On Integral Operator Defined by Convolution Involving Hybergeometric Functions

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Volume 2008, Article ID 520698,11pages doi:10.1155/2008/520698

Research Article

On Integral Operator Defined by Convolution

Involving Hybergeometric Functions

K. Al-Shaqsi and M. Darus

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor Darul Ehsan, Malaysia

Correspondence should be addressed to M. Darus,[email protected]

Received 16 September 2007; Accepted 9 January 2008

Recommended by Brigitte Forster-Heinlein

Forλ >−1 andμ≥0, we consider a liner operatorIμ_λon the classAof analytic functions in the unit disk defined by the convolutionfμ−1∗fz, wherefμ 1−μz2F1a, b, c;zμzz2F1a, b, c;z, and introduce a certain new subclass ofAusing this operator. Several interesting properties of these classes are obtained.

Copyrightq2008 K. Al-Shaqsi and M. Darus. 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.

1. Introduction

LetAdenote the class of functions of the form

fz z

∞

k2

akzk, 1.1

which are analytic in the unit diskU{z:|z|<1}. Iffz∈ Asatisfies

arg

_zf_z

fz −α

<π

2β z∈U, 0≤α <1, 0< β≤1, 1.2

thenfzis said to be strongly starlike of orderβand typeαinU, and denoted byS∗α, β. Iffz∈ Asatisfies

arg

1zf _z

fz −α

<π

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thenfzis said to be strongly convex of order βand typeαinU, and denoted byCα, β. It is obvious that fz ∈ A belongs toCα, β if and if zfz ∈ S∗α, β. Further, we note thatS∗α,1≡S∗αandCα,1≡Cαwhich are, respectively, starlike and convex univalent functions of orderα.

LetP denote the class of functions of the formpz 1p1z· · · analytic inUwhich satisfy the condition Re{Pz}>0.

For functions f given by 1.1 and gz z ∞_k₂bkzk, let f∗gz denote the Hadamard productor convolutionoffzandgz, defined by

f∗gz fz∗gz z

∞

k2

akbkzk. 1.4

Iffandgare analytic inU, we say thatfis subordinate tog, writtenf ≺gorfz≺gz, if there exists a Schwarz functionwinUsuch thatfz gwz 1.

Letf∈ A. Denote byDλ_:_{A → A}_{the operator defined by}

Dλ z

1−zλ1∗fz λ >−1. 1.5

It is obvious thatD0_f_z _f_z_,_D1_f_z _zf_z_{, and}

Dδ_f_z z

zδ−1_f_zδ

δ!

δ∈N0N∪ {0}

. 1.6

The operatorDδ_f_{is called the}_δ_{th-order Ruscheweyh derivative of}_f_{. Recently, K. I. Noor}₂ and K. I. Noor and M. A. Noor3 defined and studied an integral operatorIn : A → A, analogous toDδ_f_z_{as follows.}

Letfnz/1−zn1,n∈N0,andfn−1zbe defined such that

fnz∗fn−1z

z

1−z2. 1.7

Then

Infz fn−1z∗fz

z

1−zn1

−1

∗fz f∈ A. 1.8

We note that I0fz zfz,I1fz fz. The operator In is called the Noor integral of

nth order off see 4,5, which is an important tool in defining several classes of analytic functions. In recent years, it has been shown that Noor integral operator has fundamental and significant applications in the geometric function theory.

For real or complex numbersa,b,cother than 0,−1,−2, . . ., the hypergeometric series is defined by

2F1a, b;c;z ∞

k0

a_kb_k

c_k1_kz

k_,

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wherex_kis Pochhammer symbol defined by

x_kΓxk

Γx xx1· · ·xk−1 fork1,2,3, . . . , x∈C, x01. 1.10

We note that the series1.9converges absolutely for allz ∈Uso that it represents an analytic function inU. Also an incomplete beta functionφa, c;zis related to Gauss hyperge-ometric functionz2F1a, b;c;zas

φa, c;z z2F11, a;c;z, 1.11

and we note thatφa,1;z z/1−za, whereφ2,1;zis Koebe function. Usingφa, c;z,a convolution operator6, was defined by Carlson and Shaferr. Furthermore, Hohlov7 intro-duced a convolution operator using2F1a, b;c;z.

N. Shukla and P. Shukla8studied the mapping properties of a functionfμ to be as given in

fμa, b, cz 1−μz2F1a, b, c;z μz

z2F1a, b, c;z

μ≥0, 1.12

and investigated the geometric properties of an integral operator of the form

Iz

_z

0

fμt

t dt. 1.13

Kim and Shon9considered linear operatorLμ : A → Adefined byLμa, b, cfz

fμa, b, cz∗fz.

We now introduce a functionfμ−1 given by

fμa, b, cz∗

fμa, b, cz

−1 z

1−zλ1 μ≥0, λ >−1, 1.14

and obtain the following linear operator:

I_μλa, b, cfz fμa, b, cz

−1_∗

fz. 1.15

The operator Iλ

μ is known as the generalized integral operator. For μ 0 in 1.14, Iλa, b;

cfz:Iμλa, b;cfz, which was introduced by K. I. Noor10.

Now we find the explicit form of the functionfμ−1. It is well known thatλ >−1

z

1−zλ1

∞

k0

λ1_k

k! z

k1 _z_∈_U_. _1.16

Putting1.9and1.16in1.14, we get

∞

k0

μk1a_kb_k

c_k1_k z k1_∗_f

μ

−1 ∞

k0

λ1_k

k! z

k1_.

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Therefore, the functionfμ−1has the following form:

fμa, b, cz

−1 ∞

k0

λ1_kc_k

μk1a_kb_kz

k1 _z_∈_U_. _1.18

Now we note that

Iλ

μa, b, cfz z ∞

k1

λ1_kc_k

μk1a_kb_kak1z

k1_. _1.19

From1.19, we note that

Iλ

0a, λ1, afz fz, I01a,1, afz zfz. 1.20 Also it can easily be verified that

zI_μλa, b, cfz λ1I_μλ1a, b, cfz−λI_μλa, b, cfz, 1.21

zI_μλa1, b, cfzaI_μλa, b, cfz−a−1I_μλa1, b, cfz. 1.22

Now we introduce the following classes in term of the new operatorIλ

μa, b, c. Forλ > −1,

μ≥0, 0≤α <1, and 0< β≤1, letSλ

μa, b, c;α, βbe the class of functionsf∈ Asatisfying

arg

zIλ

μa, b, cfz

Iλ

μa, b, cfz

−α<π

2β z∈U. 1.23

Observe thatIλ

μa, b, cfz∈ S∗α, βandzIμλa, b, cfz/Iμλa, b, cfz/α. Also, forλ >

−1,μ≥0, 0≤α <1, and 0< β≤1, letCλ

μa, b, c;α, βbe the class of functionsf∈ Asatisfying

arg

1z

Iλ

μa, b, cfz

Iλ

μa, b, cfz

−α

<

π

2β z∈U. 1.24

Observe thatIλ

μa, b, cfz∈Cα, βand 1zIμλa, b, cfz/Iμλa, b, cfz/α. Clearly,f∈ Cλ

μa, b, c;α, βif and only ifzfz∈ Sμλa, b, c;α, β.

Note thatSλ₀a, λ1, a;α, β≡S∗α, β,S₀λa, λ1, a;α,1≡S∗α,C₀λa, λ1, a;α, β≡ Cα, β, andCλ

0a, λ1, a;α,1≡Cα. Finally, letKλ

μa, b, c;α, β, γ;A, Bbe the class of functionsf∈ Asatisfying

arg

zIλ

μa, b, cfz

Iλ

μa, b, cgz

−α<π

2β z∈U 1.25

forλ >−1,μ≥0, 0≤α <1, 0< β≤1, andg∈ Qλ

μa, b, c;γ;A, B, where

Qλ

μa, b, c;γ;A, B

g∈ A: 1 1−γ

zIλ

μa, b, cgz

Iμλa, b, cgz

−γ

≺ 1Az

1Bz

z∈U; 0≤γ <1, −1≤B < A≤1.

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We also note thatKλ

0a, λ1, a;α,1, γ; 1,−1andKλ0a,1, a;α,1, γ; 1,−1are the classes of qua-siconvex and close-to-convex functions of orderαand type γ, respectively, introduced and studied by Noor and Alkhorasani11and Silverman12.

2. Main results

In order to give our results, we need the following lemmas.

Lemma 2.1see13. Letβ,γbe complex numbers. Letφzbe convex univalent inUwithφ0 1 and Reβφz γ>0,z∈Uandq∈ Awithqz≺φz,z∈U. Ifp∈Pis analytic inU, then

pz zp

_z

βqz γ ≺φz z∈U 2.1

implies

pz≺φz z∈U. 2.2

Lemma 2.2see14. Letδ,ηbe complex numbers. Letφzbe convex univalent inUwithφ0 1 and Reδφz η>0,z∈U. Ifp∈Pis analytic inU, then

pz zp

_z

δpz η ≺φz z∈U 2.3

implies

pz≺φz z∈U. 2.4

Lemma 2.3see15. Letφzbe convex univalent inUand letE≥0. SupposeBzis analytic in

Uwith ReBz≥E, Ifg∈Pis analytic inU, then

Ez2_g_z _B_z_zg_z _g_z_≺_φ_z _z_∈_U _2.5

implies

gz≺φz z∈U. 2.6

Theorem 2.4. Letφzbe convex univalent inUwithφ0 1 and Reφz≥0. Iffz∈ Asatisfies the condition

1 1−γ

zIλ1

μ a, b, cfz

Iμλ1a, b, cfz

−γ

≺φz z∈U, 2.7

then

1 1−γ

zIλ

μa, b, cfz

Iλ

μa, b, cfz

−γ

≺φz z∈U 2.8

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Proof. Let

pz 1

1−γ

zIλ

μa, b, cfz

Iλ

μa, b, cfz

−γ

, 2.9

wherep∈P. By using1.21in2.9and then diﬀerentiating, we get

1 1−γ

zIλ1

μ a, b, cfz

Iλ1

μ a, b, cfz

−γ

pz zp

_z

λ1qz, 2.10

whereqz Iλ1

μ a, b, cfz/Iμλa, b, cfzandqz≺φz. Hence by applyingLemma 2.1, we obtain the required result.

1 1−γ

zIλ

μa, b, cfz

Iλ

μa, b, cfz

−γ

≺φz z∈U, 2.11

then

1 1−γ

zIλ

μa1, b, cfz

Iλ

μa1, b, cfz

−γ

≺φz z∈U 2.12

forλ >−1,μ≥0, and 0≤γ <1.

Proof. By using the same technique in the proof ofTheorem 2.4and using1.22and applying Lemma 2.2, we obtain the required result.

Takingφz 1Az/1Bz −1≤B < A≤ 1inTheorem 2.4and inTheorem 2.5, we have the following.

Corollary 2.6. It holds that

Qλ1

μ a, b, c;γ;A, B⊂Qλμa, b, c;γ;A, B,

Qλ

μa, b, c;γ;A, B⊂Qμλa1, b, c;γ;A, B

2.13

forλ >−1,μ≥0, 0≤γ <1, and Rea >1−γ.

Also, by takingφz 1z/1−zβ 0< β≤1inTheorem 2.4and inTheorem 2.5, we have the following.

Corollary 2.7. It holds that

Sλ1

μ a, b, c;γ, β⊂ Sλμa, b, c;γ, β,

Sλ

μa, b, c;γ, β⊂ Sμλa1, b, c;γ, β

2.14

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Corollary 2.8. Forλ >−1,μ≥0, 0≤γ <1, and Rea >1−β, one has

Cλ1

μ a, b, c;γ, β⊂ Cλμa, b, c;γ, β,

Cλ

μa, b, c;γ, β⊂ Cλμa1, b, c;γ, β.

2.15

Proof. We will proof the first relation and by the same method we can proof the second relation

fz∈ Cλ_μ1a, b, c;γ, β⇐⇒zfz∈ S_μλ1a, b, c;γ, β

⇐⇒zfz∈ S_μλa, b, c;γ, β

⇐⇒I_μλa, b, czfz∈S∗γ, β

⇐⇒zIλ

μa, b, cfz

_∈

S∗γ, β

⇐⇒Iμλa, b, cfz∈Cγ, β

⇐⇒fz∈ Cλ_μa, b, c;γ, β.

2.16

1 1−γ

zIλ

μa, b, cfz

Iλ

μa, b, cfz

−γ

≺φz 0≤γ <1; z∈U, 2.17

then

1 1−γ

zIλ

μa, b, cFz

Iλ

μa, b, cFz

−γ

≺φz 0≤γ <1; z∈U, 2.18

whereFbe the integral operator defined by

Fz c1 zc

z

0

tc−1ftdt c >−1. 2.19

Proof. From2.19, we have

zIλ

μa, b, cFz

c1Iλ

μa, b, cfz−cIμλa, b, cFz. 2.20 Now, let

pz 1

1−γ

zIλ

μa, b, cFz

Iμλa, b, cFz

−γ

, 2.21

wherep∈P. Then by using2.20, we get

1−γpz cγ c1I

λ

μa, b, cfz

Iλ

μa, b, cFz

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Diﬀerentiating both sides of2.22logarithmically, we obtain

pz zp

_z

cγ 1−γpz

1 1−γ

zIλ

μa, b, cfz

Iλ

μa, b, cfz

−γ

. 2.23

Then, byLemma 2.2, we obtain that

1 1−γ

zIλ

μa, b, cfz

Iλ

μa, b, cfz

−γ

≺φz 0≤γ <1; z∈U. 2.24

Now, by lettingφz 1Az/1Bz −1≤B < A≤1inTheorem 2.9, we have the following.

Corollary 2.10. Forλ > −1,μ ≥ 0,c > −γ, and 0 ≤ γ < 1. Iff ∈ Qλ

μa, b, c;γ;A, B, thenF ∈

Qλ

μa, b, c;γ;A, B, whereFgiven by2.19.

Also, by taking φz 1z/1−zβ 0 < β ≤ 1in Theorem 2.9, we have the following.

Corollary 2.11. Forλ > −1,μ≥ 0,c > −β, 0< β ≤1, and 0 ≤γ <1. Iff ∈ Sλ

μa, b, c;γ, β, then

F∈ Sλ

μa, b, c;γ, β.

Corollary 2.12. Forλ >−1,μ ≥0,c > −β, 0< β ≤1, and 0 ≤γ < 1. Iff ∈ Cλ

μa, b, c;γ, β, then

F∈ Cλ

μa, b, c;γ, β. Proof. It holds that

fz∈ Cλ

μa, b, c;γ, β⇐⇒F

zfz∈ Sλ

μa, b, c;γ, β

⇐⇒zFz∈ Sλ_μa, b, c;γ, β

⇐⇒Fz∈ Cλ_μa, b, c;γ, β.

2.25

Theorem 2.13. Letf∈ A. Then

Kλ1

μ a, b, c;α, β, γ;A, B⊂ Kμλa, b, c;α, β, γ;A, B 2.26

for Rea >1−β, 0< β≤1, 0≤α <1, 0≤γ <1, and−1≤B < A≤1.

Proof. Letf∈ Kλ1

μ a, b, c;α, β, γ;A, B, then by the definition, we can write

1 1−α

zIλ1

μ a, b, cfz

Iλ1

μ a, b, cgz

−α

≺

1z

1−z

β

z∈U 2.27

for someg∈Qλ1

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Lettinghz zIλ

μa, b, cfz/Iμλa, b, cgzand Hz zIλμa, b, cgzIμλa, b,

cgz, we observe thathz,Hz∈Pz. Now byCorollary 2.6,g∈Qλ

μa, b, c;γ;A, Band so ReHz> γ. Also, note that

zIλ

μa, b, cfz

Iλ

μa, b, cgz

hz. 2.28

Diﬀerentiating both sides in2.28yields

zIλ

μa, b, cfz

Iμλa, b, cgz

z

Iλ

μa, b, cgz

Iμλ1a, b, cgz

hz zhz Hzhz zhz. 2.29

Now by using the identity1.21, we obtain

zIλ1

μ a, b, cfz

Iμλ1a, b, cgz

I

λ1 μ a, b, c

zfz Iλ1

μ a, b, cgz

z

Iλ μa, b, c

zfzλIλ μa, b, c

zfz zIμλa, b, cgz

λIμλa, b, cgz

z

Iλ μa, b, c

zfz/Iλ

μa, b, cgz λ

Iλ μa, b, c

zfz/Iλ

μa, b, cgz

zIλ

μa, b, cgz

/Iλ

μa, b, cgz λ

Hzhz zhz λhz

Hz λ

hz zh

_z

Hz λ.

2.30

From2.27,2.28, and2.30, we conclude that

1 1−α

hz zh

_z

Hz λ −α

≺

1z

1−z

β

. 2.31

LettingE0 andBz 1/1−α1/Hz λ, we obtain

ReBz 1

1−α

1

Hz λ2Re

Hz λ>0. _2.32

The above inequality satisfies the conditions required byLemma 2.3. Henceφz≺ 1z/

1−zβ and so the proof is complete.

Theorem 2.14. Letf∈ A. Then

Kλ

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Proof. By using the same technique as in the proof ofTheorem 2.13, we get

1 1−α

hz zh

_z

Hz α−1−α

≺

1z

1−z

β

. 2.34

By lettingE0 andBz 1/1−α1/Hz a−1, we obtain

ReBz 1

1−α

1

Hz a−12Re

Hz a−1>0. _2.35

Then, by applyingLemma 2.3, we obtain the required result.

Theorem 2.15. Letc > −β, 0 < β ≤ 1, 0 ≤ α < 1, 0 ≤ γ < 1, and −1 ≤ B < A ≤ 1. If f ∈

Kλ

μa, b, c;α, β, γ;A, B, thenF∈ Kλμa, b, c;α, β, γ;A, B, whereFis given by2.19. Proof. Also, by using the same technique as in the proof ofTheorem 2.13, we get

1 1−α

_zh_z

Hz chz−α

≺1z

1−z

β

. 2.36

By lettingE0 andBz 1/1−α1/Hz c, we obtain

ReBz 1

1−α

1

_H_z _c2Re

Hz c>0. 2.37

Then, applyingLemma 2.3, we obtain the required result.

Acknowledgment

The work presented here was supported by Fundamental Research Grant Scheme UKM-ST-01-FRGS0055-2006.

References

1 C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, G ¨ottingen, Germany, 1975.

2 K. I. Noor, “On new classes of integral operators,” Journal of Natural Geometry, vol. 16, no. 1-2, pp. 71–80, 1999.

3 K. I. Noor and M. A. Noor, “On integral operators,” Journal of Mathematical Analysis and Applications, vol. 238, no. 2, pp. 341–352, 1999.

4 J. Liu, “The Noor integral and strongly starlike functions,” Journal of Mathematical Analysis and

Appli-cations, vol. 261, no. 2, pp. 441–447, 2001.

5 N. E. Cho, “The Noor integral operator and strongly close-to-convex functions,” Journal of

Mathemati-cal Analysis and Applications, vol. 283, no. 1, pp. 202–212, 2003.

6 B. C. Carlson and D. B. Shaﬀer, “Starlike and prestarlike hypergeometric functions,” SIAM Journal on

Mathematical Analysis, vol. 15, no. 4, pp. 737–745, 1984.

7 J. E. Hohlov, “Operators and operations on the class of univalent functions,” Izvestiya Vysshikh

Ucheb-nykh Zavedeni˘ı. Matematika, no. 10197, pp. 83–89, 1978,Russian.

8 N. Shukla and P. Shukla, “Mapping properties of analytic function defined by hypergeometric func-tion. II,” Soochow Journal of Mathematics, vol. 25, no. 1, pp. 29–36, 1999.

9 J. A. Kim and K. H. Shon, “Mapping properties for convolutions involving hypergeometric functions,”

(11)

10 K. I. Noor, “Integral operators defined by convolution with hypergeometric functions,” Applied

Math-ematics and Computation, vol. 182, no. 2, pp. 1872–1881, 2006.

11 K. I. Noor and H. A. Alkhorasani, “Properties of close-to-convexity preserved by some integral oper-ators,” Journal of Mathematical Analysis and Applications, vol. 112, no. 2, pp. 509–516, 1985.

12 H. Silverman, “On a class of close-to-convex functions,” Proceedings of the American Mathematical

Soci-ety, vol. 36, no. 2, pp. 477–484, 1972.

13 K. S. Padmanabhan and R. Parvatham, “Some applications of diﬀerential subordination,” Bulletin of

the Australian Mathematical Society, vol. 32, no. 3, pp. 321–330, 1985.

14 P. Eenigenburg, S. S. Miller, P. T. Mocanu, and M. O. Reade, “On a Briot-Bouquet diﬀerential subor-dination,” in General Inequalities, vol. 64 of Internationale Schriftenreihe zur Numerischen Mathematik, pp. 339–348, Birkh¨auser, Basel, Switzerland, 1983.

15 S. S. Miller and P. T. Mocanu, “Diﬀerential subordinations and inequalities in the complex plane,”