Paper19-Acta26-2011. 122KB Jun 04 2011 12:03:16 AM

Acta Universitatis Apulensis
ISSN: 1582-5329

No. 26/2011
pp. 179-188

ON P-VALENT STRONGLY STARLIKE AND STRONGLY CONVEX
FUNCTIONS

Ali muhammad
Abstract. In this paper, we define some new classes S ∗ (a, c, λ, p, α, β) and
C(a, c, λ, p, α, β) of strongly starlike and strongly convex functions or order αand
type β by using Cho- Kown -Srivastava integral operators. We also derive some
interesting properties, such as inclusion relationships of these classes.
2000 Mathematics Subject Classification: 30C45, 30C50.
Keywords: Analytic functions; Strongly srarlike functions; Strongly convex functions; Cho Kown Srivastava integral operators.
1. Introduction
Let A(p) denote the class of functions f (z) normalized by
zp +

∞

X

ap+k z p+k , p ∈ N = {1, 2, 3, ...} .

(1.1)

k=1

which are analytic and p-valent in the unit disk E = {z : z ∈ C and |z| < 1}.
A function f (z) ∈ A(p) is said to be in the class S ∗ (p, β) of p-valently starlike
function of order β in E if it satisfies the following inequality:
 ′ 
zf (z)
Re
> β, z ∈ E, 0 ≤ β < p.
(1.2)
f (z)
The class S ∗ (p, β) was introduced and studied by Goodman [4], see also [11]. For
recent investigation and more detail the interested reader are refers to the work by
[12, 13]. Also, we note that S ∗ (p, 0) = Sp∗ , where Sp∗ is the class of p-valently starlike

functions in E.

179

A. Muhammad - On P-valent strongly starlike and strongly convex functions

A function f (z) ∈ A(p) is said to be in the class C(p, β) of p-valently convex
function of order β in E if it satisfies the following inequality:


zf ′′ (z)
> β, z ∈ E, 0 ≤ β < p.
(1.3)
Re 1 + ′
f (z)
The class C(p, β) was introduced and studied by Goodman [4], see also [10]. Also
we note that C(p, 0) = Cp , where Cp is the class of p-valently convex functions in
E.
A function f (z) ∈ A(p) is said to be strongly starlike of order α and type β in
E if it satisfies the following inequality:


 ′



arg zf (z) − β  < απ , z ∈ E, 0 ≤ α < p and 0 ≤ β < p.
(1.4)


f (z)
2
We denote this class by S ∗ (p, α, β).A function f (z) ∈ A(p) is said to be strongly
convex of order α and type β in E if it satisfies the following inequality:



′′


arg 1 + zf (z) − β  < απ , z ∈ E, 0 ≤ α < p and 0 ≤ β < p.

(1.5)


f ′ (z)
2

We denote this class by C(p, α, β).
′ (z)
∈
It is obvious that f (z) ∈ A(p) belongs to C(p, α, β) if and only if zff (z)
∗
∗
∗
S (p, α, β). Also, we note that S (p, α, β) = S (p, β) and C(p, α, β) = C(p, β).
In our present investigation we shall make use of the Gauss hypergeometric
functions defined by
∞
X
(a)k (b)k z k
,

(1.6)
2 F1 (a, b; c; z) =
(c)k (1)k
k=0

where a, b, c ∈ C, c ∈
/ Z−
0 = {0, −1, −2, ...} and (k)n denote the Pochhammer symbol
(or the shifted factorial) given, in terms of the Gamma function Γ, by

Γ(k + n)
k(k + 1)...(k + n − 1)
n ∈ N,
=
(k)n =
1
n = 0.
Γ (k)

We note that the series defined by (1.6) converges absolutely for z ∈ E and hence

represents an analytic function in the open unit disk E, see [14].
We define a function φp (a, c; z) by
p

φp (a, c; z) = z +

∞
X
(a)

k z p+k

k =1

(c)k

, a ∈ R; c ∈ R\Z−
0 = {0, −1, −2, ...} (z ∈ E).

180

A. Muhammad - On P-valent strongly starlike and strongly convex functions
Using the function φp (a, c; z), we consider a function φ†p (a, c; z) defined by
φp (a, c; z) ∗ φ†p (a, c; z) =

zp
, z ∈ E,
(1 − z)λ+p

where λ > −p. This function yields the following family of linear operators
Ipλ (a, c)f (z) = φ†p (a, c; z) ∗ f (z), z ∈ E,

(1.7)

where a, c ∈ R\Z−
0 . For a function f (z) ∈ A(p), given by (1.1), it follows from (1.7)
that for λ > −p and a, c ∈ R\Z−
0
Ipλ (a, c)f (z)

∞
X
(a)k (λ + p)
= z +
ap+k z p+k
(c)k (1)k
p

(1.8)

k=1

= z P 2 F1 (c, λ + p; a; z) ∗ f (z), z ∈ E.
From equation (1.8) we deduce that
z(Ipλ (a, c)f (z))´= (λ + p)Ipλ+1 (a, c)f (z) − λIpλ (a, c)f (z),

(1.9)

z(Ipλ (a + 1, c)f (z))´= aIpλ (a, c)f (z) − (a − p)Ipλ (a + 1, c)f (z).

(1.10)

We also note that
Ip0 (p + 1, 1)f (z) = p

Rz
0

f (t)
t dt,

Ip0 (p, 1)f (z) = Ip1 (p + 1, 1)f (z) = f (z),

2zf ′ (z)+z 2 f ′′ (z)
zf ′ (z)
2
,
p , Ip (p, 1)f (z) =
p(p+1)
f (z)+zf ′ (z)

2
n
n+p−1
Ip (p + 1, 1)f (z) =
, Ip (a, a)f (z) = D
f (z), n ∈ N, n > −p,
p+1
n+p−1
where D
is the Ruscheweyh derivative of (n + p − 1)th order, see [5].

Ip1 (p, 1)f (z) =

The operator Ipλ (a, c) (λ > −p, a, c ∈ R\Z−
0 ) was recently introduced by Cho
et.al [1], who investigated (among other things) some inclusion relationships and
argument properties of various subclasses of multivalent functions in A(p), which
were defined by means of the operator Ipλ (a, c).
For λ = c = 1 and a = n + p, Cho-kown-Srivastava operator Ipλ (a, c) yields
Ip1 (n + p, 1) = In,p (n > −p),

where In,p denotes an integral operator of the (n+p-1)th order, which was studied by
Liu and Noor [6], see also [7, 8]. The linear operator I1λ (µ + 2, 1) (λ > −1, µ > −2)
was also recently introduced and studied by Choi et.al [2]. For relevant details about
further special cases of the Choi-Saigo Srivastava operator the interested reader may
refer to the works by Choi et. al [2] and Cho et. al [1], see also [3].
181

A. Muhammad - On P-valent strongly starlike and strongly convex functions

Using the operator Ipλ (a, c) we now define new subclasses of A(p) as follows:
Definition 1.1.
S ∗ (a, c, λ, p, α, β) =

(

z(Ipλ (a, c)f (z))′
6 β
=
f (z) ∈ A(p) : Ipλ (a, c)f (z) ∈ S ∗ (p, α, β),
Ipλ (a, c)f (z)

)

.

where z ∈ E.
It is easy to see that S ∗ (p, 1, 0, p, α, β) = S ∗ (p, α, β) and S ∗ (p + 1, 1, 1, p, α, β) =
is the class of strongly starlike functions of order α and type β as given
by (1.4).

S ∗ (p, α, β),

Definition 1.2.
(

C(a, c, λ, p, α, β) =

z(Ipλ (a, c)f (z))′′
6 β
=
f (z) ∈ A(p) : Ipλ (a, c)f (z) ∈ C(p, α, β), 1 +
(Ipλ (a, c)f (z))′

where z ∈ E.
′

Obviously, f (z) ∈ C(a, c, λ, p, α, β) if and only if zf p(z) ∈ S ∗ (a, c, λ, p, α, β),
C(p, 1, 0, p, α, β) = C(p, α, β) and C(p + 1, 1, 1, p, α, β) = C(p, α, β) is the class
of strongly convex functions of order α and type β as given by (1.5).
2. Preliminaries and Main results
In order to prove our main results we shall need the following result.
Lemma 2.1.[9] Let a function h(z) = 1 + c1 z + c2 z 2 + ...be analytic in E and
h(z) 6= 0, z ∈ E. If there exists a point z0 ∈ E such that
|arg h(z)| <
then we have,

where

and

π
π
α (|z| < |z0 |) and |arg h(z0 )| = α (0 ≤ α < 1),
2
2
z0 h′ (z0 )
= ikα,
h(z0 )
1
π
1
k ≥ (a + ) when arg h(z0 ) = α,
2
a
2
1
π
1
k ≤ − ((a + ) when arg h(z0 ) = α,
2
a
2
1

(h(z0 )) α = ±iα, (α > 0).
182

(2.1)

(2.2)

(2.3)
(2.4)

)

.

A. Muhammad - On P-valent strongly starlike and strongly convex functions

Theorem 2.2. S ∗ (a, c, λ+1, p, α, β) ⊂ S ∗ (a, c, λ, p, α, β) ⊂ S ∗ (a+1, c, λ, p, α, β).
Proof. Set
z(Ipλ (a, c)f (z))′
= β + (p − β)h(z).
(2.5)
Ipλ (a, c)f (z)
Then h(z) is analytic in E with h(0) = 1 and h (z) 6= 0, z ∈ E. Applying the identity
(1.9) in (2.1) and differentiating the resulting equation with respect to z we have
(p − β)zh′ (z)
z(I λ+1 (a, c)f (z))′
.
−
β
=
(p
−
β)h(z)
+
(λ + β) + (p − β)h(z)
I λ+1 (a, c)f (z)
Suppose there exists a point z0 ∈ E such that the conditions (2.1) to (2.4) of Lemma
2.1 are satisfied. Thus, if
arg h(z0 ) =

−π
α, z0 ∈ E,
2

then
z(I λ+1 (a, c)f (z))′
I λ+1 (a, c)f (z)



− β = (p − β)h(z0 ) 1 +
α − πα
2

= (p − β)a e

z0 h′ (z0 )
h(z0 )

(λ + β) + (p − β)h(z0 )


1+





ikα
.
πα
(λ + β) + (p − β)aα e− 2

This implies that




−πα
z(I λ+1 (a, c)f (z))′
−πα
ikα
=
arg
−β =
+ arg 1 +
πα
λ+1
−
2
2
I
(a, c)f (z)
(λ + β) + (p − β)aα e 2
"
#


α cos πα
kα((λ
+
β)
+
(p
−
β)a
2



.
+ tan−1
2 a2α − kα(p − β)aα sin πα
+
(p
−
β)
(λ + β)2 + 2(λ + β)(p − β)aα cos πα
2
2

This gives that

arg
since



z(I λ+1 (a, c)f (z))′
−β
I λ+1 (a, c)f (z)



≤

−πα
,
2

−1
1
(a + ) ≤ −1 and z0 ∈ E,
2
a
this contradicts the condition f (z) ∈ S ∗ (a, c, λ, p, α, β).
On the other hand if we set
k≤

arg h(z0 ) =
183

πα
,
2

A. Muhammad - On P-valent strongly starlike and strongly convex functions

then it can similarly be shown that


πα
z(I λ+1 (a, c)f (z))′
,
arg
−β ≥
λ+1
2
I
(a, c)f (z)
since

1
1
k ≥ (a + ) and z0 ∈ E,
2
a
which again contradicts the hypothesis that f (z) ∈ S ∗ (a, c, λ, p, α, β).
Thus the function defined by (2.5) has to satisfy the following inequality:
arg h(z) ≤

πα
, z ∈ E.
2

which implies that



λ+1 (a, c)f (z))′

 πα
arg z(I
, z ∈ E.
− β  <

λ+1
2
I
(a, c)f (z)

The proof of part (ii) lies on the similar lines. This completes the proof of Theorem
2.2.
Theorem 2.3.C(a, c, λ + 1, p, α, β) ⊂ C(a, c, λ, p, α, β) ⊂ C(a + 1, c, λ, p, α, β)
Proof. To prove this inclusion relationship, we observe from Theorem 2.2 that
f (z)

zf ′ (z)
∈ S ∗ (a, c, λ + 1, p, α, β)
p

∈

C(a, c, λ + 1, p, α, β) ⇔

⇒

zf ′ (z)
∈ S ∗ (a, c, λ, p, α, β) ⇔ f (z) ∈ C(a, c, p, α, β).
p

The proof of second part lies on similar lines. This completes the proof of Theorem
2.3.
For a function f (z) ∈ A(p) the integral operator, Fδ,p : A(p) −→ A(p) is defined
by
Fδ,p (f )(z) =

δ+p
zp

Zz

t

δ−1

f (t)dt =

= z

2 F1 (1, δ

z +

∞
X
k=1

0

p

p

δ+p
z p+k
δ+p+k

!

∗ f (z) (2.6)

+ p, δ + p + 1; z) ∗ f (z), z ∈ E, see [2].

It follows from (2.6) that

′
z (Ipλ (a, c)Fδ,p (f )(z) = (δ + p)Ipλ (a, c)f (z) − δIpλ (a, c)Fδ,p (f )(z).
184

(2.7)

A. Muhammad - On P-valent strongly starlike and strongly convex functions

We have the following result.
Theorem 2.4. Let δ > −β and 0 ≤ β < p.If f (z) ∈ S ∗ (a, c, λ, p, α, β) with
z(Ipλ (a, c)Fδ,p (f )(z))′
6= β, for all z ∈ E,
Ipλ (a, c)Fδ,p (f )(z)
then we have
Fδ,p (f )(z) ∈ S ∗ (a, c, λ, p, α, β).
Proof. We begin by setting
′

z ( Ipλ (a, c)Fδ,p (f )(z))
= β + (p − β)h(z), z ∈ E.
Ipλ (a, c)Fδ,p (f )(z)

(2.8)

Then h(z) is analytic in E with h(0) = 1. Using (2.7) and (2.8), we find that
(δ + p)

(Ipλ (a, c)f (z)
= (δ + p) + (p − β)h(z).
Ipλ (a, c)Fδ,p (f )(z)

(2.9)

Differentiation both sides of (2.9) logarithmically, we obtain
z( Ipλ (a, c)f (z))′
(p − β)zh′ (z)
.
−
β
=
(p
−
β)h(z)
+
(δ + β) + (p − β)h(z)
Ipλ (a, c)f (z)
Suppose now there exists a point z0 ∈ E such that
|arg h(z)| <

π
π
α (|z| < |z0 |) and |arg h(z)| = α.
2
2

Then by Lemma 2.1, we can write that
1
z0 h′ (z0 )
= ikα and (h(z0 )) α = ±iα (α > 0).
h(z0 )

If
arg h(z0 ) =

π
α, z0 ∈ E,
2

then
z(Ipλ (a, c)f (z0 ))′
Ipλ (a, c)f (z0 )



′

− β = (p − β)h(z0 ) 1 +
α i πα
2

= (p − β)a e

185

zh (z0 )
h(z0 )

(δ + β) + (p − β)h(z0 )


1+





ikα
.
πα
(δ + β) + (p − β)aα ei 2

A. Muhammad - On P-valent strongly starlike and strongly convex functions

This shows that
arg

z(Ipλ (a, c)f (z0 ))′
−β
Ipλ (a, c)f (z0 )

−1

+ tan
≥

πα
,
2

(

!



ikα
πα
πα
+ arg 1 +
=
=
πα
i
2
2
(δ + β) + (p − β)aα e 2



kα (δ + β + (p − β)) aα cos πα
2


2 2α + kα(p − β)aα sin
(δ + β)2 + 2(δ + β)(p − β)aα cos πα
2 + (p − β) a

since

1
1
k ≥ (a + ) ≥ 1 and z0 ∈ E,
2
a
which contradicts the assumption that f (z) ∈ S ∗ (a, c, λ, p, α, β).
Similarly in the case when
π
arg h(z0 ) = − α, z0 ∈ E,
2
we can prove that
′

arg

z(Ipλ (a, c)f (z0 ))
−β
Ipλ (a, c)f (z0 )

!

≤

−πα
,
2

since

1
1
k ≤ − (a + ) ≤ −1 and z0 ∈ E,
2
a
contradicting once again the condition that f (z) ∈ S ∗ (a, c, λ, p, α, β).
We thus conclude that h(z) must satisfy the following inequality
|arg h(z)| <

π
α, z ∈ E.
2

This shows that


 πα

z(Ipλ (a, c)Fδ,p (f )(z))′


− β <
, z ∈ E,
arg


2
Ipλ (a, c)Fδ,p (f )(z)

and hence the proof is complete.

Theorem 2.5. Let δ > −β and 0 ≤ β < p. If f (z) ∈ C(a, c, λ, p, α, β) and
1+

z(Ipλ (a, c)Fδ,p (f )(z))′′
6= β for all z ∈ E,
(Ipλ (a, c)Fδ,p (f )(z))′
186

πα
2




)

A. Muhammad - On P-valent strongly starlike and strongly convex functions

then we have
Fδ,p (f )(z) ∈ C(a, c, λ, p, α, β).
Proof. Since
f (z)

zf ′ (z)
∈ S ∗ (a, c, λ, p, α, β)
p
Fδ,p (f )(z)
z(Fδ,p (f )(z))′
∈ S ∗ (a, c, λ, p, α, β) ⇔
∈ S ∗ (a, c, λ, p, α, β)
⇒
p
p
⇔ Fδ,p (f )(z) ∈ C(a, c, λ, p, α, β).
∈

C(a, c, λ, p, α, β) ⇔

Acknowledgements. We would like to thank Honourable Director Finance
Sarwar Khan for providing us excellent research facilities.
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Ali Muhammad
Depatment of Basic Sciences
University of Engineering and Technology
Technology Peshawar Pakistan
E-mail: ali7887@gmail.com

188