General Mathematics Vol. 18, No. 2 (2010), 173–184
Inclusion and neighborhood properties of some
analytic p-valent functions 1
R. M. El-Ashwah , M. K. Aouf
Abstract
By means of a certain extended derivative operator of Salagean type, the
authors introduce and investigate two new subclasses of p-valently analytic
function of complex ordor. The various results obtained here for each of
these function classes include coefficient inqualities and the consequent inclusion relationships involving the neighborhoods of the p-valently analytic
functions.
2000 Mathematics Subject Classification: 30C45.
Key words and phrases: Analytic function, p-valent function, Salagean
derivative operator, neighborhood of analytic functions.
1
Introduction
Let T (j, p) denote the class of functions of the form :
f (z) = z p −
(1)
∞
X
ak z k (ak ≥ 0; p, j ∈ N = {1, 2, ...}),
k=j+p
1
Received 14 July, 2008
Accepted for publication (in revised form) 27 April, 2009
173
174
R. M. El-Ashwah , M. K. Aouf
which are analytic and p-valent in the open unit disc U = {z : |z| < 1}. For a
function f (z) in T (j, p), we define
0 f (z) = f (z),
Dλ,p
1 f (z) = D
0
Dλ,p
λ,p (Dλ,p f (z)) = (1 − λ)f (z) +
∞
P
= zp −
2 f (z)
Dλ,p
=
=
[1 + λ(
k−p
)]ak z k ,
p
k=j+p
1 f (z))
Dλ,p (Dλ,p
∞
P
k
zp −
[1 + λ(
k=j+p
and
n−1
n f (z) = D
Dλ,p
λ,p (Dλ,p f (z))
λ 0
zf (z)
p
(λ ≥ 0)
−p 2
)] ak z k
p
(n ∈ N ).
It can be easily seen that
n
Dλ,p
f (z) = z p −
(2)
∞
X
[1 + λ(
k=j+p
k−p n
)] ak z k
p
(p, j ∈ N ; n ∈ N0 = N ∪ {0}).
We note that :
n = D n was intro(i) By taking j = p = λ = 1, the differential operator D1,1
duced by Salagean[11];
n = D n was introduced
(ii) By taking j = p = 1, the differential operator Dλ,1
λ
by Al-Oboudi[1].
n f (z) given by (2), we introNow, making use of the differential operator Dλ,p
duce a new subclass Hj (n, p, λ, b, β) of the p-valent analytic function class T (j, p)
which consist of function f (z) ∈ T (j, p) satisfying the following inequality :
¯
¯
¯ 1 z(Dn f (z))0
¯
¯
¯
λ,p
,
(3)
−
p)
¯ (
¯<β
n
¯ b Dλ,p f (z)
¯
(z ∈ U ; p, j ∈ N ; n ∈ N0 ; λ ≥ 0; b ∈ C\{0}; 0 < β ≤ 1).
We note that :
Inclusion and neighborhood properties ...
175
¯
¯
¯ 1 z(Dpn f (z))0
¯
¯
(i) Hj (n, p, 1, b, β) = Hj (n, p, b, β) = {f (z) ∈ T (j, p) : ¯ ( Dn f (z) − p)¯¯ < β
p
b
(4)
(z ∈ U ; p, j ∈ N ; n ∈ N0 ; b ∈ C\{0}; 0 < β ≤ 1)};
¯
¯
¯
¯ 1 zf 0 (z)
(ii) Hj (0, p, 0, b, β) = Sj (p, b, β) = {f (z) ∈ T (j, p) : ¯¯ (
− p)¯¯ < β
b f (z)
(5)
(z ∈ U ; p, j ∈ N ; b ∈ C\{0}; 0 < β ≤ 1)};
¯
¯
¯ 1 zF 0 (z)
¯
¯
¯
λ,p
(iii) Hj (1, p, λ, b, β) = Cj (p, λ, b, β) = {f (z) ∈ T (j, p) : ¯ (
− p)¯ < β
¯ b Fλ,p (z)
¯
(6)
λ 0
(z ∈ U ; p, j ∈ N ; λ ≥ 0; b ∈ C\{0}; 0 < β ≤ 1; Fλ,p (z) = (1 − λ)f (z) + zf (z))}.
p
Now following the earlier investigations by Goodman [7], Ruscheweyh [10],
and others including Altintas and Owa [3], Altintas et al.([4] and [5]), Murugusundaramoorthy and Srivastava [8], Raina and Srivastava [9], Aouf [6] and
Srivastava and Orhan [13] (see also [2], [12] and [14]), we define the (j, δ)− neighborhood of a function f (z) ∈ T (j, p) by (see, for example, [5, p. 1668])
(7) Nj,δ (f ) = {g : g ∈ T (j, p), g(z) = z p −
∞
X
bk z k and
∞
X
k |ak − bk | ≤ δ}.
k=j+p
k=j+p
In particular, if
h(z) = z p
(8)
(p ∈ N ),
we immediately have
(9)
Nj,δ (h) = {g : g ∈ T (j, p), g(z) = z p −
∞
X
k=j+p
bk z k and
∞
X
k |bk | ≤ δ}.
k=j+p
Also, let Lj (n, p, λ, b, β, µ) denote the subclass of T (j, p) consisting of function
f (z) ∈ T (j, p) which satisfy the inequality :
¯
¯
n f (z)
n f 0 (z)
¯
¯1
D
D
¯
¯
λ,p
λ,p
+
µ
]
−
1}
¯<β
¯ {[(1 − µ)
¯
¯b
zp
pz p−1
176
R. M. El-Ashwah , M. K. Aouf
(10)
(z ∈ U ; p, j ∈ N ; n ∈ N0 ; λ ≥ 0; b ∈ C\{0}; 0 < β ≤ 1; µ ≥ 0) .
We note that :
(i) Lj (0, p, 0, b, β, µ) = Lj (p, b, β, µ)
¯ ("
(
#
)¯
0
¯1
¯
f (z)
f (z)
¯
¯
= f (z) ∈ T (j, p) : ¯
(1 − µ) p + µ p−1 − 1 ¯ < β
¯b
¯
z
pz
(11)
2
(z ∈ U ; p, j ∈ N ; b ∈ C\{0}; 0 < β ≤ 1; µ ≥ 0}.
Neighborhoods for the classes Hj (n, p, λ, b, β) and
Lj (n, p, λ, b, β, µ)
In our investigation of the inclution relations involving Nj,δ (h), we shall require
Lemmas 1 and 2 below.
Lemma 1 Let the function f (z) ∈ T (j, p) be defined by (1). Then f (z) ∈
Hj (n, p, λ, b, β) if and only if
∞
X
(12)
[1 + λ(
k=j+p
k−p n
)] (k + β |b| − p)ak ≤ β |b| .
p
Proof. Let a function f (z) of the form (1) belong to the class Hj (n, p, λ, b, β).
Then, in view of (2) and (3), we obtain the following inequality :
(13)
Re{
n f (z))0
z(Dλ,p
n f (z)
Dλ,p
− p} > −β |b|
(z ∈ U ),
or, equivalently,
−
(14)
Re{
∞
P
k−p n
)] (k − p)ak z k−p
p
k=j+p
} > −β |b|
∞
P
k−p n
[1 + λ(
)] ak z k−p
1−
p
k=j+p
[1 + λ(
(z ∈ U ).
Inclusion and neighborhood properties ...
177
Setting z = r (0 ≤ r < 1) in (14), we observe that the expression in the
denominator of the left-hand side of (14) is positive for r = 0 and also for all
r (0 < r < 1). Thus, by letting r → 1− through real values, (14) leads us to the
desired assertion (12) of Lemma 1.
Conversaly, by applying the hypothesis (12) and letting |z| = 1, we find from
(3) that
¯
¯ ∞
¯
¯ P
k−p n
k−p
¯
¯
¯
¯
[1 + λ(
)] (k − p)ak z
¯ z(Dn f (z))0
¯
¯
¯
p
k=j+p
¯
¯
¯
¯
λ,p
−
p
=
¯
¯
¯
¯
∞
n f (z)
P
k−p n
¯
¯ Dλ,p
¯
¯
k−p
¯ 1−
¯
[1 + λ(
)] ak z
¯
¯
p
k=j+p
∞
P
k−p n
)] (k − p)ak
p
k=j+p
∞
P
k−p n
1−
[1 + λ(
)] ak
p
k=j+p
∞
P
k−p n
β |b| {1 −
[1 + λ(
)] ak }
p
k=j+p
= β |b| .
∞
P
k−p n
)] ak
1−
[1 + λ(
p
k=j+p
≤
≤
[1 + λ(
Hence, by the maximum modulus theorem, we have f (z) ∈ Hj (n, p, λ, b, β),
which evidently completes the proof of Lemma 1.
Similarly, we can prove the following lemma.
Lemma 2 Let the function f (z) ∈ T (j, p) be defined by (1). Then f (z) ∈
Lj (n, p, λ, b, β, µ) if and only if
(15)
∞
X
[1 + λ(
k=j+p
k−p n
)] [p + µ(k − p)]ak ≤ pβ |b| .
p
Our first inclusion relation involving Nj,δ (h) is given in the following theorem.
Theorem 1 Let
(16)
δ=
(j + p)β |b|
λj
(1 + )n (j + β |b|)
p
(p > |b|),
178
R. M. El-Ashwah , M. K. Aouf
then
(17)
Hj (n, p, λ, b, β) ⊂ Nj,δ (h).
Proof. Let f (z) ∈ Hj (n, p, λ, b, β). Then, in view of the assertion (12) of Lemma
1, we have
∞
∞
X
X
λj n
k−p n
(18)(1+ ) (j +β |b|)
)] (k+β|b|−p)ak ≤ β |b| ,
ak ≤
[1+λ(
p
p
k=j+p
k=j+p
which readily yeilds
∞
X
(19)
ak ≤
k=j+p
β |b|
λj
(1 + )n (j + β |b|)
p
.
Making use of (12) again, in conjunction with (19), we get
(1 +
∞
∞
X
λj
λj n X
kak ≤ β |b| + (p − β |b|) (1 + )n
ak
)
p
p
k=j+p
k=j+p
≤ β |b| +
β |b| (p − β |b|)
(j + p)β |b|
=
.
(j + β |b|)
(j + β |b|)
Hence
∞
X
(20)
k=j+p
kak ≤
(j + p)β |b|
=δ
λj n
(1 + ) (j + β |b|)
p
(p > |b|),
which, by means of the definition (9), establishes the inclusion (17) asserted by
Theorem 1.
Putting (i) n = λ = 0 and (ii) n = 1 in Theorem 1, we obtain the following
results.
Corollary 1 Let
(21)
δ=
(j + p)β |b|
(p > |b|),
(j + β |b|)
then
(22)
Sj (p, b, β) ⊂ Nj,δ (h).
Inclusion and neighborhood properties ...
179
Corollary 2 Let
(23)
δ=
(j + p)pβ |b|
(p + λj)(j + β |b|)
(p > |b|),
then
(24)
Cj (p, λ, b, β) ⊂ Nj,δ (h).
In a similar manner, by applying the assertion (15) of Lemma 2 instead of
the assertion (12) of Lemma 1 to functions in the class Lj (n, p, λ, b, β, µ), we can
prove the following inclusion relationship.
Theorem 2 If
(25)
δ=
(j + p)pβ |b|
λj
(1 + )n (p + µj)
p
(µ > 1),
then
(26)
Lj (n, p, λ, b, β, µ) ⊂ Nj,δ (h).
Putting n = λ = 0 in Theorem 2, we obtain the following result.
Corollary 3 If
(27)
δ=
(j + p)pβ |b|
,
(p + µj)
then
(28)
Lj (p, b, β, µ) ⊂ Nj,δ (h).
180
3
R. M. El-Ashwah , M. K. Aouf
(α)
Neighborhoods for the classes Hj (n, p, λ, b, β) and
(α)
Lj (n, p, λ, b, β, µ)
(α)
In this section, we determine the neighborhood for the each classes Hj (n, p, λ, b, β)
(α)
and Lj (n, p, λ, b, β, µ), which we define as follows. A function f (z) ∈ T (j, p)
(α)
is said to be in the class Hj (n, p, λ, b, β) if there exists a function g(z) ∈
Hj (n, p, λ, b, β) such that
¯
¯
¯ f (z)
¯
¯
¯
(29)
¯ g(z) − 1¯ < p − α
(z ∈ U ; 0 ≤ α < p).
(α)
Analogously, a function f (z) ∈ T (j, p) is said to be in the class Lj (n, p, λ, b, β, µ)
if there exists a function g(z) ∈ Lj (n, p, λ, b, β, µ) such that the inequality (29)
holds true.
Theorem 3 If g(z) ∈ Hj (n, p, λ, b, β) and
λj n
) (j + β |b|)
p
,
α=p−
λj
(j + p)[(1 + )n (j + β |b|) − β |b|]
p
δ(1 +
(30)
then
(31)
(α)
Nj,δ (g) ⊂ Hj (n, p, λ, b, β),
where
(32)
δ ≤ p(j + p){1 − β |b| [(1 +
λj n
) (j + β |b|)]−1 }.
p
Proof. Suppose that f (z) ∈ Nj,δ (g). We find from (7) that
(33)
∞
X
k=j+p
k |ak − bk | ≤ δ,
Inclusion and neighborhood properties ...
181
which readily implies that
∞
X
(34)
|ak − bk | ≤
k=j+p
δ
j+p
(p, j ∈ N ).
Next, since g(z) ∈ Hj (n, p, λ, b, β), we have [cf. equation (19)]
∞
X
(35)
bk ≤
k=j+p
β |b|
,
λj n
(1 + ) (j + β |b|)
p
so that
¯
¯
¯ f (z)
¯
¯
(36) ¯
− 1¯¯ ≤
g(z)
∞
P
k=j+p
|ak − bk |
∞
P
1−
k=j+p
bk
δ
≤
.
j+p
(1 +
[(1 +
λj n
) (j + β |b|)
p
λj n
) (j + β |b|) − β |b|]
p
= p − α,
provided that α is given by (30).
Thus, by the above definition, f (z)
(α)
∈ Hj (n, p, λ, b, β) for α given by (30). This evidently proves Theorem 3.
Putting (i) n = λ = 0 and (ii) n = 1 in Theorem 3, we obtain the following
results.
Corollary 4 If g(z) ∈ Sj (p, b, β) and
(37)
α=p−
δ(j + β |b|)
,
(j + p)[(j + β |b|) − β |b|]
then
(38)
(α)
Nj,δ (g) ⊂ Sj (p, b, β),
where
(39)
δ ≤ p(j + p){1 − β |b| (j + β |b|)−1 }.
182
R. M. El-Ashwah , M. K. Aouf
Corollary 5 If g(z) ∈ Cj (p, λ, b, β) and
(40)
α=p−
δ(p + λj)(j + β |b|)
,
(j + p)[(p + λj)(j + β |b|) − pβ |b|]
then
(α)
Nj,δ (g) ⊂ Cj (p, λ, b, β),
where
δ ≤ p(j + p){1 − pβ |b| [(p + λj)(j + β |b|)]−1 }.
(41)
The proof of Theorem 4 below is similar to that the proof of Theorem 3 above.
We, therefore, skip its proof.
Theorem 4 If g(z) ∈ Lj (n, p, λ, b, β, µ) and
λj n
) (p + µj)
p
,
α=p−
λj n
(j + p)[(1 + ) (p + µj) − pβ |b|]
p
δ(1 +
(42)
then
(α)
(43)
Nj,δ (g) ⊂ Lj (n, p, λ, b, β, µ),
where
(44)
δ ≤ p(j + p){1 − pβ |b| [(1 +
λj n
) (p + µj)]−1 }.
p
Putting n = λ = 0 in Theorem 4, we obtain the following result.
Corollary 6 If g(z) ∈ Lj (p, b, β, µ) and
(45)
α=p−
δ(p + µj)
,
(j + p)[(p + µj) − pβ |b|]
then
(46)
(α)
Nj,δ (g) ⊂ Lj (p, b, β, µ),
where
(47)
δ ≤ p(j + p){1 − pβ |b| (p + µj)−1 }.
Inclusion and neighborhood properties ...
183
References
[1] F. M. Al-Oboudi, On univalent functions defined by agereralized Salagean
operator, Internat. J. Math. Math. Sci. 27, 2004, 1429-1436.
[2] O. P. Ahuja, M. Nunokawa, Neighborhoods of analytic functions defined by
Ruscheweyh derivatives, Math. Japon. 51, 2003, 487-492.
[3] O. Altintas, S. Owa, Neighorhoods of certain analytic functions with negative
coefficients, Internat. J. Math. Math. Sci. 19, 1996, 797-800.
[4] O. Altintas, O .Ozkan, H .M. Srivastava, Neighborhoods of class of analytic
functions with negative coefficients, Appl. Math. Letters 13 (3), 2000, 63-67.
[5] O. Altintas, O. Ozkan, H. M. Srivastava, Neighborhoods of certain family
of multivalent functions with negative coefficients, Comput, Math. Appl. 47,
2004, 1667-1673.
[6] M.K. Aouf, Neighborhoods of certain classes of analytic functions with negative coefficients, Internat. J. Math. Math. Sci., Art. ID 38258, 2006, 1-6.
[7] A. W. Goodman, Univalent functions and non-analytic curves, Proc. Amer.
Math. Soc. 8, 1957, 598-601.
[8] G. Murugusundaramoorthy, H. M. Srivastava, Neighborhoods of certain
classes of analytic functions of complex order, J. Inequal. Pure Appl. Math.
5 (2), 2004, Art. 24, 1-8.
[9] R. K. Raina, H. M. Srivastava, Inclusion and neighborhood properties of
some analytic and multivalent functions, J. Inequal. Pure. Appl. Math. 7
(1), 2006, Art. 5, 1-6.
184
R. M. El-Ashwah , M. K. Aouf
[10] S. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math.
Soc. 81, 1981, 521-527.
[11] G. S. Salagean, Subclasses of univalent function, Lecture Notes in Math.
(Springer-Verlag) 1013, 1983, 368-372.
[12] H. Silverman, Neighborhoods of classes of analytic function, Far East J.
Math. Sci. 3, 1995, 165-169.
[13] H. M. Srivastava, H. Orhan, Coefficient-inqualities and inclusion relation for
some families of analytic and multivalent function, Appl. Math. Letters 20,
2007, 686-691.
[14] H. M. Srivastava, S. Owa, Current Topics in Analytic Function Theory,
World Scientific Publishing Company, Singapore, New Jersey, London and
Hong Kong, 1992.
R. M. El-Ashwah
Mansoura University
Department of Mathematic
Mansoura 35516, Egypt
e-mail: elashwah@mans.edu.eg
M. K. Aouf
Mansoura University
Department of Mathematic
Mansoura 35516, Egypt
e-mail: mkaouf127@yahoo.com