Paper25-Acta25-2011. 111KB Jun 04 2011 12:03:16 AM
Acta Universitatis Apulensis
ISSN: 1582-5329
No. 25/2011
pp. 277-286
DIFFERENTIAL OPERATORS FOR P-VALENT FUNCTIONS
Mohamed K. Aouf
Abstract. Let f (z) = D(F (z)), where D is differential operator defined separtely in every result. Let S(p) denote the class of functions f (z) = z p +
∞
P
an z n
n=p+1
which are analytic and p-valent in the unit disc U = {z : |z| < 1}. The purpose of
this is paper to find out the disc in which the operator D transforms some classes
of p-valent functions into the same. For example, if F is p-valent starlike of order
λ(0 ≤ λ < p), then the disc in which f is also in the same class is found. We also
discuss several special cases which can be derived from our main results.
Keywords: p-Valent, analytic, starlike, close-to-convex, Bazilevic functions.
2000 Mathematics Subject Classification: 30C45.
1.Introduction
Let A(p) denote the class of functions of the form
∞
X
f (z) = z p +
an z n
(1)
n=p+1
which are analytic in the unit disc U = {z : |z| < 1}. Further let S(p) be the subclass
of A(p) consisting of functions which are p-valent in U . A function f (z) ∈ S(p) is
said to be in the class Sp∗ (λ) of p-valently starlike functions of order λ(0 ≤ λ < p) if
it satisfies
Re
The class
Sp∗ (λ)
(
′
z f (z)
f (z)
)
> λ and
Z2π
Re
0
(
′
z f (z)
f (z)
)
dθ = 2πp .
was studied recently by Owa [8] and Aouf et al. [2].
277
(2)
M. K. Aouf - Differential Operators For p-Valent Functions
Let Kp (γ, λ) denote the class of functions F (z) ∈ A(p) which satisfy
Re
(
′
z F (z)
G(z)
)
> γ (z ∈ U ),
(3)
where G(z) ∈ Sp∗ (λ) and 0 ≤ γ, λ < p. The class Kp (γ, λ) of p-valenty close-toconvex functions of order γ and type λ was studied by Aouf [1]. We note that
K1 (γ, λ) = K(γ, λ), the class of convex functions of order γ and type λ was studied
by Libera [4] and K1 (1, 1) = K, is the well-known class of close-to-convex functions,
introduced by Kaplan [3].
A function F (z) ∈ S(p) is said to be in the class Bp (β, λ) if and only if there
exists a function G(z) ∈ Sp∗ (λ), 0 ≤ λ < p, β > 0, such that
Re
(
′
zF (z)
1−β
F
(z)Gβ (z)
)
> 0 (z ∈ U ).
(4)
The class Bp (β, λ) is the subclass of p-valently Bazilevic functions in U .
We note that Bp (β, 0) = Bp (β), is the class of p-valently Bazilevic functions of type
β and B1 (β, 0) = B(β), is the class of Bazilevic functions of type β (see [10]).
2.Main Results
We shall consider some differential operators and find out the disc in which the
classes of p-valent functions defined by (2), (3) and (4), respectively, are preserved
under these operators. We prove the following:
Theorem 1. Let F (z) ∈ Sp∗ (λ)(0 ≤ λ < p), β > 0 and 0 < α ≤ 1. Let the function
f (z) be defined by the differential operator
Dα,β,p (F ) = f β (z) =
i
h
1
′
(1 − α)F β (z) + α z(F β (z)) .
αβp − α + 1
(5)
Then f (z) ∈ Sp∗ (λ) for |z| < r0 , where
r0 =
[α(βp − βλ + 1) +
The result is sharp.
Proof. We can write
f β (z) =
as
p
α2 (βp
αβp − α + 1
.
− βλ + 1)2 + (αβp − α + 1)(1 − αβp + 2αβλ − α)
(6)
i
h
1
′
(1 − α)F β (z) + α z(F β (z)) ,
αβp − α + 1
f β (z) =
h
i
1
1
1
′
α z 2− α (z α −1 F β (z)) ,
αβp − α + 1
278
(7)
M. K. Aouf - Differential Operators For p-Valent Functions
and from this it follows that
1
1
F (z) = (βp + − 1) z 1− α
α
β
Zz
1
z α −2 f β (z)d z .
0
(8)
Thus, we have
1
′
zF (z)
1
z α −1 f β (z)
.
β
= (1 − ) + Rz
1
F (z)
α
z α −2 f β (z)d z
(9)
0
Since F (z) ∈
Sp∗ (λ),
we can write (9) as
β(p − λ)h(z)
Zz
z
1
−2
α
β
f (z)d z + βλ
= (1 −
1
)
α
1
z α −2 f β (z)d z
0
0
Zz
Zz
1
1
z α −2 f β (z)d z + z α −1 f β (z),
0
where Re h(z) > 0. Differentiating again with respect to z, we obtain
′
′
z f (z)
= (p − λ)h(z) + λ +
f (z)
Rz
1
(p − λ)h (z) z α −2 f β (z)d z
0
z
1
−2
α
f β (z)
Now, using a well-known result [4],
′
|h (z)| ≤
2Re(z)
(|z| = r),
1 − r2
(11)
we have from (10
Re
(
′
z f (z)
−λ
f (z)
)
≥ Re h(z) {(p − λ)−
279
.
(10)
M. K. Aouf - Differential Operators For p-Valent Functions
z
R 1 −2 β
z α f (z)d z
2(p − λ) 0
.
1 − r2 z α1 −2 f β (z)
(12)
Also, from (7) and (8), we have
1
1
z α −1 f β (z)
Rz
=
1
′
α z(z α −1 F β (z))
1
α (z α −1 F β (z))
z α −2 f β (z)d z
0
′
z F (z)
1
− 1) + β
α
F (z)
1
= ( − 1) + β [(p − λ)h(z) + λ],
α
= (
since F (z) ∈ Sp∗ (λ). Thus we have
z α1 −1 f β (z)
Rz
1
z α −2 f β (z)d z
0
≥ Re
(
1
− 1) + β[(p − λ)h(z) + λ]
α
1
1−r
− 1) + βλ + β(p − λ)
α
1+r
( α1 − 1 + βλ)(1 + r) + (βp − βλ)(1 − r)
.
1+r
≥ (
=
(13)
Hence, from (12) and (13), we have
Re
(
′
z f (z)
−λ
f (z)
)
≥ Re h(z) {(p − λ)−
2(p − λ)
(1 + r)(1 − r)
r(1 + r)
1
( α − 1 + βλ)(1 + r) + (βp − βλ)(1 − r)
)
= (p − λ) Re h(z).
.
(
(βp +
− 1) − 2(βp − βλ + 1)r − ( α1 − βp + 2βλ − 1)r2
(1 − r)(βp + α1 − 1) + ( α1 − βp + 2βλ − 1)r
1
α
)
.
(14)
The right-hand side of (50) is positive for r < r0 , 0 < α ≤ 1, β > 0 and 0 ≤ λ < p,
where r0 is given by the relation (6).
280
M. K. Aouf - Differential Operators For p-Valent Functions
The function f0 (z) defined by:
f0β (z) =
where
i
h
1
′
(1 − α) F0β (z) + α z(F0β (z)) ,
αβp − α + 1
(15)
zp
∈ Sp∗ (λ)
(1 − z)2(p−λ)
(16)
F0 (z) =
shows that the result is sharp.
Putting β = 1 in Theorem 1, we obtain
Corollary 1. Let F (z) ∈ Sp∗ (λ)(0 ≤ λ < p), and 0 < α ≤ 1. Let the function f (z)
be defined by the differential operator
Dα,1,p (F ) = f (z) =
h
i
1
′
(1 − α)F (z) + αzF (z) .
αp − α + 1
(17)
Then f (z) ∈ Sp∗ (λ) for |z| < r0∗ , where
r0∗ =
α(p − λ + 1) +
p
α2 (p
αp − α + 1
.
− λ + 1)2 + (αp − α + 1)(1 − αp + 2αλ − α)
(18)
The result is sharp.
Putting β = 1 and λ = 0 in Theorem 1, we have
Corollary 2. Let F (z) ∈ Sp∗ and 0 < α ≤ 1. Let the function f (z) be defined by
the differential operator (53). Then f (z) ∈ Sp∗ for |z| < r0∗∗ , where
r0∗∗ =
α(p + 1) +
p
α2 (p
αp − α + 1
.
+ 1)2 + (αp − α + 1)(1 − α p − α)
(19)
The result is sharp.
Remark 1. (1) Putting p = 1 in Theorem 1, we obtain the result obtained by Noor
[6];
(2) Putting p = 1 in Corollary 2, we obtain the result obtained By Noor et al.
[7];
(3) Putting p = 1 and α = 12 in Corollary 2, we obtain the result obtained by
Livingston [5];
(4) Theorem 1 generalizes a result due to Padmanabhan [9] when we take p =
β = 1 and α = 12 .
Theorem 2. Let F (z) ∈ Kp (γ, λ)(0 ≤ γ, λ < p) and let α > 0. Then the function
f (z), defined as
′
Dα,1,p (F ) = f (z) = (1 − α)F (z) + α zF (z)
281
(20)
M. K. Aouf - Differential Operators For p-Valent Functions
belongs to the same class for |z| < R0 , R0 = min(r1 , r2 ), where r1 and r2 are the
respective least positive roots of the equations
(p − 1 +
1
1
) − 2(p + 1 − λ)r − ( + 2λ − (p + 1))r2 = 0 ,
α
α
(21)
(p − 1 +
1
1
) − 2(p + 1 − γ)r − ( + 2γ − (p + 1))r2 = 0 .
α
α
(22)
and
The result is sharp.
Proof. Since F (z) ∈ Kp (γ, λ), then there exists a function G(z) ∈ Sp∗ (λ) such
that Re
′
z F (z)
G(z)
> γ, z ∈ U . Now let
′
Dα,1,p (G) = g(z) = (1 − α)G(z) + αzG (z).
(23)
Then, from Theorem 1, it follows that g(z) ∈ Sp∗ (λ) for |z| < r1 , where r1 is the
least positive root of (57).
′
f (z)
>γ
Using the same technique of Theorem 1, we can easily show that Re z g(z)
for |z| < R0 , where R0 = min(r1 , r2 ) and r2 is given by (58). The sharpness of the
result can be seen as follows:
Let
zp
zp
F1 (z) =
and
G
(z)
=
.
1
(1 − z)2(p−γ)
(1 − z)2(p−λ)
Then
′
z F (z)
Re 1
> γ , G1 ∈ Sp∗ (λ) ⇒ F1 ∈ Kp (γ, λ).
G1 (z)
′
′
Let f1 (z) = (1 − α)F1 (z) + α z F1 (z) and g1 (z) = (1 − α) G1 (z) + α z G1 (z).
′
z f1 (z)
g1 (z)
> γ for |z| < R0 , where R0 is as given in Theorem 2 and can not be
Then Re
improved.
Remark 2. (1) Putting p = 1 in Theorem 2, we obtain the result obtained by Noor
[6];
(2) Putting p = 1 and γ = λ = 0 in Theorem 2, we obtain the result obtained By
Noor et al. [7];
(3) Putting p = 1, γ = λ = 0 and α = 12 in Theorem 2, we obtain the result
obtained by Livingston [5];
(4) Putting p = 1 and α = 21 in Theorem 2, we obtain the result obtained by
Padmanabhan[9].
Theorem 3. Let 0 < α ≤ β ≤ 1 and F (z) ∈ Sp∗ (λ)(0 ≤ λ < p). Then f (z) defined
as
282
M. K. Aouf - Differential Operators For p-Valent Functions
′
∗
(F ) = f α (z) = z α (z 1−β F β (z)) .
Dα,β,p
(24)
belongs to Sp∗ (λ1 ) for |z| < R1 , where R1 is given by
R1 =
(βp − βλ + 1) +
and
p
β 2 (p
−
1
,
+ β(1 − p)[β(1 + p − 2λ) − 2]
λ)2
λ1 = 1 −
(25)
β
(1 − λ).
α
The result is sharp.
Proof. From (2.20), we can write
β
F (z) = z
β−1
Zz
(
0
and so
f (z) α
) dz ,
z
α
z ( f (z)
zF (z)
z )
= (β − 1) + Rz
.
F (z)
f (z) α
( z ) dz
′
β
0
(26)
′
(z)
Since F (z) ∈ Sp∗ (λ), z FF(z)
= (p − λ)h(z) + λ, 0 ≤ λ < p, Re h(z) > 0, z ∈ U . Thus,
from (2.22), we obtain
β(p − λ)h(z)
Zz
0
f (z) α
(
) dz + [βλ + (1 − β)]
z
Zz
0
(
f (z) α
f (z) α
) dz = z(
) .
z
z
Differentiating again with respect to z, we obtain
f (z) α
′
) + β(p − λ)h (z)
β(p − λ)h(z)(
z
Zz
0
(
f (z) α
f (z) α
) dz + [βλ + (1 − β)](
)
z
z
′
= α z 1−α f α−1 (z) f (z) + (1 − α)(
or
′
β
z f (z)
− [1 − (1 − λ)]
f (z)
α
283
f (z) α
)
z
M. K. Aouf - Differential Operators For p-Valent Functions
=
′
β
(p − λ) h(z) +
α
h (z)
Rz
α
( f (z)
z )
0
α
( f (z)
z )
dz
.
Now, using the well-known result (11), we have from (2.23),
′
z f (z)
β
Re
− [1 − (1 − λ)] ≥
f (z)
α
z
R f (z) α
(
)
dz
β
2 0 z
(p − λ)Re h(z) 1 −
.
α
1 − r2 ( f (z) )α
z
Now
Rz
0
α
z ( f (z)
z )
α
( f (z)
z ) dz
′
=
z (z 1−β F β (z)
z 1−β F β (z)
′
= β
zF (z)
+ (1 − β)
F (z)
= β[(p − λ)h(z) + λ] + (1 − β), Re h(z) > 0,
and from this, it follows that
z( f (z) )α
z
Rz
f (z) α
( z ) dz
0
≥ β(p − λ)Re h(z) + βλ + (1 − β)
=
Hence
1−r
+ βλ + (1 − β)
1+r
β(p − λ)(1 − r) + (βλ + (1 − β))(1 + r)
.
(1 + r)
≥ β(p − λ)
′
Re
β
β
z f (z)
− [1 − (1 − λ)] ≥ (p − λ) Re h(z).
f (z)
α
α
284
(27)
M. K. Aouf - Differential Operators For p-Valent Functions
.
=
r(1 + r)
2
1−
2
1 − r β(p − λ)(1 − r) + (βλ + (1 − β))(1 + r)
β
(p − λ)Reh(z)
α
(1 − r)[β(p − λ)(1 − r) + (βλ + (1 − β))(1 + r)] − 2r
. (28)
(1 − r)[β(p − λ)(1 − r) + (βλ + (1 − β))(1 + r)]
The right-hand side of the inequality (2.24) is positive for |z| < R1 , where R1 is the
least positive root of the equation
(1 − βp − β + 2βλ)r2 + 2(βp − βλ + 1)r − (βp − β + 1) = 0,
(29)
and thus we obtain the required result. The function
zp
∈ Sp∗ (λ)
(1 − z)2(p−λ)
F1 (z) =
(30)
shows that the result is best possible.
Putting λ = 12 in Theorem 3, we obtain
Corollary 3. Let 0 < α ≤ β ≤ 1 and F (z) ∈ Sp∗ ( 21 ). Then f (z) defined by (2.20)
β
) for |z| < R2 , where R2 is given by
belongs to Sp∗ (1 − 2α
R2 =
(2βp − β + 2) +
p
β 2 (2p
2
.
− 1)2 + 8 + β(1 − p)(p − 2)
(31)
The result is sharp.
Remark 3. (1) Putting p = 1 in Theorem 3, we obtain the result obtained by Noor
[6];
(2) Putting α = β and F (z) ∈ Sp∗ (λ). Then f (z) defined by (2.20) also belongs
to Sp∗ (λ) for |z| < R1 ;
(3) Putting p = 1 in Corollary 3, we obtain the result obtained by Noor [6].
Using the same technique of Theorem 3 we can prove the following theorem.
Theorem 4. Let F (z) ∈ Bp (β, λ), 0 < β ≤ 1 and 0 ≤ λ < p. Let f (z) be defined
as
′
f β (z) = z β (z 1−β F β (z)) .
Then f (z) ∈ Bp (β, λ) for |z| < R3 , where R3 is given by
R3 =
(βp − β + 1) +
The result is sharp.
p
β 2 (p
−
λ)2
1
.
+ 2 + β(1 − p)[β(1 + p − 2λ) − 2]
285
(32)
M. K. Aouf - Differential Operators For p-Valent Functions
References
[1] M. K. Aouf, On a class of p-valent close-to-convex functions, Internat. J.
Math. Math. Sci., 11, no.2, (1988), 259-266.
[2] M. K. Aouf, H. M. Hossen and H. M. Srivastava, Some families of multivalent
functions, Comput. Math. Appl., 39, (2000), 39-48.
[3] W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J., (1952),
169-185.
[4] R. J. Libera, Some radius of convexity problems, Duke Math. J., 31, (1964),
143-158.
[5] A. E. Livingston, On the radius of univalent of certain analytic functions,
Proc. Amer. Math. Soc., 17, (1966), 352-357.
[6] K. I. Noor, Differential operators for univalent functions, Math. Japon., 32,
no.3, (1987), 427-436.
[7] K. I. Noor, F. M. Aloboudi and N. Aldihan, On the radius of univalence of
convex combinations of analytic functions, Internat J. Math. Math. Sci., 6, no.2,
(1983), 335-340.
[8] S. Owa, Some properties of certain multivalent functions, Appl. Math. Letters, 4, no.5, (1991), 79-83.
[9] K. S. Padmanabhan, On the radius of univalence of certain class of analytic
functions, J. London Math. Soc., 1, (1969), 225-231.
[10] D. K. Thomas, Bazilevic functions, Trans. Amer. Math. Soc., 132, (1968),
353-361.
Mohamed K. Aouf
Department of Mathematics, Faculty of Science
Mansoura University, Mansoura 35516, Egypt.
email: mkaouf127@yahoo.com.
286
ISSN: 1582-5329
No. 25/2011
pp. 277-286
DIFFERENTIAL OPERATORS FOR P-VALENT FUNCTIONS
Mohamed K. Aouf
Abstract. Let f (z) = D(F (z)), where D is differential operator defined separtely in every result. Let S(p) denote the class of functions f (z) = z p +
∞
P
an z n
n=p+1
which are analytic and p-valent in the unit disc U = {z : |z| < 1}. The purpose of
this is paper to find out the disc in which the operator D transforms some classes
of p-valent functions into the same. For example, if F is p-valent starlike of order
λ(0 ≤ λ < p), then the disc in which f is also in the same class is found. We also
discuss several special cases which can be derived from our main results.
Keywords: p-Valent, analytic, starlike, close-to-convex, Bazilevic functions.
2000 Mathematics Subject Classification: 30C45.
1.Introduction
Let A(p) denote the class of functions of the form
∞
X
f (z) = z p +
an z n
(1)
n=p+1
which are analytic in the unit disc U = {z : |z| < 1}. Further let S(p) be the subclass
of A(p) consisting of functions which are p-valent in U . A function f (z) ∈ S(p) is
said to be in the class Sp∗ (λ) of p-valently starlike functions of order λ(0 ≤ λ < p) if
it satisfies
Re
The class
Sp∗ (λ)
(
′
z f (z)
f (z)
)
> λ and
Z2π
Re
0
(
′
z f (z)
f (z)
)
dθ = 2πp .
was studied recently by Owa [8] and Aouf et al. [2].
277
(2)
M. K. Aouf - Differential Operators For p-Valent Functions
Let Kp (γ, λ) denote the class of functions F (z) ∈ A(p) which satisfy
Re
(
′
z F (z)
G(z)
)
> γ (z ∈ U ),
(3)
where G(z) ∈ Sp∗ (λ) and 0 ≤ γ, λ < p. The class Kp (γ, λ) of p-valenty close-toconvex functions of order γ and type λ was studied by Aouf [1]. We note that
K1 (γ, λ) = K(γ, λ), the class of convex functions of order γ and type λ was studied
by Libera [4] and K1 (1, 1) = K, is the well-known class of close-to-convex functions,
introduced by Kaplan [3].
A function F (z) ∈ S(p) is said to be in the class Bp (β, λ) if and only if there
exists a function G(z) ∈ Sp∗ (λ), 0 ≤ λ < p, β > 0, such that
Re
(
′
zF (z)
1−β
F
(z)Gβ (z)
)
> 0 (z ∈ U ).
(4)
The class Bp (β, λ) is the subclass of p-valently Bazilevic functions in U .
We note that Bp (β, 0) = Bp (β), is the class of p-valently Bazilevic functions of type
β and B1 (β, 0) = B(β), is the class of Bazilevic functions of type β (see [10]).
2.Main Results
We shall consider some differential operators and find out the disc in which the
classes of p-valent functions defined by (2), (3) and (4), respectively, are preserved
under these operators. We prove the following:
Theorem 1. Let F (z) ∈ Sp∗ (λ)(0 ≤ λ < p), β > 0 and 0 < α ≤ 1. Let the function
f (z) be defined by the differential operator
Dα,β,p (F ) = f β (z) =
i
h
1
′
(1 − α)F β (z) + α z(F β (z)) .
αβp − α + 1
(5)
Then f (z) ∈ Sp∗ (λ) for |z| < r0 , where
r0 =
[α(βp − βλ + 1) +
The result is sharp.
Proof. We can write
f β (z) =
as
p
α2 (βp
αβp − α + 1
.
− βλ + 1)2 + (αβp − α + 1)(1 − αβp + 2αβλ − α)
(6)
i
h
1
′
(1 − α)F β (z) + α z(F β (z)) ,
αβp − α + 1
f β (z) =
h
i
1
1
1
′
α z 2− α (z α −1 F β (z)) ,
αβp − α + 1
278
(7)
M. K. Aouf - Differential Operators For p-Valent Functions
and from this it follows that
1
1
F (z) = (βp + − 1) z 1− α
α
β
Zz
1
z α −2 f β (z)d z .
0
(8)
Thus, we have
1
′
zF (z)
1
z α −1 f β (z)
.
β
= (1 − ) + Rz
1
F (z)
α
z α −2 f β (z)d z
(9)
0
Since F (z) ∈
Sp∗ (λ),
we can write (9) as
β(p − λ)h(z)
Zz
z
1
−2
α
β
f (z)d z + βλ
= (1 −
1
)
α
1
z α −2 f β (z)d z
0
0
Zz
Zz
1
1
z α −2 f β (z)d z + z α −1 f β (z),
0
where Re h(z) > 0. Differentiating again with respect to z, we obtain
′
′
z f (z)
= (p − λ)h(z) + λ +
f (z)
Rz
1
(p − λ)h (z) z α −2 f β (z)d z
0
z
1
−2
α
f β (z)
Now, using a well-known result [4],
′
|h (z)| ≤
2Re(z)
(|z| = r),
1 − r2
(11)
we have from (10
Re
(
′
z f (z)
−λ
f (z)
)
≥ Re h(z) {(p − λ)−
279
.
(10)
M. K. Aouf - Differential Operators For p-Valent Functions
z
R 1 −2 β
z α f (z)d z
2(p − λ) 0
.
1 − r2 z α1 −2 f β (z)
(12)
Also, from (7) and (8), we have
1
1
z α −1 f β (z)
Rz
=
1
′
α z(z α −1 F β (z))
1
α (z α −1 F β (z))
z α −2 f β (z)d z
0
′
z F (z)
1
− 1) + β
α
F (z)
1
= ( − 1) + β [(p − λ)h(z) + λ],
α
= (
since F (z) ∈ Sp∗ (λ). Thus we have
z α1 −1 f β (z)
Rz
1
z α −2 f β (z)d z
0
≥ Re
(
1
− 1) + β[(p − λ)h(z) + λ]
α
1
1−r
− 1) + βλ + β(p − λ)
α
1+r
( α1 − 1 + βλ)(1 + r) + (βp − βλ)(1 − r)
.
1+r
≥ (
=
(13)
Hence, from (12) and (13), we have
Re
(
′
z f (z)
−λ
f (z)
)
≥ Re h(z) {(p − λ)−
2(p − λ)
(1 + r)(1 − r)
r(1 + r)
1
( α − 1 + βλ)(1 + r) + (βp − βλ)(1 − r)
)
= (p − λ) Re h(z).
.
(
(βp +
− 1) − 2(βp − βλ + 1)r − ( α1 − βp + 2βλ − 1)r2
(1 − r)(βp + α1 − 1) + ( α1 − βp + 2βλ − 1)r
1
α
)
.
(14)
The right-hand side of (50) is positive for r < r0 , 0 < α ≤ 1, β > 0 and 0 ≤ λ < p,
where r0 is given by the relation (6).
280
M. K. Aouf - Differential Operators For p-Valent Functions
The function f0 (z) defined by:
f0β (z) =
where
i
h
1
′
(1 − α) F0β (z) + α z(F0β (z)) ,
αβp − α + 1
(15)
zp
∈ Sp∗ (λ)
(1 − z)2(p−λ)
(16)
F0 (z) =
shows that the result is sharp.
Putting β = 1 in Theorem 1, we obtain
Corollary 1. Let F (z) ∈ Sp∗ (λ)(0 ≤ λ < p), and 0 < α ≤ 1. Let the function f (z)
be defined by the differential operator
Dα,1,p (F ) = f (z) =
h
i
1
′
(1 − α)F (z) + αzF (z) .
αp − α + 1
(17)
Then f (z) ∈ Sp∗ (λ) for |z| < r0∗ , where
r0∗ =
α(p − λ + 1) +
p
α2 (p
αp − α + 1
.
− λ + 1)2 + (αp − α + 1)(1 − αp + 2αλ − α)
(18)
The result is sharp.
Putting β = 1 and λ = 0 in Theorem 1, we have
Corollary 2. Let F (z) ∈ Sp∗ and 0 < α ≤ 1. Let the function f (z) be defined by
the differential operator (53). Then f (z) ∈ Sp∗ for |z| < r0∗∗ , where
r0∗∗ =
α(p + 1) +
p
α2 (p
αp − α + 1
.
+ 1)2 + (αp − α + 1)(1 − α p − α)
(19)
The result is sharp.
Remark 1. (1) Putting p = 1 in Theorem 1, we obtain the result obtained by Noor
[6];
(2) Putting p = 1 in Corollary 2, we obtain the result obtained By Noor et al.
[7];
(3) Putting p = 1 and α = 12 in Corollary 2, we obtain the result obtained by
Livingston [5];
(4) Theorem 1 generalizes a result due to Padmanabhan [9] when we take p =
β = 1 and α = 12 .
Theorem 2. Let F (z) ∈ Kp (γ, λ)(0 ≤ γ, λ < p) and let α > 0. Then the function
f (z), defined as
′
Dα,1,p (F ) = f (z) = (1 − α)F (z) + α zF (z)
281
(20)
M. K. Aouf - Differential Operators For p-Valent Functions
belongs to the same class for |z| < R0 , R0 = min(r1 , r2 ), where r1 and r2 are the
respective least positive roots of the equations
(p − 1 +
1
1
) − 2(p + 1 − λ)r − ( + 2λ − (p + 1))r2 = 0 ,
α
α
(21)
(p − 1 +
1
1
) − 2(p + 1 − γ)r − ( + 2γ − (p + 1))r2 = 0 .
α
α
(22)
and
The result is sharp.
Proof. Since F (z) ∈ Kp (γ, λ), then there exists a function G(z) ∈ Sp∗ (λ) such
that Re
′
z F (z)
G(z)
> γ, z ∈ U . Now let
′
Dα,1,p (G) = g(z) = (1 − α)G(z) + αzG (z).
(23)
Then, from Theorem 1, it follows that g(z) ∈ Sp∗ (λ) for |z| < r1 , where r1 is the
least positive root of (57).
′
f (z)
>γ
Using the same technique of Theorem 1, we can easily show that Re z g(z)
for |z| < R0 , where R0 = min(r1 , r2 ) and r2 is given by (58). The sharpness of the
result can be seen as follows:
Let
zp
zp
F1 (z) =
and
G
(z)
=
.
1
(1 − z)2(p−γ)
(1 − z)2(p−λ)
Then
′
z F (z)
Re 1
> γ , G1 ∈ Sp∗ (λ) ⇒ F1 ∈ Kp (γ, λ).
G1 (z)
′
′
Let f1 (z) = (1 − α)F1 (z) + α z F1 (z) and g1 (z) = (1 − α) G1 (z) + α z G1 (z).
′
z f1 (z)
g1 (z)
> γ for |z| < R0 , where R0 is as given in Theorem 2 and can not be
Then Re
improved.
Remark 2. (1) Putting p = 1 in Theorem 2, we obtain the result obtained by Noor
[6];
(2) Putting p = 1 and γ = λ = 0 in Theorem 2, we obtain the result obtained By
Noor et al. [7];
(3) Putting p = 1, γ = λ = 0 and α = 12 in Theorem 2, we obtain the result
obtained by Livingston [5];
(4) Putting p = 1 and α = 21 in Theorem 2, we obtain the result obtained by
Padmanabhan[9].
Theorem 3. Let 0 < α ≤ β ≤ 1 and F (z) ∈ Sp∗ (λ)(0 ≤ λ < p). Then f (z) defined
as
282
M. K. Aouf - Differential Operators For p-Valent Functions
′
∗
(F ) = f α (z) = z α (z 1−β F β (z)) .
Dα,β,p
(24)
belongs to Sp∗ (λ1 ) for |z| < R1 , where R1 is given by
R1 =
(βp − βλ + 1) +
and
p
β 2 (p
−
1
,
+ β(1 − p)[β(1 + p − 2λ) − 2]
λ)2
λ1 = 1 −
(25)
β
(1 − λ).
α
The result is sharp.
Proof. From (2.20), we can write
β
F (z) = z
β−1
Zz
(
0
and so
f (z) α
) dz ,
z
α
z ( f (z)
zF (z)
z )
= (β − 1) + Rz
.
F (z)
f (z) α
( z ) dz
′
β
0
(26)
′
(z)
Since F (z) ∈ Sp∗ (λ), z FF(z)
= (p − λ)h(z) + λ, 0 ≤ λ < p, Re h(z) > 0, z ∈ U . Thus,
from (2.22), we obtain
β(p − λ)h(z)
Zz
0
f (z) α
(
) dz + [βλ + (1 − β)]
z
Zz
0
(
f (z) α
f (z) α
) dz = z(
) .
z
z
Differentiating again with respect to z, we obtain
f (z) α
′
) + β(p − λ)h (z)
β(p − λ)h(z)(
z
Zz
0
(
f (z) α
f (z) α
) dz + [βλ + (1 − β)](
)
z
z
′
= α z 1−α f α−1 (z) f (z) + (1 − α)(
or
′
β
z f (z)
− [1 − (1 − λ)]
f (z)
α
283
f (z) α
)
z
M. K. Aouf - Differential Operators For p-Valent Functions
=
′
β
(p − λ) h(z) +
α
h (z)
Rz
α
( f (z)
z )
0
α
( f (z)
z )
dz
.
Now, using the well-known result (11), we have from (2.23),
′
z f (z)
β
Re
− [1 − (1 − λ)] ≥
f (z)
α
z
R f (z) α
(
)
dz
β
2 0 z
(p − λ)Re h(z) 1 −
.
α
1 − r2 ( f (z) )α
z
Now
Rz
0
α
z ( f (z)
z )
α
( f (z)
z ) dz
′
=
z (z 1−β F β (z)
z 1−β F β (z)
′
= β
zF (z)
+ (1 − β)
F (z)
= β[(p − λ)h(z) + λ] + (1 − β), Re h(z) > 0,
and from this, it follows that
z( f (z) )α
z
Rz
f (z) α
( z ) dz
0
≥ β(p − λ)Re h(z) + βλ + (1 − β)
=
Hence
1−r
+ βλ + (1 − β)
1+r
β(p − λ)(1 − r) + (βλ + (1 − β))(1 + r)
.
(1 + r)
≥ β(p − λ)
′
Re
β
β
z f (z)
− [1 − (1 − λ)] ≥ (p − λ) Re h(z).
f (z)
α
α
284
(27)
M. K. Aouf - Differential Operators For p-Valent Functions
.
=
r(1 + r)
2
1−
2
1 − r β(p − λ)(1 − r) + (βλ + (1 − β))(1 + r)
β
(p − λ)Reh(z)
α
(1 − r)[β(p − λ)(1 − r) + (βλ + (1 − β))(1 + r)] − 2r
. (28)
(1 − r)[β(p − λ)(1 − r) + (βλ + (1 − β))(1 + r)]
The right-hand side of the inequality (2.24) is positive for |z| < R1 , where R1 is the
least positive root of the equation
(1 − βp − β + 2βλ)r2 + 2(βp − βλ + 1)r − (βp − β + 1) = 0,
(29)
and thus we obtain the required result. The function
zp
∈ Sp∗ (λ)
(1 − z)2(p−λ)
F1 (z) =
(30)
shows that the result is best possible.
Putting λ = 12 in Theorem 3, we obtain
Corollary 3. Let 0 < α ≤ β ≤ 1 and F (z) ∈ Sp∗ ( 21 ). Then f (z) defined by (2.20)
β
) for |z| < R2 , where R2 is given by
belongs to Sp∗ (1 − 2α
R2 =
(2βp − β + 2) +
p
β 2 (2p
2
.
− 1)2 + 8 + β(1 − p)(p − 2)
(31)
The result is sharp.
Remark 3. (1) Putting p = 1 in Theorem 3, we obtain the result obtained by Noor
[6];
(2) Putting α = β and F (z) ∈ Sp∗ (λ). Then f (z) defined by (2.20) also belongs
to Sp∗ (λ) for |z| < R1 ;
(3) Putting p = 1 in Corollary 3, we obtain the result obtained by Noor [6].
Using the same technique of Theorem 3 we can prove the following theorem.
Theorem 4. Let F (z) ∈ Bp (β, λ), 0 < β ≤ 1 and 0 ≤ λ < p. Let f (z) be defined
as
′
f β (z) = z β (z 1−β F β (z)) .
Then f (z) ∈ Bp (β, λ) for |z| < R3 , where R3 is given by
R3 =
(βp − β + 1) +
The result is sharp.
p
β 2 (p
−
λ)2
1
.
+ 2 + β(1 − p)[β(1 + p − 2λ) − 2]
285
(32)
M. K. Aouf - Differential Operators For p-Valent Functions
References
[1] M. K. Aouf, On a class of p-valent close-to-convex functions, Internat. J.
Math. Math. Sci., 11, no.2, (1988), 259-266.
[2] M. K. Aouf, H. M. Hossen and H. M. Srivastava, Some families of multivalent
functions, Comput. Math. Appl., 39, (2000), 39-48.
[3] W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J., (1952),
169-185.
[4] R. J. Libera, Some radius of convexity problems, Duke Math. J., 31, (1964),
143-158.
[5] A. E. Livingston, On the radius of univalent of certain analytic functions,
Proc. Amer. Math. Soc., 17, (1966), 352-357.
[6] K. I. Noor, Differential operators for univalent functions, Math. Japon., 32,
no.3, (1987), 427-436.
[7] K. I. Noor, F. M. Aloboudi and N. Aldihan, On the radius of univalence of
convex combinations of analytic functions, Internat J. Math. Math. Sci., 6, no.2,
(1983), 335-340.
[8] S. Owa, Some properties of certain multivalent functions, Appl. Math. Letters, 4, no.5, (1991), 79-83.
[9] K. S. Padmanabhan, On the radius of univalence of certain class of analytic
functions, J. London Math. Soc., 1, (1969), 225-231.
[10] D. K. Thomas, Bazilevic functions, Trans. Amer. Math. Soc., 132, (1968),
353-361.
Mohamed K. Aouf
Department of Mathematics, Faculty of Science
Mansoura University, Mansoura 35516, Egypt.
email: mkaouf127@yahoo.com.
286