http://dx.doi.org/10.12988/ijma.2013.35122
Coefficient Inequalities for Certain Subclass of p− Valent Functions of Complex Order
Loh Part Leam and Aini Janteng
School of Science and Technology, Universiti Malaysia Sabah Jalan UMS, 88400 Kota Kinabalu, Sabah, Malaysia
travis−[email protected], aini−[email protected]
Copyright c 2013 Loh Part Leam and Aini Janteng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
This paper introduces a new subclass of p−valent functions of complex order which is denoted bySp(b, λ, α) with 0≤λ≤1 andα >1. The coefficient inequality and Fekete-Szeg¨o inequality for functions f belongs to this class are obtained.
Mathematics Subject Classification: Primary 30C45
Keywords: p−valent functions, complex order, coefficient inequality, Fekete-Szeg¨o inequality
1 Introduction
LetAp be the class of p−valent functionsf(z) of the form f(z) =zp+
∞ n=1
ap+nzp+n (1)
which are analytic in the open unit disc D={z:|z|<1} andp∈N={1,2,3, ...}.
In 2012, Sharma and Saroja introduced some subclasses ofp−valent function of complex order and obtained the coefficient inequality and Fekete-Szeg¨oinequality for functions in these classes.
Definition 1.1 ([3]) Let b be a non-zero complex number and α > 1. A function f(z) of the form (1) is said to be in the class Mp(b, α) if
Re
1 +1 b
zf(z) f(z) −1
< α, z∈D.
It is noted that
Mp(1, α) =Mp(α) defined by Srivastava and Owa in [4]
M1(1, α) =M(α) defined by Owa and Nishiwaki in [1]
Definition 1.2 ([3]) Let b be a non-zero complex number and α > 1. A function f(z) of the form (1) is said to be in the class Np(b, α) if
Re
1 +1 b
zf(z) f(z)
< α, z∈D.
It is noted that
Np(1, α) =Np(α) defined by Srivastava and Owa in [4]
N1(1, α) =N(α) defined by Owa and Nishiwaki in [1]
In this paper, we define a subclass of p−valent functions of complex order. We determine the coefficient inequality and Fekete-Szeg¨o inequality for functions f belongs to this class.
Definition 1.3 Let b be a non-zero complex number and α > 1. A function f(z) of the form (1) is said to be in the class Sp(b, λ, α) if and only if
Re
1 + 1 b
λz2f(z) +zf(z) λzf(z) + (1−λ)f(z) −1
< α, 0≤λ≤1, z∈D. (2) It is noted that
Sp(b,0, α) =Mp(b, α) and Sp(b,1, α) =Np(b, α) defined by Sharma and Saroja in [3]
Sp(1,0, α) =Mp(α) and Sp(1,1, α) =Np(α) defined by Srivastava and Owa in [4]
2 Preliminary Results
To prove our results, we require the following lemma.
Lemma 2.1 ([2])Ifp(z) = 1 +c1z+c2z2+... is a function with positive real part andp(0) = 1 then for any complex number v, we have
|c2−vc21| ≤2Max{1,|2v−1|}
The result is sharp for the functions
p(z) = 1 +z2
1−z2 and p(z) =1 +z 1−z.
3 Main Results
The coefficient inequality for functions f ∈Sp(b, λ, α) is given by the following theorem.
Theorem 3.1 Let f(z)∈Sp(b, λ, α),0≤λ≤1, α >1 and be given by (1) then
|ap+n| ≤ λp+ (1−λ) n![λ(p+n) + (1−λ)]
n−1
j=0
[2[|b|(α−1) + (p−1)] +j] (3)
Proof: Sincef(z)∈Sp(b, λ, α) then from the Definition 1.3, we have Re
1 + 1
b
λz2f(z) +zf(z) λzf(z) + (1−λ)f(z) −1
< α.
Define a functionp(z) such that
p(z) = α− 1 +1b
λz2f(z)+zf(z) λzf(z)+(1−λ)f(z) −1 α−
1 + 1b(p−1) = 1 + ∞ n=1
cnzn, z∈D (4)
Here p(z) is a function with positive real part withp(0) = 1.
Replacing f(z), zf(z), z2f(z) with their equivalent expressions on both sides, we get
1 + ∞ n=1
cnzn
[b(α−1)−(p−1)]
zp+
∞ n=1
ap+nzp+n
= [b(α−1) + 1]
zp+
∞ n=1
ap+nzp+n
−
pzp+ ∞ n=1
(p+n)ap+nzp+n
(5)
Comparing the coeffiecient ofzp+n on both sides of equations (5), we get
−n[λ(p+n) + (1−λ)]ap+n
= [b(α−1)−(p−1)]
[λp+ (1−λ)]cn+ [λ(p+ 1) + (1−λ)]ap+1cn−1
+ [λ(p+ 2) + (1−λ)]ap+2cn−2+· · ·+ [λ(p+n−2) + (1−λ)]ap+n−2c2
+ [λ(p+n−1) + (1−λ)]ap+n−1c1
(6)
Taking modulus on both sides of (6) and applying|cn| ≤2 ∀n≥1, we get
|ap+n| ≤2
|b|(α−1) + (p−1)
n(λ(p+n) + (1−λ)) [λp+ (1−λ)] + [λ(p+ 1) + (1−λ)]|ap+1| + [λ(p+ 2) + (1−λ)]|ap+2|+· · ·+ [λ(p+n−2) + (1−λ)]|ap+n−2|
+ [λ(p+n−1) + (1−λ)]|ap+n−1| (7)
Forn= 1
|ap+1| ≤ λp+ (1−λ) λ(p+ 1) + (1−λ)
2[|b|(α−1) + (p−1)]
Thus the result (3) is true for n= 1.
Forn= 2
|ap+2| ≤ λp+ (1−λ) 2![λ(p+ 2) + (1−λ)]
1 j=0
2[|b|(α−1) + (p−1)] +j
Thus the result (3) holds true for n=2.
Suppose the result (3) is true for n=k Now for n=k+1, we have
|ap+k+1| ≤
[λp+ (1−λ)]
[λ(p+k+ 1) + (1−λ)]
2
[|b|(α−1) + (p−1)]
(k+ 1)
1 + 2[|b|(α−1) + (p−1)]
+
2
[|b|(α−1) + (p−1)]
2
1 + 2[|b|(α−1) + (p−1)]
+· · ·
+1 k!
k−1
j=0
2[|b|(α−1) + (p−1)] +j
|ap+k+1| ≤ λp+ (1−λ)
(k+ 1)![λ(p+k+ 1) + (1−λ)]
k j=0
2[|b|(α−1) + (p−1)] +j
Thus the result (3) is true for n=k+ 1.
By mathematical induction the result (3) is true for all values of n. This completes the proof of Theorem 3.1.
Next, the result of Fekete-Szeg¨o inequality is given by following theorem.
Theorem 3.2 If f(z)∈Sp(b, λ, α) then for any complex number μ
|ap+2−μa2p+1| ≤ [λp+ (1−λ)][|b|(α−1) + (p−1)]
[λ(p+ 2) + (1−λ)]
max
1,
2[b(1−α) + (p−1)]
2μ(λp+ (1−λ))
λ(p+ 2) + (1−λ) [λ(p+ 1) + (1−λ)]2
−1
−1 The result obtained is sharp.
Proof: Sincef(z)∈Sp(b, λ, α) then from (6), we obtain
ap+1 = [λp+ (1−λ)][b(1−α) + (p−1)]c1
λ(p+ 1) + (1−λ) and
ap+2 = [λp+ (1−λ)][b(1−α) + (p−1)]
2[λ(p+ 2) + (1−λ)]
c2+ [b(1−α) + (p−1)]c21
For any complex number μ
ap+2−μa2p+1
= [λp+ (1−λ)][b(1−α) + (p−1)]
2[λ(p+ 2) + (1−λ)]
c2+ [b(1−α) + (p−1)]c21
−μ
[b(1−α) + (p−1)]2[λ(p) + (1−λ)]2c21 [λ(p+ 1) + (1−λ)]2
= [λp+ (1−λ)][b(1−α) + (p−1)]
2[λ(p+ 2) + (1−λ)]
c2−[b(1−α) + (p−1)]
2μ[λp+ (1−λ)]
λ(p+ 2) + (1−λ) (λ(p+ 1) + (1−λ))2
−1
c21
= [λp+ (1−λ)][b(1−α) + (p−1)]
2[λ(p+ 2) + (1−λ)]
c2−vc21
where
v= [b(1−α) + (p−1)]
2μ
λp+ (1−λ) λ(p+ 2) + (1−λ) [λ(p+ 1) + (1−λ)]2
−1
Taking modulus on both sides and applying Lemma 2.1, we get
|ap+2−μa2p+1|
≤
[λp+ (1−λ)][b(1−α) + (p−1)]
2[λ(p+ 2) + (1−λ)]
c2−vc21
≤ [λp+ (1−λ)][|b|(α−1) + (p−1)]
[λ(p+ 2) + (1−λ)] max{1,|2v−1|}
≤ [λp+ (1−λ)][|b|(α−1) + (p−1)]
[λ(p+ 2) + (1−λ)]
max
1,
2[b(1−α) + (p−1)]
2μ(λp+ (1−λ))
λ(p+ 2) + (1−λ) [λ(p+ 1) + (1−λ)]2
−1
−1
This proves Theorem 3.2. The result is sharp.
|ap+2−μa2p+1|
= [λp+ (1−λ)][|b|(α−1) + (p−1)]
[λ(p+ 2) + (1−λ)] if p(z) = 1 +z2 1−z2 and
|ap+2−μa2p+1|
= [λp+ (1−λ)][|b|(α−1) + (p−1)]
[λ(p+ 2) + (1−λ)]
2[b(1−α) + (p−1)]
2μ(λp+ (1−λ))
λ(p+ 2) + (1−λ) [λ(p+ 1) + (1−λ)]2
−1
−1
if p(z) = 1 +z 1−z
By takingλ= 0 in Theorem 3.1 and Theorem 3.2, we obtain the following corollaries respectively Corollary 3.1 ([3])If f(z) ∈Mp(b, α) then
|an+p| ≤ 1 n!
n−1
j=0
[2[|b|(α−1) + (p−1)] +j]
Corollary 3.2 ([3])If f(z) ∈Mp(b, α) then for any complex number μ, we have
|ap+2−μa2p+1| ≤[|b|(α−1) + (p−1)]max{1,|2[[b(1−α) + (p−1)][2μ−1]]−1|}
The result is sharp.
By takingλ= 1 in Theorem 3.1 and Theorem 3.2, we obtain the following corollaries respectively Corollary 3.3 ([3])If f(z) ∈Np(b, α) then
|an+p| ≤ p n!(n+p)
n−1
j=0
[2[|b|(α−1) + (p−1)] +j]
Corollary 3.4 ([3])If f(z) ∈Np(b, α) then for any complex number μ, we have
|ap+2−μa2p+1| ≤ p
(p+ 2)[|b|(α−1) + (p−1)]max
1,
2[[b(1−α) + (p−1)]
2μp
p+ 2 (p+ 1)2
−1
−1
The result is sharp.
ACKNOWLEDGEMENTS. This work was supported by FRG0268-ST-2/2010 Grant, Malaysia. The authors express their gratitude to the referee for his valuable comments.
References
[1] Owa, S. and Nishiwaki, J., Coefficients estimates for certain classes of analytic functions, JIPAM,3(5)(2002).
[2] Ravichandran, V., Polatoglu, Y., Bolcol, M. and Sen, A., Certain subclasses of starlike and convex functions of complex order.Hacettepe Journal of Mathematics and Statistics, (34)(2005):9-15.
[3] Sharma, R.B. and Saroja, K., Coefficient inequalities for certain subclasses ofp−valent func- tions,International Journal of Mathematical and Engineering and Science1(8)(2012):13-25.
[4] Srivastava, H.M. and Owa, S., Some generalized convolution properties associated with certain subclasses of analytic fucntions,J. Ineq. Pure Appl. Math3(3)(2002)
Received: May 21, 2013
