International Journal of Analysis and Applications
ISSN 2291-8639
Volume 5, Number 1 (2014), 20-26
http://www.etamaths.com

SOME NEW INEQUALITIES OF QI TYPE FOR DEFINITE

INTEGRALS

BO-YAN XI1 AND FENG QI2,∗

Abstract. In the paper, the authors establish some new integral inequalities,

from which some integral inequalities of Qi type may be derived.

1. Introduction

In [11] and its preprint [12], an interesting integral inequality below was obtained.

Theorem 1.1 ([11, 12]). Let n ∈ N and the n-th order derivative of f be continuous
on [a,b] ⊆ R = (−∞,∞), satisfying f(i)(a) ≥ 0 and f(n)(x) ≥ n! for 0 ≤ i ≤ n−1.
Then

(1.1)

∫ b
a

fn+2(x) d x ≥
[∫ b

a

f(x) d x

]n+1
.

At the end of [11, 12], the following open problem was posed.

Open Problem 1.1 ([11, 12]). Under what conditions does the inequality

(1.2)

∫ b
a

ft(x) d x ≥
[∫ b

a

f(x) d x

]t−1
hold for some t > 1?

Thereafter, the following answer to Open Problem 1.1 was confirmed.

Theorem 1.2 ([14, 15]). Let t ≥ 1 and f be a continuous function on [a,b] ⊆ R
such that

(1.3)

∫ b
a

f(x) d x ≥ (b−a)t−1.

Then the inequality (1.2) is valid.

To the best of our knowledge, till now there have been many mathematician-
s and articles devoted to generalizing and applying the integral inequality (1.1)
and to answering Open Problem 1.1. In these investigations, different and various
tools, ideas, methods, and techniques, such as Jensen’s inequality [6], convexity
method [4], functional inequalities in abstract spaces [1, 4, 6], probability measures
viewpoint [1, 7, 8], Hölder inequality and its reversed variants [10, 19], analytical

2010 Mathematics Subject Classification. 26D15.
Key words and phrases. generalization; integral inequality; Qi type integral inequality; convex

function; Jensen’s inequality; r-mean convex function; geometrically convex function; logarithmi-
cally convex function; open problem.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

20



SOME NEW INEQUALITIES OF QI TYPE FOR DEFINITE INTEGRALS 21

methods [15], Cauchy’s mean value theorem [3, 13], and q-integral [2, 9, 20], have
been created. Recently, this type of inequalities were generalized in [18] to double
integrals. Importantly, the mathematical meanings in probability and statistics was
found in [4]. For a much complete list of references appeared in recent years on this
topic, please refer to [20].

The aim of this paper is to establish some new integral inequalities, from which
some integral inequalities of Qi type may be derived. In other words, the integral
inequality (1.1) will be generalized and some more answers to Open Problem 1.1
will be supplied in this paper.

2. Definitions and lemmas

Before establishing some new inequalities of Qi type, we state several definitions
and lemmas.

Let I ⊆ R be an interval and n ∈ N. For f : I → R+ = (0,∞), xk ∈ I for
1 ≤ k ≤ n, and λk ≥ 0 satisfying

∑n
i=1 λk = 1, let

(2.1) Mn(f(x),λ,r) =




[
n∑
k=1

λkf
r(xk)

]1/r
, r 6= 0,

n∏
k=1

fλk (xk), r = 0.

Especially, for xk ∈ I ⊆ R+, let

(2.2) Mn(x,λ,r) =




(
n∑
k=1

λkx
r
k

)1/r
, r 6= 0,

n∏
k=1

x
λk
k , r = 0.

Definition 2.1 ([5, p. 348]). Let I ⊆ R be an interval. A function f : I → R is
said to be convex if

(2.3) f(λx + (1 −λ)y) ≤ λf(x) + (1 −λ)f(y)
for all x,y ∈ I and λ ∈ [0, 1]. If the above inequality is reversed, then f is said to
be concave on I.

Definition 2.2 ([16, 17]). Let I ⊆ R+ be an interval and r ∈ R. A function
f : I → R+ is said to be r-mean convex on I if
(2.4) f(M2(x,λ,r)) ≤ M2(f(x),λ,r)
for all x1,x2 ∈ I and λ ∈ [0, 1]. If the above inequality is reversed, then we say
that the function f is r-mean concave on I.

When r = 0, the r-mean convex (r-mean concave, respectively) functions are
called geometrically convex (geometrically concave, respectively) functions.

Definition 2.3 ([5, p. 349]). Let I ⊆ R be an interval. A function f : I → R+ is
said to be logarithmically convex on I if

(2.5) f(λx + (1 −λ)y) ≤ [f(x)]λ[f(y)]1−λ

for all x,y ∈ I and λ ∈ [0, 1]. If the above inequality is reversed, then the function
f is said to be logarithmically concave on I.



22 XI AND QI

Lemma 2.1 (Jensen’s Inequality). Let I ⊆ R+ be an interval, r ∈ R, and f : I →
R+. Then f is r-mean convex (r-mean concave, respectively) on I if and only if
(2.6) f(Mn(x,λ,r)) ≤ Mn(f(x),λ,r)
holds for all x = (x1,x2, . . . ,xn) ∈ In and λk ≥ 0 satisfying

∑n
k=1 λk = 1.

Proof. This may be found in [16, 17]. �

The following lemmas are useful for us.

Lemma 2.2. For x,y ∈ R+, if either xy ≤ 4, or x ≤ 1, or y ≤ 1, then xy ≤ x + y.

Proof. The proof is elementary. �

3. Some new integral inequalities of Qi type

Now we are in a position to establish some new integral inequalities of Qi type.

Theorem 3.1. Suppose I ⊆ R0 = [0,∞) is an interval, f : [a,b] → I is continuous
and not identically zero, g : I → R0 is convex (or concave, respectively), and
(3.1) g((b−a)u) Q g(b−a)g(u)
for u ∈ I, and

(3.2)

∫ b
a

f(x) d x R
g(b−a)
b−a

.

Then we have

(3.3)

∫ b
a

g(f(x)) d x R
g
(∫ b
a
f(x) d x

)
∫ b
a
f(x) d x

.

Proof. Let

(3.4) xk = a +
k

n
(b−a), 1 ≤ k ≤ n.

If g(u) is a convex function on I, then it is easy to see that Mn
(
f(x), 1

n
, 1
)
∈ I,

and, by Jensen’s inequality (2.6) and corresponding conditions,

g

[∫ b
a

f(x) d x

]
= g

(
(b−a) lim

n→∞
Mn

(
f(x),

1

n
, 1

))
≤ g(b−a) lim

n→∞
g

(
Mn

(
f(x),

1

n
, 1

))
≤ g(b−a) lim

n→∞
Mn

(
g

(
f(x),

1

n
, 1

))
=
g(b−a)
b−a

∫ b
a

g(f(x)) d x.

Therefore, it follows that∫ b
a

f(x) d x

∫ b
a

g(f(x)) d x−g
[∫ b

a

f(x) d x

]
=

∫ b
a

g(f(x)) d x

[∫ b
a

f(x) d x−
g(b−a)
b−a

]
≥ 0.



SOME NEW INEQUALITIES OF QI TYPE FOR DEFINITE INTEGRALS 23

Thus, the inequality (3.3) in the direction ≥ is true.
If g(u) is a concave function on I, the proof is similar. This completes the proof

of Theorem 3.1. �

Applying Theorem 3.1 to special cases of g(u) result in the following corollaries,
which show that Theorem 3.1 and Theorem 3.2 and 3.3 below are generalizations
of the inequality (1.1) and answers of Open Problem 1.1.

Corollary 3.1. Let f(x) is a positive continuous function on an interval [a,b] ⊆ R.
(1) If t 6∈ [0, 1) and

∫ b
a
f(x) d x ≥ (b−a)t−1, then the inequality (1.2) is valid;

(2) If 0 < t ≤ 1 and
∫ b
a
f(x) d x ≤ (b − a)t−1, then the inequality (1.2) is

reversed.

Corollary 3.2. Let f(x) be a positive continuous function on [a,b] ⊆ R.
(1) If t 6∈ [0, 1) and f(x) ≥ (b−a)t−2, then the inequality (1.2) is valid;
(2) If 0 < t ≤ 1 and f(x) ≤ (b−a)t−2, then the inequality (1.2) is reversed.

Corollary 3.3. Suppose f(x) is a positive continuous function on [a,b] ⊆ R.
(1) If t ≥ 2 and f(x) ≥ (t− 1)(x−a)t−2, then the inequality (1.2) is valid;
(2) If 2 < t ≤ 3 and f′(x) ≥ (b − a)(t − 1)(x − a)t−2 on [a,b], then the

inequality (1.2) is also valid;
(3) If t > 3 and f′(x) ≥ (t−1)(t−2)(x−a)t−3 on [a,b], then the inequality (1.2)

is still valid.

Corollary 3.4. Suppose f(x) is a positive continuous function on [a,b] ⊆ R, and
suppose that either 0 < f(x) ≤ 4

b−a, or 0 < f(x) ≤ 1, or 0 < b−a ≤ 1. If c > 1
and ∫ b

a

f(x) d x ≥
cb−a

b−a
,

then ∫ b
a

cf(x) d x ≥
c
∫

b
a
f(x) d x∫ b

a
f(x) d x

.(3.5)

In particular, ∫ b
a

ef(x) d x ≥
exp
[∫ b
a
f(x) d x

]
∫ b
a
f(x) d x

.(3.6)

Proof. From Lemma 2.2, when x,y > 0 and either xy ≤ 4 or x ≤ 1, it follows that
cx+y ≥ cxy. By choosing g(u) = cu in Theorem 3.1, Corollary 3.4 follows. �

Theorem 3.2. Suppose I ⊆ R+ is an interval, f : [a,b] → I is a continuous
function and not identically zero, and g : I → R+.

(1) For r 6= 0, if g(u) is r-mean convex (or r-mean concave, respectively) on I,
and

(3.7) g
(
(b−a)1/ru

)
Q g
(
(b−a)1/r

)
g(u)

for u ∈ I, and

(3.8)

∫ b
a

f(x) d x R
g
(
(b−a)1/r

)
(b−a)1/r

,



24 XI AND QI

then [∫ b
a

gr(f(x)) d x

]1/r
R
g
((∫ b

a
fr(x) d x

)1/r)
∫ b
a
f(x) d x

.(3.9)

(2) If g(u) is a geometrically convex (or geometrically concave, respectively) on
I, satisfying

(3.10) g
(
e(b−a)u

)
Q g
(
eb−a

)
g(eu), u ∈ I

and

(3.11)

∫ b
a

f(x) d x R g
(
eb−a

)
,

then

exp

(
1

b−a

∫ b
a

ln g(f(x)) d x

)
R
g
(
exp
(∫ b
a

ln f(x) d x
))

∫ b
a
f(x) d x

.(3.12)

Proof. Let g(u) be a r-mean convex function on I and adopt the notations in (3.4).
Utilizing Mn

(
f(x), 1

n
, 1
)
∈ I and Jensen’s inequality (2.6) leads to

g

([∫ b
a

fr(x) d x

]1/r)
= g

(
(b−a)1/r lim

n→∞
Mn

(
f(x),

1

n
,r

))
≤ g
(
(b−a)1/r

)
lim
n→∞

g

(
Mn

(
f(x),

1

n
,r

))
≤ g
(
(b−a)1/r

)
lim
n→∞

Mn

(
g(f(x)),

1

n
,r

)
=
g
(
(b−a)1/r

)
(b−a)1/r

[∫ b
a

gr(f(x)) d x

]1/r
,

hence, the inequality (3.9) is true.
Let g(u) be a geometrically convex function on I. Making use of Jensen’s in-

equality (2.6) results in

g

(
exp

(∫ b
a

ln f(x) d x

))
= g

(
exp

(
(b−a) lim

n→∞
Mn

(
ln f(x),

1

n
, 1

)))
≤ g
(
eb−a

)
lim
n→∞

g

(
Mn

(
f(x),

1

n
, 0

))
≤ g
(
eb−a

)
lim
n→∞

Mn

(
g(f(x)),

1

n
, 0

)
= g
(
eb−a

)
exp

(
1

b−a

∫ b
a

ln g(f(x)) d x

)
,

therefore, the inequality (3.12) is true.
The rest can be proved similarly. The proof of Theorem 3.2 is complete. �

Theorem 3.3. Suppose I ⊆ R+ is an interval, f : [a,b] → I is a continuous
function and not identically zero, and g : I → R+ is a logarithmically convex (or
logarithmically concave, respectively) function, satisfying

(3.13) g((b−a)u) Q g(b−a)g(u), u ∈ I



SOME NEW INEQUALITIES OF QI TYPE FOR DEFINITE INTEGRALS 25

and

(3.14)

∫ b
a

f(x) d x R g(b−a).

Then we have

(3.15) exp

(
1

b−a

∫ b
a

ln g(f(x)) d x

)
R
g
(∫ b
a
f(x) d x

)
∫ b
a
f(x) d x

.

Proof. When g(u) is a logarithmically convex function on I, Jensen’s inequali-
ty (2.6) gives

g

(∫ b
a

f(x) d x

)
= g

(
(b−a) lim

n→∞
Mn

(
f(x),

1

n
, 1

))
≤ g(b−a) lim

n→∞
g

(
Mn

(
f(x),

1

n
, 1

))
≤ g(b−a) lim

n→∞
Mn

(
g(f(x)),

1

n
, 0)

)
= g
(
eb−a

)
exp

(
lim
n→∞

Mn

(
ln g(f(x)),

1

n
, 1

))
= g
(
eb−a

)
exp

(
1

b−a

∫ b
a

ln g(f(x)) d x

)
,

as a result, the inequality (3.15) is true.
The rest can be proved similarly. The proof of Theorem 3.3 is complete. �

Acknowledgements

The author was partially supported by the NNSF under Grant No. 11361038 of
China and by the Foundation of the Research Program of Science and Technology at
Universities of Inner Mongolia Autonomous Region under Grant No. NJZY13159,
China.

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1College of Mathematics, Inner Mongolia University for Nationalities, Tongliao

City, Inner Mongolia Autonomous Region, 028043, China

2Department of Mathematics, School of Science, Tianjin Polytechnic University,

Tianjin City, 300387, China

∗Corresponding author