11-21.2010. 109KB Jun 04 2011 12:07:23 AM

Acta Universitatis Apulensis
ISSN: 1582-5329

No. 21/2010
pp. 105-112

COEFFICIENT INEQUALITIES FOR CERTAIN CLASSES
OF ANALYTIC FUNCTIONS INVOLVING
SALAGEAN OPERATOR

Yong Sun, Wei-Ping Kuang and Zhi-Gang Wang
Abstract. The main purpose of this paper is to derive some coefficient inequalities for certain classes of analytic functions which are defined by means of the
Salagean operator. Relevant connections of the results presented here with those
given in earlier works are also pointed out.
2000 Mathematics Subject Classification: 30C45.
Keywords and phrases: Analytic functions; Coefficient inequalities; Salagean
operator.
1. Introduction
Let A denote the class of functions of the form:
f (z) = z +

∞
X

ak z k

(z ∈ U),

(1)

k=2

which are analytic in the open unit disk
U := {z : z ∈ C

and |z| < 1}.

Let S denote the subclass of A whose members are analytic and univalent functions in U. For 0 ≦ β < 1, We denote by S ∗ (β) and K(β) the usual subclasses of
A consisting of functions which are, respectively, starlike of order β and convex of
order β in U.
Salagean [1] once introduced the following operator which called the Salagean

operator:
D0 f (z) = f (z), D1 f (z) = Df (z) = zf ′ (z),
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Y. Sun, W. Kuang, Z. Wang - Coefficient inequalities for certain classes...

and
Dn f (z) = D(Dn−1 f (z))

(n ∈ N := {1, 2, . . .}) .

We note that
Dn f (z) = z +

∞
X

k n ak z k

(n ∈ N0 := N ∪ {0}) .

(2)

k=2

A function f ∈ A is said to be in the class Nm,n (α, β) if it satisfies the inequality:

 m

 m

 D f (z)
D f (z)

− 1 + β (α ≧ 0, 0 ≦ β < 1; m ∈ N, n ∈ N0 ). (3)
> α n
ℜ
n
D f (z)
D f (z)

Also let Msm,n (α, β)(s ∈ N0 ) be the subclass of A consisting of functions f which
are satisfied the condition:
f ∈ Msm,n (α, β) ⇐⇒ Ds f ∈ Nm,n (α, β).

(4)

It is easy to see that if s = 0, then M0m,n (α, β) ≡ Nm,n (α, β). We observe that,
by specializing the parameters m, n, α and β, we obtain the following subclasses
studied by various authors.
(1) N1,0 (0, β) ≡ S ∗ (β) and N2,1 (0, β) ≡ K(β) (see Silverman [2]).
(2) N1,0 (α, β) ≡ SD(α, β) and N2,1 (α, β) ≡ KD(α, β) (see Shams et al. [3], Owa
et al. [4]).
(3) Nm,n (0, β) ≡ Km,n (β) and Msm,n (0, β) ≡ Msm,n (β) (see Eker and Owa [5]).
The function classes Nm,n (α, β) and Msm,n (α, β) were introduced and investigated by Eker and Owa [6] (see also Srivastava and Eker [7]). They obtained many
interesting results associated with them. In this paper, we aim at proving some coefficient inequalities for the function classes Nm,n (α, β) and Msm,n (α, β). Relevant
connections of the results presented here with those given in earlier works are also
pointed out.
2. Main Results
In order to prove our main results, we need the following lemma.

Lemma 1. Let δ > 0, m ∈ N and n ∈ N0 . Suppose also that the sequence {Bk }∞
k=1
is defined by
δ
B1 = 1,
B2 = m
,
|2 − 2n |

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Y. Sun, W. Kuang, Z. Wang - Coefficient inequalities for certain classes...

and

k−1

X
δ
j n Bj

Bk = m
|k − k n |

(k ∈ N \ {1, 2}).

(5)

j=1

Then


k−1
Y
δj n
δ
1+ m
Bk = m
|k − k n |
|j − j n |

(k ∈ N \ {1, 2}).

(6)

j=2

Proof. We make use of the principle of mathematical induction to prove the assertion
(6).
For k = 3, we know that

2 
Y
δ
δ
δj n
n
B3 = m
(1 + 2 B2 ) = m
,

1+ m
|3 − 3n |
|3 − 3n |
|j − j n |
j=2

which implies that (6) holds for k = 3.
We now suppose that (6) holds for k = 3, 4, · · · , r. Then

r−1
Y
δ
δj n
Br = m
1+ m
.
|r − rn |
|j − j n |
j=2

Combining (5) and (7), we find that
r

Br+1

X
δ
=
j n Bj
|(r + 1)m − (r + 1)n |
j=1

=

δ
|(r + 1)m − (r + 1)n |

r−1
X

j n Bj +

j=1

δrn
Br
|(r + 1)m − (r + 1)n |

δ
|rm − rn |
δrn
=
B
+
Br
r
|(r + 1)m − (r + 1)n |
δ
|(r + 1)m − (r + 1)n |


r−1
Y
δj n
δ
|rm − rn | + δrn
=
1+ m
|(r + 1)m − (r + 1)n | |rm − rn |
|j − j n |
j=2

 r−1

Y
δrn
δj n
1+ m
1+ m
|r − rn |
|j − j n |
j=2

r 
Y
δ
δj n
=
1+ m
,
|(r + 1)m − (r + 1)n |
|j − j n |
δ
=
m
|(r + 1) − (r + 1)n |



j=2

107

(7)

Y. Sun, W. Kuang, Z. Wang - Coefficient inequalities for certain classes...

which shows that (6) holds for k = r + 1. The proof of Lemma 1 is evidently
completed.
Theorem 1. If f ∈ Nm,n (α, β), then
2(1 − β)
,
|1 − α| |2m − 2n |

|a2 | ≦

(8)

and

k−1
Y
2(1 − β)j n
2(1 − β)
1+
|ak | ≦
|1 − α| |k m − k n |
|1 − α| |j m − j n |

(k ∈ N \ {1, 2}).

(9)

j=2

Proof. Let f ∈ Nm,n (α, β) and suppose that
f (z)
(1 − α) D
Dn f (z) − (β − α)
m

p(z) :=

(z ∈ U) .

1−β

(10)

Then p is analytic in U with
p(0) = 1

and ℜ(p(z)) > 0

(z ∈ U).

We now set
p(z) = 1 + p1 z + p2 z 2 + · · · .

(11)

Combining (10) and (11), we find that
∞

1−β X
pk z k
D f (z) = D f (z) 1 +
1−α
m

n

k=1

!

,

which implies that
(k m −k n )ak =

1−β
[pk−1 + 2n a2 pk−2 + · · · + (k − 1)n ak−1 p1 ]
1−α

(k ∈ N\{1}). (12)

On the other hand, it is well known that
|pk | ≦ 2

(k ∈ N).

(13)

Combining (12) and (13), we obtain
k−1

X
2(1 − β)
j n |aj |
|ak | ≦
|1 − α| |k m − k n |
j=1

108

(a1 = 1; k ∈ N \ {1}).

(14)

Y. Sun, W. Kuang, Z. Wang - Coefficient inequalities for certain classes...

Suppose that δ =
{Bk }∞
k=1 by

2(1−β)
|1−α|

> 0, m ∈ N and n ∈ N0 . We define the sequence

B1 = 1,
and

B2 =

|2m

δ
,
− 2n |

k−1

X
δ
j n Bj
Bk = m
|k − k n |

(k ∈ N \ {1, 2}).

(15)

j=1

In order to prove that
|ak | ≦ Bk

(k ∈ N \ {1}),

we use the principle of mathematical induction. By noting that
|a2 | ≦

2(1 − β)
= B2 .
|1 − α| |2m − 2n |

(16)

Thus, assuming that
|ak | ≦ Bk

(k ∈ {2, 3, . . . , r}),

we find from (14) and (15) that
r

|ar+1 | ≦

X
2(1 − β)
j n |aj |
m
n
|1 − α| |(r + 1) − (r + 1) |
j=1

≦

|(r +

1)m

δ
− (r + 1)n |

r
X

j n Bj = Br+1 .

j=1

Therefore, we have
|ak | ≦ Bk

(k ∈ N \ {1})

(17)

as desired.
By virtue of Lemma 1 and (15), we know that

k−1
Y
2(1 − β)
2(1 − β)j n
Bk =
1+
|1 − α| |k m − k n |
|1 − α| |j m − j n |

(k ∈ N \ {1, 2}).

(18)

j=2

Combining (16), (17) and (18), we readily arrive at the coefficient inequalities (8)
and (9) asserted by Theorem 1.
By virtue of Theorem 1 and (4), we get the following result.

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Y. Sun, W. Kuang, Z. Wang - Coefficient inequalities for certain classes...

Corollary 1. If f (z) ∈ Msm,n (α, β), then
|a2 | ≦

2(1 − β)
,
|1 − α| |2m+s − 2n+s |

and

k−1
Y
2(1 − β)j n
2(1 − β)
1+
|ak | ≦
|1 − α| |k m+s − k n+s |
|1 − α| |j m − j n |

(k ∈ N \ {1, 2}).

j=2

Remark 1. By specializing the parameters m and n, we get the corresponding
results obtained by Owa et al. [4].
By means of Theorem 1 and Corollary 1, we easily get the following distortion
theorems.
Corollary 2. If f ∈ Nm,n (α, β), then



Qk−1 
2(1−β)j n
∞


1
+
X
m
n
j=2
|1−α||j −j |
2(1 − β)
k
|z|
max 0, |z| −


|1 − α|
|k m − k n |
k=2

Qk−1 
2(1−β)j n
∞
1
+
X
m
n
j=2
|1−α||j −j |
2(1 − β)
≦ |f (z)| ≦ |z| +
|z|k ,
m
|1 − α|
|k − k n |
k=2

and



Qk−1 
2(1−β)j n
∞


1
+
X
j=2
|1−α||j m −j n |
2(1 − β)
k−1
max 0, 1 −
|z|


|1 − α|
|k m−1 − k n−1 |
k=2


Qk−1
2(1−β)j n
∞
X
 ′ 
j=2 1 + |1−α||j m −j n |
2(1
−
β)
|z|k−1 .
≦ f (z) ≦ 1 +
|1 − α|
|k m−1 − k n−1 |
k=2

Corollary 3. If f ∈ Msm,n (α, β), then




Qk−1 
2(1−β)j n
∞


1
+
X
m
n
j=2
|1−α||j −j |
2(1 − β)
k
max 0, |z| −
|z|


|1 − α|
|k m+s − k n+s |
k=2

Qk−1 
2(1−β)j n
∞
1
+
X
m
n
j=2
|1−α||j −j |
2(1 − β)
≦ |f (z)| ≦ |z| +
|z|k ,
m+s
|1 − α|
|k
− k n+s |
k=2

110

Y. Sun, W. Kuang, Z. Wang - Coefficient inequalities for certain classes...

and



Qk−1 
2(1−β)j n
∞


1
+
X
m
n
j=2
|1−α||j −j |
2(1 − β)
k−1
|z|
max 0, 1 −


|1 − α|
|k m+s−1 − k n+s−1 |
k=2

Qk−1 
2(1−β)j n
∞
1
+
X
m
n
 ′ 
j=2
|1−α||j −j |
2(1 − β)
≦ f (z) ≦ 1 +
|z|k−1 .
m+s−1
n+s−1
|1 − α|
|k
−k
|
k=2

Acknowledgements. The present investigation was supported by the Science
Fund of Huaihua University under Grant HHUQ2009-03 and the Scientific Research
Fund of Hunan Provincial Education Department under Grant 08C118 of the People’s Republic of China.
References
[1] G. S. Salagean, Subclasses of univalent functions, Lecture Notes in Math.
Springer-Verlag, 1013 (1983), 362–372.
[2] H. Silverman, Univalent functions with negative coefficients, Proc. Amer.
Math. Soc. 51 (1975), 109–116.
[3] S. Shams, S. R. Kulkarni and J. M. Jahangiri, Classes of uniformly starlike
and convex functions, Int. J. Math. Math. Sci. 2004 (2004), 2959–2961.
[4] S. Owa. Y. Polatoˇglu and E. Yavuz, Coefficient inequalities for class of
uniformly starlike and convex functions, J. Inequal. Pure Appl. Math. 7 (2006),
Article 160, pp. 1–5 (electronic).
[5] S. S. Eker and S. Owa, New applications of classes of analytic functions involving the Salagean operator, Proceedings of the International Symposium on Complex Function Theory and Applications, Transilvania University of Brasov Printing
House, Brasov, Romania, 2006, 21–34.
[6] S. S. Eker and S. Owa, Certain classes of analytic functions involving Salagean
operator, J. Inequal. Pure Appl. Math. 10 (2009), Article 22, pp. 1–12 (electronic).
[7] H. M. Srivastava and S. S. Eker, Some applications of a subordination theorem
for a class of analytic functions, Appl. Math. Lett. 21 (2008), 394–399.
Yong Sun and Wei-Ping Kuang
Department of Mathematics,
Huaihua University,
Huaihua 418008, Hunan,
People’s Republic of China

111

Y. Sun, W. Kuang, Z. Wang - Coefficient inequalities for certain classes...

email: sy785153@126.com
email: kuangweipingppp@163.com
Zhi-Gang Wang
School of Mathematics and Computing Science,
Changsha University of Science and Technology,
Yuntang Campus, Changsha 410114, Hunan,
People’s Republic of China
email: zhigwang@163.com

112