133 (2008)
MATHEMATICA BOHEMICA
No. 3, 241–245
ON STRONG STARLIKENESS CRITERIA OF
p-VALENT FUNCTIONS
V. B. L. Chaurasia, Hari Singh Parihar, Jaipur
(Received December 18, 2006)
Abstract. H. Silverman (1999) investigated the properties of functions defined in terms of
the quotient of the analytic representations of convex and starlike functions. Many research
workers have been working on analytic functions to be strongly starlike like Obradović and
Owa (1989), Takahashi and Nunokawa (2003), Lin (1993) etc. In this paper we obtain a
sufficient condition for p-valent functions to be strongly starlike of order α.
Keywords: analytic function, strongly starlike function
MSC 2000 : 30C45
Introduction
Let A denote the class of functions that are analytic in the open unit disc U =
{z ∈ C : |z| < 1} and let An,p be the subclass of A consisting of the functions fn,p
of the form
(0.1)
p
fn,p (z) = z +
∞
X
k=n+p
ak z k
(n ∈ N )
where p is a positive integer and fn,p is analytic and p-valent in U . Then a function
fn,p ∈ An,p is said to be in class Sn (p, δ) if and only if
(0.2)
Re
zf ′ (z) n,p
> δ (z ∈ U )
fn,p (z)
for some δ (0 6 δ < p).
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A function fn,p ∈ Sn (p, δ) is called p-valent starlike of order δ. On the other hand,
a function fn,p ∈ An,p is said to be in the class Kn (p, δ) iff
(0.3)
′′
zfn,p
(z) Re 1 + ′
> δ (z ∈ U )
fn,p (z)
for some δ (0 6 δ < p). A function fn,p ∈ Kn (p, δ) is called a p-valent convex function
of order δ. On the other hand, a functionfn,p (z) in An,p is said to be strongly starlike
of order α if it satisfies
n zf ′ (z) o πα
n,p
arg
<
fn,p (z)
2
(0.4)
for some α (0 < α 6 p) and for all z ∈ U . We write fn,p (z) ∈ An,p S ∗ (α) if fn,p (z) is
strongly starlike of order α in U . Silverman [1] investigated the properties of functions
defined in terms of the quotient of the analytic representation of convex and starlike
functions. Let Gb be the subclass of An,p consisting of functions fn,p ∈ An,p which
satisfy
(0.5)
′′
1 + zfn,p (z)
′ (z)
fn,p
−
p
<b
′
(z)
zfn,p
pfn,p (z)
(z ∈ U )
for some real b. For this class Gb , Silverman obtained the following result.
Theorem 0.1 [1]. If 0 < b 6 1, then
(0.6)
Gb ⊂ S ∗
n
o
2
√
.
1 + (1 + 8b)
The result is sharp for all b.
In this paper we consider the strong starlikeness for p-valent functions fn,p (z)
belonging to Gb .
1. Strong starlikeness
For discussing the strong starlikeness of a function fn,p (z) in Gb , we have to recall
the following result by Nunokawa [3].
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Lemma 1.2. Let p(z) be analytic in U with p(0) = 1 and let p(z) 6= 0 (z ∈ U ).
Suppose that there exists a point z0 ∈ U such that
(1.7)
|arg p(z)| <
πβ
(|z| < |z0 |)
2
and
(1.8)
|arg p(z0 )| =
πβ
2
where β > 0. Then we have
z p′ (z ) 0
0
= ιKβ
p(z0 )
(1.9)
where K > 21 {a + 1/a} when arg{p(z0 )} = 12 πβ, and K 6 − 21 {a + 1/a} when
arg{p(z0 )} = − 21 πβ where p{z0 }1/β = ±ιa and a > 0.
Theorem 1.3. If fn,p (z) belong to the class Gb(α) with
(1.10)
b(α) =
α
q
p+α
p−α
p2 (p − α) p (p + α) p
(0 < α 6 p)
then fn,p (z) ∈ An,p S ∗ (α).
P r o o f. Let us define a function ϕn,p (z) by
(1.11)
ϕn,p (z) =
zf ′ (z) n,p
.
pfn,p (z)
Then it follows that
(1.12)


′′
zfn,p
(z)






 1 + f ′ (z)
 zϕ′ (z) n,p
n,p
−p =
.
′
2 [ϕ
2

zf
(z)

p
n,p (z)]
n,p






pfn,p (z)
If there exists a point z0 ∈ U such that
(1.13)
(1.14)
πα
for |z| < |z0 | ,
2p
πα
|arg {ϕn,p (z0 )}| =
2p
|arg{ϕn,p (z)}| <
243
then, applying Lemma 1.2 we obtain that
(1.15)
zϕ′ (z ) 1 ιKα α|K|a−α/p
α[a(p−α)/p + a−(p+α)/p ]
p
n,p 0
>
.
2
= 2
=
2
3
α/p
p [ϕn,p (z0 )]
p
2p3
p (±ιa)
Define a function Fp (a) by
Fp (a) = [a(p−α)/p + a−(p+α)/p ] (a > 0; 0 < α 6 p),
h (p − α)a2 − (p + α) i
1
.
Fp′ (a) = (2p+α)/p
p
a
(1.16)
(1.17)
Fp (a) assumes its minimum value at a =
(1.18)
p
(p + α)/(p − α). This implies that
zϕ′ (z ) r p + α α
n,p 0
>
F
2
p
p [ϕn,p (z0 )]2
2p3
p−α
h
p + α (p−α)/2p p − α (p+α)/2p i
α
+
= 3
2p
p−α
p+α
α
= p
,
p2 (p − α)(p−α)/p (p + α)(p+α)/p
which contradicts our condition fn,p (z) ∈ Gb(α) of the theorem. Thus we complete
the proof of the theorem.
Corollary 1.4. If we take p = 1 then the function fn,p (z) reduces to a univalent
function fn,1 (z) = fn (z) and we get a sufficient condition of strong starlikeness of
univalent functions.
Considering the case of α = 1 in Corollary 1.4, we have
Corollary 1.5. If fn (z) ∈ Gb with b =
strongly starlike in U
Taking α =
1
2
1
2,
then fn (z) ∈ AS ∗ (1) or fn (z) is
in Corollary 1.4 ,we have
p √
Corollary 1.6. If fn (z) ∈ Gb with b = 1/ 3 3 = 0.438691 . . ., then fn (z) ∈
AS ∗ ( 12 ).
A c k n o w l e d g m e n t . The authors are thankful to Professor H. M. Srivastava,
University of Victoria, Canada for his kind help and valuable suggestions in the
preparation of this paper, and also to the worthy referee for his/her fruitful suggestions.
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References
[1] H. Silverman: Convex and starlike criteria. Internet J. Math. Math. Sci. 22 (1999), 75–79.
[2] L. J. Lin: On a criterion of starlikeness. Math. Japonica 38 (1993), 897–899.
[3] M. Nunokawa: On the order of strongly starlikeness of strongly convex functions. Proc.
Japan Acad. 69 (1993), 234–237.
[4] M. Obradović, S. Owa: A criterion for starlikeness. Math. Nachr. 140 (1989), 97–102.
[5] N. Takahashi, M. Nunokawa: A certain connection between starlike and convex functions. Appl. Math. Lett. 66 (2003), 653–655.
Authors’ address: V. B. L. Chaurasia, Hari Singh Parihar, Department of Mathematics,
University of Rajashthan, Jaipur-302004, India, e-mail: harisingh.p@rediffmail.com.
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