On Some Integral Operators for Certain Classes of

(1)

Volume 2011, Article ID 783084,10pages doi:10.1155/2011/783084

Research Article

On Some Integral Operators for Certain Classes of

p

-Valent Functions

Lokesh Vijayvargy,

1

_{Pranay Goswami,}

2

_{and Bushra Malik}

3

1_{Faculty of Management, Jaipuria Institute of Management, Jaipur 302033, India} 2_{Departement of Mathematics, Amity University Rajasthan, Jaipur 302002, India}

3_{Mathematics Department, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan}

Correspondence should be addressed to Pranay Goswami,[email protected]

Received 3 December 2010; Revised 9 February 2011; Accepted 3 March 2011

Academic Editor: H. S. V. De Snoo

Copyrightq2011 Lokesh Vijayvargy et al. 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.

We study some generalized integral operators for the classes of p-valent functions with bounded radius and boundary rotation. Our work generalizes many previously known results. Many of our results are best possible.

1. Introduction

LetApdenote the class of functions of the form

fz zp

∞

np1

anzn, p∈N{1,2, . . .}, 1.1

which are analytic in the open unit discU{z:|z|<1}.

Letfandgbe analytic functions inUwe say thatfis subordinate tog, written as

f ≺g; 1.2

if there exists a Schwarz functionwzinU, withw0 0 and|wz|<1z∈U, such that

(2)

In particular, whengis univalent, then the above subordination is equivalent to

f0 0, fU⊆gU. 1.4

For functionsf, g∈Ap, given by

fz zp

∞

np1

anzn, gz zp

∞

np1

bnzn, z∈U, 1.5

we define the Hadamard productor convolutionoffandgby

f∗gz zp

∞

np1

anbnzn, z∈U. 1.6

Janowski1defined the classPA, Bas follows.

Lethbe a function, analytic inU, withh0 1. Thenhis said to belong to the class

PA, B,−1≤B < A≤1, if and only if, forz∈U,

hz 1Awz

1Bwz, wherewzis a Schwarz function. 1.7

Or equivalently, we can say thath∈PA, B,−1≤B < A≤1, if and only if,

hz≺ 1Az

1Bz, z∈U. 1.8

Geometrically,hzis in the classPA, B, if and only if,h0 1 and the image ofhUlies inside the open disc centered on the real axis with diameter end points,

D1h−1 1−A

1−B, D2h1

1A

1B, 0< D1<1< D2. 1.9

ClearlyPA, B⊂P1−A/1−B.

In the recent paper, Noor2introduced the classPkα. We define it as follows. Let

Pkα,0≤α < p, be the class of functionspzwithp0 1 and satisfying the property

pz 1

2

2π

0

1 1−2αze−it

1−ze−it dμt, 1.10

whereμtis a real-valued function of bounded variation on0,2πand₀2πdμt 2 and

₂_π

(3)

The classesVkαandRkαare related to the classPkαand can be defined as

f∈Vkα, iﬀ

zfz

pfz ∈Pkα, z∈U,

f∈Rkα, iﬀ

zfz

pfz ∈Pkα, z∈U.

1.11

We define a classPkA, Bas follows.

LetPkA, B,k≥2,−1≤B < A≤1, denote the class ofp-valent analytic functionshz that are represented by

hz

k

4

1 2

h1z−

k

4 − 1 2

h2z, z∈U, 1.12

whereh1, h2 ∈PA, B. ForA1−2α0 ≤α < pandB −1,it reduces to the classPkα andP2α Pαis the class ofp-valent analytic functions hzwith Rehz > α,z ∈ U.

TakingA 1,B −1, andp 1, we havePk1,−1 Pk see3, andP21,−1 P is the

class of functions with positive real part.

Definition 1.1. A functionf,analytic inU, and given by1.1is said to be in the classRkA, B;

−1≤B < A≤1,k≥2, if and only if,

zfz

pfz ∈PkA, B, z∈U. 1.13

Forp1,RkA, Bis introduced and studied by Noor4. We note that

RkA, B⊂Rk

1−A

1−B

⊂Rk, 1.14

whereRkis the class of functions with bounded radius rotationsee5. Fork2, we have

R2A, B≡S∗pA, B⊂S∗p

1−A

1−B

⊂S∗_p, 1.15

whereS∗_pis the class ofp-valent starlike functions. Similarly, we can define the classVkA, B as follows.

Definition 1.2. A functionf,analytic inU, and given by1.1is said to be in the classVkA, B;

−1≤B < A≤1,k≥2, if and only if,

zfz

(4)

It is clear that

f ∈VkA, B, iﬀ

zfz

p ∈RkA, B, z∈U. 1.17

Forp1,VkA, Bis the class introduced and studied by Noor4. It is easy to see that,

VkA, B⊂Vk

1−A

1−B

⊂Vk, 1.18

whereVkis the class of functions with bounded boundary rotation see5. Also

V2A, B≡CpA, B⊂Cp

1−A

1−B

⊂Cp, 1.19

whereCpis the class ofp-valent convex functions.

Very recently, Frasin6, introduced the following general integral operators forp -valent functions,

Fpz

_z

0

ptp−1

_f

1t

tp

α1 · · ·

_f

nt

tp

αn

dt, 1.20

Gpz

z

0

ptp−1 f1t ptp−1

α1

· · · fnt

ptp−1

αn

dt, whereαi∈C, z∈U. 1.21

Clearly, we may see that forp1, these operators become the general integral operators

F1z Fnz, G1z Fα1,α2,...,αnz, 1.22

introduced and studied by Breaz and Breaz7and Breaz et al.8,see also9,10. Forp n 1,α1 α∈0,1in1.20, we obtain the integral operator

_z

0 ft/t

α_dt

studied in11 and forp n 1,α1 δ ∈ C,|δ| < 1/4 in1.21, we obtain the integral

operator₀zftδdt, studied in12.

2. Main Results

Lemma 2.1. Letβ >0,βγ >0,α∈α0,1, withα0max{β−γ−1/2β,−γ/β}. If

pz zpz βpz γ ≺

1 1−2αz

1−z , 2.1

then

pz≺Qz≺ 1 1−2αz

(5)

whereQz 1/βGz−γ/β,

Gz

₁

0

1−z

1−tz

2β1−α

tβγ−1_dt

2F1

2β1−α,1, βγ1; z 1−z

, 2.3

ρρα, β, γ βγ β2F1

2β1−α,1, βγ1; 1/2−

γ

β, 2.4

2F1 denotes the Gauss hypergeometric function. From 2.2, we can deduce the sharp result p ∈

Pρ,whereρis defined in2.4. This result is a special case of one given in [11].

Proof. To prove this Lemma we use Theorem 3.2j of 11, page 97. Take hz 1 1 −

2αz/1−z, 0≤α <1 and

Hz βhz γ abz

1−z , 2.5

whereaβγandbβ1−2α γ.

SinceHis convex to apply Theorem 3.2j of11, page 97we only need to determine condition ReHz>0.

The range of |z| ≤ 1 under Hz is a half plane. In order to satisfy the required condition this half plane needs to lie in the right half plane. This requirement will be satisfied if ReH−1 ReHiand ReH0>ReH−1≥0. Or we can write it as

β1−α>0, βαγ≥0. 2.6

When β > 0,βγ > 0, these conditions imply thatα ∈ −γ/β,1, and ifβγ > 1, then

α∈β−γ−1/2β,1. Hence all the conditions of Theorem 3.2j of11, page 97are satisfied forα∈α0,1,withα0max{β−γ−1/2β,−γ/β}, thus we have the required result.

To show that the solution Qz can be represented in terms of hypergeometric functions we takeA1−2α,B−1,n1 in Theorem 3.3d of11, page 109.

Lemma 2.2. Letf ∈Vkα, 0≤α < p,k≥2.Thenf ∈RkρinU, where

ρρα, p 1

2F1

2p1−α,1, p1; 1/2. 2.7

This result is sharp.

Proof. Let fork≥2,z∈U,we have

zfz pfz hz

k

4

1 2

h1z−

k

4 − 1 2

h2z, 2.8

(6)

We define

φpz z

∞

n1 1

p1 n−1z

n_, _z_∈_U. _2.9

By using2.8, with convolution technique, see13, we have

φpz

z ∗hz

k

4

1 2

φpz

z ∗h1z

−

k

4 − 1 2

φpz

z ∗h2z

. 2.10

This implies that,

hz zhz phz

k

4

1

2 h1z

zh₁z ph1z

−

k

4 − 1

2 h2z

zh₂z ph2z

. 2.11

Logarithmic diﬀerentiation of2.8yields,

zfz pfz hz

zhz phz

k

4

1

2 h1z

zh₁z ph1z

−

k

4 − 1

2 h2z

zh₂z ph2z

.

2.12

Sincezfz/pfz∈Pkα, 0≤α < p, thus

hiz

zh_iz phiz ∈

Pα, i1,2. 2.13

By usingLemma 2.1forβ pandγ 0, we deduce thathi ∈ Pρ,whereρis given in

2.7. This estimate is best possible because of the best dominant property of functionQz, where

Qz 1

2F1

2p1−α,1, p1;z/1−z, z∈U. 2.14

Forp1,we have the sharp result proved in14. We begin with the following theorem.

Theorem 2.3. iLetαi>0,fi∈RkA, Bfor alli1,2, . . . , n, and,ni1αi1. Then the integral

operatorFp∈VkA, BinU, where−1≤B < A≤1,k≥2.

ii Let fi ∈ Rkα,αi > 0 for all i 1,2, . . . , n with α 1−A/1−B,k ≥ 2. If

n

i1αi 1, then the integral operatorFpdefined by 1.20 also belongs to the classRkρin U,

(7)

Proof (i). From1.20, we can see thatFp∈ApinU, and

F_pz pzp−1

_f

1z

zp

α1 · · ·

_f

nz

zp

αn

. 2.15

Diﬀerentiating logarithmically and multiplying byz, we obtain,

zF _pz Fpz

p−1 n

i1

αi

zf_iz f_iz −p

, z∈U. 2.16

Thus, we have

1zF

pz

Fpz

n

i1

αi

zf_iz f_iz

, 2.17

or

zF_pz pFpz

n

i1

αi

zf_iz pf_iz

k 4 1 2 n i1

αipiz

− k 4 − 1 2 n i1

αihiz

,

2.18

wherehi, pi∈PA, B,for alli1,2, . . . , n.

SincePA, Bis a convex set, see15, it follows that,

zF_pz pFpz

k 4 1 2

H1z−

k 4 − 1 2

H2z, 2.19

whereH1, H2∈PA, Band therefore,

zF_pz pFpz

∈PkA, B, z∈U. 2.20

This proves the result.

Substitutingp1, inTheorem 2.3i, we have the following corollary.

Corollary 2.4. Letαi >0,fi ∈RkA, Bfor alli1,2, . . . , n,−1 ≤B < A ≤1,k ≥ 2. Then the

integral operatorFp∈VkA, BinU.

Remark 2.5. Lettingα1α,α2β, andn2 inCorollary 2.4, we obtain a result due to Noor

(8)

Forn1,α1α1, andf1finTheorem 2.3i, we have the following.

Corollary 2.6. Letf ∈ RkA, B in U, −1 ≤ B < A ≤ 1, k ≥ 2. Then the integral operator

z

0 pft/tdt∈VkA, B,z∈U.

Proof (ii). TakingA1−2α,B−1, withα 1−A/1−B, we have for alli1,2, . . . , n,

fi∈Rk1−2α,−1 Rkα, 2.21

using partiofTheorem 2.3, we have

Fp∈Vk1−2α,−1 Vkα inU. 2.22

Now usingLemma 2.2forFp∈Vkα,α 1−A/1−Bimplies that

Fp∈Rk

ρ, whereρρα, pis defined in2.7. 2.23

The sharpness of the result is clear from the functionQzdefined by2.14.

Forp1, we have the following corollary.

Corollary 2.7. Letαi>0,fi∈Rkαfor alli1,2, . . . , n, withα 1−A/1−BandA1−2α,

B−1. Then the integral operatorFpdefined by1.20also belongs to the classRkρinU,where

ρρα

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

2α−1

2−221−α, ifα / 1 2 1

2 ln 2, ifα

1 2.

2.24

Remark 2.8. Lettingα1μ,α2η, andn2 inCorollary 2.7, we have the sharp result proved

in14.

ForA1,B−1, andp1,we have

fi∈Rk0 implies thatFp∈Vk

1 2

inU. 2.25

Theorem 2.9. iLetαi > 0,fi ∈VkA, Bfor alli 1,2, . . . , n. Ifni1αi 1, then the integral

operatorGpdefined by1.21, also belongs to the classVkA, BinU,where−1≤B < A≤1,k≥2.

ii Let forαi > 0,

_n

i1αi 1 and fi ∈ Vkα,for all i 1,2, . . . , nwith 0 ≤ α < p,

α 1−A/1−B,k≥2. Then the integral operatorGp∈RkρinU, whereρρα, pis defined

(9)

Proof (i). From definition1.20, we have

1zG

pz

Gpz

p

n

i1

αi

zf_i z f_iz −p1

n

i1

αi

zf_iz f_iz ,

2.26

or

zG_pz pGpz

n

i1

αi

zf_iz pf_iz

k

4

1 2

n

i1

αipiz

−

k

4 − 1 2

n

i1

αihiz

,

2.27

wherehi, pi∈PA, B,for alli1,2, . . . , n.

SincePA, Bis a convex set, see15, it follows that,

zG_pz pGpz

k

4

1 2

H1z−

k

4 − 1 2

H2z, z∈U, 2.28

whereH1, H2∈PA, Band therefore,

zG_pz pGpz

∈PkA, B inU. 2.29

This implies thatGp∈VkA, B.

Lettingp1 inTheorem 2.9i, we have the following corollary.

Corollary 2.10. Letαi > 0,fi ∈ VkA, Bfor alli 1,2, . . . , nand−1 ≤ B < A ≤ 1,k ≥ 2. If

_n

i1αi1, thenGp∈VkA, BinU.

Proof (ii). TakingA1−2α,B−1,we have for alli1,2, . . . , n

fi∈Vk1−2α,−1 Vkα, whereα 1−A

1−B. 2.30

Now using partiofTheorem 2.9, we have

(10)

Now usingLemma 2.2, forα 1−A/1−B,we have

Gp∈Vkαimplies thatGp∈Rk

ρ, inU, where ρρα, pis defined in2.7. 2.32

The sharpness of the result is clear from the functionQzdefined by2.14.

Forp1, we have the following corollary.

Corollary 2.11. iLetαi>0,fi ∈Vkα,i1,2, . . . , n, withα 1−A/1−BandA1−2α,

B−1. ThenGp∈RkρinU, whereρραand defined in2.24.

Also forA1,B−1,we have.

iiIffi∈Vk0for alli1,2, . . . , n, thenGp∈Rk1/2inU.

Acknowledgments

Authors are thankful to anonymous referees for their very constructive comments to improve this paper. Second author P. Goswami is also grateful to Professor S. P. Goyal for his guidance and constant encouragement.

References

1 W. Janowski, “Some extremal problems for certain families of analytic functions. I,” Annales Polonici

Mathematici, vol. 28, pp. 297–326, 1973.

2 K. I. Noor, “Genealized integral operator and multivalent functions,” Journal of Inequalities in Pure and

Applied Mathematics, vol. 6, no. 2, Article 51, 2005.

3 B. Pinchuk, “Functions of bounded boundary rotation,” Israel Journal of Mathematics, vol. 10, pp. 6–16,

1971.

4 K. Inayat Noor, “Some properties of analytic functions with bounded radius rotation,” Complex

Variables and Elliptic Equations, vol. 54, no. 9, pp. 865–877, 2009.

5 A. W. Goodman, Univalent Functions, vol. 1 and 2, Mariner, Tampa, Fla, USA, 1983.

6 B. A. Frasin, “Convexity of integral operators of p-valent functions,” Mathematical and Computer

Modelling, vol. 51, no. 5-6, pp. 601–605, 2010.

7 D. Breaz and N. Breaz, “Some convexity properties for a general integral operator,” Journal of

Inequalities in Pure and Applied Mathematics, vol. 7, no. 5, Article 177, 2006.

8 D. Breaz, S. Owa, and N. Breaz, “A new integral univalent operator,” Acta Universitatis Apulensis, no.

16, pp. 11–16, 2008.

9 D. Breaz, “A convexity property for an integral operator on the class Spβ,” General Mathematics, vol.

15, no. 2, pp. 177–183, 2007.

10 D. Breaz and H. G ¨uney, “The integral operator on the classes Sαband Cαb,” Journal of Mathematical

Inequalities, vol. 2, no. 1, pp. 97–100, 2008.

11 S. S. Miller and P. T. Mocanu, Diﬀerential Subordinations: Theory and Applications, vol. 225 of Monographs

and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000.

12 N. N. Pascu and V. Pescar, “On the integral operators of Kim-Merkes and Pfaltzgraﬀ,” Mathematica,

vol. 3255, no. 2, pp. 185–192, 1990.

13 K. I. Noor, “On some diﬀerential operators for certain classes of analytic functions,” Journal of

Mathematical Inequalities, vol. 2, no. 1, pp. 129–137, 2008.

14 K. I. Noor, W. Ul-Haq, M. Arif, and S. Mustafa, “On bounded boundary and bounded radius

rotations,” Journal of Inequalities and Applications, vol. 2009, Article ID 813687, 12 pages, 2009.

15 K. I. Noor, “On subclasses of close-to-convex functions of higher order,” International Journal of

(11)

Submit your manuscripts at

http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

Journal of

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations International Journal of

Volume 2014

Applied MathematicsJournal of

Mathematical PhysicsAdvances in

Complex Analysis

Journal of

Optimization

Journal of Hindawi Publishing Corporation

Combinatorics

International Journal of

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014

The Scientific

World Journal

Discrete Dynamics in Nature and Society

http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation

Discrete Mathematics

Journal of

On Some Integral Operators for Certain Classes of

Research Article