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 thep-valent functions that satisfy the differen-tial 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 onAp(r, j, β, a, A, B).
1. INTRODUCTION
Denote byA the class of functions f(z) =z−P∞
n=2anzn analytic inD =
{z∈C:|z|<1}. Also denoteAp the class of all analytic functions of the form
f(z) =kzp+ 2p−1
X
n=p−1
tn−p+1zn−p+1− 2F1(a, b;c;z), |z|<1 (1)
where
2F1(a, b;c;z) = ∞
X
n=0
(a, n)(b, n) (c, n)n! z
n
(a, n) = Γ(a+n)
Γ(a) =a(a+ 1, n−1), c > b >0, c > a+b and
tn−p+1 =
(a, n−p+ 1)(b, n−p+ 1)
(c, n−p+ 1)(n−p+ 1)! , k >0. 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.
These functions are analytic in the unit diskD (For details see [1], [5]).
Definition 1.1. A function f ∈ Ap is said to be in the class S∗
p(α) , p-valently
starlike functions of orderα, if it satisfiesRenzf ′ Minda [6]. But results about the convex class can be obtained easily from the corresponding result of functions inS∗
p(h). If
h(z) = 1 +z
1−z, (3)
then the classes reduce to the usual classes of starlike and convex functions. If h(z) = 1+(1−2α)z
1−z , 0≤α < p,then the classes reduce to the usual classes of starlike and convex functions of order α. Ifh(z) = 1+Az
1+Bz, −1 ≤B < A ≤1, then the classes reduce to the class of Janowski starlike functionS∗
p[A, B] defined by starlike and convex function of orderαthat consists of univalent functions f ∈A satisfying
or equivalently we have
SS∗
Obradoviˇcand Owa [8], Silverman [11], Obradoviˇcand Tuneski [9] and Tuneski [12] have studied the properties of classes of functions defined in terms of the ratio of 1 + zff′′′((zz)) and
zf′
Definition 1.2. A function f ∈Ap is said to be p-valent Bazilevic of type η and order αif there exists a function g∈S∗
p such that
Re
½ zf′(z) f1−η(z)gη(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 orderαin ∆.
Definition 1.3. [7] For two functions f and g, analytic in ∆ we say f is sub-ordinate 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 partic-ular, 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 tof.
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 operatorIp(r, λ) on the infinite series f(z) =zp− P∞
n=p+1
anzn as
Ip(r, λ)f(z) =zp− ∞
X
n=1+p
µn+λ
p+λ
¶r
anzn, (λ≥0). (7)
We note that Sˇalˇagean derivative operators [9] is closely related to the op-erators Ip(r, λ) when the coefficient of f(z) is positive. Also note that the class I1(r,1) =Ir [12] ,I1(r, λ) =Iλ
r the classes studied in [4] and [3].
Definition 1.5. For eachf(z) =zp− ∞
P
n=p+1
anzn we have
f(j)(z) = p! (p−i)!z
p−j− ∞
X
n=p+1 n! (n−j)!anz
n−j (8)
wheren, p∈N, p > j,andj∈N0={0} ∪N. Forj= 0 we havef(0)(z) =f(z).
Definition 1.6. A functionf ∈Apis said to be in the classAp(r, j;h)if it satisfies
z(Ip(r, λ)f(z))(j+1)
where
h(z) = 1 + A−B a
βz
1 +Bz, z∈∆,
and−1≤B < A≤1,0< β < p, a >0, we denoteAp(r, j;h) =Ap(r, j, β, a, A, B).
We say thatf(z) is superordinate toh(z) iff(z) satisfies the following
h(z)≺ z(Ip(r, λ)f(z))
(j+1)
(p−j)(Ip(r, λ)f(z))(j)
whereh(z) is analytic in ∆ andh(0) = 1. We note that if
z(Ip(r, λ)f(z))(j+1) (Ip(r, λ)f(z))(j) ≺
(p−j)[a+ (aB+ (A−B)β)z] a(1 +Bz)
so h(0) = 1 by choosingj =r= 0, p= 1, then f(z)∈S∗
p(h). Also a=A=β = 1, B = −1, f(z) ∈ S∗
p(1). But if a = β = 1 and −1 ≤ B < A ≤1 then f(z)∈ S∗[A, B], class of Janowski starlike function. If we putp=a=β=A= 1, B=−1 thenf(z)∈SS∗
(1) classes of strongly starlike.
By Definition 1.2., ifg(z)∈S∗, univalent starlike andj=r= 0 andp=a= A=β = 1, B=−1 and ifRen zf
′
(z) f(z)−1g2(z)
o
>1, thenf(z)∈B(2,1) class Bazilevic function of typeη= 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
m=p+1
γr(m, p)
·
a(1 +B)(δ(m, j)(p−j)−δ(m, j+ 1)) βκ(A−B)δ(p, j+ 1) −
δ(m, j)(p−j) κδ(p, j+ 1)
¸
km<1
(10)
where
γr(m, p) =
µm+λ
p+λ
¶r
, λ≥0, p, r∈N
and
δ(m, j) = m!
Proof. The function f(z) of the form (1) can be expressed as f(z) = κzp −
But we have
¯ ¯ ¯ ¯
az(Ip(r, λ)f(z))(j+1)−a(p−j)(Ip(r, λ)f(z))(j) (p−j)R(Ip(r, λ)f(z))(j)−Baz(Ip(r, λ)f(z))(j+1)
¯ ¯ ¯ ¯
=|[a ∞
X
m=p+1
γr(m, p)(δ(m, j)(p−j)−δ(m, j+ 1))kmzm−j]/
[β(A−B)δ(p, j+ 1)κzp−j
−
∞
X
m=p+1
γr(m, p)(aB(δ(m, j)(p−j)−δ(m, j+ 1))−
δ(m, j)(A−B)β(p−j))km]|
<{[a ∞
X
m=p+1
γr(m, p)(δ(m, j)(p−j)−δ(m, j+ 1))km]/
[βκ(A−B)δ(p, j+ 1)
−
∞
X
m=p+1
γr(m, p)(aB(δ(m, j)(p−j)−δ(m, j+ 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) =zp− βκ(A−B)δ(p, j+ 1)
γr(q, p)[a(1 +B)(δ(q, j)(p−j)−δ(q, j+ 1))−δ(q, j)(A−B)β(p−j)]z q,
withq≥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)
γr(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 classAp(r, j, β, α, A, B) is closed under linear combination.
Theorem 2.3. Letfq(z) =κzp− P∞
m=p+1
km,qzm(q= 1,2,· · ·, t)be inAp(r, j, β, a, A, B).
Then the functionF(z) = t
P
q=1
dqfq(z)where
t
P
q=1
Proof. We have and this completes the proof.
Then the functionf(z)∈Ap(r, j, β, a, A, B)if and only if it can be expressed in the
Proof. Suppose thatf can be expressed in the form (15) then we have
f(z) =
Therefore we conclude the result.
= κzp− ∞ X
m=1+p
µm(κzp−fm(z))
= κzp(1
− ∞ X
m=1+p
µm)− ∞ X
m=1+p
µmfm(z)
= ∞ X
m=p
µmfm(z).
Remark 2.6. The extreme points of the classAp(r, j, β, a, A, B) are the function fp(z), fm+p(z), m≥1 +pas in Theorem 2.1.
In the following theorem, we obtain the integral representation for Ap(r, j, β, a, A, B).
Theorem 2.7. Letf(z)∈Ap(r, j, β, a, A, B)then
f(z) = exp
·Z z
0
p(ψ(t)R+a) t(1 +Bψ(t))dt
¸
where|ψ(z)|<1, z∈U andR=aB+ (A−B)β. Also
f(z) =zpexp
·Z
X
log(1−Bxz)p(AaB−B)βdµ( x)
¸
whereµ(x) is the probability measure onX={x:|x|= 1}.
Proof. Set
z(Ip(r, λ)f(z))(j+1)
(p−j)(Ip(r, λ)f(z))(j) =Q(z)
Sincef(z)∈Ap(r, j, β, a, A, B) so¯¯ ¯
a(Q(z)−1) R−aBQ(z)
¯ ¯
¯<1 whereR=aB+ (A−B)β.
Consequently we put Ra(−QaBQ(z)−1)(z) =ψ(z),|ψ(z)|<1. Finally we can writeQ(z) = aψ+(aBψz)R+(za) or
(Ip(r, λ)f(z))(j+1) (Ip(r, λ)f(z))(j) =
(p−j)(ψ(z)R+a) az(1 +Bψ(z))
Then we have
log(Ip(r, λ)f(z))(j)=
Z z
0
(p−j)(ψ(t)R+a) t(1 +Bψ(t)) dt
(Ip(r, λ)f(z))(j)= exp·Z z 0
(p−j)(ψ(t)R+a) t(1 +Bψ(t)) dt
forr=j= 0 we have
Now, we introduce an integral operator due to Bernardi [2]
Sincem > pthen mp++cc ≤1 so we have
P∞
m=1+pγr(m, p)
ha(1+B)(δ(m,j)(p−j)−δ(m,j+1))
βκ(A−B)δ(p,j+1) −
δ(m,j)(p−j) κδ(p,j+1)
i ³
p+c m+c
´
km
≤P∞
m=1+pγr(m, p)
h
a(1+B)(δ(m,j)(p−j)−δ(m,j+1)) βκ(A−B)δ(p,j+1) −
δ(m,j)(p−j) κδ(p,j+1)
i
km<1.
ThusLc(f(z))∈Ap(r, j, β, a, A, B).
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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.
