General Mathematics Vol. 15, Nr. 2–3(2007), 154-163
On a class of multivalent functions defined
by Salagean operator
Sevtap Sümer Eker
and
Bilal Şeker
Abstract
The present paper investigates new subclasses of multivalent
functions involving Salagean operator. Coefficient inequalities and
other interesting properties of these classes are studied.
2000 Mathematical Subject Classification: Primary 30C45
Keywords : Multivalent functions, Salagean operator, Coefficient
Inequalities, Extreme Points, Integral Means.
1
Introduction and definitions
Let A denote the class of functions f (z) of the form
(1)
f (z) = z +
∞
X
j=2
154
aj z j
On a class of multivalent functions defined by Salagean operator
155
which are analytic in the open disc U = {z : |z| < 1}.
For f (z) ∈ A, Salagean [1] introduced the following operator:
D0 f (z) = f (z)
D1 f (z) = Df (z) = zf 0 (z)
Dn f (z) = D(Dn−1 f (z))
(n ∈ N = 1, 2, 3, ...).
We note that,
n
D f (z) = z +
∞
X
j n aj z j
(n ∈ N0 = N ∪ {0}).
j=2
Let Ap denote the class of functions f (z) of the form
p
(2)
f (z) = z +
∞
X
aj z j
(p ≥ 1)
j=p+1
which are analytic and p-valent in the open disc U. We can write the
following equalities for the functions f (z) ∈ Ap :
D0 f (z) = f (z)
∞ µ ¶
X
j
z 0
p
D f (z) = Df (z) = f (z) = z +
aj z j
p
p
j=p+1
1
..
.
n
D f (z) = D(D
..
.
n−1
∞ µ ¶n
X
j
f (z)) = z +
aj z j
p
j=p+1
p
(n ∈ N0 = N ∪ {0}).
Let Np (m, n, α, β) denote the subclass of Ap consisting of functions f (z)
which satisfies the inequality
¯
¯ m
½ m
¾
¯
¯ D f (z)
D f (z)
¯ + α.
¯
Re
−
1
>
β
¯
¯ Dn f (z)
Dn f (z)
156
Sevtap Sümer Eker
and
Bilal Şeker
for some 0 ≤ α < 1, β ≥ 0, m ∈ N, n ∈ N0 and all z ∈ U.
Special cases of our classes are following:
(i)N1 (m, n, α, β) = Nm,n (α, β) which was studied by Eker and Owa [5].
(ii)N1 (1, 0, α, β) = SD(α, β) which was studied by Shams at all [3].
(iii) N1 (1, 0, α, 0) = S ∗ (α) and N1 (2, 1, α, 0) = K(α) which was studied
by Silverman [2].
(iv) N1 (m, n, α, 0) = Km,n (α) which was studied by Eker and Owa [4].
2
Coefficient inequalities for classes
Np(m, n, α, β)
Theorem 1. If f (z) ∈ Ap satisfies
(3)
∞
X
Ψp (m, n, j, α, β) |aj | ≤ 2(1 − α)
j=2
where
¯
µ ¶n µ ¶m ¯ µ
µ ¶n µ ¶m ¶
¯
¯
j
j
j
¯ + (1 − α) j
(4) Ψp (m, n, j, α, β) = ¯¯(1 + α)
−
+
¯
p
p
p
p
¯µ ¶m µ ¶n ¯
¯ j
j ¯¯
+2β ¯¯
−
p
p ¯
for some α(0 ≤ α < 1), β ≥ 0, m ∈ N and n ∈ N0 then f (z) ∈ Np (m, n, α, β).
Proof. Suppose that (3) is true for α(0 ≤ α < 1), β ≥ 0, m ∈ N , n ∈ N0 .
Using the fact that Rew > α if and only if |1 − α + w| > |1 + α − w|, it
suffices to show that
On a class of multivalent functions defined by Salagean operator
157
¯
¯
¯(1 − α)Dn f (z) + Dm f (z) − βeiθ |Dm f (z) − Dn f (z)|¯
(5)
¯
¯
− ¯(1 + α)Dn f (z) − Dm f (z) + βeiθ |Dm f (z) − Dn f (z)|¯ > 0
Substituting for Dn f (z) and Dm f (z) in (5) yields,
¯
¯
¯(1 − α)Dn f (z) + Dm f (z) − βeiθ |Dm f (z) − Dn f (z)|¯
¯
¯
− ¯¯(1 + α)Dn f (z) − Dm f (z) + βeiθ |Dm f (z) − D¯ n f (z)|¯
¯¯
µ ¶n µ ¶m ¸
∞ ·
∞ ·µ ¶m µ ¶n ¸
¯
¯X
¯¯
X
j
j
j
j
¯
¯
¯¯
(1−α)
= ¯(2−α)z p +
+
aj z j −βeiθ ¯
−
aj z j ¯¯
¯
¯
¯¯
p
p
p
p
j=p+1
j=p+1
¯
¯
¯
¯
µ ¶n µ ¶m ¸
∞ ·
∞ ·µ ¶m µ ¶n ¸
¯
¯X
¯¯
j
j
j
j
¯ p X
¯¯
j
iθ ¯
−¯αz +
(1+α)
−
aj z +βe ¯
−
aj z j ¯¯
¯
¯
¯¯
p
p
p
p
j=p+1
j=p+1
µ ¶n µ ¶m¯
∞ ¯
∞ ¯µ ¶m µ ¶n ¯
X
¯
¯
¯ iθ ¯ X
¯ j
j
j ¯¯
j
j
¯|aj | |z| −β ¯e ¯
¯
+
−
|aj | |z|j
≥ (2−α)|z| − ¯¯(1−α)
¯
¯
¯
p
p
p
p
j=p+1
j=p+1
p
µ ¶n µ ¶m ¯
∞ ¯
∞ ¯µ ¶m µ ¶n ¯
X
¯
¯
¯ iθ ¯ X
¯ j
j
j ¯¯
j
j
¯(1+α)
¯ |aj | |z| −β ¯e ¯
¯
−
−
−α|z| −
|aj | |z|j
¯
¯
¯
¯
p
p
p
p
p
j=p+1
≥ 2(1−α)−
j=p+1
¯µ ¶m µ ¶n ¯¸
µ ¶n µ ¶m ¶
µ ¶n µ ¶m ¯ µ
∞ ·¯
X
¯
¯ j
¯
j ¯¯
¯ + (1−α) j + j
¯
¯(1+α) j − j
+2β
−
|aj |
¯
¯
¯
p
p
p
p
p
p ¯
j=p+1
≥0
Example 1.The function f (z) given by
p
f (z) = z +
∞
X
2(p + 1 + δ)(1 − α)²j
zj
(j
+
δ)(j
+
1
+
δ)Ψ
(m,
n,
j,
α,
β)
p
j=p+1
belongs to the class Np (m, n, α, β) for δ > −p − 1, 0 ≤ α < 1, β ≥ 0, ²j ∈ C
and |²j | = 1.
158
3
Sevtap Sümer Eker
and
Bilal Şeker
ep(m, n, α, β)
Relation for N
ep (m, n, α, β) which
In view of Theorem 1, we now introduce the subclass N
consist of functions f (z) ∈ Ap whose Taylor-Maclaurin coefficients satisfy
ep (m, n, α, β)
the inequality (3). By the coefficient inequality for the class N
we see,
Theorem 2. If f (z) ∈ Ap , then
ep (m, n, α, β2 ) ⊂ N
ep (m, n, α, β1 )
N
for some β1 and β2 , 0 ≤ β1 ≤ β2 .
Proof. For 0 ≤ β1 ≤ β2 we obtain
∞
X
Ψp (m, n, j, α, β1 ) |aj | ≤
j=p+1
∞
X
Ψp (m, n, j, α, β2 ) |aj | .
j=p+1
ep (m, n, α, β2 ), then f (z) ∈ N
ep (m, n, α, β1 ). Hence we
Therefore, if f (z) ∈ N
get the required result.
4
Extreme points
The determination of the extreme points of a family F of univalent functions
enables us to solve many extremal problems for F .
Theorem 3. Let fp (z) = z p and
fj (z) = z p +
2(1 − α)²j
zj
Ψp (m, n, j, α, β)
(j = p + 1, p + 2, ... ; |²j | = 1).
On a class of multivalent functions defined by Salagean operator
159
ep (m, n, α, β) if and only if it can be expressed in the form
Then f ∈ N
f (z) = λp fp (z) +
∞
X
λj fj (z),
j=p+1
∞
X
where λj > 0 and λp = 1 −
λj .
j=p+1
Proof. Suppose that
f (z) = λp fp (z) +
∞
X
p
λj fj (z) = z +
j=p+1
∞
X
λj
j=p+1
2(1 − α)²j
zj
Ψp (m, n, j, α, β)
Then
∞
X
j=p+1
¯
¯
∞
X
¯ 2(1 − α)²j
¯
Ψp (m, n, j, α, β) ¯¯
λj ¯¯ =
2(1−α)λj
Ψp (m, n, j, α, β)
j=p+1
∞
X
= 2(1 − α)
λj
j=p+1
= 2(1 − α)(1 − λp )
≤ 2(1 − α)
ep (m, n, α, β) from the definition of the class of N
ep (m, n, α, β).
Thus, f (z) ∈ N
ep (m, n, α, β). Since
Conversely, suppose that f (z) ∈ N
|aj | ≤
2(1 − α)
Ψp (m, n, j, α, β)
(j = p + 1, p + 2, ...),
we may set
λj =
Ψp (m, n, j, α, β)
aj
2(1 − α)²j
and
λp = 1 −
∞
X
j=p+1
(|²j | = 1)
λj .
160
Sevtap Sümer Eker
Then,
f (z) = λp fp (z) +
∞
X
and
Bilal Şeker
λj fj (z).
j=p+1
This completes the proof of theorem.
ep (m, n, α, β) are the functions
Corollary 1. The extreme points of N
fp (z) = z p and
2(1 − α)²j
zj
Ψp (m, n, j, α, β)
(6)
fj (z) = z p +
5
Integral means inequalities
(j = p + 1, p + 2, ... ; |²j | = 1).
Definition 1. (Subordination Principle) For two functions f and g,
analytic in U, we say that the function f (z) is subordinate to g(z) in U,
and write
f (z) ≺ g(z)
(z ∈ U),
if there exists a Schwarz function w(z), analytic in U with
w(0) = 0
and
|w(z)| < 1 ,
such that
f (z) = g(w(z))
(z ∈ U).
In particular, if the function g is univalent in U, the above subordination is
equivalent to
f (0) = g(0) and f (U) ⊂ g(U).
In 1925, Littlewood [6] proved the following subordination theorem. (See
also Duren [7])
On a class of multivalent functions defined by Salagean operator
Theorem 4. (Littlewood [6])
161
If f and g are analytic in U with f ≺ g,
then for µ > 0 and z = reiθ (0 < r < 1)
Z 2π
Z
µ
|f (z)| dθ 5
0
2π
|g (z)|µ dθ.
0
We will make use of Theorem 5 to prove
ep (m, n, α, β) and supposed thatfj (z) is defined by
Theorem 5. Let f (z) ∈ N
(6). If there exists an analytic function w(z) given by
{w(z)}
j−p
∞
Ψp (m, n, j, α, β) X
=
aj z j−p ,
2(1 − α)²j
j=p+1
then for z = reiθ and 0 < r < 1,
Z 2π
Z
¯
¯µ
iθ
¯f (re )¯ dθ ≤
0
2π
¯
¯
¯fj (reiθ )¯µ dθ
(µ > 0).
0
Proof We must show that
¯µ
¯µ
Z 2π ¯¯
Z 2π ¯
∞
¯
X
¯
¯
2(1
−
α)²
¯
¯
j
j−p ¯
¯1 +
z
aj z j−p ¯ dθ ≤
¯1 +
¯ dθ.
¯
¯
¯
Ψp (m, n, j, α, β)
0
0
j=p+1
By applying Littlewood’s subordination theorem, it would suffice to show
that
1+
∞
X
aj z j−p ≺ 1 +
j=p+1
2(1 − α)²j
z j−p .
Ψp (m, n, j, α, β)
By setting
1+
∞
X
aj z j−p = 1 +
j=p+1
we find that
{w(z)}
j−p
2(1 − α)²j
{w(z)}j−p
Ψp (m, n, j, α, β)
∞
Ψp (m, n, j, α, β) X
=
aj z j−p
2(1 − α)²j
j=p+1
162
Sevtap Sümer Eker
and
Bilal Şeker
which readily yields w(0) = 0.
Furthermore, using (3), we obtain
¯
¯
∞
¯ Ψ (m, n, j, α, β) X
¯
¯ p
j−p
j−p ¯
|{w(z)}|
=¯
aj z ¯
¯ 2(1 − α)²j
¯
j=p+1
≤
∞
Ψp (m, n, j, α, β) X
|aj | |z|j−p
2(1 − α) |²j | j=p+1
≤ |z| < 1.
This completes the proof of the theorem.
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(1) (1975), 109-116.
[3] S.Shams, S.R.Kulkarni, J.M.Jahangiri, Classes of uniformly starlike
and convex functions, International Journal of Mathematics and Mathematical Sciences, Vol. 2004 (2004), Issue 55, 2959-2961.
[4] S.S.Eker, S.Owa, New applications of classes of analytic functions involving Salagean operator, International Symposium on Complex Function Theory and Applications, Brasov, Romania 1-5 September 2006
[5] S.S.Eker, S.Owa, Certain Classes Of Analytic Functions Involving
Salagean Operator, J. Inequal. Pure Appl. Math. (in course of publication).
On a class of multivalent functions defined by Salagean operator
163
[6] J. E. Littlewood, On inequalities in the theory of functions, Proc. London Math. Soc. 23(1925), 481-519.
[7] P. L. Duren, Univalent Functions, Springer-Verlag, New York ,1983.
Department of Mathematics
Faculty of Science and Letters
Dicle University
Diyarbakır, Turkey
Email address: sevtaps@dicle.edu.tr and bseker@dicle.edu.tr