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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 ). For the class ( p, m, α, β), the following characterization was given by Ali et al. .
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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 ).

For the class ���(p,m,α,β), the following characterization was given by Ali et al. .

• 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. . 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  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  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  and Mγ( f ) was defined by Silverman.

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

 H. Silverman, “Univalent functions with negative coefficients,” Proceedings of the AmericanMathematical Society, vol. 51, no. 1, pp. 109–116, 1975.

 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.

 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.

 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

 S. Ruscheweyh, “Neighborhoods of univalent functions,” Proceedings of the American Mathe-matical Society, vol. 81, no. 4, pp. 521–527, 1981.

 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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