Journal of Mathematical Analysis
ISSN: 2217-3412, URL: http://www.ilirias.com
Volume 3 Issue 2(2012), Pages 10-17.
EXTENSIONS OF STEFFENSEN’S INEQUALITY
W. T. SULAIMAN
Abstract. In this paper we generalized the result of Pecaric concerning Steffensen’s inequality. For another result we extend the scope of this result. As
well as we presented the reverse inequality. Other results are also deduced as
corollaries.
1. Introduction
Steffensen’s inequality reads as follows :
Theorem 1.1. Assume that two integrable functions 𝑓 (𝑡) and 𝑔(𝑡) are defined on
the interval (𝑎, 𝑏), that 𝑓 (𝑡) non-increasing and that 0 ≤ 𝑔(𝑡) ≤ 1 in (𝑎, 𝑏). Then
∫𝑏
∫𝑏
𝑓 (𝑡)𝑑𝑡 ≤
where 𝜆 =
𝑓 (𝑡)𝑔(𝑡)𝑑𝑡 ≤
𝑎
𝑎−𝜆
∫𝑏
𝑎+𝜆
∫
𝑓 (𝑡)𝑑𝑡,
(1.1)
𝑎
𝑔(𝑡)𝑑𝑡.
𝑎
2. Known Results
Bellman [1] gives the following generalization via new proof
Theorem 2.1. Let 𝑓 (𝑡) be non-negative and monotone decreasing in [𝑎, 𝑏] and ∈
∫𝑏
𝐿𝑝 [𝑎, 𝑏] and let 𝑔(𝑡) ≥ 0 in [𝑎, 𝑏] and 𝑔(𝑡)𝑑𝑡 ≤ 1, where 𝑝 > 1 and (1/𝑝)+(1/𝑞) = 1.
𝑎
Then
⎛ 𝑏
⎞𝑝
𝑎+𝜆
∫
∫
⎝ 𝑓 (𝑡)𝑔(𝑡)𝑑𝑡⎠ ≤
𝑓 𝑝 (𝑡)𝑑𝑡
𝑎
⎛
⎛
∫𝑏
⎝𝜆 = ⎝
𝑎
⎞𝑝 ⎞
𝑔(𝑡)𝑑𝑡⎠ ⎠ .
(2.1)
𝑎
It may be mentioned that Bellman’s result is not correct as has been mentioned
by Levin [4]. A generalization in a different sense is made for 𝑝 ≤ 1. Inequality for
𝑝 ≥ 1, which similar to the inequality (2.1) given in [3].
Pecaric [6], however, through some modification, gives the following
2000 Mathematics Subject Classification. 26D15, 26D10.
Key words and phrases. Steffensen’s inequality, integral inequality.
c
⃝2012
Ilirias Publications, Prishtinë, Kosovë.
Submitted May 6, 2012. Published Jun 20, 2012.
10
EXTENSIONS OF STEFFENSEN’S INEQUALITY
11
Theorem 2.2. Let 𝑓 : [0, 1] → ℜ be nonnegative and non-increasing function and
let 𝑔 : [0, 1] → ℜ be an integrable function such that 0 ≤ 𝑔(𝑡) ≤ 1 for all 𝑡 ∈ [0, 1] .
If 𝑝 ≥ 1, then
⎛ 1
⎞𝑝
∫
∫𝜆
⎝ 𝑓 (𝑡)𝑔(𝑡)𝑑𝑡⎠ ≤ 𝑓 𝑝 (𝑡)𝑑𝑡
(2.2)
0
where 𝜆 =
(1
∫
0
)𝑝
𝑔(𝑡)𝑑𝑡 .
0
A mapping 𝜙 : ℜ → ℜ is said to be convex on [𝑎, 𝑏] if
𝜙(𝑡𝑥 + (1 − 𝑡)𝑦) ≤ 𝑡𝜙(𝑥) + (1 − 𝑡)𝜙(𝑦),
𝑥, 𝑦 ∈ [𝑎, 𝑏], 0 ≤ 𝑡 ≤ 1.
(2.3)
If (2.3) reverses, then 𝜙 is called concave.
We have proved the following (see [9])
Theorem 2.3. Let 𝑓, 𝑔, ℎ : [𝑎, 𝑏] → ℜ be non-negative, 𝑓 is non-increasing, 𝑔 and
ℎ are integrable functions satisfying
(
)𝑝
∫𝑏
1−1/𝑝
(1) 𝜆
𝑔(𝑡) ≤ ℎ(𝑡), where 𝜆 =
𝑔(𝑡)𝑑𝑡 , 𝑝 > 0,
𝑎
(2)
∫𝑏
𝑘1∫(𝜆)
ℎ(𝑡)𝑑𝑡 ≤ 𝜆 ≤
ℎ(𝑡)𝑑𝑡, for some 𝑘1 and 𝑘2 functions of 𝜆 with
𝑎
𝑘2 (𝜆)
𝑎 ≤ 𝑘1 (𝜆); 𝑘2 (𝜆) ≤ 𝑏. Then we have
∫𝑏
𝑓 𝑝 (𝑡)ℎ(𝑡)𝑑𝑡 ≤ 𝜆1−1/𝑝
∫𝑏
𝑓 𝑝 (𝑡)𝑔(𝑡)𝑑𝑡 ≤
𝑎
𝑘2 (𝜆)
𝑘∫
1 (𝜆)
𝑓 𝑝 (𝑡)ℎ(𝑡)𝑑𝑡.
𝑎
Very recently Srivastava [8] proved the following
Theorem 2.4. Let 𝑓, 𝑔 and ℎ be integrable functions defined on [𝑎, 𝑏] with 𝑓 nonincreasing . Also let 0 ≤ 𝑔(𝑡) ≤ ℎ(𝑡), 𝑡 ∈ [𝑎, 𝑏]. Then the following integral inequalities hold
∫𝑏
∫𝑏
𝑓 (𝑡)ℎ(𝑡)𝑑𝑡 ≤
[𝑓 (𝑡)ℎ(𝑡) − [𝑓 (𝑡) − 𝑓 (𝑏 − 𝜆)][ℎ(𝑡) − 𝑔(𝑡)]]𝑑𝑡
𝑏−𝜆
𝑏−𝜆
∫𝑏
≤
𝑓 (𝑡)𝑔(𝑡)𝑑𝑡
𝑎
𝑎+𝜆
∫
≤
[𝑓 (𝑡)ℎ(𝑡) − [𝑓 (𝑡) − 𝑓 (𝑎 + 𝜆)][ℎ(𝑡) − 𝑔(𝑡)]]𝑑𝑡
𝑎
𝑎+𝜆
∫
≤
𝑓 (𝑡)ℎ(𝑡)𝑑𝑡,
𝑎
where 𝜆 is given by
𝑎+𝜆
∫
∫𝑏
ℎ(𝑡)𝑑𝑡 =
𝑎
∫𝑏
𝑔(𝑡)𝑑𝑡 =
𝑎
ℎ(𝑡)𝑑𝑡.
𝑏−𝜆
12
W. T. SULAIMAN
The aim of this paper is to give a generalization of Theorem 1.2, as well as other
results including a reverse of Steffensen’s inequality.
The following Lemma is needed
Lemma 2.5. The mapping 𝜙(𝑥) = 𝑥𝑝 is convex for 𝑝 ≥ 1, and concave for 0 ≤
𝑝 ≤ 1.
Proof. As
𝜙′′(𝑥) = 𝑝(𝑝 − 1)𝑥𝑝−2 ≥ 0,
then 𝜙 is convex . Also for 0 < 𝑝 ≤ 1, the concavity of 𝜙 follows from the inequality
𝜙′′(𝑥) ≤ 0.
□
3. New Results
The following gives a generalization of Theorem 1.2 [6].
Theorem 3.1. Let 𝑓, 𝑔, 𝜙 ≥ 0, 0 ≤ 𝑔 ≤ 1, 𝑝 ≥ 1, 𝜙(𝜆1/𝑝 ) ≤ 𝜆 , where 𝜆 =
(1
)𝑝
∫
𝑔(𝑡)𝑑𝑡 , 𝑓 is non-increasing , 𝜙 is non-decreasing. Then
0
∫𝜆
𝜙𝑜𝑓 (𝑥)𝑑𝑥 ≥ 𝜆
−1/𝑝
𝜙(𝜆
1/𝑝
∫1
)
0
𝜙𝑜𝑓 (𝑥)𝑔(𝑥)𝑑𝑥.
(3.1)
0
Proof. As 𝜙 ≥ 0 and 𝑓 is non-increasing, then 𝜙𝑜𝑓 is non-increasing. Also
𝜆−1/𝑝 𝜙(𝜆1/𝑝 )𝑔(𝑥) ≤ 𝜆−1/𝑝 𝜙(𝜆1/𝑝 ) ≤ 𝜆1−1/𝑝 ≤ 1,
then
∫𝜆
−1/𝑝
𝜙𝑜𝑓 (𝑥)𝑑𝑥 − 𝜆
1/𝑝
𝜙(𝜆
∫1
)
0
∫𝜆
=
𝜙𝑜𝑓 (𝑥)𝑔(𝑥)𝑑𝑥
0
⎞
⎛ 𝜆
∫
∫1
𝜙𝑜𝑓 (𝑥)𝑑𝑥 − 𝜆−1/𝑝 𝜙(𝜆1/𝑝 ) ⎝ + ⎠ 𝜙𝑜𝑓 (𝑥)𝑔(𝑥)𝑑𝑥
0
∫𝜆
=
0
𝜆
∫1
(
)
𝜙𝑜𝑓 (𝑥) 1 − 𝜆−1/𝑝 𝜙(𝜆1/𝑝 )𝑔(𝑥) 𝑑𝑥 − 𝜆−1/𝑝 𝜙(𝜆1/𝑝 ) 𝜙𝑜𝑓 (𝑥)𝑔(𝑥)𝑑𝑥
0
𝜆
⎛
≥
𝜙𝑜𝑓 (𝜆) ⎝
∫𝜆 (
)
1 − 𝜆−1/𝑝 𝜙(𝜆1/𝑝 )𝑔(𝑥) 𝑑𝑥 − 𝜆−1/𝑝 𝜙(𝜆1/𝑝 )
0
∫1
⎞
𝑔(𝑥)𝑑𝑥⎠
𝜆
⎛
⎛ 𝜆
⎞
⎞
∫
∫1
= 𝜙𝑜𝑓 (𝜆) ⎝𝜆 − 𝜆−1/𝑝 𝜙(𝜆1/𝑝 ) ⎝ + ⎠ 𝑔(𝑥)𝑑𝑥⎠
0
⎛
= 𝜙𝑜𝑓 (𝜆) ⎝𝜆 − 𝜆−1/𝑝 𝜙(𝜆1/𝑝 )
∫1
𝜆
⎞
𝑔(𝑥)𝑑𝑥⎠
0
(
1/𝑝
= 𝜙𝑜𝑓 (𝜆) 𝜆 − 𝜙(𝜆
)
) ≥ 0.
□
EXTENSIONS OF STEFFENSEN’S INEQUALITY
13
The following result is deals with Steffensen’s inequality for 𝑝 > 0.
Theorem 3.2. Let 𝑓, 𝑔, 𝜙 ≥ 0, 0 ≤ 𝑔 ≤ 1, 𝑝 > 0, 𝑓 is non-increasing. 𝜙(𝑝) > 0,
∫1
𝜆𝜙(𝑝) 𝑔 ≤ 1, 𝑔(𝑥)𝑑𝑥 ≤ 𝜆1−𝜙(𝑝) , 𝜙 is non-decreasing .Then
0
∫𝜆
𝜙𝑜𝑓 (𝑥)𝑑𝑥 ≥ 𝜆
𝜙(𝑝)
0
∫1
𝜙𝑜𝑓 (𝑥)𝑔(𝑥)𝑑𝑥.
(3.2)
0
Proof. As 𝜙 ≥ 0 and 𝑓 is non-increasing, then 𝜙𝑜𝑓 is non-increasing.
∫1
Also, 𝜆𝜙(𝑝) 𝑔 ≤ 1 =⇒ 𝜆𝜙(𝑝) 𝑔(𝑥)𝑑𝑥 =⇒ 𝜆 ≤ 1. Then, we have
0
∫𝜆
𝜙𝑜𝑓 (𝑥)𝑑𝑥 − 𝜆
𝜙(𝑝)
0
∫1
𝜙𝑜𝑓 (𝑥)𝑔(𝑥)𝑑𝑥
0
∫𝜆
=
⎞
⎛ 𝜆
∫
∫1
𝜙𝑜𝑓 (𝑥)𝑑𝑥 − 𝜆𝜙(𝑝) ⎝ + ⎠ 𝜙𝑜𝑓 (𝑥)𝑔(𝑥)𝑑𝑥
0
0
∫𝜆
=
𝜆
∫1
(
)
𝜙𝑜𝑓 (𝑥) 1 − 𝜆𝜙(𝑝) 𝑔(𝑥) 𝑑𝑥 − 𝜆𝜙(𝑝) 𝜙𝑜𝑓 (𝑥)𝑔(𝑥)𝑑𝑥
0
𝜆
⎛ 𝜆
⎞
∫ (
∫1
)
≥ 𝜙𝑜𝑓 (𝑥) ⎝
1 − 𝜆𝜙(𝑝) 𝑔(𝑥) 𝑑𝑥 − 𝜆𝜙(𝑝) 𝑔(𝑥)𝑑𝑥⎠
0
𝜆
⎛
⎛ 𝜆
⎞
⎞
∫
∫1
= 𝜙𝑜𝑓 (𝜆) ⎝𝜆 − 𝜆𝜙(𝑝) ⎝ + ⎠ 𝑔(𝑥)𝑑𝑥⎠
0
⎛
= 𝜙𝑜𝑓 (𝜆) ⎝𝜆 − 𝜆𝜙(𝑝)
𝜆
⎞
∫1
𝑔(𝑥)𝑑𝑥⎠
0
≥ 𝜙𝑜𝑓 (𝜆) (𝜆 − 𝜆) = 0.
□
Theorem 3.3. Let 𝑓, 𝑔, ℎ, 𝜙 ≥ 0, 0 ≤ 𝑔 ≤ 1, 𝑓 is non-increasing,
( 1𝜙 is non)𝑝
∫𝜆
∫
−1/𝑝
1/𝑝
1/𝑝
𝜙(𝜆 ) ≤ ℎ, where 𝜆 =
𝑔(𝑡)𝑑𝑡 .
decreasing 𝑝 ≥ 1, 𝜙(𝜆 ) ≤ ℎ(𝑥)𝑑𝑥, 𝜆
0
0
Then
∫𝜆
−1/𝑝
𝜙𝑜𝑓 (𝑥)ℎ(𝑥)𝑑𝑥 ≥ 𝜆
0
1/𝑝
𝜙(𝜆
∫1
)
𝜙𝑜𝑓 (𝑥)𝑔(𝑥)𝑑𝑥.
0
(3.3)
14
W. T. SULAIMAN
Proof. As before, 𝜙𝑜𝑓 is non-increasing. Therefore
∫𝜆
−1/𝑝
𝜙𝑜𝑓 (𝑥)ℎ(𝑥)𝑑𝑥 − 𝜆
1/𝑝
𝜙(𝜆
0
∫1
𝜙𝑜𝑓 (𝑥)𝑔(𝑥)𝑑𝑥
)
0
⎛ 𝜆
⎞
∫𝜆
∫
∫1
=
𝜙𝑜𝑓 (𝑥)ℎ(𝑥)𝑑𝑥 − 𝜆−1/𝑝 𝜙(𝜆1/𝑝 ) ⎝ + ⎠ 𝜙𝑜𝑓 (𝑥)𝑔(𝑥)𝑑𝑥
0
∫𝜆
0
(
𝜙𝑜𝑓 (𝑥) ℎ(𝑥) − 𝜆
=
−1/𝑝
𝜙(𝜆
1/𝑝
𝜆
∫1
)
−1/𝑝
1/𝑝
)𝑔(𝑥) 𝑑𝑥 − 𝜆
𝜙(𝜆 ) 𝜙𝑜𝑓 (𝑥)𝑔(𝑥)𝑑𝑥
0
𝜆
⎛ 𝜆
⎞
∫ (
∫1
)
≥ 𝜙𝑜𝑓 (𝜆) ⎝
ℎ(𝑥) − 𝜆−1/𝑝 𝜙(𝜆1/𝑝 )𝑔(𝑥) 𝑑𝑥 − 𝜆−1/𝑝 𝜙(𝜆1/𝑝 ) 𝑔(𝑥)𝑑𝑥⎠
0
𝜆
⎛ 𝜆
⎛ 𝜆
⎞
⎞
∫
∫
∫1
= 𝜙𝑜𝑓 (𝜆) ⎝ ℎ(𝑥)𝑑𝑥 − 𝜆−1/𝑝 𝜙(𝜆1/𝑝 ) ⎝ + ⎠ 𝑔(𝑥)𝑑𝑥⎠
0
=
0
𝜆
⎛ 𝜆
⎞
∫
𝜙𝑜𝑓 (𝜆) ⎝ ℎ(𝑥)𝑑𝑥 − 𝜙(𝜆1/𝑝 )⎠ ≥ 0.
0
□
The following gives a reverse inequality
Theorem 3.4. Let 𝑓, 𝑔, 𝜙 ≥ 0, 𝜙 is concave with 𝜙(0) = 0, 𝑓 is non-decreasing, 𝜙
(1
)𝑝
∫
is non-decreasing , 0 ≤ 𝑔 ≤ 1, 𝑝 ≥ 1 and 𝜆 =
𝑔(𝑡)𝑑𝑡
. Then
0
∫𝜆
0
⎛
𝜙𝑜𝑓 (𝑥)𝑑𝑥 ≤ 𝜙 ⎝𝜆1−1/𝑝
∫1
⎞
𝑓 (𝑥)𝑔(𝑥)𝑑𝑥⎠ .
0
(3.4)
EXTENSIONS OF STEFFENSEN’S INEQUALITY
15
Proof.
∫𝜆
⎛
𝜙𝑜𝑓 (𝑥)𝑑𝑥 − 𝜙 ⎝𝜆1−1/𝑝
⎛
𝜙𝑜𝑓 (𝑥)𝑑𝑥 − 𝜙 ⎝𝜆
≤
⎞
∫1
1
𝑓 (𝑥)𝑔(𝑥)𝑑𝑥⎠
𝜆1/𝑝
0
0
⎛
⎞
1
𝜙𝑜𝑓 (𝑥)𝑑𝑥 − 𝜆𝜙 ⎝ 1/𝑝
𝑓 (𝑥)𝑔(𝑥)𝑑𝑥⎠ (as 𝜙 is concave with 𝜙(0) = 0)
𝜆
0
0
⎛ 1
⎞
∫𝜆
∫
𝜙𝑜𝑓 (𝑥)𝑑𝑥 − 𝜆1−1/𝑝 ⎝ 𝜙𝑜𝑓 (𝑥)𝑔(𝑥)𝑑𝑥⎠ (by Jensen’s inequality)
∫𝜆
≤
𝑓 (𝑥)𝑔(𝑥)𝑑𝑥⎠
0
0
∫𝜆
=
⎞
∫1
∫1
0
0
∫𝜆
⎛ 𝜆
⎞
∫
∫1
𝜙𝑜𝑓 (𝑥)𝑑𝑥 − 𝜆1−1/𝑝 ⎝ + ⎠ 𝜙𝑜𝑓 (𝑥)𝑔(𝑥)𝑑𝑥
0
0
=
∫𝜆
=
(
1−1/𝑝
1−𝜆
𝜆
∫1
)
1−1/𝑝
𝑔(𝑥) 𝑑𝑥 − 𝜆
𝜙𝑜𝑓 (𝑥)𝑔(𝑥)𝑑𝑥
0
≤
𝜆
⎛ 𝜆
⎞
∫ (
∫1
)
𝜙𝑜𝑓 (𝜆) ⎝
1 − 𝜆1−1/𝑝 𝑔(𝑥) 𝑑𝑥 − 𝜆1−1/𝑝 𝑔(𝑥)𝑑𝑥⎠
0
𝜆
⎛
=
⎛ 𝜆
⎞
⎞
∫
∫1
𝜙𝑜𝑓 (𝜆) ⎝𝜆 − 𝜆1−1/𝑝 ⎝ + ⎠ 𝑔(𝑥)𝑑𝑥⎠
0
⎛
=
𝜙𝑜𝑓 (𝜆) ⎝𝜆 − 𝜆1−1/𝑝
=
𝜙𝑜𝑓 (𝜆) (𝜆 − 𝜆) = 0.
𝜆
⎞
∫1
𝑔(𝑥)𝑑𝑥⎠
0
□
4. Applications
For an application of Theorem 3.1, we have the following
(1) If we are put 𝜙(𝑥) = 𝑥𝑝 , 𝑝 ≥ 1, we obtain the inequality (2.2) as follows
⎛ 1
⎞𝑝
⎛ 1
⎞𝑝−1 1
∫
∫
∫𝜆
∫
𝑓 𝑝 (𝑥)𝑑𝑥 ≥ ⎝ 𝑔(𝑡)𝑑𝑡⎠
𝑓 𝑝 (𝑥)𝑔(𝑥)𝑑𝑥 ≥ ⎝ 𝑓 (𝑥)𝑔(𝑥)𝑑𝑥⎠ .
0
0
0
0
(2) If we are take 𝜙(𝑥) = 𝑠𝑖𝑛𝑥, 0 ≤ 𝑥 ≤ 𝜋/2, we obtain
∫𝜆
𝑠𝑖𝑛 (𝑓 (𝑥)) 𝑑𝑥 ≥ 𝜆
0
−1/𝑝
(
1/𝑝
sin 𝜆
) ∫1
0
sin (𝑓 (𝑥)) 𝑔(𝑥)𝑑𝑥.
16
W. T. SULAIMAN
(3) If we are take 𝜙(𝑥) = 𝑙𝑛𝑥, 𝑥 > 0, we obtain
∫𝜆
) ∫1
(
(ln 𝑓 (𝑥)) 𝑔(𝑥)𝑑𝑥.
ln 𝑓 (𝑥)𝑑𝑥 ≥ 𝜆−1/𝑝 ln 𝜆1/𝑝
0
0
1/𝑝−1
Corollary 4.1. Let 𝑓, 𝑔, 𝜙 ≥ 0, 𝜆
𝑔 ≤ 1, 𝑓 is non-increasing, 𝑝 > 0, where
𝑝
(1
) 2𝑝−1
∫
𝜆=
𝑔(𝑡)𝑑𝑡
.𝜙 is non-decreasing. Then
0
∫𝜆
𝜙𝑜𝑓 (𝑥)𝑑𝑥 ≥ 𝜆
0
1/𝑝−1
∫1
𝜙𝑜𝑓 (𝑥)𝑔(𝑥)𝑑𝑥.
(4.1)
0
Proof. The proof follows from Theorem 2.2, by putting 𝜙(𝑝) = 1/𝑝 − 1, 0 < 𝑝 <
1.
□
The following gives another extension of Theorem 2.1
Corollary
)𝑝 𝑓, 𝑔 ≥ 0, 𝑓 is non-decreasing, 0 ≤ 𝑔 ≤ 1, 𝑝 ≥ 1, 0 < 𝑞 < 1,
( 14.2. Let
∫
𝑔(𝑡)𝑑𝑡 .𝑇 ℎ𝑒𝑛
and 𝜆 =
0
∫𝜆
0
⎛ 1
⎞𝑞
∫
𝑓 𝑞 (𝑥)𝑑𝑥 ≤ 𝜆𝑞−𝑞/𝑝 ⎝ 𝑓 (𝑥)𝑔(𝑥)𝑑𝑥⎠ .
(4.2)
0
Proof. The proof follows from Theorem 2.4 via Lemma 1.5, by putting 𝜙(𝑥) = 𝑥𝑞 ,
0 < 𝑞 < 1.
□
5. Conclusion
Several results are proved in the present paper. The first gives a generalization of
the main result of [6] in which 𝑝 ≥ 1. The second result is dealing with Steffensen’s
inequality with 𝑝 > 0. The third result present the reverse of Steffensen’s inequality.
Other results give some examples.
Acknowledgments. The authors would like to thank the anonymous referee for
his/her comments that helped us improve this article.
References
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[2] E. F. Beckenbach and R. Bellman, “Inequalities”, Springer –Verlag, Berlin, 1965.
[3] J. Berch, A generalization of Steffensen’s inequality, J. Math. Anal. Appl. 41 (1973), 187-191.
[4] E. K. Godunova and V. I. Levin, A general class of inequalities concerning Steffensen’s inequality, Mat. Zametki, 3 (1968), 339-344 (Russian).
[5] D. S. Mitrinovic(in cooperation withP. M. Vasic) “Analytic Inequalities”, Springer –Verlag,
Berlin-Heidelber-New York, !970.
[6] J. E. Pecaric, On the Bellman generalization of Steffencens’s inequality, J. Math. Anal. Appl.,
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[7] J. F. Steffensen, On certain inequalities between mean value and their application to actuarial
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[8] R. Srivastava, Some families of integral, trigonometric and other related inequalltes, Appl.
Math. Inform. Sci., 5 (3) (2011), 342-360.
EXTENSIONS OF STEFFENSEN’S INEQUALITY
17
[9] W.T. Sulaiman, Some generalizations of Steffensen’s inequality, Aust. J. Math. Anal, 2(1),
(2005), 1-88.
W. T. Sulaiman, Department of Computer Engineering, College of Engineering, University of Mosul, Iraq.
E-mail address: [email protected]