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Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2007, Article ID 34017, 7 pagesdoi:10.1155/2007/34017
Research ArticleFurther Properties of β-Pascu Convex Functions of Order α
H. Özlem Güney and Grigore Stefan Sâlâgean
Received 2 March 2007; Accepted 17 April 2007
Recommended by Narendra K. Govil
We obtain several further properties of β-Pascu convex functions of order α which wererecently introduced and studied by Ali et al. in (2006).
Copyright © 2007 H. Ö. Güney and G. S. Sâlâgean. This is an open access article distribu-ted under the Creative Commons Attribution License, which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properlycited.
1. Introduction
LetN := {1,2, . . .}, for m, p ∈N, m≥ p+ 1, let �(p,m) be the class of all p-valent analyticfunctions f (z)= zp +∑∞n=manzn defined on the open unit disk U= {z ∈ C : |z| < 1} andlet � :=�(1,2).
Let �(p,m) be the subclass of �(p,m) consisting of functions of the form
f (z)= zp−∞∑
n=manz
n, an ≥ 0 for n≥m, (1.1)
and let � :=�(1,2).A function f ∈A(p,m) is β-Pascu convex function of order α if
1p
Re
{(1−β)z f ′(z) + (β/p)z(z f ′(z))′
(1−β) f (z) + (β/p)z f ′(z)
}
> α (β ≥ 0, 0≤ α < 1). (1.2)
We denote by ���(p,m,α,β) the subclass of �(p,m) consisting of β-Pascu convex func-tion of order α. Clearly, ��∗(α) :=���(1,2,α,0) is the class of starlike functions withnegative coefficients of order α and ��(α) :=���(1,2,α,1) is the class of convex func-tions with negative coefficients of order α (studied by Silverman [1]).
For the class ���(p,m,α,β), the following characterization was given by Ali et al. [2].
2 International Journal of Mathematics and Mathematical Sciences
Lemma 1.1. Let the function f be defined by (1.1). Then f is in the class ���(p,m,α,β) ifand only if
∞∑
n=m(n− pα)[(1−β)p+βn]an ≤ p2(1−α). (1.3)
The result is sharp.
Lemma 1.2. Let f (z) be given by (1.1). If f ∈���(p,m,α,β), then
an ≤ p2(1−α)
(n− pα)[(1−β)p+βn] (1.4)
with equality only for functions of the form
fn(z)= zp− p2(1−α)
(n− pα)[(1−β)p+βn]zn. (1.5)
Many interesting properties such as coefficient estimate and distortion theorems forthe class ���(p,m,α,β) were given by Ali et al. [2]. In the present sequel to these earlierworks, we will derive several interesting properties and characteristic of the δ-neighborhood associated with the class ���(p,m,α,β).
2. Integral properties of the class ���(p,m,α,β)
We recall the following definition of integral operator before we give integral propertiesof the class ���(p,m,α,β).
Let �c : �(p,m) → �(p,m) be integral operator defined by g = �c( f ), where c ∈(−p,∞), f ∈�(p,m) and
g(z)= c+ pzc
∫ z
0tc−1 f (t)dt. (2.1)
We note that if f ∈�(p,m) is a function of the form (1.1), then
g(z)=�c( f )(z)= zp−∞∑
n=m
c+ pc+n
anzn. (2.2)
Theorem 2.1. Let p,m ∈ N, m ≥ p + 1, α ∈ [0,1), β ∈ [0,∞), and c ∈ (−p,∞). If f ∈���(p,m,α,β) and g =�c( f ), then g ∈���(p,m,λ,β), where
λ= λ(p,m,α,c)= 1− (1−α)(c+ p)(m− p)(m− pα)(c+m)− (1−α)(c+ p)p (2.3)
and α < λ. The result is sharp.
Proof. From Lemma 1.1 and (2.2), we have g ∈���(p,m,λ,β) if and only if∞∑
n=m
(n− pλ)[(1−β)p+βn](c+ p)p2(1− λ)(c+n) an ≤ 1. (2.4)
H. Ö. Güney and G. S. Sâlâgean 3
We find the largest λ such that (2.4) holds. We note that the inequalities
(n− pλ)[(1−β)p+βn](c+ p)p2(1− λ)(c+n) ≤
(n− pα)[(1−β)p+βn]p2(1−α) (2.5)
imply (2.4), because f ∈ ���(p,m,α,β) and satisfy (1.3). But inequalities (2.5) areequivalent to
(n− pλ)(c+ p)(1− λ)(c+n) ≤
(n− pα)(1−α) . (2.6)
Since (n− pα) > p(1−α) and c+n > c+ p, we obtain λ≤ λ(p,n,α,c), where
λ(p,n,α,c)= (n− pα)(c+n)− (1−α)(c+ p)n(n− pα)(c+n)− (1−α)(c+ p)p . (2.7)
Now we show that λ(p,n,α,c) is an increasing function of n, n≥m. Indeed,
λ(p,n,α,c)= 1− (1−α)(c+ p)E(p,n,α,c), (2.8)
where
E(p,n,α,c)= (n− p)(n− pα)(c+n)− (1−α)(c+ p)p , (2.9)
and λ(p,n,α,c) increases when n increases if and only if E(p,n,α,c) is a strictly decreasingfunction of n.
Let h(x)= E(p,x,α,c), x ∈ [m,∞)⊂ [p+ 1,∞), we have
h′(x)=− (x− p)2
[(x− pα)(c+ x)− (1−α)(c+ p)p]2
< 0. (2.10)
We obtained
λ= λ(p,m,α,c)≤ λ(p,n,α,c), n≥m. (2.11)
The result is sharp because
�c(fα)= fλ, (2.12)
where
fα(z)= zp− p2(1−α)
(m− pα)[(1−β)p+βm]zm,
fλ(z)= zp− p2(1− λ)
(m− pλ)[(1−β)p+βm]zm
(2.13)
are extremal functions of ���(p,m,α,β) and ���(p,m,λ,β), respectively, and λ =λ(p,m,α,c).
4 International Journal of Mathematics and Mathematical Sciences
Indeed, we have
�c(fα(z)
)= zp− p2(1−α)(c+ p)
(m− pα)[(1−β)p+βm](c+m)zm. (2.14)
We deduce
p2(1− λ)(m− pλ) =
p2(1−α)(c+ p)(m− pα)(c+m) , (2.15)
and this implies (2.14).From λ= 1− (1−α)(c+ p)(m− p)/((m− pα)(c+m)− (1−α)(c+ p)p) we obtain λ <
1 and also λ > α. Indeed,
λ−α= (1−α){
1− (c+ p)(m− p)(m− pα)(c+m)− (1−α)(c+ p)p
}
= (1−α) (m− pα)(m− p)(m− pα)(c+m)− (1−α)(c+ p)p > 0.
(2.16)
�
3. Integral means inequalities for the class ���(p,m,α,β)
An analytic function g is said to be subordinate to an analytic function f (written g ≺f ) if g(z) = f (w(z)), z ∈ U, for some analytic function w with |w(z)| ≤ |z|. In 1925,Littlewood [3] proved the following subordination result which will be required in ourpresent investigation.
Lemma 3.1. If f and g are analytic in U with g ≺ f , then∫ 2π
0
∣∣g(reiθ
)∣∣δdθ ≤
∫ 2π
0
∣∣ f(reiθ
)∣∣δdθ, (3.1)
where δ > 0, z = reiθ , and 0 < r < 1.Applying Lemmas 1.1 and 3.1, we prove the following.
Theorem 3.2. Let δ>0. If f ∈���(p,m,α,β) and fm(z)=zp−(p2(1−α)/(m−pα)[(1−β)p+βm])zm, then for z = reiθ and 0 < r < 1,
∫ 2π
0
∣∣ f(reiθ
)∣∣δdθ ≤
∫ 2π
0
∣∣ fm
(reiθ
)∣∣δdθ. (3.2)
Proof. Let
f (z)= zp−∞∑
n=manz
n, an ≥ 0, n≥m,
fm(z)= zp− p2(1−α)
(m− pα)[(1−β)p+βm]zm,
(3.3)
H. Ö. Güney and G. S. Sâlâgean 5
then we must show that
∫ 2π
0
∣∣∣∣∣
1−∞∑
n=manz
n−p∣∣∣∣∣
δ
dθ ≤∫ 2π
0
∣∣∣∣∣
1− p2(1−α)
(m− pα)[
(1−β)p+βm]zm−p
∣∣∣∣∣
δ
dθ. (3.4)
By Lemma 3.1, it suffices to show that
1−∞∑
n=manz
n−p ≺ 1− p2(1−α)
(m− pα)[(1−β)p+βm]zm−p. (3.5)
Set
1−∞∑
n=manz
n−p = 1− p2(1−α)
(m− pα)[(1−β)p+βm]w(z)m−p. (3.6)
From (3.6) and (1.3), we obtain
∣∣w(z)
∣∣m−p =
∣∣∣∣
(m− pα)[(1−β)p+βm]p2(1−α)
∣∣∣∣
∣∣∣∣∣
∞∑
n=manz
n−p∣∣∣∣∣
≤ ∣∣zm−p∣∣∞∑
n=m
(n− pα)[(1−β)p+βn]p2(1−α) an ≤
∣∣zm−p
∣∣≤ |z|.
(3.7)
This completes the proof of the theorem. �
The proof for the first derivative is similar.
Theorem 3.3. Let δ > 0. If f ∈���(p,m,α,β) and fm(z)=zp−(p2(1−α)/(m−pα)[(1−β)p+βm])zm, then for z = reiθ and 0 < r < 1,
∫ 2π
0
∣∣ f ′
(reiθ
)∣∣δdθ ≤
∫ 2π
0
∣∣ f ′m(re
iθ)∣∣δdθ. (3.8)
Proof. It suffices to show that
1−∞∑
n=m
n
panz
n−p ≺ 1− mp(1−α)(m− pα)[(1−β)p+βm]z
m−p. (3.9)
This follows because
∣∣w(z)
∣∣m−p =
∣∣∣∣∣
∞∑
n=m
(n− pα)[(1−β)p+βn]p2(1−α) anz
n−p∣∣∣∣∣
≤ |z|n−p∞∑
n=m
(n− pα)[(1−β)p+βn]p2(1−α) an ≤ |z|
n−p ≤ |z|.(3.10)
�
6 International Journal of Mathematics and Mathematical Sciences
4. Neighborhoods of the class ���(p,m,α,β)
For f ∈�(p,m) and γ ≥ 0, Frasin [4] defined
Mqγ ( f )=
{
g ∈�(p,m) : g(z)= zp−∞∑
n=mbnz
n,∞∑
n=mnq+1
∣∣an− bn
∣∣≤ γ
}
, (4.1)
which was called q-γ-neighborhood of f . So, for e(z)= z, we see that
Mqγ (e)=
{
g ∈�(p,m) : g(z)= zp−∞∑
n=mbnz
n,∞∑
n=mnq+1
∣∣bn∣∣≤ γ
}
, (4.2)
where q is a fixed positive integer. Note that M0γ( f )≡Nγ( f ) and M1γ( f )≡Mγ( f ). Nγ( f )is called a γ-neighborhood of f by Ruscheweyh [5] and Mγ( f ) was defined by Silverman[6].
Now, we consider q-γ-neighborhood for function in the class ���(p,m,α,β).
Theorem 4.1. Let
γ = mq+1p2(1−α)
(m− pα)[(1−β)p+βm] , (4.3)
then ���(p,m,α,β)⊂Mqγ (e).Proof. If f ∈���(p,m,α,β), then
∞∑
n=mnq+1an ≤ m
q+1p2(1−α)(m− pα)[(1−β)p+βm] = γ. (4.4)
This gives that ���(p,m,α,β)⊂Mqγ (e). �Putting p = 1, m= 2 and β = 0 in Theorem 4.1, we have the following.
Corollary 4.2. ��∗(α)⊂Mqγ (e), where γ = 2q+1(1−α)/(2−α).Putting p = 1,m= 2, and β = 1 in Theorem 4.1, we have the following.
Corollary 4.3. ��(α)⊂Mqγ (e), where γ = 2q(1−α)/(2−α).
References
[1] H. Silverman, “Univalent functions with negative coefficients,” Proceedings of the AmericanMathematical Society, vol. 51, no. 1, pp. 109–116, 1975.
[2] R. M. Ali, M. H. Khan, V. Ravichandran, and K. G. Subramanian, “A class of multivalent func-tions with negative coefficients defined by convolution,” Bulletin of the Korean MathematicalSociety, vol. 43, no. 1, pp. 179–188, 2006.
[3] J. E. Littlewood, “On inequalities in the theory of functions,” Proceedings of the London Mathe-matical Society, vol. 23, no. 1, pp. 481–519, 1925.
[4] B. A. Frasin, “Family of analytic functions of complex order,” Acta Mathematica. AcademiaePaedagogicae Nýıregyháziensis, vol. 22, no. 2, pp. 179–191, 2006.
H. Ö. Güney and G. S. Sâlâgean 7
[5] S. Ruscheweyh, “Neighborhoods of univalent functions,” Proceedings of the American Mathe-matical Society, vol. 81, no. 4, pp. 521–527, 1981.
[6] H. Silverman, “Neighborhoods of classes of analytic functions,” Far East Journal of MathematicalSciences, vol. 3, no. 2, pp. 165–169, 1995.
H. Özlem Güney: Department of Mathematics, Faculty of Science and Arts, University of Dicle,21280 Diyarbakir, TurkeyEmail address: [email protected]
Grigore Stefan Sâlâgean: Faculty of Mathematics and Computer Science, Babes-Bolyai University,1 M. Kogalniceanu Street, 400084 Cluj-Napoca, RomaniaEmail address: [email protected]
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