General Mathematics Vol. 15, Nr. 2–3(2007), 184-189
Some preserving properties of a integral
operator
Ana-Maria Acu and Eugen Constantinescu
Abstract
In this paper we will give some preserving properties of some
subclasses of functions with negative coefficients by using a integral
operator.
2000 Mathematical Subject Classification: 30C45
Key words and phrases: Function with negative coefficients, integral operator,
Sălăgean operator
1
Introduction and Preliminaries
Let H(U ) be the set of functions which are regular in the unit disc U ,
A = {f ∈ H(U ) : f (0) = f 0 (0) − 1 = 0}, Hu (U ) = {f ∈ H(U ) :
f is univalent in U } and S = {f ∈ A : f is univalent in U }.
184
Some preserving properties of a integral operator
185
We denote with T the subset of the functions f ∈ S, which have the
form
(1)
f (z) = z −
T
∞
X
aj z j , aj ≥ 0 , j ≥ 2 , z ∈ U
j=2
and with T ∗ = T S ∗ , T ∗ (α) = T
T
T S c (α), where 0 ≤ α < 1 .
T
S ∗ (α), T c = T
T
S c and T c (α) =
Theorem 1. [6] For a function f having the form (1) the following assertions are equivalents:
∞
X
(i)
jaj ≤ 1
j=2
(ii) f ∈ T
(iii) f ∈ T ∗ .
Regarding the classes T ∗ (α) and T c (α) with 0 ≤ α < 1 , we recall here
the following result:
Theorem 2. [6] A function f having the form (1) is in the class T ∗ (α) if
and only if:
(2)
∞
X
j−α
j=2
1−α
aj ≤ 1 ,
and is in the class T c (α) if and only if:
(3)
∞
X
j(j − α)
j=2
1−α
aj ≤ 1 .
Definition 1. [2] Let S ∗ (α, β) denote the class of functions having the form
(1) which are starlike and satisfy
¯
¯
zf 0 (z)
¯
¯
−
1
f (z)
¯
¯
(4)
¯ zf 0 (z)
¯<β
¯
+ (1 − 2α) ¯
f (z)
186
Ana-Maria Acu and Eugen Constantinescu
for 0 ≤ α < 1 and 0 < β ≤ 1. And let C ∗ (α, β) denote the class of functions
such that zf 0 (z) is in the class S ∗ (α, β).
Theorem 3. [2] A function f having the form (1) is in the class S ∗ (α, β)
if and only if:
(5)
∞
X
{(j − 1) + β(j + 1 − 2α)} aj ≤ 2β(1 − α) ,
j=2
and is in the class C ∗ (α, β) if and only if:
(6)
∞
X
j {(j − 1) + β(j + 1 − 2α)} aj ≤ 2β(1 − α) .
j=2
Let Dn be the Sălăgean differential operator (see [3]) defined as:
Dn : A → A ,
n ∈ N and D0 f (z) = f (z)
D1 f (z) = Df (z) = zf 0 (z) ,
Dn f (z) = D(Dn−1 f (z)).
In [5] the author define the class Tn (α, β), from which by choosing different values for the parameters we obtain variously subclasses of analytic
functions with negative coefficients (for example Tn (α, 1) = Tn (α) which is
the class of n-starlike of order α functions with negative coefficients and
T0 (α, β) = S ∗ (α, β) is the class defined by (4)).
Definition 2. [5] Let α ∈ [0, 1), β ∈ (0, 1] and n ∈ N . We define the class
Sn (α, β) of the n-starlike of order α and type β through
Sn (α, β) = {f ∈ A ; |J(f, n, α; z)| < β}
Some preserving properties of a integral operator
187
Dn+1 f (z) − Dn f (z)
where J(f, n, α; z) = n+1
, z ∈ U . Consequently
n
T D f (z) + (1 − 2α)D f (z)
Tn (α, β) = Sn (α, β) T .
Theorem 4. [5] Let f be a function having the form (1). Then f ∈
Tn (α, β) if and only if
∞
X
(7)
j n [j − 1 + β(j + 1 − 2α)] aj ≤ 2β(1 − α) .
j=2
2
Main results
In [1] the authors consider the integral operator Ic+δ : A → A, 0 < u ≤ 1,
1 ≤ δ < ∞, 0 < c < ∞, defined by
Z1
(8)
uc+δ−2 F (uz)du.
f (z) = Ic+δ (F (z)) = (c + δ)
0
Remark 1. For F (z) = z +
∞
P
aj z j , from (8) we obtain
j=2
f (z) = z +
∞
X
j=2
Also, we notice that 0 <
c+δ
aj z j .
c+j+δ−1
c+δ
< 1, where 0 < c < ∞, j ≥ 2,
c+j+δ−1
1 ≤ δ < ∞.
Remark 2. It is easy to prove that for F (z) ∈ T and f (z) = Ic+δ (F (z)),
we have f (z) ∈ T , where Ic+δ is the integral operator defined by (8).
188
Ana-Maria Acu and Eugen Constantinescu
Theorem 5. Let F (z) be in the class T ∗ (α), α ∈ [0, 1), F (z) = z −
∞
P
aj z j ,
j=2
aj ≥ 0, j ≥ 2. Then f (z) = Ic+δ (F (z)) ∈ T ∗ (α), where Ic+δ is the integral
operator defined by (8).
Proof. From Remark 2 we obtain f (z) = Ic+δ (F (z)) ∈ T .
∞
X
∞
P
c+δ
j−α
We have
aj ≤ 1 and f (z) = z− bj z j , where bj =
aj .
1−α
c+j+δ−1
j=2
j=2
c+δ
By using the fact that 0 <
< 1, where 0 < c < ∞, j ≥ 2,
c+j+δ−1
∞
X
j−α
j−α
j−α
1 ≤ δ < ∞, we obtain
bj <
aj and thus
bj ≤ 1. This
1−α
1−α
1−α
j=2
mean (see Theorem 2) that f (z) = Ic+δ (F (z)) ∈ T ∗ (α).
Similarly (by using Theorems 2 , 3 and 4) we obtain:
Theorem 6. Let F (z) be in the class T c (α), α ∈ [0, 1), F (z) = z −
∞
P
aj z j ,
j=2
aj ≥ 0, j ≥ 2. Then f (z) = Ic+δ (F (z)) ∈ T c (α), where Ic+δ is the integral
operator defined by (8).
Theorem 7. Let F (z) be in the class C ∗ (α, β), α ∈ [0, 1), β ∈ (0, 1],
∞
P
aj z j , aj ≥ 0, j ≥ 2. Then f (z) = Ic+δ (F (z)) ∈ C ∗ (α, β),
F (z) = z −
j=2
where Ic+δ is the integral operator defined by (8).
Theorem 8. Let F (z) be in the class Tn (α, β), α ∈ [0, 1), β ∈ (0, 1], n ∈ N,
∞
P
F (z) = z −
aj z j , aj ≥ 0, j ≥ 2. Then f (z) = Ic+δ (F (z)) ∈ Tn (α, β),
j=2
where Ic+δ is the integral operator defined by (8).
Remark 3. By choosing β = 1, respectively n = 0, in the above theorem,
we obtain the similarly result for the classes Tn (α) and S ∗ (α, β).
Some preserving properties of a integral operator
189
References
[1] M. Acu and all, On some starlike functions with negative coefficients,
(to appear).
[2] V.P. Gupta, P.K. Jain, Certain classes of univalent functions with negative coefficients, Bull. Austral. Math. Soc., 14(1976), 409-416.
[3] G.S. Sălăgean, Geometria planului complex, Ed. Promedia Plus, ClujNapoca, 1997.
[4] G.S. Sălăgean, Subclasses of univalent functions, Complex Analysis.
Fifth Roumanian-Finnish Seminar, Lectures Notes in Mathematics,
1013, Springer-Verlag, 1983, 362-372.
[5] G.S. Sălăgean, Analytic functions with negative coefficients, Mathematica, 36(59), 2(1994), 219-224.
[6] H. Silverman, Univalent functions with negative coefficients, Proc.
Amer. Math. Soc., 51(1975), 109-116.
Department of Mathematics
Faculty of Science
University ”Lucian Blaga” of Sibiu
Str. I. Ratiu, no. 5-7,
550012 - Sibiu, Romania
Email addresses: acuana77@yahoo.com , egnconst68@yahoo.com