International Journal of Analysis and Applications
ISSN 2291-8639
Volume 11, Number 1 (2016), 19-22
http://www.etamaths.com

ON | C, 1 |k INTEGRABILITY OF IMPROPER INTEGRALS

H. N. ÖZGEN∗

Abstract. In this paper, we introduce the concept of | C, 1 |k, k ≥ 1, integrability of improper
integrals and we prove a known theorem of Mazhar [3] by using this definition.

1. Introduction

Throughout this paper we assume that f is a real valued function which is continuous on [0,∞)
and s(x) =

∫ x
0
f(t)dt. The Cesàro mean of s(x) is defined by

σ(x) =
1

x

∫ x
0

s(t)dt.

The integral
∫∞
0
f(t)dt is said to be integrable | C, 1 |k,k ≥ 1, in the sense of Flett [2], if∫ ∞

0

xk−1 | σ′(x) |k dx(1.1)

is convergent. The Kronecker identity (see [1]): s(x) − σ(x) = v(x), where v(x) = 1
x

∫ x
0
tf(t)dt is

well-known and will be used in the various steps of proofs.
Since σ′(x) = 1

x
v(x), condition (1.1) can also be written as

(1.2)

∫ ∞
0

1

x
| v(x) |k dx

is convergent.

We note that for infinite series, an analogous definition was introduced by Flett [2]. Using this
definition, Mazhar [3] established the following theorem for | C, 1 |k summability factors of infinite
series.

Given any functions f,g, it is customary to write g(x) = O(f(x)), if there exist η and N, for every

x > N, | g(x)
f(x)
|≤ η.

Theorem 1.1. If (Xn) is a positive monotonic non-decreasing sequence such that

(1.3) λmXm = O(1) as m →∞,

(1.4)

m∑
n=1

nXn | ∆2λn |= O(1),

(1.5)

m∑
n=1

1

n
| tn |k= O(Xm) as m →∞,

then the series
∑
anλn is summable | C, 1 |k,k ≥ 1.

2010 Mathematics Subject Classification. 40F05, 40D25, 35A23.

Key words and phrases. absolute summability; summability factors; improper integral; inequalities for integrals.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

19



20 ÖZGEN

2. The Main Result

The aim of this paper is to prove Mazhar’s theorem for | C, 1 |k integrability of improper integrals.
Now, we shall state the following theorem.

Theorem 2.1. If γ(x) is a positive monotonic non-decreasing function such that

(2.1) λ(x)γ(x) = O(1) as x →∞,

(2.2)

∫ x
0

u | λ′′(u) | γ(u)du = O(1),

(2.3)

∫ x
0

| v(u) |k

u
du = O(γ(x)) as x →∞,

then the integral
∫∞
0
f(t)dt is integrable | C, 1 |k,k ≥ 1.

We need the following lemma for the proof of our theorem.

Lemma 2.2. Under the conditions of the theorem we have that

(2.4)

∫ ∞
0

γ(t) | λ′(t) | dt is convergent,

(2.5) xγ(x) | λ′(x) |= O(1) as x →∞.

Proof. Since λ′(t) =
∫ t
0
λ′′(u), we have

∫ x
0

γ(t) | λ′(t) | dt =
∫ x
0

γ(t) |
∫ t
0

λ′′(u)du | dt

≤
∫ x
0

γ(t)

∫ t
0

| λ′′(u) | dudt

=

∫ x
0

| λ′′(u) | du
∫ x
u

γ(t)dt

≤
∫ x
0

uγ(u) | λ′′(u) | du = O(1) as x →∞

by (2.2).
Since xγ(x) is a non decreasing function, we get

xγ(x) | λ′(x) | = xγ(x) |
∫ x
0

λ′′(u)du |

≤ xγ(x)
∫ x
0

| λ′′(u) | du

=

∫ x
0

uγ(u) | λ′′(u) | du = O(1)

≤
∫ x
0

uγ(u) | λ′′(u) | du = O(1) as x →∞

This completes the proof of Lemma 2.2. �



ON | C, 1 |k INTEGRABILITY OF IMPROPER INTEGRALS 21

3. Proof of the Theorem

Let A(x) be the function of (C, 1) means of the integral
∫∞
0
f(t)dt. Then, by definition, we have

A(x) =
1

x

∫ x
0

∫ t
0

λ(u)f(u)dudt

=
1

x

∫ x
0

λ(u)f(u)du

∫ x
u

dt

=
1

x

∫ x
0

(x−u)λ(u)f(u)du

=

∫ x
0

(
1 −

u

x

)
λ(u)f(u)du

Differentiating the function A(x) and later integrating by parts, we obtain

A′(x) =
1

x2

∫ x
0

uλ(u)f(u)du

=
v(x)λ(x)

x
−

1

x2

∫ x
0

λ′(u)uv(u)du

= A1(x) + A2(x), say.

To complete the proof of the theorem, it is sufficient to show that

(3.1)

∫ x
0

tk−1 | Ar(t) |k dt = O(1) as x →∞, for r = 1, 2.

First, applying Hölder’s inequality, we have∫ x
0

tk−1 | A1(t) |k dt =
∫ x
0

tk−1
| v(t) |k| λ(t) |k

tk
dt

=

∫ x
0

1

t
| v(t) |k| λ(t) |k−1| λ(t) | dt

≤
∫ x
0

| v(t) |k

t
| λ(t) | dt

= | λ(x) |
∫ x
0

| v(t) |k

t
dt−

∫ x
0

| λ′(t) |
∫ t
0

| v(u) |k

u
dudt

= | λ(x) | γ(x) −
∫ x
0

| λ′(t) | γ(t)dt

= O(1) as x →∞

by virtue of the hypotheses of Theorem 2.1 and Lemma 2.2.
Now, as in A1(x), we have that∫ x

0

tk−1 | A2(t) |k dt =
∫ x
0

tk−1
1

t2k
|
∫ t
0

uλ′(u)v(u)du |k dt

≤
∫ x
0

1

t2

{∫ t
0

| λ′(u) |k uk | v(u) |k du
}
x

{
1

t

∫ t
0

du

}k−1
dt

=

∫ x
0

| uλ′(u) |k−1| uλ′(u) || v(u) |k du
∫ x
u

dt

t2

=

∫ x
0

| uλ′(u) || v(u) |k
(

1

u
−

1

x

)
du

≤
∫ x
0

| uλ′(u) |
| v(u) |k

u
du



22 ÖZGEN

Integrating by parts,we get∫ x
0

tk−1 | A2(t) |k dt = x | λ′(x) |
∫ x
0

| v(u) |k

u
du +

∫ x
0

(u | λ′(u) |)′
∫ u
0

| v(t) |k

t
dtdu

= x | λ′(x) | γ(x) −
∫ x
0

(u | λ′(u) |)′γ(u)du

= x | λ′(x) | γ(x) −
∫ x
0

| λ′(u) | γ(u)du−
∫ x
0

u | λ′′(u) | γ(u)du

= O(1) as x →∞
by virtue of the hypotheses of Theorem 2.1 and Lemma 2.2.
Thus, we obtain ∫ x

0

tk−1 | A′(t) |k dt = O(1) as x →∞.

This completes the proof of the theorem.

References

[1] İ. Çanak and Ü. Totur, A Tauberian theorem for Cesàro summability factors of integrals, Appl. Math. Lett. 24
(2011), 391–395.

[2] T. M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London

Math. Soc. 7 (1957), 113–141.
[3] S. M. Mazhar, On | C, 1 |k summability factors of infinite series, Indian J. Math. 14 (1972), 45–48.

Department of Mathematics, Faculty of Education, University of Mersin, TR-33169 Mersin, Turkey

∗Corresponding author: nogduk@gmail.com