Univalence of Certain Linear Operators Defined by Hypergeometric Function

Volume 2009, Article ID 807943,12pages doi:10.1155/2009/807943

Research Article

Univalence of Certain Linear Operators Defined by

Hypergeometric Function

R. Aghalary and A. Ebadian

Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran

Correspondence should be addressed to A. Ebadian,[email protected]

Received 11 January 2009; Accepted 22 April 2009

Recommended by Vijay Gupta

The main object of the present paper is to investigate univalence and starlikeness of certain integral operators, which are defined here by means of hypergeometric functions. Relevant connections of the results presented here with those obtained in earlier works are also pointed out.

Copyrightq2009 R. Aghalary and A. Ebadian. 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 and Preliminaries

LetHdenote the class of all analytic functionsf in the unit diskD {z ∈C:|z|< 1}. For

n≥0, a positive integer, let

An

f∈ H:fz z ∞

k1

ankznk

, 1.1

withA1:A, whereAis referred to as the normalized analytic functions in the unit disc. A functionf∈ Ais called starlike inDiffDis starlike with respect to the origin. The class of all starlike functions is denoted byS∗:S∗0. Forα <1, we define

S∗α

f ∈ A: Re

zfz fz

> α, z∈D

, 1.2

and it is called the class of all starlike functions of orderα. Clearly,S∗α⊆ S∗for 0< α <1. For functionsfjz, given by

fjz

∞

k0

ak,jzk, j 1,2

we define the Hadamard productor convolutionoff1zandf2zby

f1∗f2z: ∞

k0

ak,1ak,2zk: f2∗f1

z. 1.4

An interesting subclass of S the class of all analytic univalent functions is denoted by

Uα, μ, λand is defined by

U α, μ, λ

f ∈ A:1−α

z fz

μ

α

z fz

μ1

fz−1< λ, z∈D

, 1.5

where 0< α≤1, 0≤μ < αn,andλ >0.

The special case of this class has been studied by Ponnusamy and Vasundhra1and Obradovi´c et al.2.

For a,b,c∈C and c/0,-1,-2,. . ., the Gussian hypergeometric series Fa,b;c;zis defined as

Fa, b;c;z ∞

n0

anbn

cn

zn

n!, z∈D, 1.6

wherea_n aa1a2· · ·an−1anda₀ 1. It is well-known thatFa, b;c;zis analytic inD. As a special case of the Euler integral representation for the hypergeometric function, we have

F1, b;c;z Γc

ΓbΓc−b

1

0

1 1−tzt

b−1_1−tc−b−1_{dt, z∈D, Rec >Reb >0. 1.7}

Now by letting

φa;c;z:F1, a;c;z, 1.8

it is easily seen that

zφa;c1;zcφa;c;z−cφa;c1;z. 1.9

Forf∈ A, Owa and Srivastava3introduced the operatorΩλ_{:A → Adefined by}

Ωλ_fz Γ2−λ

Γ1−λz

λ d

dz

z

0 ft

which is extensions involving fractional derivatives and fractional integrals. Using definition ofφa;c;z:F1, a;c;zwe may write

Ωλ_{fz zφ2; 2−λ;z∗fz. 1.11}

This operator has been studied by Srivastava et al.4and Srivastava and Mishra5. Also forλ <1,Reα >0,andfz z∞_k2akzk, let us define the functionFby

Fz:λz1−λ α

₁

0

t1/α−2ftzdt

z 1−λ ∞

k2 ak

k−1α1z

k_.

1.12

This operator has been investigated by many authors such as Trimble 6, and Obradovi´c et al.7.

If we take

ψ m, γ, z1 1−m ∞

k2

1

k−1γ1z

k_{, 1.13}

then we can rewrite operatorFdefined by1.11as

Fz z

ψλ, α, z∗fz z

. 1.14

From the definition ofψm, γ, zit is easy to check that

zψ m, γ, z 1

γψ m, γ, z

1 γ

1 1−m z

1−z

. 1.15

Forf ∈Uα, μ, λwithz/fzμ∗φa;c1;z/0 for allz∈Dwe define the transform

Gby

Gz z

1

z/fzμ_∗φa

;c1;z 1/μ

, 1.16

wherea, c∈Candc /0,−1,−2, . . . .

Also for f ∈ Uα, μ, λ with z/fzμ ∗ψm, γ, z/0 for all z ∈ D we define the transformHby

Hz z

1

z/fzμ_{∗ψm, γ, z}

1/μ

, 1.17

In this investigation we aim to find conditions on α, μ, λ such that f ∈ Uα, μ, λ

implies that the functionfto be starlike. Also we find conditions onα, μ, λ, m, γ, a, cfor each

f∈Uα, μ, λ; the transformsGandHbelong toUα, μ, λandS∗. For proving our results we need the following lemmas.

Lemma 1.1cf. Hallenbeck and Ruscheweyh8. Lethzbe analytic and convex univalent in the unit diskDwithh0 1. Also let

gz 1b1zb2z2· · · 1.18

be analytic inD. If

gz zgz

c ≺hz z∈U;c /0, 1.19

then

gz≺ψz c zc

_z

0

tc−1_{htdt≺hz z∈D;Rec≥0;c /0. 1.20}

andψzis the best dominant of 1.20.

Lemma 1.2 cf. Ruscheweyh and Stankiewicz 8. If f andg are analytic and F and G are convex functions such thatf ≺F, g≺G,thenf∗g≺F∗G.

Lemma 1.3cf. Ruscheweyh and Sheil-Small9. LetFandGbe univalent convex functions in D. Then the Hadamard productF∗Gis also univalent convex inD.

2. Main Results

We follow the method of proof adopted in1,10.

Theorem 2.1. Let n be positive integer withn≥2. Also letn1/2n < α≤1andn1−α< μ < αn. Iffz zan1zn1· · · belongs toUα, μ, λ, Thenf ∈S∗γwhenever0< λ≤ λα, μ, n, γ,

where

λ α, μ, n, γ: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

αn−μ2α 1−γ−1

αn−μ2μ2_{2α 1−γ−1}

, 0≤γ≤ μ−n1−α μ1n ,

αn−μ 1−γ nμγ −μ ,

μ−n1−α

μ1n < γ <1.

2.1

Proof. Let us define

pz

z fz

μ

Sincef ∈Uα, μ, λ, we have

1−α

z fz

μ

α

z fz

μ1

fz pz−α μzf

_z

1 αn−μan1zn· · · 1λωz,

2.3

whereωzis an analytic function with|ωz|<1 andω0 ω0 · · ·ωn−1_{0 0.By}

Schwarz lemma, we have|ωz| ≤ |z|n_{. By2.3, it is easy to check that}

pz 1−μλ

α

1

0 ωtz tμ/α1dt,

1−α αzf _z

fz

1λωz

1−μλ/α1₀ωtz/ tμ/α1_dt.

2.4

Therefore

1 1−γ

zfz fz −γ

α−1−αγ

/ 1−γα−μλ1₀ ωtz/tμ/α1_{dt α/ 1−γ1λωz} αα−μλ1₀ ωtz/tμ/α1_dt .

2.5

We need to show thatf ∈S∗γ. To do this, according to a well-known result9and2.5it suffices to show that

α−1−αγ/ 1−γα−μλ₀1 ωtz/tμ/α1_{dt α/1−γ1λωz}

αα−μλ1₀ ωtz/tμ/α1_dt /−iT, T ∈R, 2.6

which is equivalent to

λ

⎡

⎣ωz μ

αγ1−α/α−i 1−γT1₀ ωtz/tμ/α1_dt α 1−γ1iT

⎤

⎦_{/ −}1, T∈R. 2.7

Suppose thatBndenote the class of all Schwarz functionsωsuch thatω0 ω0

· · ·ωn−1_{0 0,and let}

M sup

z∈D,ω∈Bn,T∈R

ωz μ

αγ1−α/α−i 1−γT1₀ ωtz/tμ/α1_dt α 1−γ1iT

then,f ∈S∗γifλM ≤1. This observation shows that it suffices to findM. First we notice that

M≤sup

T∈R

⎧ ⎪ ⎨ ⎪ ⎩

1 μ/ n−μ/α αγ1−α2/α2 ₁−_γ2_T2 α 1−γ1T2

⎫ ⎪ ⎬ ⎪

⎭. 2.9

Defineφ:0,∞ →Rby

φx αn−μ

μ αγ1−α2 1−γ2α2_x

αn−μα 1−γ√1x . 2.10

Differentiatingφwith respect tox, we get

φx μ αn−μ

α3 _1−γ3√_{1x/2 αγ1−α}2 _1−γ2_α2_x αn−μ2α2 _1−γ2_1x

−

αn−μα 1−γ αn−μμ αγ1−α2 1−γ2α2_x

/2√1x αn−μ2α2 _1−γ2_1x .

2.11

Case 1. Let 0< γ <μ−n1−α/μ1n.Then we see thatφhas its only critical point in the positive real line at

x0

1

1−γ2α2

#

μ2 _{2α1−γ−1}2

αn−μ2 − αγ1−γ

2

$

. 2.12

Furthermore, we can see thatφx >0 for 0≤ x < x0 andφx <0 forx > x0. Henceφx attains its maximum value atx0and

φx≤φx0 αn−μ

2_μ2_{2α 1−γ−1}

αn−μ2α 1−γ−1 αn−μ2μ2₂α₁−γ−₁2

. _2.13

Case 2. Letγ >μ−n1−α/μ1n, then it is easy to see thatφx<0,and soφxattains its maximum value at 0 and

φx≤φ0 nμγ−μ

αn−μ 1−γ, ∀x≥0. 2.14

By puttingγ0 inTheorem 2.1we obtain the following result.

Corollary 2.2. Letnbe the positive integer withn≥2. Also letn1/2n < α≤1andn1−α< μ < αn. Iffz zan1zn1· · · belongs toUα, μ, λ, thenf ∈S∗ whenever0 < λ ≤αn− μ√2α−1/

αn−μ2_μ22α−1.

Remark 2.3. Takingα1, μ1 inTheorem 2.1andCorollary 2.2we get results of10. We follow the method ofproof adopted in11.

Theorem 2.4. Letn≥2, a /0, c∈Cwith Re c≥0/cand the functionϕz 1b1zb2z2_{· · ·} withbn/0be univalent convex inD. Iffz zan1zn1· · · ∈Uα, μ, λandφa;c;zdefined by1.8satisfy the conditions

z fz

μ

∗φa;c1;z/0 ∀z∈D, φa;c;z≺ϕz,

2.15

then the transformGdefined by1.16has the following: 1G∈Uα, μ, λ|bn||c|/|cn|,

2G∈S∗whenever

0< λ≤ |cn| αn−μ

√

2α−1

|bn||c| αn−μ

2

μ2_2α−1

. 2.16

Proof. From the definition ofGwe obtain

z Gz

μ

z fz

μ

∗φa;c1;z. 2.17

Differentiatingz/Gzμshows that

z

z Gz

μ

μ

z Gz

μ

−μ

z Gz

_μ1

Gz. 2.18

It is easy to see that

z

z fz

μ

∗φa;c1;z z

z fz

μ

∗φa;c1;z

. 2.19

From1.9and2.19we deduce that

z

z fz

μ

∗φa;c1;z

c

z fz

μ

∗φa;c;z−c

z fz

μ

or z z Gz μ c z Gz μ c z fz μ

∗φa;c;z. 2.21

Let us define

pz 1−α z Gz μ α z Gz μ1

Gz:1dnzn· · · , 2.22

thenpzis analytic inD, withp0 1 andp0 · · ·pn−1_{0 0.Combining2.18with}

2.21, one can obtain

pz

1αc

μ z Gz μ − αc μ z fz μ

∗φa;c;z. 2.23

Differentiatingpzyields

zpz

1αc

μ z z Gz μ −αc μz z fz μ

∗φa;c;z. 2.24

In view of2.21,2.23, and2.24, we obtain

cpz zpz c

1αc

μ

z Gz

μ

−αc2 μ

z fz

μ

∗φa;c;z

1αc

μ z z Gz μ −αc μ z z fz μ

∗φa;c;z

c

1αc

μ

z fz

μ

∗φa;c;z

−αc2 μ

z fz

μ

∗φa;c;z−αc

μ

z fz

μ

∗φa;c;z

c

z fz

μ

∗φa;c;z−cα

# _z fz μ − z fz _μ1 fz

$

∗φa;c;z

c

#

1−α z fz μ α z fz _μ1 fz

$

∗φa;c;z.

2.25

Hence

pz 1 czp

_z

#

1−α z fz μ α z fz μ1 fz

$

Since 1λzn_{andϕz 1b1zb2z}2_{· · · are convex and}

1−α

z fz

μ

α

z fz

μ1

fz≺1λzn, φa;c;z≺ϕz, 2.27

by using Lemmas1.2and1.3, from2.26we deduce that

pz 1 czp

_z≺1b

nλzn. 2.28

It now follows fromLemma 1.1that

pz≺ψz c zc

_z

0

tc−11bnλzndt. 2.29

Therefore

pz≺1 λbnc

cnz

n_{, 2.30}

and the result follows from the last subordination andCorollary 2.2.

It is well-known thatsee,12ifc, a >0 andc≥max{2, a}, thenφa;c;zis univalent convex function in D. So if we take ϕz φa;c;z in the Theorem 2.4, we obtain the following.

Corollary 2.5. Forn ≥2, c, a >0,andc≥ max{2, a}, let the functionfz zanzn1· · · ∈

Uα, μ, λandφa;c;zdefined by1.8satisfy the condition

z fz

μ

∗φa;c1;z/0 ∀z∈D. 2.31

Then the transformGdefined by1.16has the following: 1G∈Uα, μ, λ|a_n|c/|c_n|cn;

2G∈S∗whenever

0< λ≤ cn|cn| αn−μ

√

2α−1

|an|c αn−μ

2

μ2_2α−1

. 2.32

Corollary 2.6. Forn≥2, c∈Cwith Re c≥0/c, let the functionfz zanzn1· · · ∈Uα, μ, λ

andφa;c;zdefined by1.8satisfy the condition

z fz

μ

∗φa;c1;z/ ∀z∈D. 2.33

Then the transformGdefined by1.16has the following: 1G∈Uα, μ, λ|c|/|cn|;

2G∈S∗whenever

0< λ≤ |cn| αn−μ

√

2α−1

|c| αn−μ2μ2_2α−1

. 2.34

Remark 2.7. Takingα1 andμ1 onCorollary 2.6, we get a result of11. By puttingc1−Manda2 onTheorem 2.10we obtain the following.

Corollary 2.8. Letn≥2andϕz 1∞_k1bkzkwithbn/0be univalent convex function inD.

Also letM∈Cwith ReM <1andfz zan1zn1· · · ∈Uα, μ, λ, satisfy

ΩM

z fz

μ

/

0 z∈D, 2.35

and letGbe the function which is defined by

Gz z

1

ΩM z/fzμ

1/μ

. 2.36

If

φ2; 1−M;z≺ϕz, 2.37

then we have the following:

1G∈Uα, μ, λ|bn||1−M|/|n1−M|;

2G∈S∗whenever

0< λ≤ |1−Mn| αn−μ

√

2α−1

|bn||1−M| αn−μ

2

μ2_2α−1

. 2.38

In13, Pannusamy and Sahoo have also considered the classUα, μ, λfor the case

α1 withμn.

Theorem 2.10. Form <1, γ /0;Reγ >0, n≥2,letfz zan1zn1· · · ∈Uα, μ, λand ψm, γ, zdefined by1.13satisfy the condition

z fz

μ

∗ψ m, γ, z/0 ∀z∈D. 2.39

Then the transformHdefined by1.17has the following: 1H∈Uα, μ, λ1−m/|1nγ|;

2H∈S∗whenever

0< λ≤ 1nγ αn−μ

√

2α−1

1−m αn−μ2μ2_2α−1

. 2.40

Proof. Let us define

pz 1−α

z Hz

μ

α

z Hz

_μ1

Hz, 2.41

thenpzis analytic inD, withp0 1 andp0 · · ·pn−1_{0 0.Using the same method}

as onTheorem 2.4we get

pz γzpz

#

1−α

z fz

μ

α

z fz

_μ1 fz

$

∗

1 1−m z

1−z

. 2.42

Since 1λzn_{andhz 1 1−mz/1−zare convex,}

1−α

z fz

μ

α

z fz

_μ1

fz≺1λzn_{. 2.43}

Using Lemmas1.2and1.3, from2.42it yields

pz γzpz≺1−mλzn. 2.44

It now follows fromLemma 1.1that

pz≺ 1 γz1/γ

z

0

Therefore

_pz−1≤ λ 1−m 1nγ|z|

n_{, 2.46}

and the result follows from2.46andCorollary 2.2.

References

1 S. Ponnusamy and P. Vasundhra, “Criteria for univalence, starlikeness and convexity,” Annales Polonici Mathematici, vol. 85, no. 2, pp. 121–133, 2005.

2 M. Obradovi´c, S. Ponnusamy, V. Singh, and P. Vasundhra, “Univalency, starlikeness and convexity applied to certain classes of rational functions,”Analysis, vol. 22, no. 3, pp. 225–242, 2002.

3 S. Owa and H. M. Srivastava, “Univalent and starlike generalized hypergeometric functions,” Canadian Journal of Mathematics, vol. 39, no. 5, pp. 1057–1077, 1987.

4 H. M. Srivastava, A. K. Mishra, and M. K. Das, “A nested class of analytic functions defined by fractional calculus,”Communications in Applied Analysis, vol. 2, no. 3, pp. 321–332, 1998.

5 H. M. Srivastava and A. K. Mishra, “Applications of fractional calculus to parabolic starlike and uniformly convex functions,”Computers & Mathematics with Applications, vol. 39, no. 3-4, pp. 57–69, 2000.

6 S. Y. Trimble, “The convex sum of convex functions,”Mathematische Zeitschrift, vol. 109, pp. 112–114, 1969.

7 M. Obradovi´c, S. Ponnusamy, and P. Vasundhra, “Univalence, strong starlikeness and integral transforms,”Annales Polonici Mathematici, vol. 86, no. 1, pp. 1–13, 2005.

8 St. Ruscheweyh and J. Stankiewicz, “Subordination under convex univalent functions,”Bulletin of the Polish Academy of Sciences, Mathematics, vol. 33, no. 9-10, pp. 499–502, 1985.

9 St. Ruscheweyh and T. Sheil-Small, “Hadamard products of Schlicht functions and the P ¨olya-Schoenberg conjecture,”Commentarii Mathematici Helvetici, vol. 48, pp. 119–135, 1973.

10 S. Ponnusamy and P. Sahoo, “Geometric properties of certain linear integral transforms,”Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 12, no. 1, pp. 95–108, 2005.

11 M. Obradovi´c and S. Ponnusamy, “Univalence and starlikeness of certain transforms defined by convolution of analytic functions,”Journal of Mathematical Analysis and Applications, vol. 336, no. 2, pp. 758–767, 2007.

12 Y. Ling, F. Liu, and G. Bao, “Some properties of an integral transform,”Applied Mathematics Letters, vol. 19, no. 8, pp. 830–833, 2006.