On Certain Classess of Multivalent Functions

IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 6 Ver. III (Nov. - Dec. 2015), PP 44-49
www.iosrjournals.org
DOI: 10.9790/5728-11634449 www.iosrjournals.org 44 | Page
On Certain Classess of Multivalent Functions
P. N. Kamble, M.G.Shrigan
1
Department of Mathematics Dr. Babasaheb Ambedkar Marathwada University, Aurangabad – 431004, (M.S.),
India
2
Department of Mathematics Dr D Y Patil School of Engineering & Technology, Pune - 412205, (M.S.), India
Abstract: In this we defined certain analytic p-valent function with negative type denoted by 𝜏 𝑝 . We obtained
sharp results concerning coefficient bounds, distortion theorem belonging to the class 𝜏 𝑝 .
Keywords: p-valent function, distortion theorem, convexity.
Mathematics Subject Classification (2000): 30C45, 30C50
I. Introduction
Let A (p) denote the class of f normalized univalent functions of the form
f z = zp
+ an+pzn+p
∞
n=1
, (an+p ≥ 0, p ∈ ℕ = {1,2,3, … }) (1.1)
analytic and p-valent in the unit disc E = {z : z  C;|z| < 1}.
A function f(z) Є A (p) is said to in the class of Sp
∗
() p-valently starlike function of order α (0 ≤ α < p)
if it satisfies, for z  E, the condition
1.2)(





f(z)
(z)fz
Re
'
Furthermore, a function f(z)  A (p) is said to in the class 𝒦p() of p-valently convex function of
order α (0 ≤ α < p) if it satisfies, for z  E, the condition
1.3)(1 








(z)f
(z)fz
Re '
''
It follows from the definition (1.2) and (1.3) that
f(z) 𝒦p() ⇔
zf ′ z
p
 Sp
∗
() (0 ≤ α < p) (1.4)
whose special case, when  = 0 is the familiar Alexander theorem (see for example [1] p.43, Theorem 2.12). We
also note that
𝒦p   Sp
∗
 (0 ≤ α < p)
Sp
∗
  Sp
∗
0  Sp
∗
( 0 ≤ α < p)
and
𝒦p   𝒦p 0  𝒦p (0 ≤ α < p)
Where Sp
∗
and 𝒦p denote the subclasses of A (p) consisting of p-valently starlike and convex functions
in unit disk E respectively.
Let τp(, ) denote the subclass of A (p) consisting of functions analytic and p-valent which can be express in
the form
f z = zp
− an+pzn+p
∞
n=1
, an+p ≥ 0
The subclass τp ,  of p-valent functions with negative coefficients is studied by H. M. Srivastava
and M. K. Auof [2].
Following S. Owa [3], we say that a function f(z) τp is in the subclass τp(, ) if and only if

f′
z − pz1−p
f′ z + pz1−p(1 − 2α)
< β
The subclass τp ,  was studied by Goel and Sohi [4]. Moreover S. Owa studied several
interesting results on radius of convexity for p-valent function with negative coefficients. In this present paper
we investigate sharp results concerning coefficient inequalities, distortion theorem and radius of convexity for
class the τp ,  .
II. Main Result
Theorem 2.1 A function
f z = zp
− an+pzn+p
∞
n=1
, an+p ≥ 0
is in the class τp ,  if and only if
n + p (1 + β) an+p
∞
n=1
≤ 2β 1 − α p (2.1)
The result is sharp.
Proof: Assume (2.1) holds. We show that f z  τp(α, β).
Let z = 1. We have,
f z = zp
− an+pzn+p
∞
n=1
(2.2)
f′ z = pzp−1
− an+p n + p zn+p−1
∞
n=1
(2.3)
Now,
f′
z − pzp−1
= pzp−1
− an+p n + p zn+p−1
∞
n=1
− pzp−1
= − an+p n + p zn+p−1
∞
n=1
Also,
β f′
z + pzp−1
(1 − 2α)
= βpzp−1
−  an+p n + p zn+p−1
∞
n=1
+ βpzp−1
(1 − 2α)
= − an+p n + p zn+p−1
∞
n=1
+ 2βpzp−1
− 2αβpzp−1
Then,
f′
z − pzp−1
− β f′
z + pzp−1
(1 − 2α)
= − an+p n + p zn+p−1
∞
n=1
− − an+p n + p zn+p−1
∞
n=1
+ 2βpzp−1
− 2αβpzp−1
since 𝑧 = 1
𝑓′
𝑧 − 𝑝𝑧 𝑝−1
− 𝛽 𝑓′
𝑧 + 𝑝𝑧 𝑝−1
(1 − 2𝛼)
≤ 𝑛 + 𝑝 𝑎 𝑛+𝑝 + 
∞
𝑛=1
𝑛 + 𝑝 𝑎 𝑛+𝑝
∞
𝑛=1
− 2𝛽𝑝 + 2𝛼 𝑝
≤ (1 +  ) 𝑛 + 𝑝 𝑎 𝑛+𝑝
∞
𝑛=1
− 2𝛽𝑝 + 2𝛼𝛽𝑝
≤ (1 +  ) 𝑛 + 𝑝 𝑎 𝑛+𝑝
∞
𝑛=1
− 2𝛽(1 − )𝑝
≤ 0

Hence by maximum modulus theorem, 𝑓 𝑧  𝜏 𝑝(𝛼, 𝛽).
Conversely, suppose that
𝑓′
𝑧 − 𝑝𝑧 𝑝−1
𝑓′ 𝑧 + 𝑝𝑧 𝑝−1(1 − 2𝛼)
=
𝑝𝑧 𝑝−1
− 𝑎 𝑛+𝑝 𝑛 + 𝑝 𝑧 𝑛+𝑝−1∞
𝑛=1 − 𝑝𝑧 𝑝−1
𝑝𝑧 𝑝−1 − 𝑎 𝑛+𝑝 𝑛 + 𝑝 𝑧 𝑛+𝑝−1∞
𝑛=1 + 𝑝𝑧 𝑝−1(1 − 2)
=
− 𝑎 𝑛+𝑝 𝑛 + 𝑝 𝑧 𝑛+𝑝−1∞
𝑛=1
2𝑧 𝑝−1 1 −  𝑝 − 𝑎 𝑛+𝑝 𝑛 + 𝑝 𝑧 𝑛+𝑝−1∞
𝑛=1
since 𝑅𝑒(𝑧) ≤ 𝑧 for all z , we have
𝑅𝑒
𝑎 𝑛+𝑝 𝑛 + 𝑝 𝑧 𝑛+𝑝−1∞
𝑛=1
2𝑧 𝑝−1 1 −  𝑝 − 𝑎 𝑛+𝑝 𝑛 + 𝑝 𝑧 𝑛+𝑝−1∞
𝑛=1
<  (2.4)
Choose value of z on real axis so that 𝑓 ′ 𝑧 is real. Upon clearing the denominator in (2.4) and letting 𝑧 → 1
through real values, we have
𝑛 + 𝑝 𝑎 𝑛+𝑝 
∞
𝑛=1
2 1 −  𝑝 −  𝑛 + 𝑝 𝑎 𝑛+𝑝
∞
𝑛=1
𝑛 + 𝑝 𝑎 𝑛+𝑝 +  𝑛 + 𝑝 𝑎 𝑛+𝑝
∞
𝑛=1

∞
𝑛=1
2 1 −  𝑝
𝑛 + 𝑝 (1 + 𝛽) 𝑎 𝑛+𝑝
∞
𝑛=1
≤ 2𝛽 1 − 𝛼 𝑝
This completes the proof.
III. Distortion Theorem
Theorem 3.1 If 𝑓 𝑧  𝜏 𝑝 𝛼, 𝛽 , then
𝑟 𝑝
−
2𝛽(1 − 𝛼)𝑝
(1 + 𝑝)(1 + 𝛽)
𝑟 𝑝+1
≤ 𝑓(𝑧) ≤ 𝑟 𝑝
+
2𝛽(1 − 𝛼)𝑝
(1 + 𝑝)(1 + 𝛽)
𝑟 𝑝+1
, 𝑧 = 𝑟 (3.1)
and
𝑝𝑟 𝑝−1
−
2𝛽(1 − 𝛼)𝑝
(1 + 𝛽)
𝑟 𝑝
≤ 𝑓′(𝑧) ≤ 𝑝𝑟 𝑝−1
+
2𝛽(1 − 𝛼)𝑝
(1 + 𝛽)
𝑟 𝑝
, 𝑧 = 𝑟 (3.2)
Proof: from Theorem 1, we have
𝑛 + 𝑝 (1 + 𝛽) 𝑎 𝑛+𝑝
∞
𝑛=1
≤ 2𝛽 1 − 𝛼 𝑝
1 + 𝑝 1 + 𝛽 𝑎 𝑛+𝑝
∞
𝑛=1
 𝑛 + 𝑝 (1 + 𝛽) 𝑎 𝑛+𝑝
∞
𝑛=1
≤ 2𝛽 1 − 𝛼 𝑝
This implies that
𝑎 𝑛+𝑝
∞
𝑛=1
≤
2𝛽 1 − 𝛼 𝑝
1 + 𝑝 (1 + 𝛽)
Hence
𝑓(𝑧) ≤ 𝑧 𝑝
+ 𝑎 𝑛+𝑝
∞
𝑛=1
𝑧 𝑛+𝑝
≤ 𝑟 𝑝
+ 𝑎 𝑛+𝑝
∞
𝑛=1
𝑟 𝑛+𝑝
∵ 𝑧 = 𝑟
≤ 𝑟 𝑝
+
2𝛽(1 − 𝛼)𝑝
(1 + 𝑝)(1 + 𝛽)
𝑟 𝑝+1
(3.3)
and
𝑓(𝑧) ≥ 𝑧 𝑝
− 𝑎 𝑛+𝑝
∞
𝑛=1
𝑧 𝑛+𝑝

≥ 𝑟 𝑝
− 𝑎 𝑛+𝑝
∞
𝑛=1 𝑟 𝑛+𝑝
∵ 𝑧 = 𝑟
≥ 𝑟 𝑝
−
2𝛽(1 − 𝛼)𝑝
(1 + 𝑝)(1 + 𝛽)
𝑟 𝑝+1
(3.4)
From (3.3) and (3.4) we get,
𝑟 𝑝
−
2𝛽(1 − 𝛼)𝑝
(1 + 𝑝)(1 + 𝛽)
𝑟 𝑝+1
≤ 𝑓(𝑧) ≤ 𝑟 𝑝
+
2𝛽(1 − 𝛼)𝑝
(1 + 𝑝)(1 + 𝛽)
𝑟 𝑝+1
, 𝑧 = 𝑟
Thus (3.1) holds.
Also
𝑓′(𝑧) ≤ 𝑝 𝑧 𝑝−1
+ 𝑎 𝑛+𝑝 (𝑛 + 𝑝)
∞
𝑛=1
𝑧 𝑛+p−1
≤ 𝑟 𝑝−1
𝑝 + 𝑟 𝑎 𝑛+𝑝
∞
𝑛=1
(𝑛 + 𝑝) ∵ 𝑧 = 𝑟
≤ 𝑟 𝑝−1
𝑝 + 𝑟
2𝛽(1 − 𝛼)𝑝
(1 + 𝛽)
≤ 𝑝𝑟 𝑝−1
+
2𝛽(1 − 𝛼)𝑝
(1 + 𝛽)
𝑟 𝑝
(3.5)
Also,
𝑓′(𝑧) ≥ 𝑝 𝑧 𝑝−1
− 𝑎 𝑛+𝑝 (𝑛 + 𝑝)
∞
𝑛=1
𝑧 𝑛+𝑝−1
≥ 𝑟 𝑝−1
𝑝 − 𝑟 𝑎 𝑛+𝑝
∞
𝑛=1
𝑛 + 𝑝 ∵ 𝑧 = 𝑟
≥ 𝑟 𝑝−1
𝑝 − 𝑟
2𝛽(1 − 𝛼)𝑝
(1 + 𝛽)
≥ 𝑝𝑟 𝑝−1
−
2𝛽(1 − 𝛼)𝑝
(1 + 𝛽)
𝑟 𝑝
(3.6)
Thus from (3.5) and (3.6) we get,
𝑝𝑟 𝑝−1
−
2𝛽(1 − 𝛼)𝑝
(1 + 𝛽)
𝑟 𝑝
≤ 𝑓′(𝑧) ≤ 𝑝𝑟 𝑝−1
+
2𝛽(1 − 𝛼)𝑝
(1 + 𝛽)
𝑟 𝑝
, 𝑧 = 𝑟
Thus (3.1) holds.
IV. Radius of convexity
Theorem 4.1 If 𝑓 𝑧  𝜏 𝑝 𝛼, 𝛽 is p-valently convex in the disc then
𝑧 ≤
1 + 𝛽 𝑝
2𝛽 1 − 𝛼 (𝑛 + 𝑝)
1
𝑛
, 𝑛 = 1, 2, 3, … (4.1)
Proof: Let
𝑓 𝑧 = 𝑧 𝑝
− 𝑎 𝑛+𝑝 𝑧 𝑛+𝑝
∞
𝑛=1
Then,
𝑓′ 𝑧 = 𝑝𝑧 𝑝−1
− 𝑎 𝑛+𝑝 𝑛 + 𝑝 𝑧 𝑛+𝑝−1
∞
𝑛=1
𝑓′′ 𝑧 = 𝑝(𝑝 − 1)𝑧 𝑝−2
− 𝑎 𝑛+𝑝 𝑛 + 𝑝 (𝑛 + 𝑝 − 1)𝑧 𝑛+𝑝−2
∞
𝑛=1
Now,

1 +
𝑧𝑓′′
(𝑧)
𝑓′ (𝑧)
= 1 +
𝑝(𝑝 − 1)𝑧 𝑝−1
− 𝑎 𝑛+𝑝 𝑛 + 𝑝 (𝑛 + 𝑝 − 1)𝑧 𝑛+𝑝−1∞
𝑛=1
𝑝𝑧 𝑝−1 − 𝑎 𝑛+𝑝 𝑛 + 𝑝 𝑧 𝑛+𝑝−1∞
𝑛=1
=
𝑝𝑧 𝑝−1
− 𝑎 𝑛+𝑝 𝑛 + 𝑝 𝑧 𝑛+𝑝−1∞
𝑛=1 + 𝑝(𝑝 − 1)𝑧 𝑝−1
− 𝑎 𝑛+𝑝 𝑛 + 𝑝 (𝑛 + 𝑝 − 1)𝑧 𝑛+𝑝−1∞
𝑛=1
𝑝𝑧 𝑝−1 − 𝑎 𝑛+𝑝 𝑛 + 𝑝 𝑧 𝑛+𝑝−1∞
𝑛=1
=
𝑝2
𝑧 𝑝−1
− 𝑎 𝑛+𝑝 𝑛 + 𝑝 2
𝑧 𝑛+𝑝−1∞
𝑛=1
𝑝𝑧 𝑝−1 − 𝑎 𝑛+𝑝 𝑛 + 𝑝 𝑧 𝑛+𝑝−1∞
𝑛=1
To prove the theorem it is sufficient to show,
1 +
𝑧𝑓′′
(𝑧)
𝑓′ (𝑧)
− 𝑝 ≤ 𝑝
Now,
1 +
𝑧𝑓′′
(𝑧)
𝑓′ (𝑧)
− 𝑝
=
𝑝2
𝑧 𝑝−1
− 𝑎 𝑛+𝑝 𝑛 + 𝑝 2
𝑧 𝑛+𝑝−1∞
𝑛=1
𝑝𝑧 𝑝−1 − 𝑎 𝑛+𝑝 𝑛 + 𝑝 𝑧 𝑛+𝑝−1∞
𝑛=1
− 𝑝
=
𝑝2
𝑧 𝑝−1
− 𝑎 𝑛+𝑝 𝑛 + 𝑝 2
𝑧 𝑛+𝑝−1
− 𝑝2
𝑧 𝑝−1
+ 𝑝 𝑎 𝑛+𝑝 𝑛 + 𝑝 𝑧 𝑛+𝑝−1∞
𝑛=1
∞
𝑛=1
𝑝𝑧 𝑝−1 − 𝑎 𝑛+𝑝 𝑛 + 𝑝 𝑧 𝑛+𝑝−1∞
𝑛=1
=
− 𝑎 𝑛+𝑝 𝑛 𝑛 + 𝑝 𝑧 𝑛+𝑝−1∞
𝑛=1
𝑝𝑧 𝑝−1 − 𝑎 𝑛+𝑝 𝑛 + 𝑝 𝑧 𝑛+𝑝−1∞
𝑛=1
≤
|𝑎 𝑛+𝑝 |𝑛 𝑛 + 𝑝 |𝑧| 𝑛∞
𝑛=1
𝑝 − |𝑎 𝑛+𝑝 | 𝑛 + 𝑝 |𝑧| 𝑛∞
𝑛=1
Thus
1 +
𝑧𝑓′′
(𝑧)
𝑓′ (𝑧)
− 𝑝 ≤ 𝑝
if
|𝑎 𝑛+𝑝 | 𝑛 𝑛 + 𝑝 |𝑧| 𝑛∞
𝑛=1
𝑝 − |𝑎 𝑛+𝑝| 𝑛 + 𝑝 |𝑧| 𝑛∞
𝑛=1
≤ 𝑝
𝑛 𝑛 + 𝑝 𝑎 𝑛+𝑝 𝑧 𝑛
∞
𝑛=1
≤ 𝑝2
− 𝑝 𝑛 + 𝑝 𝑎 𝑛+𝑝 𝑧 𝑛
∞
𝑛=1
𝑛 𝑛 + 𝑝 + 𝑝 𝑛 + 𝑝 𝑎 𝑛+𝑝 𝑧 𝑛
∞
𝑛=1
≤ 𝑝2
𝑛2
+ 2𝑛𝑝 + 𝑝2
𝑎 𝑛+𝑝 z n
∞
𝑛=1
≤ p2
(n + p)2
an+p z n
∞
n=1
≤ p2

n + p
p
2
an+p z n
∞
n=1
≤ 1
But from Theorem 1, we get
n + p (1 + β) an+p
2β(1 − α)p
∞
n=1
≤ 1
hence f(z) is p-valently convex if
n + p
p
2
an+p z n
≤
n + p (1 + β) an+p
2β(1 − α)p
n + p
p
2
z n
≤
n + p (1 + β)
2β(1 − α)p
or
z ≤
1 + β p
2β 1 − α (n + p)
1
n
, n = 1, 2, 3, …
References
[1]. P. L. Duren, Univalent Functions, Grundlehren Math. Wiss., Vol. 259, Springer, New York 1983.
[2]. H. M. Srivastava, M. K. Aouf, A certain derivative operator and its applications to a new class of analytic and multivalent functions
with negative coefficients, J. Math. Anal.Appl.171, (1992),1-13.
[3]. S. Owa, On certain subclasses of analytic p-valent functions, J. Korean Math. Soc. 20, (1983), 41-58.
[4]. R. M. Goel, N. S. Sohi, Multivalent functions with negative coefficients, Indian J. Pure Appl. Math. 12(7), (1981), 844-853.

On Certain Classess of Multivalent Functions

More Related Content

What's hot (20)

Viewers also liked (12)

Similar to On Certain Classess of Multivalent Functions (20)

More from iosrjce (20)

Recently uploaded (20)

On Certain Classess of Multivalent Functions