General Mathematics Vol. 17, No. 4 (2009), 205–210
On some integral operators on analytic
functions
M. Acu, A. Branga, D. Breaz, N. Breaz, E. Constantinescu,
A. Totoi
Abstract
2000 Mathematics Subject Classification: Primary 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 }.
205
206 M. Acu, A. Branga, D. Breaz, N. Breaz, E. Constantinescu, A. Totoi
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. [5] 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. [5] 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. [1] 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)
On some integral operators on analytic functions
207
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. [1] 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 [2]) 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 [3] 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
T
T0 (α, β) = S ∗ (α, β) T , where S ∗ (α, β) is the class defined by (4)).
Definition 2. [3] 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}
208 M. Acu, A. Branga, D. Breaz, N. Breaz, E. Constantinescu, A. Totoi
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. [3] 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
From [4] we have the following definitions:
Z
∞
P
n
an z , satisfies Vµ (f )(z) =
Let f (z) ∈ T , f (z) = z −
1
f (tz)
dt,
t
n=2
0
where
µ is a real-valued, non-negative weight function normalized so that
Z 1
µ(t)dt = 1.
0
µ
¶δ−1
(c + 1)δ c
1
If µ(t) =
t log
(c > −1 ; δ > 0), which gives the Koµ(δ)
t
matu operator. Then we have
(8)
Vµ (f )(z) = z −
¶δ
∞ µ
X
c+1
n=2
µ
Remark 1. We notice that 0 <
c+1
c+n
c+n
µ(t)
an z n .
¶δ
< 1, where c > −1 , δ > 0 and
j ≥ 2.
Remark 2. It is easy to prove, by using Theorem 1 and Remark 1, that for
F (z) ∈ T and f (z) = Vµ (F )(z), we have f (z) ∈ T , where Vµ is the integral
operator defined by (8).
On some integral operators on analytic functions
Theorem 5. Let F (z) be in the class T ∗ (α), α ∈ [0, 1), F (z) = z −
209
∞
P
aj z j ,
j=2
aj ≥ 0, j ≥ 2. Then f (z) = Vµ (F )(z) ∈ T ∗ (α), where Vµ is the integral
operator defined by (8).
Proof. From Remark 2 we obtain f (z) = Vµ (F )(z) ∈ T .
¶δ
µ
∞
P
c+1
j
aj z j . By using Remark
We have f (z) = z− bj z , where bj =
c+j
j=2
j−α
j−α
1 we obtain
bj <
aj , for j = 2, 3, . . . , 0 ≤ α < 1,
1−α
1−α
∞
∞
X
X
j−α
j−α
and thus
bj ≤
aj ≤ 1. This mean (see Theorem 2)
1−α
1−α
j=2
j=2
that f (z) = Vµ (F )(z) ∈ T ∗ (α).
Similarly (by using Remark 2 and the 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) = Vµ (F )(z) ∈ T c (α), where Vµ 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) = Vµ (F )(z) ∈ C ∗ (α, β),
F (z) = z −
j=2
where Vµ 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) = Vµ (F )(z) ∈ Tn (α, β), where
j=2
Vµ is the integral operator defined by (8).
Remark 3. By choosing β = 1, respectively n = 0, in the above theorem,
we obtain the similarly results for the classes Tn (α) and S ∗ (α, β).
210 M. Acu, A. Branga, D. Breaz, N. Breaz, E. Constantinescu, A. Totoi
References
[1] V.P. Gupta, P.K. Jain, Certain classes of univalent functions with negative coefficients, Bull. Austral. Math. Soc., 14(1976), 409-416.
[2] G.S. Sălăgean, Subclasses of univalent functions, Complex Analysis.
Fifth Roumanian-Finnish Seminar, Lectures Notes in Mathematics,
1013, Springer-Verlag, 1983, 362-372.
[3] G.S. Sălăgean, Analytic functions with negative coefficients, Mathematica, 36(59), 2(1994), 219-224.
[4] S. Shams et. al., Classes of Ruscheweyh-type analytic univalent functions, Southwest J. Pure Appl. Math., 2(2003), 105-110.
[5] H. Silverman, Univalent functions with negative coefficients, Proc.
Amer. Math. Soc., 51(1975), 109-116.
M. Acu, A. Branga, E. Constantinescu, A. Totoi
Department of Mathematics, ”Lucian Blaga” University from Sibiu
550012 Sibiu, Romania
Corresponding e-mail: acu mugur@yahoo.com Daniel Breaz, Nicoleta Breaz
Department of Mathematics, ”1 Decembrie 1918” University of Alba Iulia
510009 Alba, Romania
E-mails: dbreaz@uab.ro, nbreaz@uab.ro