IJMMS 26:11 (2001) 701–706
PII. S0161171201006020
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ANALYTIC FUNCTIONS OF NON-BAZILEVIČ
TYPE AND STARLIKENESS
MILUTIN OBRADOVIĆ and SHIGEYOSHI OWA
(Received 23 October 2000)
Abstract. Two classes Ꮾ̄n (µ, α, λ) and ᏼ̄n (µ, α, λ) of analytic functions which are not
Bazilevič type in the open unit disk U are introduced. The object of the present paper is to
consider the starlikeness of functions belonging to the classes Ꮾ̄n (µ, α, λ) and ᏼ̄n (µ, α, λ).
2000 Mathematics Subject Classification. 30C45.
1. Introduction. Let U = {z ∈ C : |z| < 1} denote the open unit disk in the complex
plane C. For n 1, we define
Ꮽn = f : f (z) = z +
∞
j
aj z analytic in U .
(1.1)
n+1
Also, we need the following notations and definitions. Let
zf (z)
> 0, z ∈ U
᏿∗ = f ∈ Ꮽ1 : Re
f (z)
(1.2)
be the class of starlike functions (with respect to the origin) in U, and let
zf (z)
< λ, 0 < λ 1, z ∈ U
−
1
᐀λ = f ∈ Ꮽ1 : f (z)
be the subclass of ᏿∗ . Further we define
f (z) µ−1
Ꮾ(µ, λ) = f ∈ Ꮽ1 : f (z)
− 1 < λ, µ > 0, 0 < λ 1, z ∈ U
z
(1.3)
(1.4)
which is the subclass of Bazilevič class of univalent functions (cf. [1]). Ponnusamy [8]
has considered the starlikeness and other properties of functions f (z) in the class
Ꮾ(µ, λ). For negative µ, that is, for −1 < µ < 0, which is better to write (with 0 < µ < 1)
in the form
1+µ
z
f (z)
− 1 < λ, 0 < µ < 1, 0 < λ < 1, z ∈ U , (1.5)
Ꮾ̄(µ, λ) = f ∈ Ꮽ1 : f (z)
we obtain the class which was considered earlier by Obradović [3, 4], Obradović and
Owa [5], and Obradović and Tuneski [6].
For the limit case µ = 0, this class becomes the class ᐀λ . When µ = 1, this class
702
M. OBRADOVIĆ AND S. OWA
becomes the class of univalent functions f (z) satisfying
2 z f (z)
f (z)2 − 1 < λ,
(1.6)
which was studied by Ozaki and Nunokawa [7].
Next, for α = (α1 , α2 , . . . , αk ) ∈ Rk with αj ∈ R, we define the operator Dα by
D α = 1 + α1 z
d2
dk
d
+ α2 z 2
+ · · · + αk z k
,
2
dz
dz
dzk
(1.7)
and, by virtue of the operator Dα , the subclasses
1+µ z
Ꮾ̄n (µ, α, λ) = f ∈ Ꮽn : Dα f (z)
−1 < λ, 0 < µ < 1, 0 < λ < 1, z ∈ U ,
f (z)
(1.8)
1+µ z
< λ, 0 < µ < 1, 0 < λ < 1, z ∈ U
ᏼ̄n (µ, α, λ) = f ∈ Ꮽn : D
f (z)
α
f (z)
(1.9)
of Ꮽn .
Samaris [9] has investigated the appropriate classes for the case (1.4), and has obtained results which are stronger than those given earlier and in several cases sharp
ones. By using the method by Samaris [9], we will generalize some results given in
[3, 4], and we will obtain some new results. We also note that we cannot directly apply
some nice estimates given by Samaris [9].
For α = (α1 , α2 , . . . , αk ) ∈ Rk , we define the polynomial Pα (x) by
Pα (x) = 1 + α1 x + α2 x(x − 1) + · · · + αk x(x − 1) · · · (x − k + 1).
(1.10)
In this paper, we will use the classes such that α = 0(Pα (x) = 1) or Pα (x) has nonpositive real zeros given by ρj (j = 1, 2, 3, . . . , k). In this case, we can write
Pα (x) = αk
k
x − ρj
(1.11)
j=1
with
αk
k
− ρj = 1.
(1.12)
j=1
Further, for t = (t1 , t2 , . . . , tk ) ∈ (0, 1)k such that tj ∈ (0, 1), we denote tα by
−1/ρ1 −1/ρ2
t2
t α = t1
−1/ρk
· · · tk
=
k
−1/ρj
tj
.
(1.13)
j=1
If Pα (x) = 1, then we define tα = 1. Also if n = 1, 2, 3, . . . , then we denote by ᐃα the
class of analytic functions w(z) in U for which |w(z)| |z|n (z ∈ U).
2. Starlikeness of the classes Ꮾ̄n (µ, α, λ) and ᏼ̄n (µ, α, λ). Our first result is contained in the following theorem.
ANALYTIC FUNCTIONS OF NON-BAZILEVIČ TYPE AND STARLIKENESS
703
Theorem 2.1. Let Ꮾ̄n (µ, α, λ) be the class defined by (1.8) for which (n−µ)Pα (n)−
λn > 0 (n = 1, 2, 3, . . .), 0 < µ < 1.
(i) If λn/((n − µ)Pα (n) − λµ) r , then Ꮾ̄n (µ, α, λ) ⊂ ᐀r ,
(ii) if λ Pα (n)((n − µ)/(n + µ)), then Ꮾ̄n (µ, α, λ) ⊂ ᐀1 .
Proof. From
1+µ ∞
z
Dα f (z)
wn zn
= 1 + λw(z) = 1 + λ
f (z)
n=1
we obtain
w ∈ ᐃn ,
1+µ
∞
z
zn
,
= 1+λ
wn
f (z)
f (z)
Pα (n)
n=1
Since
1
=
Pα (n)
z
f (z)
µ
= 1−λ
n=1
[0,1]k
∞
tαn ,
wn
(2.2)
zn
µ
.
n − µ Pα (n)
µ
=
Pα (n)(n − µ)
n/µ−2
[0,1]k+1
(2.1)
tαn tk+1
,
(2.3)
using (2.2), we have
1+µ
z
f (z)
= 1+λ
w tα z ,
k
f (z)
[0,1]
µ
z
1/µ −2
= 1−λ
w tα tk+1 z tk+1
.
f (z)
[0,1]k+1
(2.4)
From (2.4), we easily obtain that
1 + λ [0,1]k w tα z
zf (z)
=
1/µ −2 ,
f (z)
1 − λ [0,1]k+1 w tα tk+1 z tk+1
(2.5)
and from here
1/µ t −2
w tα z +
zf (z)
[0,1]k+1 w tα tk+1 z
k+1
λ [0,1]k
−
1
f (z)
1/µ
−2
1 − λ [0,1]k+1 w tα tk+1 z tk+1
<λ
n
n n/µ−2
[0,1]k tα + [0,1]k+1 tα tk+1
n/µ−2
1 − λ [0,1]k+1 tαn tk+1
(2.6)
λn
,
=
(n − µ)Pα (n) − λµ
from which the conclusion of the theorem easily follows.
Remark 2.2. For n = 1 and Pα = 1 from Theorem 2.1, we have the result given by
Obradović [3].
Remark 2.3. From (2.5), similar to the proof of Theorem 2.1, for the class
Ꮾ̄n (µ, α, λ) we have
zf (z)
λn
|1 − t| +
−
t
f (z)
(n − µ)Pα (n) − λµ
(t > 0).
(2.7)
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M. OBRADOVIĆ AND S. OWA
If λ1 = λn/((n − µ)Pα (n) − λµ) < 1 and t (1 + λ1 )/2, then we have
Ꮾ̄n (µ, α, λ) ⊂ ᏿∗ t ,
where
᏿∗
t
zf (z)
= f ∈ Ꮽ1 : f (z) − t < t, z ∈ U .
(2.8)
(2.9)
Especially, for n = 1 and α = 0, we have
Ꮾ̄(µ, λ) ⊂ ᏿∗ t
(2.10)
for λ/(1 − µ − λ) < 1 and t 1/2 + λ/(2(1 − µ − λ)).
In the next theorem for the class Ꮾ̄(µ, λ) ∩ Ꮽn = Ꮾ̄n (µ, 0, λ), we will prove that the
appropriate results are the best possible.
Theorem 2.4. Let Ꮾ̄(µ, λ) ∩ Ꮽn be the class for which n − µ − λµ > 0. Then
(i) Ꮾ̄(µ, λ) ∩ Ꮽn ⊂ ᐀r if and only if λn/(n − µ − λµ) r and
(ii) Ꮾ̄(µ, λ) ∩ Ꮽn ⊂ ᏿∗ if and only if λ (n − µ)/ (n − µ)2 + µ 2 .
Proof. For t1 ∈ R and t2 ∈ R, with the lemma by Fournier [2], there exists a sequence of functions φk (z) analytic in the closed unit disk Ū such that |φk (z)| |z|,
limk→∞ φk (z) = zeit1 , limk→∞ φk (1) = eit2 uniformly on compact subsets of U. If wk (z) =
zn−1 φk (z), then we consider the sequence fk (z) ∈ Ꮾ̄(µ, λ) ∩ Ꮽn which is given by
fk (z)
or
z
fk (z)
1+µ
= 1 + λwk (z),
(2.11)
zfk (z)
1 + λwk (z)
.
=
1
fk (z)
1 − λ 0 wk t 1/µ z t −2 dt
(2.12)
Since limk→∞ wk (z) = zn eit1 and limk→∞ wk (1) = eit2 , we see that
zfk (z)
1 + λeit1
=
.
1 − λ µ/(n − µ) eit2
k→∞ z→1 fk (z)
lim lim
(2.13)
For t1 = 0 and t2 = 0, we get
zfk (z)
λn
=
−
1
lim lim n − µ − λµ .
fk (z)
k→∞ z→1
(2.14)
Let 0 < λ < 1 and q1 ∈ [0, π /2], q2 ∈ [0, π /2] such that sin q1 = λ, sin q2 = λµ/(n−µ).
If we choose t1 = q1 + π /2, t2 = π /2 − q2 , then we obtain
arg 1 + λeit1 = q1 ,
and therefore
λµ it2
e
arg 1 −
= −q2 ,
n−µ
zfk (z) = q1 + q2 .
lim lim arg
fk (z) k→∞ z→1 (2.15)
(2.16)
ANALYTIC FUNCTIONS OF NON-BAZILEVIČ TYPE AND STARLIKENESS
705
From cos(q1 + q2 ) 0 or sin2 q1 + sin2 q2 − 1 0, we have the statement (ii) (one
part) of the theorem. We note that the “if” part of the theorem follows from the result
of Theorem 2.1 and from the result given by Obradović and Owa [5].
Theorem 2.5. Let ᏼ̄n (µ, α, λ) be the class defined by (1.9) such that n(n−µ)Pα (n−1)
−λn > 0 (n = 1, 2, 3, . . .), 0 < µ < 1. If
λn
r,
n(n − µ)Pα (n − 1) − λµ
(2.17)
ᏼ̄n (µ, α, λ) ⊂ ᐀r .
(2.18)
then
Proof. The proof of this theorem is similar to that of Theorem 2.1. By virtue of
1+µ z
f (z)
= λw(z) w ∈ ᐃn−1 ,
f (z)
Dα
we have
(2.19)
1+µ z
f (z)
=λ
w tα z ,
f (z)
[0,1]k
1+µ
z
f (z)
= 1+λ
w tα tk+1 z z.
k+1
f (z)
[0,1]
(2.20)
From the last relation, we see that
and so
z
f (z)
µ
= 1−λ
[0,1]k+2
1/µ −2
w tα tk+1 tk+2 z tk+2
z,
1 + λ [0,1]k+1 w tα tk+1 z z
zf (z)
=
.
1/µ −2
f (z)
1 − λ [0,1]k+2 w tα tk+1 tk+2 z tk+2
z
(2.21)
(2.22)
By using (2.22), as in the proof of Theorem 2.1, we easily obtain that
zf (z)
λn
−
1
n(n − µ)P (n − 1) − λµ .
f (z)
α
(2.23)
The statement of this theorem follows from the above inequality.
Finally we give the following example of the theorem.
Example 2.6. For Pα = 1 + t and 0 < µ < 1, 0 < λ < 1, if f (z) ∈ Ꮽn satisfies
1+µ 1+µ z
z
f (z)
<λ
+ z f (z)
f (z)
f (z)
implies that
zf (z)
λn
−
1
n2 (n − µ) − λµ .
f (z)
(2.24)
(2.25)
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M. OBRADOVIĆ AND S. OWA
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Milutin Obradović: Department of Mathematics, Faculty of Technology and Metallurgy, University of Belgrade, 4 Karnegijeva Street, 11000 Belgrade, Yugoslavia
E-mail address: obrad@elab.tmf.bg.ac.yu
Shigeyoshi Owa: Department of Mathematics, Kinki University, Higashi-Osaka,
Osaka 577-8502, Japan
E-mail address: owa@math.kindai.ac.jp