ON CERTAIN CLASSES OF P-VALENT FUNCTIONS DEFINED BY MULTIPLIER TRANSFORMATION AND DIFFERENTIAL OPERATOR | Tehranchi | Journal of the Indonesian Mathematical Society 72 289 1 PB
J. Indones. Math. Soc. (MIHMI)
Vol. 13, No. 1 (2007), pp. 27–38.
ON CERTAIN CLASSES OF P -VALENT
FUNCTIONS DEFINED BY
MULTIPLIER TRANSFORMATION
AND DIFFERENTIAL OPERATOR
A. Tehranchi , S. R. Kulkarni
and G. Murugusundaramoorthy
Abstract. In this paper, we discuss the p-valent functions that satisfy the differential subordinations
z(Ip (r,λ)f (z))(j+1)
(p−j)(Ip (r,λ)f (z))(j)
≺
a+(aB+(A−B)β)z
.
a(1+Bz)
We also obtain coefficient
inequalities, extreme points, integral representation and arithmetic mean. Further we
investigate some interesting properties of operators defined on Ap (r, j, β, a, A, B).
1. INTRODUCTION
P∞
Denote by A the class of functions f (z) = z − n=2 an z n analytic in D =
{z ∈ C : |z| < 1}. Also denote Ap the class of all analytic functions of the form
f (z) = kz p +
2p−1
X
tn−p+1 z n−p+1 −
2 F1 (a, b; c; z),
|z| < 1
(1)
n=p−1
where
2 F1 (a, b; c; z)
∞
X
(a, n)(b, n) n
z
=
(c, n)n!
n=0
(a, n)
=
tn−p+1
=
Γ(a + n)
= a(a + 1, n − 1), c > b > 0, c > a + b and
Γ(a)
(a, n − p + 1)(b, n − p + 1)
, k > 0.
(c, n − p + 1)(n − p + 1)!
Received 14 August 2005, revised 28 July 2006, accepted 2 August 2006 .
2000 Mathematics Subject Classification: 30C45, 30C50.
Key words and Phrases: Multivalent function, Differential subordinations, multiplier transformation,
differential operator.
27
28
A. Tehranchi , S. R. Kulkarni and G.Murugusundaramoorthy
These functions are analytic in the unit disk D (For details see [1], [5]).
Definition 1.1. A function f ∈ Ap is said to be in the class Sp∗ (α) , p-valently
n ′ o
(z)
starlike functions of order α, if it satisfies Re zff (z)
> α, (0 ≤ α < p, z ∈ D).
∗
∗
We note that Sp (0) = Sp , the class of p-valently starlike functions in D. A function
f ∈nAp is said to
o be in the class Cp (α) of p-valently convex of order α, if it satisfies
Re 1 +
zf ′′ (z)
f ′ (z)
> α, (0 ≤ α < p, z ∈ D).
Let h(z) be analytic and h(0) = 1. A function f ∈ Ap is in the class Sp∗ (h) if
zf ′ (z)
≺ h(z),
f (z)
z ∈ ∆.
(2)
The class Sp∗ (h) and a corresponding convex class Cp (h) are defined by Ma and
Minda [6]. But results about the convex class can be obtained easily from the
corresponding result of functions in Sp∗ (h). If
h(z) =
1+z
,
1−z
(3)
then the classes reduce to the usual classes of starlike and convex functions. If
, 0 ≤ α < p, then the classes reduce to the usual classes of starlike
h(z) = 1+(1−2α)z
1−z
1+Az
and convex functions of order α. If h(z) = 1+Bz
,
− 1 ≤ B < A ≤ 1, then the
classes reduce to the class of Janowski starlike function Sp∗ [A, B] defined by
Sp∗ [A, B]
=
½
1 + Az
zf ′
≺
,
f ∈ Ap :
f
1 + Bz
¾
− 1 ≤ B < A ≤ 1, z ∈ D .
(4)
³
´α
1+z
If h(z) = 1−z
, 0 < α ≤ 1, then the classes reduce to the classes of strongly
starlike and convex function of order α that consists of univalent functions f ∈ A
satisfying
¯
µ ′ ¶¯
¯
¯
¯arg zf (z) ¯ < απ , 0 < α ≤ 1, z ∈ ∆
¯
f (z) ¯
z
or equivalently we have
SS ∗ (α) =
¶α
¾
µ
½
1+z
zf ′
, 0 < α ≤ 1, z ∈ ∆ .
≺
f ∈ Ap :
f
1−z
(5)
Obradovič and Owa [8], Silverman [11], Obradovič and Tuneski [9] and Tuneski [12]
have studied the properties of classes of functions defined in terms of the ratio of
′′
′
(z)
(z)
1 + zff ′ (z)
and zff (z)
.
29
On certain classes of p-valent functions
Definition 1.2. A function f ∈ Ap is said to be p-valent Bazilevic of type η and
order α if there exists a function g ∈ Sp∗ such that
Re
½
zf ′ (z)
1−η
f
(z)g η (z)
¾
> α,
(z ∈ ∆)
(6)
for some η(η ≥ 0) and α(0 ≤ α < p). We denote by Bp (η, α), the subclass of Ap
consisting of all such functions. In particular, a function in Bp (1, α) = Bp (α) is
said to be p-valently close-to-convex of order α in ∆.
Definition 1.3. [7] For two functions f and g, analytic in ∆ we say f is subordinate to g denoted by f ≺ g if there exists a Schwarz function w(z), analytic
in ∆ with w(0) = 1 and |w(z)| < 1, such that f (z) = g(w(z)), z ∈ ∆. In particular, if the function g is univalent in ∆, the above subordination is equivalent to
f (0) = g(0), f (∆) ⊂ g(∆). Also, we say that g is superordinate to f .
Using the techniques of Cho and Srivastava[4],Cho and Kim[3] and Uralegadi
and Somanatha[12] we define the following transformation.
Definition 1.4. We define the multiplier transformation operator Ip (r, λ) on the
∞
P
an z n as
infinite series f (z) = z p −
n=p+1
¶r
∞ µ
X
n+λ
Ip (r, λ)f (z) = z −
an z n , (λ ≥ 0).
p
+
λ
n=1+p
p
(7)
We note that Sǎlǎgean derivative operators [9] is closely related to the operators Ip (r, λ) when the coefficient of f (z) is positive. Also note that the class
I1 (r, 1) = Ir [12] ,I1 (r, λ) = Irλ the classes studied in [4] and [3].
Definition 1.5. For each f (z) = z p −
∞
P
an z n we have
n=p+1
f (j) (z) =
∞
X
n!
p!
z p−j −
an z n−j
(p − i)!
(n
−
j)!
n=p+1
(8)
where n, p ∈ N, p > j, and j ∈ N0 = {0} ∪ N . For j = 0 we have f (0) (z) = f (z).
Definition 1.6. A function f ∈ Ap is said to be in the class Ap (r, j; h) if it satisfies
z(Ip (r, λ)f (z))(j+1)
≺ h(z)
(p − j)(Ip (r, λ)f (z))(j)
(9)
30
A. Tehranchi , S. R. Kulkarni and G.Murugusundaramoorthy
where
h(z) = 1 +
A − B βz
,
a 1 + Bz
z ∈ ∆,
and −1 ≤ B < A ≤ 1, 0 < β < p, a > 0 , we denote Ap (r, j; h) = Ap (r, j, β, a, A, B).
We say that f (z) is superordinate to h(z) if f (z) satisfies the following
h(z) ≺
z(Ip (r, λ)f (z))(j+1)
(p − j)(Ip (r, λ)f (z))(j)
where h(z) is analytic in ∆ and h(0) = 1.
We note that if
z(Ip (r, λ)f (z))(j+1)
(p − j)[a + (aB + (A − B)β)z]
≺
(j)
a(1 + Bz)
(Ip (r, λ)f (z))
so h(0) = 1 by choosing j = r = 0, p = 1, then f (z) ∈ Sp∗ (h). Also a = A = β =
1, B = −1, f (z) ∈ Sp∗ (1). But if a = β = 1 and −1 ≤ B < A ≤ 1 then f (z) ∈
S ∗ [A, B], class of Janowski starlike function. If we put p = a = β = A = 1, B = −1
then f (z) ∈ SS ∗ (1) classes of strongly starlike.
∗
By Definition 1.2., if g(z)
starlike and j = r = 0 and p = a =
n ∈ S ′ , univalent
o
zf (z)
A = β = 1, B = −1 and if Re f (z)−1 g2 (z) > 1, then f (z) ∈ B(2, 1) class Bazilevic
function of type η = 2 and order α = 1.
2. MAIN RESULTS
In this section we obtain sharp coefficient estimates for functions in
Ap (r, j, β, a, A, B).
Theorem 2.1. Let f (z) be of the form (1). Then f ∈ Ap (r, j, β, a, A, B) if and
only if
∞
X
¸
a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1)) δ(m, j)(p − j)
−
km < 1
γ (m, p)
βκ(A − B)δ(p, j + 1)
κδ(p, j + 1)
m=p+1
r
·
(10)
where
γ r (m, p) =
and
δ(m, j) =
m!
,
(m − j)!
µ
m+λ
p+λ
¶r
, λ ≥ 0, p, r ∈ N
− 1 ≤ B < A ≤ 1, 0 < β < p, j < p.
On certain classes of p-valent functions
31
Proof. The function f (z) of the form (1) can be expressed as f (z) = κz p −
∞
P
kn−p+1 z n−p+1 , or
n=2p
∞
X
f (z) = κz p −
km z m
(11)
m=p+1
where m = n − p + 1 and km =
(j)
(Ip (r, λ)f (z))
(a,m)(b,m)
(c,m)m! ,
and also we have for all r, j ∈ N0
=
µ
¶r
∞
X
m+λ
m!
κp!
p−j
z
−
km z m−j
(p − j)!
p
+
λ
(m
−
j)!
m=p+1
=
κδ(p, j)z p−j −
∞
X
γ r (m, p)δ(m, j)km z m−j .
(12)
m=p+1
Let f (z) ∈ Ap (r, j, a, β, A, B) then
¯
¯
¯ az(Ip (r, λ)f (z))(j+1) − a(p − j)(Ip (r, λ)f (z))(j) ¯
¯
¯
¯ R(p − j)(I (r, λ)f (z))(j) − Baz(I (r, λ)f (z))(j+1) ¯ < 1
p
p
(13)
where R = aB + (A − B)β. Now, we can write
Re
a
∞
P
γ r (m, p)(δ(m, j)(p − j) − δ(m, j + 1))km z m−j
m=p+1
< 1.
∞
P
γ r (m, p)(δ(m, j)(p − j)R − δ(m, j + 1)Ba)km z m−j
β(A − B)δ(p, j + 1)κz p−j −
m=p+1
We choose the values of z on the real axis and letting z → 1− then we have
a
∞
P
γ r (m, p)(δ(m, j)(p − j) − δ(m, j + 1))km
m=p+1
βκ(A − B)δ(p, j + 1) −
∞
P
γ r (m, p)(aB(δ(m, j)(p − j) − δ(m, j + 1)) − δ(m, j)(A − B)β(p − j))km
m=p+1
and
∞
X
γ r (m, p)[a(1 + B)(δ(m, j)(p − j)
m=p+1
−δ(m, j + 1)) − δ(m, j)(A − B)β(p − j)]km < βκ(A − B)δ(p, j + 1)
Conversely, we assume that the condition (10) holds true. Hence it is sufficient
to show that f ∈ Ap (r, j, β, a, A, B), that is to prove that
¯
¯
¯ az(Ip (r, λ)f (z))(j+1) − a(p − j)(Ip (r, λ)f (z))(j) ¯
¯
¯
¯ (p − j)R(I (r, λ)f (z))(j) − Baz(I (r, λ)f (z))(j+1) ¯ < 1.
p
p
< 1,
32
A. Tehranchi , S. R. Kulkarni and G.Murugusundaramoorthy
But we have
¯
¯
¯ az(Ip (r, λ)f (z))(j+1) − a(p − j)(Ip (r, λ)f (z))(j) ¯
¯
¯
¯ (p − j)R(I (r, λ)f (z))(j) − Baz(I (r, λ)f (z))(j+1) ¯
p
= |[a
∞
X
p
γ r (m, p)(δ(m, j)(p − j) − δ(m, j + 1))km z m−j ]/
m=p+1
[β(A − B)δ(p, j + 1)κz p−j
∞
X
−
γ r (m, p)(aB(δ(m, j)(p − j) − δ(m, j + 1)) −
m=p+1
δ(m, j)(A − B)β(p − j))km ]|
∞
X
< {[a
γ r (m, p)(δ(m, j)(p − j) − δ(m, j + 1))km ]/
m=p+1
[βκ(A − B)δ(p, j + 1)
∞
X
−
γ r (m, p)(aB(δ(m, j)(p − j) − δ(m, j + 1)) −
m=p+1
δ(m, j)(A − B)β(p − j))km ]} < 1
and so proof is complete.
The inequality (10) is sharp for the function
f (z) = z p −
βκ(A − B)δ(p, j + 1)
zq ,
γ r (q, p)[a(1 + B)(δ(q, j)(p − j) − δ(q, j + 1)) − δ(q, j)(A − B)β(p − j)]
with q ≥ 1 + p.
Corollary 2.2. Let f ∈ Ap (r, j, β, a, A, B) then we have
km <
κβ(A−B)δ(p,j+1)
γ r (m,p)[a(1+B)(δ(m,j)(p−j)−δ(m,j+1))−δ(m,j)(A−B)β(p−j)] ,
(14)
with m ≥ p + 1
In the next theorem we prove that the class Ap (r, j, β, α, A, B) is closed under
linear combination.
Theorem 2.3. Let fq (z) = κz p −
∞
P
km,q z m (q = 1, 2, · · · , t) be in Ap (r, j, β, a, A, B).
m=p+1
Then the function F (z) =
t
P
q=1
dq fq (z) where
t
P
q=1
dq = 1, is also in Ap (r, j, β, a, A, B).
33
On certain classes of p-valent functions
Proof. We have
q
X
F (z) =
dk
k=1
p
= κz −
Ã
p
κz −
∞
X
km,q z
m
m=1+p
∞
X
m=1+p
Ã
t
X
dq km,q
q=1
!
!
p
= κz −
γ (m, p)
=
t
X
q=1
<
t
X
Ã
∞
X
βκ(A − B)δ(p, j + 1)
∞
X
km,q z
m
m=1+p
!
δ(m, j)(p − j)
−
κδ(p, j + 1)
à t
i X
dq dm,q
q=1
!
a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1))
δ(m, j)(p − j)
γ (m, p)
km,q
−
βκ(A − B)δ(p, j + 1)
κδ(p, j + 1)
r
m=1+p
dq
q=1
Ã
zm.
Hence we obtain
∞
h
X
a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1)
r
m=1+p
t
X
h
i
!
dq
dq = 1.
q=1
Now we prove that the class Ap (r, j, β, a, A, B) is closed under arithmetic mean.
Theorem 2.4. Let fj (z) = κz p −
∞
P
m=1+p
∞
P
Then the function F (z) = κz p −
km,q z m (q = 1, 2, · · · , s) are in Ap (r, j, β, a, A, B).
m=1+p
Ap (r, j, β, a, A, B).
bm z m where bm =
1
s
s
P
km,q , is also in
q=1
Proof. Since fj (z) ∈ Ap (r, j, β, a, A, B), then by (10) we have
·
¸
∞
X
a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1)) δ(m, j)(p − j)
γ r (m, p)
−
βκ(A − B)δ(p, j + 1)
κδ(p, j + 1)
m=1+p
km,q < 1, q = 1, 2, 3, · · · s.
Therefore
h
i
γ r (m, p) a(1+B)(δ(m,j)(p−j)−δ(m,j+1))
− δ(m,j)(p−j)
bm
βκ(A−B)δ(p,j+1)
κδ(p,j+1)
i
³
´
h
Ps
P∞
δ(m,j)(p−j)
1
−
k
= m=1+p γ r (m, p) a(1+B)(δ(m,j)(p−j)−δ(m,j+1))
m,q
q=1
βκ(A−B)δ(p,j+1)
κδ(p,j+1)
s
h
i
´
Ps ³P∞
δ(m,j)(p−j)
a(1+B)(δ(m,j)(p−j)−δ(m,j+1))
1
r
− κδ(p,j+1) km,q
≤ s q=1
m=1+p γ (m, q)
βκ(A−B)δ(p,j+1)
Ps
≤ q=1 1s = 1
P∞
m=1+p
and this completes the proof.
Theorem 2.5. (Extreme Points) Let fp (z) = z p and for m ≥ 1 + p
fm (z)
=
κz p −
βκ(A − B)δ(p, j + 1)
zm ,
γ r (m, p)[a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1)) − δ(m, j)(A − B)β(p − j)]
34
A. Tehranchi , S. R. Kulkarni and G.Murugusundaramoorthy
Then the function f (z) ∈ Ap (r, j, β, a, A, B) if and only if it can be expressed in the
form
∞
X
f (z) =
µm fm (z)
(15)
m=p
∞
P
where µm ≥ 0 and
µm = 1.
m=p
Proof. Suppose that f can be expressed in the form (15) then we have
f (z)
=
∞
X
µm fm (z)
m=p
=
∞
X
µp fp (z) +
µm fm (z)
m=p+1
=
µp κz p +
∞
X
µm (κz p −
m=p+1
βκ(A − B)δ(p, j + 1)
zm )
γ r (m, p)[a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1)) − δ(m, j)(A − B)β(p − j)]
=
∞
X
κz p −
m=1+p
Consequently
∞
X
βκ(A − B)δ(p, j + 1)
z m µm
γ r (m, p)[a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1)) − δ(m, j)(A − B)β(p − j)]
γ r (m, p)
m=p+1
h
δ(m, j)(p − j)
a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1))
−
βκ(A − B)δ(p, j + 1)
κδ(p, j + 1)
i
βκ(A − B)δ(p, j + 1)
µn
γ r (m, p)[a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1)) − δ(m, j)(A − B)β(p − j)]
=
∞
X
µm = 1 − µp < 1.
m=1+p
Therefore we conclude the result.
Conversely, let f ∈ Ap (r, j, β, a, A, B) since by (10) we may set
µm = γ r (m, p)km
a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1)) − δ(m, j)(A − B)β(p − j)
,
βκ(A − B)δ(p, j + 1)
with m ≥ 1 + p. Therefore µm ≥ 0 and if we set µp = 1 −
∞
P
µn then we can
m=1+p
write
f (z)
=
κz p −
∞
X
km z m
m=1+p
=
κz p −
∞
X
m=1+p
βκ(A − B)δ(p, j + 1)
µm z m
γ r (m, p)[a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1)) − δ(m, j)(A − B)β(p − j)]
35
On certain classes of p-valent functions
=
κz p −
∞
X
µm (κz p − fm (z))
m=1+p
=
κz p (1 −
∞
X
m=1+p
=
∞
X
∞
X
µm ) −
µm fm (z)
m=1+p
µm fm (z).
m=p
Remark 2.6. The extreme points of the class Ap (r, j, β, a, A, B) are the function
fp (z), fm+p (z), m ≥ 1 + p as in Theorem 2.1.
In the following theorem, we obtain the integral representation for
Ap (r, j, β, a, A, B).
Theorem 2.7. Let f (z) ∈ Ap (r, j, β, a, A, B) then
¸
·Z z
p(ψ(t)R + a)
dt
f (z) = exp
0 t(1 + Bψ(t))
where |ψ(z)| < 1, z ∈ U and R = aB + (A − B)β. Also
·Z
¸
p(A−B)β
p
aB
log(1 − Bxz)
f (z) = z exp
dµ(x)
X
where µ(x) is the probability measure on X = {x : |x| = 1}.
Proof. Set
z(Ip (r, λ)f (z))(j+1)
= Q(z)
(p − j)(Ip (r, λ)f (z))(j)
¯
¯
¯ a(Q(z)−1) ¯
Since f (z) ∈ Ap (r, j, β, a, A, B) so ¯ R−aBQ(z)
¯ < 1 where R = aB + (A − B)β.
Consequently we put
Finally we can write
a(Q(z)−1)
R−aBQ(z) = ψ(z), |ψ(z)|
ψ(z)R+a
or
Q(z) = a+aBψ(z)
< 1.
(Ip (r, λ)f (z))(j+1)
(p − j)(ψ(z)R + a)
=
az(1 + Bψ(z))
(Ip (r, λ)f (z))(j)
Then we have
log(Ip (r, λ)f (z))
(j)
Z
z
(p − j)(ψ(t)R + a)
dt
t(1 + Bψ(t))
·Z
z
(p − j)(ψ(t)R + a)
dt
t(1 + Bψ(t))
=
0
(j)
(Ip (r, λ)f (z))
= exp
0
¸
36
A. Tehranchi , S. R. Kulkarni and G.Murugusundaramoorthy
for r = j = 0 we have
f (z) = exp
·Z
z
¸
p(ψ(t)R + a)
dt .
t(1 + Bψ(t))
0
For obtaining the second representation let X = {x : |x| = 1} then we have
= xz, z ∈ ∆ and then we conclude that
a(Q(z)−1)
R−aBQ(z)
(Ip (r, λ)(f (z))(j+1)
(Ip (r, λ)f (z))(j)
(p − j) x(p − j)(A − B)β
(p − j)(Rxz + a)
=
+
az(1 + Bxz)
z
a(1 + Bxz)
µ
¶
1
x(A − B)β
= (p − j)
+
z
a(1 + Bxz)
=
(j)
log(Ip (r, λ)f (z))
¶
µ
(A − B)β
log(1 + Bxz)
= (p − j) log z +
aB
(p − j)(A − B)β
(Ip (r, λ)f (z))(j)
=
log(1 + Bxz)
p−j
z
aB
·Z
¸
(p−j)(A−B)β
(j)
p−j
aB
log(1 − Bxz)
(Ip (r, λ)f (z)) = z
exp
dµ(x)
log
X
where µ(x) is probability measure on X. For j = r = 0 we have
p
f (z) = z exp
·Z
log(1 − Bxz)
p(A−B)β
aB
¸
dµ(x) .
X
Now, we introduce an integral operator due to Bernardi [2]
Lc (f (z)) =
p+c
zc
Z
z
f (t)tc−1 dt, (c > −p)
0
and we study the effect of this operator on class Ap (r, j, β, a, A, B).
Theorem 2.8. If f ∈ Ap (r, j, β, a, A, B) then Lc (f (z)) is also in Ap (r, j, β, a, A, B).
Proof. If f (z) = κz p −
∞
P
km z m then
m=1+p
Lc (f (z)) =
=
p+c
zc
κz p −
Z
z
0
Ã
∞
X
p
κt −
∞
X
km t
m=1+p
p+c
km z m .
m
+c
m=1+p
m
!
tc−1 dt
On certain classes of p-valent functions
Since m > p then
p+c
m+c
37
≤ 1 so we have
h
i³
´
a(1+B)(δ(m,j)(p−j)−δ(m,j+1))
δ(m,j)(p−j)
p+c
r
γ
(m,
p)
−
m=1+p
βκ(A−B)δ(p,j+1)
κδ(p,j+1)
m+c km
i
h
P∞
− δ(m,j)(p−j)
km < 1.
≤ m=1+p γ r (m, p) a(1+B)(δ(m,j)(p−j)−δ(m,j+1))
βκ(A−B)δ(p,j+1)
κδ(p,j+1)
P∞
Thus Lc (f (z)) ∈ Ap (r, j, β, a, A, B).
REFERENCES
1. C. E. Andrews, R. Askey and R. Roy, Special functions, Cambridge University
Press, Cambridge, 1991.
2. S. D. Bernardi, “Convex and starlike univalent functions”, Trans. Amer. Math. Soc.
135 (1969), 429-446.
3. N. E. Cho and T. H. Kim,, “Multiplier transformations and strongly close-to-convex
functions ”, Bull. Korean Math. Soc. 40(3) (2003), 399-410.
4. N. E. Cho and H. M. Srivastava,, “Argument estimates of certain analytic functions
defined by a class of multiplier transformations ”, Math. Comput. Modelling. 37 (1-2)
(2003), 39-49.
5. Y. C. Kim and F. Rønning,, “Integral transforms of certain subclasses of analytic
functions ”, J. Math. Anal. Appl. 258 (2001), 466-489.
6. W. C. Ma and D. Minda., A unified treatment of some special classes of univalent
functions., Proceedings of the Conference on Complex Analysis (Tianjin, 1992), Conf.
Proc. Lecture Notes Anal. I. pages 157-169, Cambridge, MA, 1994, Internat. Press.
7. S. S. Miller and P. T. Mocanu, “Subordinants of differential superordinations”,
Complex Var. Theory Appl. 48(10) (2003), 815-826.
8. M. Obradovič and S. Owa, “A criterion for starlikeness”, Math. Nachr. 140 (1989),
97-102.
9. M. Obradovič, N. Tuneski, “On the starlike criteria defined Silverman”, Zesz, Nauk.
Politech. Rzeszowskiej Mat. 181(24) (2000), 59-64.
10. G. S. Sǎlǎgean, “Subclasses of univalent functions in Complex analysis” Fifth Romanian - Finnish Seminar, Part 1 (Bucharest, 1981), 362-372, Lecture Notes in Math.,
1013, Springer, Berlin.
11. H. Silverman, “Convex and starlike criteria”, Int. J. Math. Math. Sci. 22 (1999),
75-79.
12. N. Tuneski, “On the quotient of the representations of convexity and starlikeness”,
Math. Nachr., 248-249 (2003), 200-203.
13. B.A. Uralegaddi and C. Somanatha, “Certain classes of univalent functions”, Current topics in analytic function theory, 371-374, World Sci. Publishing, River Edge,
NJ, 1992.
38
A. Tehranchi , S. R. Kulkarni and G.Murugusundaramoorthy
A. Tehranchi and S. R. Kulkarni : Department of Mathematics, Fergusson College,
Pune University, Pune - 411004, India.
E-mail: Tehranchiab@yahoo.co.uk and kulkarni− ferg@yahoo.com
G.Murugusundaramoorthy: School of Science and Humanities, Vellore Institute of
Technology, Deemed University, Vellore-632 014 ,India.
E-mail: gmsmoorthy@yahoo.com
Vol. 13, No. 1 (2007), pp. 27–38.
ON CERTAIN CLASSES OF P -VALENT
FUNCTIONS DEFINED BY
MULTIPLIER TRANSFORMATION
AND DIFFERENTIAL OPERATOR
A. Tehranchi , S. R. Kulkarni
and G. Murugusundaramoorthy
Abstract. In this paper, we discuss the p-valent functions that satisfy the differential subordinations
z(Ip (r,λ)f (z))(j+1)
(p−j)(Ip (r,λ)f (z))(j)
≺
a+(aB+(A−B)β)z
.
a(1+Bz)
We also obtain coefficient
inequalities, extreme points, integral representation and arithmetic mean. Further we
investigate some interesting properties of operators defined on Ap (r, j, β, a, A, B).
1. INTRODUCTION
P∞
Denote by A the class of functions f (z) = z − n=2 an z n analytic in D =
{z ∈ C : |z| < 1}. Also denote Ap the class of all analytic functions of the form
f (z) = kz p +
2p−1
X
tn−p+1 z n−p+1 −
2 F1 (a, b; c; z),
|z| < 1
(1)
n=p−1
where
2 F1 (a, b; c; z)
∞
X
(a, n)(b, n) n
z
=
(c, n)n!
n=0
(a, n)
=
tn−p+1
=
Γ(a + n)
= a(a + 1, n − 1), c > b > 0, c > a + b and
Γ(a)
(a, n − p + 1)(b, n − p + 1)
, k > 0.
(c, n − p + 1)(n − p + 1)!
Received 14 August 2005, revised 28 July 2006, accepted 2 August 2006 .
2000 Mathematics Subject Classification: 30C45, 30C50.
Key words and Phrases: Multivalent function, Differential subordinations, multiplier transformation,
differential operator.
27
28
A. Tehranchi , S. R. Kulkarni and G.Murugusundaramoorthy
These functions are analytic in the unit disk D (For details see [1], [5]).
Definition 1.1. A function f ∈ Ap is said to be in the class Sp∗ (α) , p-valently
n ′ o
(z)
starlike functions of order α, if it satisfies Re zff (z)
> α, (0 ≤ α < p, z ∈ D).
∗
∗
We note that Sp (0) = Sp , the class of p-valently starlike functions in D. A function
f ∈nAp is said to
o be in the class Cp (α) of p-valently convex of order α, if it satisfies
Re 1 +
zf ′′ (z)
f ′ (z)
> α, (0 ≤ α < p, z ∈ D).
Let h(z) be analytic and h(0) = 1. A function f ∈ Ap is in the class Sp∗ (h) if
zf ′ (z)
≺ h(z),
f (z)
z ∈ ∆.
(2)
The class Sp∗ (h) and a corresponding convex class Cp (h) are defined by Ma and
Minda [6]. But results about the convex class can be obtained easily from the
corresponding result of functions in Sp∗ (h). If
h(z) =
1+z
,
1−z
(3)
then the classes reduce to the usual classes of starlike and convex functions. If
, 0 ≤ α < p, then the classes reduce to the usual classes of starlike
h(z) = 1+(1−2α)z
1−z
1+Az
and convex functions of order α. If h(z) = 1+Bz
,
− 1 ≤ B < A ≤ 1, then the
classes reduce to the class of Janowski starlike function Sp∗ [A, B] defined by
Sp∗ [A, B]
=
½
1 + Az
zf ′
≺
,
f ∈ Ap :
f
1 + Bz
¾
− 1 ≤ B < A ≤ 1, z ∈ D .
(4)
³
´α
1+z
If h(z) = 1−z
, 0 < α ≤ 1, then the classes reduce to the classes of strongly
starlike and convex function of order α that consists of univalent functions f ∈ A
satisfying
¯
µ ′ ¶¯
¯
¯
¯arg zf (z) ¯ < απ , 0 < α ≤ 1, z ∈ ∆
¯
f (z) ¯
z
or equivalently we have
SS ∗ (α) =
¶α
¾
µ
½
1+z
zf ′
, 0 < α ≤ 1, z ∈ ∆ .
≺
f ∈ Ap :
f
1−z
(5)
Obradovič and Owa [8], Silverman [11], Obradovič and Tuneski [9] and Tuneski [12]
have studied the properties of classes of functions defined in terms of the ratio of
′′
′
(z)
(z)
1 + zff ′ (z)
and zff (z)
.
29
On certain classes of p-valent functions
Definition 1.2. A function f ∈ Ap is said to be p-valent Bazilevic of type η and
order α if there exists a function g ∈ Sp∗ such that
Re
½
zf ′ (z)
1−η
f
(z)g η (z)
¾
> α,
(z ∈ ∆)
(6)
for some η(η ≥ 0) and α(0 ≤ α < p). We denote by Bp (η, α), the subclass of Ap
consisting of all such functions. In particular, a function in Bp (1, α) = Bp (α) is
said to be p-valently close-to-convex of order α in ∆.
Definition 1.3. [7] For two functions f and g, analytic in ∆ we say f is subordinate to g denoted by f ≺ g if there exists a Schwarz function w(z), analytic
in ∆ with w(0) = 1 and |w(z)| < 1, such that f (z) = g(w(z)), z ∈ ∆. In particular, if the function g is univalent in ∆, the above subordination is equivalent to
f (0) = g(0), f (∆) ⊂ g(∆). Also, we say that g is superordinate to f .
Using the techniques of Cho and Srivastava[4],Cho and Kim[3] and Uralegadi
and Somanatha[12] we define the following transformation.
Definition 1.4. We define the multiplier transformation operator Ip (r, λ) on the
∞
P
an z n as
infinite series f (z) = z p −
n=p+1
¶r
∞ µ
X
n+λ
Ip (r, λ)f (z) = z −
an z n , (λ ≥ 0).
p
+
λ
n=1+p
p
(7)
We note that Sǎlǎgean derivative operators [9] is closely related to the operators Ip (r, λ) when the coefficient of f (z) is positive. Also note that the class
I1 (r, 1) = Ir [12] ,I1 (r, λ) = Irλ the classes studied in [4] and [3].
Definition 1.5. For each f (z) = z p −
∞
P
an z n we have
n=p+1
f (j) (z) =
∞
X
n!
p!
z p−j −
an z n−j
(p − i)!
(n
−
j)!
n=p+1
(8)
where n, p ∈ N, p > j, and j ∈ N0 = {0} ∪ N . For j = 0 we have f (0) (z) = f (z).
Definition 1.6. A function f ∈ Ap is said to be in the class Ap (r, j; h) if it satisfies
z(Ip (r, λ)f (z))(j+1)
≺ h(z)
(p − j)(Ip (r, λ)f (z))(j)
(9)
30
A. Tehranchi , S. R. Kulkarni and G.Murugusundaramoorthy
where
h(z) = 1 +
A − B βz
,
a 1 + Bz
z ∈ ∆,
and −1 ≤ B < A ≤ 1, 0 < β < p, a > 0 , we denote Ap (r, j; h) = Ap (r, j, β, a, A, B).
We say that f (z) is superordinate to h(z) if f (z) satisfies the following
h(z) ≺
z(Ip (r, λ)f (z))(j+1)
(p − j)(Ip (r, λ)f (z))(j)
where h(z) is analytic in ∆ and h(0) = 1.
We note that if
z(Ip (r, λ)f (z))(j+1)
(p − j)[a + (aB + (A − B)β)z]
≺
(j)
a(1 + Bz)
(Ip (r, λ)f (z))
so h(0) = 1 by choosing j = r = 0, p = 1, then f (z) ∈ Sp∗ (h). Also a = A = β =
1, B = −1, f (z) ∈ Sp∗ (1). But if a = β = 1 and −1 ≤ B < A ≤ 1 then f (z) ∈
S ∗ [A, B], class of Janowski starlike function. If we put p = a = β = A = 1, B = −1
then f (z) ∈ SS ∗ (1) classes of strongly starlike.
∗
By Definition 1.2., if g(z)
starlike and j = r = 0 and p = a =
n ∈ S ′ , univalent
o
zf (z)
A = β = 1, B = −1 and if Re f (z)−1 g2 (z) > 1, then f (z) ∈ B(2, 1) class Bazilevic
function of type η = 2 and order α = 1.
2. MAIN RESULTS
In this section we obtain sharp coefficient estimates for functions in
Ap (r, j, β, a, A, B).
Theorem 2.1. Let f (z) be of the form (1). Then f ∈ Ap (r, j, β, a, A, B) if and
only if
∞
X
¸
a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1)) δ(m, j)(p − j)
−
km < 1
γ (m, p)
βκ(A − B)δ(p, j + 1)
κδ(p, j + 1)
m=p+1
r
·
(10)
where
γ r (m, p) =
and
δ(m, j) =
m!
,
(m − j)!
µ
m+λ
p+λ
¶r
, λ ≥ 0, p, r ∈ N
− 1 ≤ B < A ≤ 1, 0 < β < p, j < p.
On certain classes of p-valent functions
31
Proof. The function f (z) of the form (1) can be expressed as f (z) = κz p −
∞
P
kn−p+1 z n−p+1 , or
n=2p
∞
X
f (z) = κz p −
km z m
(11)
m=p+1
where m = n − p + 1 and km =
(j)
(Ip (r, λ)f (z))
(a,m)(b,m)
(c,m)m! ,
and also we have for all r, j ∈ N0
=
µ
¶r
∞
X
m+λ
m!
κp!
p−j
z
−
km z m−j
(p − j)!
p
+
λ
(m
−
j)!
m=p+1
=
κδ(p, j)z p−j −
∞
X
γ r (m, p)δ(m, j)km z m−j .
(12)
m=p+1
Let f (z) ∈ Ap (r, j, a, β, A, B) then
¯
¯
¯ az(Ip (r, λ)f (z))(j+1) − a(p − j)(Ip (r, λ)f (z))(j) ¯
¯
¯
¯ R(p − j)(I (r, λ)f (z))(j) − Baz(I (r, λ)f (z))(j+1) ¯ < 1
p
p
(13)
where R = aB + (A − B)β. Now, we can write
Re
a
∞
P
γ r (m, p)(δ(m, j)(p − j) − δ(m, j + 1))km z m−j
m=p+1
< 1.
∞
P
γ r (m, p)(δ(m, j)(p − j)R − δ(m, j + 1)Ba)km z m−j
β(A − B)δ(p, j + 1)κz p−j −
m=p+1
We choose the values of z on the real axis and letting z → 1− then we have
a
∞
P
γ r (m, p)(δ(m, j)(p − j) − δ(m, j + 1))km
m=p+1
βκ(A − B)δ(p, j + 1) −
∞
P
γ r (m, p)(aB(δ(m, j)(p − j) − δ(m, j + 1)) − δ(m, j)(A − B)β(p − j))km
m=p+1
and
∞
X
γ r (m, p)[a(1 + B)(δ(m, j)(p − j)
m=p+1
−δ(m, j + 1)) − δ(m, j)(A − B)β(p − j)]km < βκ(A − B)δ(p, j + 1)
Conversely, we assume that the condition (10) holds true. Hence it is sufficient
to show that f ∈ Ap (r, j, β, a, A, B), that is to prove that
¯
¯
¯ az(Ip (r, λ)f (z))(j+1) − a(p − j)(Ip (r, λ)f (z))(j) ¯
¯
¯
¯ (p − j)R(I (r, λ)f (z))(j) − Baz(I (r, λ)f (z))(j+1) ¯ < 1.
p
p
< 1,
32
A. Tehranchi , S. R. Kulkarni and G.Murugusundaramoorthy
But we have
¯
¯
¯ az(Ip (r, λ)f (z))(j+1) − a(p − j)(Ip (r, λ)f (z))(j) ¯
¯
¯
¯ (p − j)R(I (r, λ)f (z))(j) − Baz(I (r, λ)f (z))(j+1) ¯
p
= |[a
∞
X
p
γ r (m, p)(δ(m, j)(p − j) − δ(m, j + 1))km z m−j ]/
m=p+1
[β(A − B)δ(p, j + 1)κz p−j
∞
X
−
γ r (m, p)(aB(δ(m, j)(p − j) − δ(m, j + 1)) −
m=p+1
δ(m, j)(A − B)β(p − j))km ]|
∞
X
< {[a
γ r (m, p)(δ(m, j)(p − j) − δ(m, j + 1))km ]/
m=p+1
[βκ(A − B)δ(p, j + 1)
∞
X
−
γ r (m, p)(aB(δ(m, j)(p − j) − δ(m, j + 1)) −
m=p+1
δ(m, j)(A − B)β(p − j))km ]} < 1
and so proof is complete.
The inequality (10) is sharp for the function
f (z) = z p −
βκ(A − B)δ(p, j + 1)
zq ,
γ r (q, p)[a(1 + B)(δ(q, j)(p − j) − δ(q, j + 1)) − δ(q, j)(A − B)β(p − j)]
with q ≥ 1 + p.
Corollary 2.2. Let f ∈ Ap (r, j, β, a, A, B) then we have
km <
κβ(A−B)δ(p,j+1)
γ r (m,p)[a(1+B)(δ(m,j)(p−j)−δ(m,j+1))−δ(m,j)(A−B)β(p−j)] ,
(14)
with m ≥ p + 1
In the next theorem we prove that the class Ap (r, j, β, α, A, B) is closed under
linear combination.
Theorem 2.3. Let fq (z) = κz p −
∞
P
km,q z m (q = 1, 2, · · · , t) be in Ap (r, j, β, a, A, B).
m=p+1
Then the function F (z) =
t
P
q=1
dq fq (z) where
t
P
q=1
dq = 1, is also in Ap (r, j, β, a, A, B).
33
On certain classes of p-valent functions
Proof. We have
q
X
F (z) =
dk
k=1
p
= κz −
Ã
p
κz −
∞
X
km,q z
m
m=1+p
∞
X
m=1+p
Ã
t
X
dq km,q
q=1
!
!
p
= κz −
γ (m, p)
=
t
X
q=1
<
t
X
Ã
∞
X
βκ(A − B)δ(p, j + 1)
∞
X
km,q z
m
m=1+p
!
δ(m, j)(p − j)
−
κδ(p, j + 1)
à t
i X
dq dm,q
q=1
!
a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1))
δ(m, j)(p − j)
γ (m, p)
km,q
−
βκ(A − B)δ(p, j + 1)
κδ(p, j + 1)
r
m=1+p
dq
q=1
Ã
zm.
Hence we obtain
∞
h
X
a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1)
r
m=1+p
t
X
h
i
!
dq
dq = 1.
q=1
Now we prove that the class Ap (r, j, β, a, A, B) is closed under arithmetic mean.
Theorem 2.4. Let fj (z) = κz p −
∞
P
m=1+p
∞
P
Then the function F (z) = κz p −
km,q z m (q = 1, 2, · · · , s) are in Ap (r, j, β, a, A, B).
m=1+p
Ap (r, j, β, a, A, B).
bm z m where bm =
1
s
s
P
km,q , is also in
q=1
Proof. Since fj (z) ∈ Ap (r, j, β, a, A, B), then by (10) we have
·
¸
∞
X
a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1)) δ(m, j)(p − j)
γ r (m, p)
−
βκ(A − B)δ(p, j + 1)
κδ(p, j + 1)
m=1+p
km,q < 1, q = 1, 2, 3, · · · s.
Therefore
h
i
γ r (m, p) a(1+B)(δ(m,j)(p−j)−δ(m,j+1))
− δ(m,j)(p−j)
bm
βκ(A−B)δ(p,j+1)
κδ(p,j+1)
i
³
´
h
Ps
P∞
δ(m,j)(p−j)
1
−
k
= m=1+p γ r (m, p) a(1+B)(δ(m,j)(p−j)−δ(m,j+1))
m,q
q=1
βκ(A−B)δ(p,j+1)
κδ(p,j+1)
s
h
i
´
Ps ³P∞
δ(m,j)(p−j)
a(1+B)(δ(m,j)(p−j)−δ(m,j+1))
1
r
− κδ(p,j+1) km,q
≤ s q=1
m=1+p γ (m, q)
βκ(A−B)δ(p,j+1)
Ps
≤ q=1 1s = 1
P∞
m=1+p
and this completes the proof.
Theorem 2.5. (Extreme Points) Let fp (z) = z p and for m ≥ 1 + p
fm (z)
=
κz p −
βκ(A − B)δ(p, j + 1)
zm ,
γ r (m, p)[a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1)) − δ(m, j)(A − B)β(p − j)]
34
A. Tehranchi , S. R. Kulkarni and G.Murugusundaramoorthy
Then the function f (z) ∈ Ap (r, j, β, a, A, B) if and only if it can be expressed in the
form
∞
X
f (z) =
µm fm (z)
(15)
m=p
∞
P
where µm ≥ 0 and
µm = 1.
m=p
Proof. Suppose that f can be expressed in the form (15) then we have
f (z)
=
∞
X
µm fm (z)
m=p
=
∞
X
µp fp (z) +
µm fm (z)
m=p+1
=
µp κz p +
∞
X
µm (κz p −
m=p+1
βκ(A − B)δ(p, j + 1)
zm )
γ r (m, p)[a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1)) − δ(m, j)(A − B)β(p − j)]
=
∞
X
κz p −
m=1+p
Consequently
∞
X
βκ(A − B)δ(p, j + 1)
z m µm
γ r (m, p)[a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1)) − δ(m, j)(A − B)β(p − j)]
γ r (m, p)
m=p+1
h
δ(m, j)(p − j)
a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1))
−
βκ(A − B)δ(p, j + 1)
κδ(p, j + 1)
i
βκ(A − B)δ(p, j + 1)
µn
γ r (m, p)[a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1)) − δ(m, j)(A − B)β(p − j)]
=
∞
X
µm = 1 − µp < 1.
m=1+p
Therefore we conclude the result.
Conversely, let f ∈ Ap (r, j, β, a, A, B) since by (10) we may set
µm = γ r (m, p)km
a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1)) − δ(m, j)(A − B)β(p − j)
,
βκ(A − B)δ(p, j + 1)
with m ≥ 1 + p. Therefore µm ≥ 0 and if we set µp = 1 −
∞
P
µn then we can
m=1+p
write
f (z)
=
κz p −
∞
X
km z m
m=1+p
=
κz p −
∞
X
m=1+p
βκ(A − B)δ(p, j + 1)
µm z m
γ r (m, p)[a(1 + B)(δ(m, j)(p − j) − δ(m, j + 1)) − δ(m, j)(A − B)β(p − j)]
35
On certain classes of p-valent functions
=
κz p −
∞
X
µm (κz p − fm (z))
m=1+p
=
κz p (1 −
∞
X
m=1+p
=
∞
X
∞
X
µm ) −
µm fm (z)
m=1+p
µm fm (z).
m=p
Remark 2.6. The extreme points of the class Ap (r, j, β, a, A, B) are the function
fp (z), fm+p (z), m ≥ 1 + p as in Theorem 2.1.
In the following theorem, we obtain the integral representation for
Ap (r, j, β, a, A, B).
Theorem 2.7. Let f (z) ∈ Ap (r, j, β, a, A, B) then
¸
·Z z
p(ψ(t)R + a)
dt
f (z) = exp
0 t(1 + Bψ(t))
where |ψ(z)| < 1, z ∈ U and R = aB + (A − B)β. Also
·Z
¸
p(A−B)β
p
aB
log(1 − Bxz)
f (z) = z exp
dµ(x)
X
where µ(x) is the probability measure on X = {x : |x| = 1}.
Proof. Set
z(Ip (r, λ)f (z))(j+1)
= Q(z)
(p − j)(Ip (r, λ)f (z))(j)
¯
¯
¯ a(Q(z)−1) ¯
Since f (z) ∈ Ap (r, j, β, a, A, B) so ¯ R−aBQ(z)
¯ < 1 where R = aB + (A − B)β.
Consequently we put
Finally we can write
a(Q(z)−1)
R−aBQ(z) = ψ(z), |ψ(z)|
ψ(z)R+a
or
Q(z) = a+aBψ(z)
< 1.
(Ip (r, λ)f (z))(j+1)
(p − j)(ψ(z)R + a)
=
az(1 + Bψ(z))
(Ip (r, λ)f (z))(j)
Then we have
log(Ip (r, λ)f (z))
(j)
Z
z
(p − j)(ψ(t)R + a)
dt
t(1 + Bψ(t))
·Z
z
(p − j)(ψ(t)R + a)
dt
t(1 + Bψ(t))
=
0
(j)
(Ip (r, λ)f (z))
= exp
0
¸
36
A. Tehranchi , S. R. Kulkarni and G.Murugusundaramoorthy
for r = j = 0 we have
f (z) = exp
·Z
z
¸
p(ψ(t)R + a)
dt .
t(1 + Bψ(t))
0
For obtaining the second representation let X = {x : |x| = 1} then we have
= xz, z ∈ ∆ and then we conclude that
a(Q(z)−1)
R−aBQ(z)
(Ip (r, λ)(f (z))(j+1)
(Ip (r, λ)f (z))(j)
(p − j) x(p − j)(A − B)β
(p − j)(Rxz + a)
=
+
az(1 + Bxz)
z
a(1 + Bxz)
µ
¶
1
x(A − B)β
= (p − j)
+
z
a(1 + Bxz)
=
(j)
log(Ip (r, λ)f (z))
¶
µ
(A − B)β
log(1 + Bxz)
= (p − j) log z +
aB
(p − j)(A − B)β
(Ip (r, λ)f (z))(j)
=
log(1 + Bxz)
p−j
z
aB
·Z
¸
(p−j)(A−B)β
(j)
p−j
aB
log(1 − Bxz)
(Ip (r, λ)f (z)) = z
exp
dµ(x)
log
X
where µ(x) is probability measure on X. For j = r = 0 we have
p
f (z) = z exp
·Z
log(1 − Bxz)
p(A−B)β
aB
¸
dµ(x) .
X
Now, we introduce an integral operator due to Bernardi [2]
Lc (f (z)) =
p+c
zc
Z
z
f (t)tc−1 dt, (c > −p)
0
and we study the effect of this operator on class Ap (r, j, β, a, A, B).
Theorem 2.8. If f ∈ Ap (r, j, β, a, A, B) then Lc (f (z)) is also in Ap (r, j, β, a, A, B).
Proof. If f (z) = κz p −
∞
P
km z m then
m=1+p
Lc (f (z)) =
=
p+c
zc
κz p −
Z
z
0
Ã
∞
X
p
κt −
∞
X
km t
m=1+p
p+c
km z m .
m
+c
m=1+p
m
!
tc−1 dt
On certain classes of p-valent functions
Since m > p then
p+c
m+c
37
≤ 1 so we have
h
i³
´
a(1+B)(δ(m,j)(p−j)−δ(m,j+1))
δ(m,j)(p−j)
p+c
r
γ
(m,
p)
−
m=1+p
βκ(A−B)δ(p,j+1)
κδ(p,j+1)
m+c km
i
h
P∞
− δ(m,j)(p−j)
km < 1.
≤ m=1+p γ r (m, p) a(1+B)(δ(m,j)(p−j)−δ(m,j+1))
βκ(A−B)δ(p,j+1)
κδ(p,j+1)
P∞
Thus Lc (f (z)) ∈ Ap (r, j, β, a, A, B).
REFERENCES
1. C. E. Andrews, R. Askey and R. Roy, Special functions, Cambridge University
Press, Cambridge, 1991.
2. S. D. Bernardi, “Convex and starlike univalent functions”, Trans. Amer. Math. Soc.
135 (1969), 429-446.
3. N. E. Cho and T. H. Kim,, “Multiplier transformations and strongly close-to-convex
functions ”, Bull. Korean Math. Soc. 40(3) (2003), 399-410.
4. N. E. Cho and H. M. Srivastava,, “Argument estimates of certain analytic functions
defined by a class of multiplier transformations ”, Math. Comput. Modelling. 37 (1-2)
(2003), 39-49.
5. Y. C. Kim and F. Rønning,, “Integral transforms of certain subclasses of analytic
functions ”, J. Math. Anal. Appl. 258 (2001), 466-489.
6. W. C. Ma and D. Minda., A unified treatment of some special classes of univalent
functions., Proceedings of the Conference on Complex Analysis (Tianjin, 1992), Conf.
Proc. Lecture Notes Anal. I. pages 157-169, Cambridge, MA, 1994, Internat. Press.
7. S. S. Miller and P. T. Mocanu, “Subordinants of differential superordinations”,
Complex Var. Theory Appl. 48(10) (2003), 815-826.
8. M. Obradovič and S. Owa, “A criterion for starlikeness”, Math. Nachr. 140 (1989),
97-102.
9. M. Obradovič, N. Tuneski, “On the starlike criteria defined Silverman”, Zesz, Nauk.
Politech. Rzeszowskiej Mat. 181(24) (2000), 59-64.
10. G. S. Sǎlǎgean, “Subclasses of univalent functions in Complex analysis” Fifth Romanian - Finnish Seminar, Part 1 (Bucharest, 1981), 362-372, Lecture Notes in Math.,
1013, Springer, Berlin.
11. H. Silverman, “Convex and starlike criteria”, Int. J. Math. Math. Sci. 22 (1999),
75-79.
12. N. Tuneski, “On the quotient of the representations of convexity and starlikeness”,
Math. Nachr., 248-249 (2003), 200-203.
13. B.A. Uralegaddi and C. Somanatha, “Certain classes of univalent functions”, Current topics in analytic function theory, 371-374, World Sci. Publishing, River Edge,
NJ, 1992.
38
A. Tehranchi , S. R. Kulkarni and G.Murugusundaramoorthy
A. Tehranchi and S. R. Kulkarni : Department of Mathematics, Fergusson College,
Pune University, Pune - 411004, India.
E-mail: Tehranchiab@yahoo.co.uk and kulkarni− ferg@yahoo.com
G.Murugusundaramoorthy: School of Science and Humanities, Vellore Institute of
Technology, Deemed University, Vellore-632 014 ,India.
E-mail: gmsmoorthy@yahoo.com