Scientia Magna, book series, Vol. 3, No. 2

2007·3.3 MB·English

by  Zhang, Wenpeng.

Z W
HANG ENPENG
editor

Scientia Magna
International Book Series
Vol. 3, No. 2

2007
Editor:

Dr. Zhang Wenpeng
Department of Mathematics
Northwest University
Xi’an, Shaanxi, P. R. China

Scientia Magna
- international book series (Vol. 3, No. 2) -

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2007
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ii
Contributing to Scientia Magna book series
Authors of papers in science (mathematics, physics, engineering, philosophy,
psychology, sociology, linguistics) should submit manuscripts to the main editor:
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Shaanxi, P. R. China, E-mail: [email protected].
Associate Editors
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P.R.China, E-mail: [email protected].
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Shaanxi, P.R.China, E-mail: [email protected].
Dr. Xu Zhefeng, Department of Mathematics, Northwest University, Xi’an, Shaanxi,
P.R.China, E-mail: [email protected].
Dr. Zhang Tianping, College of Mathematics and Information Science, Shaanxi
Normal University, Xi’an, Shaanxi, P.R.China, E-mail: [email protected].
iii
Contents
S. M. Khairnar and M. More : Subclass of analytic and univalent functions
in the unit disk 1
Y. Ma, etc. : Large quaternary cyclic codes of length 85 and related quantum
error-correcting 9
J. Chen : Value distribution of the F.Smarandache LCM function 15
M. Bencze : About a chain of inequalities 19
C. Tian and N. Yuan : On the Smarandache LCM dual function 25
J. Caltenco, etc. : Riemannian 4-spaces of class two 29
Y. Xue : On a conjecture involving the function SL∗(n) 41
Y. Wang : Some identities involving the near pseudo Smarandache function 44
A. Gilani and B. Waphare : On pseudo a-ideal of pseudo-BCI algebras 50
J. Chen : An equation involving the F.Smarandache multiplicative function 60
M. Enciso-Aguilar, etc. : Matrix elements for the morse and coulomb interactions 66
Y. Xu and S. Li : On s∗−Supplemented Subgroups of Finite Groups 73
C. Tian : Two equations involving the Smarandache LCM dual function 80
W. Zheng and J. Dou : Periodic solutions of impulsive periodic Competitor-
Competitor-Mutualist system 86
X. Zhang : On the fourth power mean value of B(χ) 93
Z. Lv : On the F.Smarandache function and its mean value 104
J. Ge : Mean value of F. Smarandache LCM function 109
iv
Scientia Magna
Vol. 3 (2007), No. 2, 1-8
Subclass of analytic and univalent functions in
the unit disk
S. M. Khairnar and Meena More
Department of Applied Mathematics, Maharashtra Academy of Engineering
Alandi - 412 105, Pune, M.S., India
Email: smkhairnar123@rediffmail.com
Received January 5, 2007
Abstract InthispaperwehavestudiedafewpropertiesoftheclassH (α,β,γ)andsome
λ,µ
ofitssubclasses. WehavealsoobtainedsomegeneralisationbyusingRuschewayhderivatives
on the lines of K. S. Padmanabhan and R. Bharati [1].
Keywords Ruschewayh derivatives, analytic and univalent functions, quasi-subordinate.
§1. Introduction
Let T denote the class of functions f which are analytic and univalent in the unit disc
E ={z;|z|<1} with f(0)=f(cid:48)(0)−1=0.
Let g(z) and G(z) be analytic in E, then g(z) is said to be quasi-subordinate to G(z),
written as g(z)≺G(z),z ∈E, if there exist functions φ and ψ are analytic in E with |φ(z)|≤
1,|ψ(z)|<1 in E and g(z)=φ(z)G(ψ(z)) in E.
Definition 1. A function h(z) analytic in E with h(0)=1, is said to belong to the class
J(α,β) if
(cid:175) (cid:175)
(cid:175) h(z)−1 (cid:175)
(cid:175) (cid:175)<1, z ∈E, (1)
(cid:175) (cid:175)
βh(z)−[β+(1−α)(1−β)]
where 0≤α<1, 0<β ≤1.
Definition 2. Let H (α,β,γ) denote the class of all functions f ∈T for which
λ,µ
Dλf(z) (Dλf(z))(cid:48)
(1−γ) +γ ∈J(α,β)
Dµg(z) (Dµg(z))(cid:48)
for some g where
(cid:181) (cid:182)
(Dµg(z))(cid:48)
Re z >0
Dµg(z)
and λ≥0, µ≥0, γ ≥0.
Note that the subclass H (0,β,λ) was studied by Padmanabhan and Bharati [1].
1,0
In this paper we investigate a few properties of the class H (α,β,γ) and some of its
λ,µ
subclasses.
2 S.M.KhairnarandMeenaMore No. 2
§2. Lemmas
We need the following some preliminary lemmas for our study.
Lemma 1. If
(cid:88)∞ (cid:88)∞
g(z)= a zn ≺G(z)= A zn,
n n
n=0 n=0
then
(cid:88)k (cid:88)k
|a |2 ≤ |A |2 (k =0,1,2,···).
n n
n=0 n=0
This lemma is due to Robertson [2].
(cid:88)∞
Lemma 2. Let P(z) = 1+ P zn be analytic in E and belongs to J(α,β), then for
n
n=1
n≥1, |P |≤(β−1)(1−α).
n
Proof. Condition (1) is equivalent to
1+[β+(1−α)(1−β)]ψ(z)
P(z)= ,
1+βψ(z)
where ψ(0)=0 and |ψ(z)|<1 in E. After simplification we have
(cid:88)∞ (cid:88)∞
P Zn =[(1−α)(1−β)−β P zn]ψ(z).
n n
n=1 n=1
By the application of Lemma 1, we obtain
(cid:88)n n(cid:88)−1
|P |2 ≤(1−α)2(1−β)2+β2 |P |2.
k k
k=1 k=1
So
n(cid:88)−1
|P |2 ≤(1−α)2(1−β)2−(1−β)2 |P |2.
n k
k+1
Hence |P |≤(1−α)(β−1).
n
Lemma 3. Let P(z) and P (z) belongs to J(α,β) then
1
(1−ρ)P(z)+ρP (z)∈J(α,β),
1
where 0<ρ<1. Itiseasytoseethatconditionψ ∈J(α,β)isequivalentto|ψ−b|<c,where
1−[β2+β(1−α)(1−β)]
b= ,
1−β2
and
(1−α)(1−β) (1−α)
c= = .
(1−β2) (1+β)
So it is sufficient to show that
|(1−ρ)P(z)+ρP (z)−b|≤(1−ρ)|P(z)−b|+ρ|P (z)−b|<c.
1 1
The last inequality is true, since P(z),P (z)∈J(α,β).
1
Vol. 3 Subclassofanalyticandunivalentfunctionsintheunitdisk 3
Hence the proof is complete.
Lemma 4. Let γ ≥ 0 and V(z) be a starlike function in the unit disk E. Let U(z) be
U(z)
analytic in E with U(0)=V(0)=0=U(cid:48)(0)−1. Then ∈J(α,β), whenever
V(z)
U(z) U(cid:48)(z)
(1−γ) +γ ∈J(α,β).
V(z) V(cid:48)(z)
Proof. Suppose ψ(z) be a function in E that is defined by
U(z) 1+[β+(1−α)(1−β)]ψ(z)
= ,
V(z) 1+βψ(z)
where 0 ≤ α ≤ 1, 0 < β ≤ 1. It is clear that ψ(z) is analytic, ψ(0) = 0. We shall prove
|ψ(z)| < 1 in E. For, if not, there exists z ∈ E, by Jack’s Lemma [2], such that |ψ(z )| = 1
0 0
and zψ(cid:48)(z )=kψ(z ), k ≥1.
0 0
U(z) U(cid:48)(z)
Let Q(z)=(1−γ) +γ , then on simplification we have
V(z) V(cid:48)(z)
U(z ) kγV(z )(1−β)(1−α)ψ(z )
Q(z )= 0 + 0 0 .
0 V(z ) zV(cid:48)(z ) (1+βψ(z ))2
0 0 0
We consider
(cid:175) (cid:179) (cid:180)(cid:175)
(cid:175)(cid:175)(cid:175) Q(z0)−1 (cid:175)(cid:175)(cid:175) = (cid:175)(cid:175)(cid:175)(1−β1+)(β1−ψ(αz)0ψ)(z0)(cid:179) 1+ kzγγV(cid:48)((zz00)) + 1+βψ1(cid:180)(z0) (cid:175)(cid:175)(cid:175)
(cid:175) (cid:175) (cid:175) (cid:175)
βQ(z0)−[β+(1−α)(1−β)] (cid:175) (1−α)(1−β) 1− kγV(z0) βψ(z0) (cid:175)
(cid:175) 1+βψ(z0) z(cid:175)V(cid:48)(z0) 1+βψ(z0)
(cid:175)(cid:175)1+ kγVz0) (cid:175)(cid:175)
= (cid:175) zV(cid:48)(z0)(1+βψ(z0))(cid:175).
(cid:175) (cid:175)
(cid:175)1− kγV(z0)βψ(z0) (cid:175)
zV(cid:48)(z0)(1+βψ(z0))
Now (cid:175) (cid:175) (cid:175) (cid:175)
(cid:175) γkV(z ) (cid:175) (cid:175) kγV(z )βψ(z ) (cid:175)
(cid:175)1+ 0 (cid:175)>(cid:175)1− 0 0 (cid:175),
(cid:175) zV(cid:48)(z )(1+βψ(z ))(cid:175) (cid:175) zV(cid:48)(z )(1+βψ(z ))(cid:175)
0 0 0 0
provided
|1+D(z )|2 >|1−βψ( )D(z )|2,
0 0 0
where
γkV(z )
D(z )= 0 .
0 zV(cid:48)(z )(1+βψz ))
0 0
This condition reduces to the following
1+|D(z )|2+2ReD(z )>1+β2|D(z )|2−2Re(βψ(z )D(z )),
0 0 0 0 0
or equivalently
V(z )
(1−β2)|D(z )|2+2Reγk 0 >0.
0 zV(cid:48)(z )
0
But this is true since V is starlike and γ ≥0, k ≥0. Hence
(cid:175) (cid:175)
(cid:175) Q(z )−1 (cid:175)
(cid:175) 0 (cid:175)>1,
(cid:175) (cid:175)
βQ(z )−[β+(1−α)(1−β)]
0
which is contradiction with hypothesis.
This proves the lemma.
4 S.M.KhairnarandMeenaMore No. 2
§3. Properties of the Class H (α,β,γ)
λ,µ
Theorem 1. If f ∈H (α,β,γ), then f ∈H (α,β,0).
λ,µ λ,µ
Proof. It is sufficient to choose V(z) = Vµg(z) and U(z) = Dλf(z) in Lemma 4. This
leads to the required result.
f(cid:48)(z)
Corollary 1. If 0 ≤ α < 1, 0 ≤ β < 1 and f, g ∈ T, then ∈ J(α,β) implies
g(cid:48)(z)
f(z)
∈J(α,β) whenever g(z) is starlike in E. The special case β =1, α=0 is obtained by [3].
g(z
Proof. Take µ=1, λ=0, µ=0 in Theorem 1.
Corollary 2. If 0≤α<1, 0≤β <1 and f, g ∈T, then
f(cid:48)(z)+zf(cid:48)(cid:48)(z)
∈J(α,β),
g(cid:48)(z)+zg(cid:48)(cid:48)(z)
f(cid:48)(z)
implies ∈J(α,β), whenever g is convex in E.
g(cid:48)(z)
Proof. Take µ=1,λ=1, µ=1 in Theorem 1.
Corollary 3. If 0≤α<1 and f, g ∈T, then
f(cid:48)(z)+zf(cid:48)(cid:48)(z) f(cid:48)(z)
∈J(α,β)=⇒z ∈J(α,β),
g(cid:48)(z) g(z)
whenever g is starlike in E.
Proof. Take γ =1, λ=1, µ=0 in Theorem 1.
Theorem 2. For 0≤γ <γ, H (α,β,γ)⊂H (α,β,γ ).
1 λ,µ λ,µ 1
Proof. If γ = 0, the result follows from Theorem 1. Assume therefore γ (cid:54)= 0 and
1 1
f ∈H (α,β,γ). Then there exist a starlike function (Vµ) in E such that
λ,µ g
Vλf(z) (Vλf(z))(cid:48)
P(γ,f)=(1−γ) +γ ∈J(α,β),
Uµg(z) (Vµg(z))(cid:48)
Vλf(z)
and ∈J(α,β). Now the result follows from the identity
Vµg(z)
(cid:181) (cid:182)
γ γ Vλf(z)
P(γ,f)= 1P(γ,f)+ 1− 1 , γ <γ,
γ γ Vµg(z) 1
and the lemma 3.
Theorem 3. A function f is in H (α,β,γ), if and only if there exist a function g in E
(cid:181) (cid:182) λ,µ
z(Dµg(z))(cid:48)
with Re > 0, g(0) = g(cid:48)(0)−1 = 0, and analytic function P,P(0) = 1, belongs
Dµg(z)
to H (α,β,γ) such that
λ,µ
(cid:90)
1 z
Dλf(z)= [Dµg(t)]γ1−1(Dµg(t))(cid:48)P(t)dt,γ >0.
γ[Dµg(z)]γ1−1 0
If γ =0 then Dλf(z)=Dµg(z)P(z).
(cid:181) (cid:182)
z(Dµg(z))(cid:48)
Proof. Let f ∈ H (α,β,γ), then there exist a function g where Re > 0
λ,µ Dµg(z)
in E and
Dλf(z) (Dλf(z))(cid:48)
P(z)=(1−γ) +γ ∈H(α,β,γ). (2)
dµg(z) (Dµg(z))(cid:48)