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Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2007, Article ID 34017, 7 pages doi:10.1155/2007/34017 Research Article Further Properties of β-Pascu Convex Functions of Order α H. ¨ Ozlem G ¨ uney and Grigore Stefan Sˆ alˆ agean 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 were recently introduced and studied by Ali et al. in (2006). Copyright © 2007 H. ¨ O. G ¨ uney and G. S. Sˆ alˆ agean. 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 properly cited. 1. Introduction Let N :={1,2, ... }, for m, p N, m p + 1, let ( p, m) be the class of all p-valent analytic functions f (z) = z p + n=m a n z n deﬁned on the open unit disk U ={z C : |z| < 1} and let := (1, 2). Let ( p, m) be the subclass of ( p, m) consisting of functions of the form f (z) = z p n=m a n z n , a n 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 1 p Re (1 β)zf (z)+(β/p)z ( zf (z) ) (1 β) f (z)+(β/p)zf (z) (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 with negative coecients of order α and (α):= (1,2, α, 1) is the class of convex func- tions with negative coecients of order α (studied by Silverman [1]). For the class ( p, m, α, β), the following characterization was given by Ali et al. [2].
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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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