c>0, let{xi}ni=1be a finite sequence of real numbers such thatxi>cfor alli,AandBbe positive

real numbers,

n

P

i=1

xi6A, and n

P

i=1

ϕ(xi)>ϕ(B). If, for any k∈ {i}ni=1 satisfyingk>τ(AB), there

arek numbers among{xi}ni=1 whose sum is not less thanB, then we call (ϕ, τ) a Steffensen pair

on [c,∞).

In [1] and [2], the following Steffensen pairs were found:

(xα_{, x}1/(α−1)₎_, _α_>₂_, _x_∈_[0_,_∞_);

(xexp(xα−1),(1 + lnx)1/α), α>1, x∈[1,∞).

It was verified in [5] that: Letaandbbe real numbers satisfyingb > a >1 orb > a−1_>_{1, and}

√

ab>e, then

x

Z b

a

tlnx−1dt, x1/ln√ab !

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is a Steffensen pair on [1,+∞). Ifaandbare real numbers satisfyingb > a >1 and√ab>e, then

x

Z b

a

(lnt)ntlnx−1dt, x

n+2 n+1·

(lnb)n+1−(lna)n+1 (lnb)n+2₋(lna)n+2

!

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are Steffensen pairs on [1,+∞) for any positive integern.

In this article, we will establish more general Steffensen pairs, that is

Theorem 1. Let a, b∈R, letp6≡0 be a nonnegative and integrable function andf a positive and integrable function on the interval [a, b].

(i) If inequality

Z b

a

p(u)du6 Z b

a

p(u) lnf(u)du (5)

holds, then

x

Z b

a

p(u)[f(u)]lnx_{du, x}

Rb a p(u)du

Rb

a p(u) lnf(u)du

!

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is a Steffensen pair on [1,+∞).

(ii) If f(u)>1 and inequality (5) holds, then

x Z b

a

p(u)[f(u)]lnx[lnf(u)]ndu, x Rb

a p(u)[lnf(u)]n du

R_b

a p(u)[lnf(u)]n+1du

!

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are Steffensen pairs on[1,+∞)for any positive integer n.

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2. Lemmas

Lemma 1([2]). Let ψ: [c,∞)→[0,∞)be increasing and convex, c>0. Assume thatψ satisfies ψ(xy) >ψ(x)g(y) for all x>c andy >1, whereg : [1,∞)→ [0,∞) is strictly increasing. Set ϕ(x) =xψ(x),τ(x) =g−1₍_x₎_{, where}_g−1 _{is the inverse function of}_{g. Then} ₍_{ϕ, τ}₎_{is a Steffensen} pair on[c,∞).

Define

h(t) =

Z b

a

p(u)ft₍_u₎_du, _t_∈_R_, ₍₈₎

where p(u) is a nonnegative and continuous function, f(u) a positive and continuous function on the interval [a, b], anda, b∈R.

It is clear [4, 7] that, iff(u)>1 on [a, b], then

h(n)(t) =

Z b

a

p(u)ft(u)[lnf(u)]ndu>0, (9)

that is,h(t) is an absolutely monotonic function, see [3, 4].

By the Cauchy-Schwarz-Buniakowski inequality, it is easy to obtain Lemma 2. Forn>0, iff(u)>1 on [a, b], then we have

[h(n+1)(x)]26h(n)(x)h(n+2)(x), x∈R. (10)

Let a, b, r, s ∈ R, let p 6≡ 0 be a nonnegative and integrable function and f a positive and integrable function on the interval betweena andb. Then the generalized weighted mean values

Mp,f(r, s;a, b) of the functionf with weight pand two parametersrandsare defined in [4] by

Mp,f(r, s;a, b) =

Rb a p(u)f

s₍_u_)d_u

Rb

a p(u)fr(u)du

!1/(s−r)

=

h(s)

h(r)

1/(s−r)

, (r−s)(a−b)6= 0; (11)

Mp,f(r, r;a, b) = exp

Rb

ap(u)fr(u) lnf(u)du

Rb

ap(u)fr(u)du

!

= exp

h0₍_r₎

h(r)

, a−b6= 0; (12)

M(r, s;a, a) =f(a).

From the Cauchy-Schwarz-Buniakowski inequality again and standard argument, we have Lemma 3 ([7]). The generalized weighted mean values Mp,f(r, s;a, b) are increasing with both r

andsfor any given continuous nonnegative weightpand continuous positive functionf.

Lemma 4. Forn>0 andx>0, if Z b

a

p(u)du6 Z b

a

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then we have

h(n+1)₍_x₎_>_h(n)₍_x₎_. ₍₁₄₎

Proof. By Lemma 3, the mean values

h(x+y)

h(x)

1/y

are increasing with respect toxandy, then the function

F(x) =h(x+y)

h(x) is increasing withxfor fixedy>0. Therefore

F0(x) =h0(x+y)h(x)−h(x+y)h0(x) [h(x)]2 >0.

Hence, the inequality

h0(x+y)h(x)−h(x+y)h0(x)>0 (15)

holds for allxand ally>0.

Note that the inequality (15) can also be obtained from the Lemma in [6]. Takingx= 0 in inequality (15), we obtain

h0(y)h(0)−h(y)h0(0)>0 (16)

for ally>0, and

h(0) =

Z b

a

p(u)du, (17)

h0_{(0) =}

Z b

a

p(u) lnf(u)du. (18)

Since inequality (13) means thath0₍₀₎_>_h_{(0), thus}_h0₍_y₎_>_h₍_y_{) for all} _y_>_0.

By mathematical induction, assume thath(n+1)₍_x₎_>_h(n)₍_x_{) for}_n_>_{2 and}_x_>_{0. Then, from}

Lemma 2, we obtain

h(n)(x)h(n+1)(x)6[h(n+1)(x)]26h(n)(x)h(n+2)(x), (19)

therefore

h(n+1)(x)6h(n+2)(x).

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3. New Steffensen Pairs

Now we give a proof of Theorem 1.

Setψ(x) =h(n)_(ln_x_{) for}_x_>_{1 and}_n_>_{0. Direct computation yields that}_ψ0₍_x_{) =} h(n+1)(lnx)

x >

0 andψ00₍_x_{) =}h(n+2)(lnx)−h(n+1)(lnx)

x2 >0. Henceψ(x) is increasing and convex. Sincef(u)>1, forn>1, by Lemma 3, we have

h(n)₍_x₊_y₎ h(n)₍_x₎ =

Rb

a p(u)[f(u)]

x+y_[ln_f₍_u_)]n_du

Rb

ap(u)[f(u)]x[lnf(u)]ndu

>exp

y·

Rb

ap(u)[lnf(u)] n+1_du Rb

ap(u)[lnf(u)]ndu

!

. (20)

Therefore, forx, y>1,

ψ(xy)

ψ(x) =

h(n)_(ln(_xy₎₎ h(n)_(ln_x₎ =

h(n)_(ln_x_{+ ln}_y₎ h(n)_(ln_x₎ >y

Rb

a p_R (u)[lnf(u)]n+1du b

a p(u)[lnf(u)]ndu . (21)

Letg(x) =x Rb

a p_R (u)[lnf(u)]n+1du b

a p(u)[lnf(u)]ndu forx>1, theng−1(x) =x

Rb

a p(u)[lnf(u)]n du

Rb

a p(u)[lnf(u)]n+1du,x∈[1,+∞). By Lemma 1, (ϕ, τ), whereϕ(x) =xψ(x) = xh(n)_(ln_x_{) =} _xRb

ap(u)[f(u)]

lnx_[ln_f₍_u_)]n_du _and

τ(x) =x Rb

a p(u)[lnf(u)]ndu

Rb

a p(u)[lnf(u)]n+1du forx>1 andn>1, are Steffensen pairs on [1,+∞) for any givenn>1. Forn= 0, by Lemma 3, we have

h(x+y)

h(x) =

Rb

a p(u)[f(u)] x+y_du

Rb

ap(u)[f(u)]xdu

>exp

y·

Rb

ap(u) lnf(u)du

Rb ap(u)du

!

. (22)

Therefore, forx, y>1,

ψ(xy)

ψ(x) =

h(ln(xy))

h(lnx) =

h(lnx+ lny)

h(lnx) >y

Rb

a p_R(u) lnf(u)du b

a p(u)du . (23)

Letg(x) =x Rb

a p_R(u) lnf(u)du b

a p(u)du forx>1, theng−1(x) =x

Rb a p(u)du

Rb

a p(u) lnf(u)du,x∈[1,+∞).

By Lemma 1, (ϕ, τ), where ϕ(x) = xψ(x) = xh(lnx) = xR_abp(u)[f(u)]lnx_du _and _τ₍_x_{) =}

x Rb

a p(u)du

R_b

a p(u) lnf(u)du forx>1 andn>1, are Steffensen pairs on [1,+∞) for any givenn>1. The proof is complete.

References

[1] J.-C. Evard and H. Gauchman,Steffensen type inequalities over general measure spaces, Analysis17(1997), 301-322.

[2] H. Gauchman,Steffensen pairs and associated inequalities, Journal of Inequalities and Applications5(2000), no. 1, 53–61

[3] D. S. Mitrinovi´c, J. E. Peˇcaric and A. M. Fink,Classical and New Inequalities in Analysis, Kluwer Academic Publishers, 1993.

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[5] Feng Qi and Jun-Xiang Cheng, New Steffensen pairs, RGMIA Research Report Collection 3 (2000). http://rgmia.vu.edu.au.

[6] Feng Qi, Jia-Qiang Mei, Da-Feng Xia, and Sen-Lin Xu,New proofs of weighted power mean inequalities and monotonicity for generalized weighted mean values, Mathematical Inequalities and Applications3(2000), no. 3, in the press.

[7] Feng Qi and Shi-Qin Zhang,Note on monotonicity of generalized weighted mean values, Proceedings of the Royal Society of London Series A—Mathematical, Physical and Engineering Sciences 455(1999), no. 1989, 3259–3260.

[8] J. F. Steffensen,On certain inequalities and methods of approximation, J. Inst. Actuaries51(1919), 274-297.

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

People’s Republic of China

E-mail address:[email protected]

URL:http://rgmia.vu.edu.au/qi.htmlor http://rgmia.vu.edu.au/authors/FQi.htm

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