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## CERTAIN CLASSES OF ANALYTIC FUNCTIONS OF COMPLEX ORDER

C.Selvaraj and C.Santhosh Moni

Abstract.By making use of the Hadamard product, we define a new class of analytic functions of complex order. Coefficient inequalities, sufficient condition and an interesting subordination result are obtained.

2000Mathematics Subject Classification: 30C45, 30C50.

Keywords and Phrases: analytic functions, Hadamard product, generalized hy-pergeometric functions, coefficient inequalities, subordinating sequence.

1. Introduction, Definitions And Preliminaries Let Adenote the class of functions of the form

f(z) =z+ ∞ X

k=2

akzk ak ≥0, (1)

which are analytic in the open disc U = {z ∈ C\ | z |< 1} and S be the class of

function f ∈ Awhich are univalent inU.

The Hadamard product of two functions f(z) = z+P∞k=2akzk and g(z) =

z+P∞k=2bkzk is given by

(f∗g)(z) =z+ ∞ X

k=2

akbkzk. (2)

For a fixed functiong∈ Adefined by

g(z) =z+ ∞ X

k=2

bkzk (bk≥0 fork≥2) (3)

We now define the following linear operator Dm

λ, gf : A −→ A by

(2)

Dλ, g1 f(z) = (1−λ)(f(z)∗g(z)) +λ z(f(z)∗g(z))′, (4)

Dλ, gm f(z) =D1λ, g(Dm−λ, g1f(z)). (5) If f ∈ A, then from (4) and (5) we may easily deduce that

Dmλ, gf(z) =z+ ∞ X

k=2

1 + (k−1)λmakbk

zk

(k−1)!, (6)

where m∈N0 =N∪ {0} and λ0.

Remark 1. It is interesting to note that several integral and differential operator follows as a special case ofDm

λ, gf(z), here we list few of them. 1. When g(z) = z/(1−z),Dm

λ, gf(z) reduces to an operator introduced recently by F. Al-Oboudi [1].

2. Let the coefficients bk be of the form

bk=

(α1)k−1 . . .(αq)k−1

(β1)k−1 . . . (βs)k−1(k−1)!

(7) (α1, . . . αq, β1, . . . βs∈C andβj 6= 0,−1,−2, . . . for j={1,2,. . . ,s}) and ifm= 0, thenDλ, gm f(z) reduces to the well-known Dziok-Srivastava opera-tor. It is well known that Dziok-Srivastava operator includes as its special cases various other linear operator which were introduced and studied by Hohlov, Carlson and Shaffer and Ruscheweyh. For details we refer to [6, 7, 8]

Apart from these, the operator Dm

λ, gf(z) generalizes the well-known operators like S˘al˘agean operator [14] and Bernardi-Libera-Livingston operator.

Using the operatorDm

λ, gf(z), we defineHmλ(g, δ;A, B) to be the class of functions

f ∈ Asatisfying the inequality

1 +1

δ

Dm+1

λ, g f(z)

Dm λ, gf(z)

−1

≺ 1 +Az

1 +Bz, z∈ U, (8)

whereδ ∈C\ {0},A andB are arbitrary fixed numbers,1B < A1,mN0.

(3)

1. If we letg(z) = z

1−z, λ= 1 in (8), then the class H

m

λ (g, δ;A, B) reduces to the well- known class

Hm(δ;A, B) :=

f : 1 + 1

δ

Dm+1f(z) Dmf(z) −1

≺ 1 +Az

1 +Bz

where Dmf is the well- known S˘al˘agean operator. The class Hm(δ;A, B) was introduced and studied by Attiya [4].

2. If we letg(z) =z+ ∞ X

n=2

(α1)k−1 . . . (αq)k−1

(β1)k−1 . . .(βs)k−1(k−1)!

akzk in (8), then we have following class of functions

Hmλ(α1, β1;A, B) :=

f : 1 + 1

δ

Dm+1(α

1, β1)f(z) Dm(α

1, β1)f(z) −1

≺ 1 +Az

1 +Bz

introduced and studied by C. Selvaraj and K.R.Karthikeyan in [17]. 3. If we letg(z) = z

1−z, λ= 1, m= 0 in (8), then the class H

m

λ(g, δ;A, B) reduces to the well- known class of starlike functions of complex order [12].

Further, we note that several subclass of analytic functions can be obtained by specializing the parameters in Hm

λ(g, δ;A, B) (see for example [3, 11, 12]).

We use Ω to denote the class of bounded analytic functions w(z) in U which satisfy the conditions w(0) = 1 and|w(z)|<1 for z∈ U.

2.Coefficient estimates

Theorem 1. Let the function f(z) defined by (1) be in the class Hm

λ(g, δ;A, B). (a) If (A−B)2|δ |2>(k1)2B(AB)λ Re{δ}+ (1B2)λ2(k1) , let

G= (A−B)

2|δ |2

(k−1)2B(A−B)λ Re{δ}+ (1−B2)λ2(k1) , k= 2,3, . . . , m−1, M = [G] (Gauss symbol) and [G] is the greatest integer not greater than G. Then, for j= 2,3, . . . , M + 2

|aj |≤

1

[1 + (j−1)λ]mλj−1(j1)!bj

j Y

k=2

(4)

and for j > M + 2

|aj |≤

1

[1 + (j−1)λ]mλj−1(j1) (M+ 1)!b

j MY+3

k=2

|(A−B)δ−(k−2)B|.

(10)

(b) If (A−B)2|δ |2≤(k−1)2B(A−B)λ Re{δ}+ (1−B2)λ2(k−1) , then

|aj |≤

(A−B)|δ| λ(j−1) [1 + (j−1)λ]mb

j

j≥2. (11)

The bounds in (9) and (11) are sharp for all admissible A, B, δ ∈ C\ {0} and for

each j.

Proof. Since f(z)∈ Hm

λ(g, δ;A, B), the inequality (8) gives

|Dλ, gm+1f(z)−Dλ, gm f(z)|=[(A−B)δ+B]Dmλ, gf(z)−BDλ, gm+1f(z) w(z). (12) Equation (12) may be written as

∞ X

k=2

[1 + (k−1)λ]mλ(k−1)bkakzk (13)

=

(A−B)δz+ ∞ X

k=2

[(A−B)δ−B(k−1)λ][1 + (k−1)λ]mbkakzk

w(z).

Or equivalently j

X

k=2

[1 + (k−1)λ]mλ(k−1)bkakzk+ ∞ X

k=j+1 ckzk

=

(A−B)δz+ j−1

X

k=2

[(A−B)δ−B(k−1)λ][1+(k−1)λ]mbkakzk

w(z), (14) for certain coefficients ck. Explicitlyck = [1 + (k−1)λ]mλ(k−1)bkak−[(A−B)δ−

B(k−2)λ][1 + (k−2)λ]mb

k−1ak−1z−1. Since |w(z)|<1, we have

j X

k=2

[1 + (k−1)λ]mλ(k−1)bkakzk+ ∞ X

k=j+1 ckzk

(5)

(A−B)δz+ j−1

X

k=2

[(A−B)δ−B(k−1)λ][1 + (k−1)λ]mbkakzk .

Let z=reiθ, r <1, applying the Parseval’s formula (see [9] p.138) on both sides of the above inequality and after simple computation , we get

j X

k=2

[1 + (k−1)λ]2mλ2(k−1)2b2k|ak|2r2k+ ∞ X

k=j+1

|ck|2r2k

≤(A−B)2 |δ |2 r2+ j−1

X

k=2

|(A−B)δ−B(k−1)λ|2 [1 + (k−1)λ]2mb2k |ak |2 r2k. Let r−→1−, then on some simplification we obtain

[1 + (j−1)λ]2mλ2(j−1)2bj2 |aj |2≤(A−B)2 |δ|2 +

j−1

X

k=2

|(A−B)δ−B(k−1)λ|2 −(k−1)2λ2 [1 + (k−1)λ]2mb2k |ak|2 j ≥2. (16) Now the following two cases arises:

(a) Let (A−B)2|δ |2>(k−1)λ2B(A−B)Re(δ) + (1−B2)λ(k−1) ,suppose that j≤M + 2, then forj = 2, (16) gives

|a2 |≤

(A−B)|δ |

(1 +λ)mλ b

2 ,

which gives (9) for j = 2. We establish (9) for j ≤ M + 2, from (16), by mathematical induction. Suppose (9) is valid for j = 2,3, . . . , (k−1). Then it follows from (16)

[1 + (j−1)λ]2mλ2(j−1)2b2j |aj |2

≤(A−B)2|δ |2

+ j−1

X

k=2

|(A−B)δ−B(k−1)λ|2 −(k−1)2λ2 [1 + (k−1)λ]2mb2k |ak|2

≤(A−B)2|δ |2

+ j−1

X

k=2

(6)

×

1 [1 + (k−1)λ]2mb2

k{λk−1(k−1)!}2 k Y

n=2

|(A−B)δ−(n−2)B |2

= (A−B)2 |δ |2 + j−1

X

k=2

|(A−B)δ−B(k−1)λ|2−(k−1)2λ2

×

1

{λk−1(k1)!}2

k Y

n=2

|(A−B)δ−(n−2)B|2

= (A−B)2 |δ|2 +

|(A−B)δ−Bλ|2 −λ2 1

λ2(1!)2(A−B)

2|δ |2 + |(A−B)δ−2Bλ|2−4λ2 1

λ4(2!)2 (A−B)

2 |δ |2|(AB)δ|2

+. . .up to (k=j−1)

= 1

{λj−2(j2)!}2

j Y

k=2

|(A−B)δ−(k−2)B |2.

Thus, we get

|aj |≤

1

[1 + (j−1)λ]mλj−1(j1)!b

j j Y

k=2

|(A−B)δ−(k−2)B |,

which completes the proof of (9).

Next, we suppose j > M+ 2. Then (16) gives

[1 + (j−1)λ]2mλ2(j−1)2bj2 |aj |2≤(A−B)2|δ |2

+ MX−2

k=2

|(A−B)δ−B(k−1)λ|2 −(k−1)2λ2 [1 + (k−1)λ]2mb2k |ak |2

+ j−1

X

k=M+3

|(A−B)δ−B(k−1)λ|2−(k−1)2λ2 [1 + (k−1)λ]2mb2k |ak|2 . On substituting upper estimates fora2, a3, . . . , aM+2obtained above and

(7)

(b) Let (A−B)2 |δ |2≤(k−1)λ2B(A−B)Re(δ) + (1−B2)λ(k−1) ,then it follows from (16)

[1 + (j−1)λ]2mλ2(j−1)2b2j |aj |2≤(A−B)2|δ |2 (j≥2), which proves (11).

The bounds in (9) are sharp for the functions f(z) given by

Dλ, gm f(z) = (

z(1 +Bz)(A−BB)δ if B6= 0,

z exp(Aδz) if B= 0.

Also, the bounds in (11) are sharp for the functionsfk(z) given by

Dλ, gm fk(z) =     

z(1 +Bz) (A−B)δ

Bλ(k−1) if B6= 0,

z exp

Aδ λ(k−1)zk−1

if B= 0.

Remark 2. If we letλ= 1, g(z) = z

1−z in Theorem 1, we get the result

due to Attiya [4].

3. A sufficient condition for a function to be in Hm

λ(g, δ;A, B) Theorem 2. Let the functionf(z) defined by (1) and let

∞ X

k=2

[1 + (k−1)λ]m(k−1)+|(A−B)δ−B(k−1)| λ bk |ak |≤(A−B)|δ | (17)

holds, then f(z) belongs to Hm

λ(g, δ;A, B).

Proof. Suppose that the inequality holds. Then we have for z∈ U |Dmλ, g+1f(z)−Dmλ, gf(z)| −(A−B)δDλ, gm f(z)−

BDmλ, g+1f(z)−Dmλ, gf(z) =

∞ X

k=2

[1 + (k−1)λ]mλ(k−1)bkakzk −

(A−B)δ

z+ ∞ X

k=2

[1 + (k−1)λ]mbkakzk

B

∞ X

k=2

(8)

∞ X

k=2

[1 + (k−1)λ]m(k−1)λ+|(A−B)δ−B(k−1)λ| bk|ak|rk−(A−B)|δ|r. Letting r−→1−, then we have

|Dmλ, g+1f(z)−Dmλ, gf(z)| −(A−B)δDλ, gm f(z)−

BDmλ, g+1f(z)−Dmλ, gf(z) ≤

∞ X

k=2

[1 + (k−1)λ]m(k−1)λ+|(A−B)δ−B(k−1)λ| bk|ak| −(A−B)|δ|≤0. Hence it follows that

Dm+1λ, g f(z)

Dm

λ, gf(z) −1

B

Dλ, gm+1f(z)

Dm

λ, gf(z) −1

−(A−B)δ

<1, z∈ U.

Letting

w(z) =

Dλ, gm+1f(z)

Dm

λ, gf(z) −1

B

Dm+1λ, g f(z)

Dm

λ, gf(z) −1

−(A−B)δ ,

then w(0) = 0, w(z) is analytic in|z|<1 and|w(z)|<1. Hence we have

Dλ, gm+1f(z)

Dm λ, gf(z)

= 1 + [B+δ(A−B)]w(z) 1 +Bw(z) which shows that f belongs to Hm

λ (g, δ;A, B).

3. Subordination Results For the Class Hm

λ(g, δ;A, B) Definition 1. A sequence {bk}

k=1 of complex numbers is called a subordinating

factor sequence if, whenever f(z) is analytic, univalent and convex in U, we have the subordination given by

∞ X

k=1

(9)

Lemma 1. The sequence{bk}

k=1 is a subordinating factor sequence if and only

if

Re

1 + 2

∞ X

k=1 bkzk

>0 (z∈ U). (19)

For convenience, we shall henceforth

σk(δ, λ, m, A, B) (20)

= [1 + (k−1)λ]mλ(k−1)+|(A−B)δ−B(k−1)| bk to be real.

LetHem

λ(g, δ;A, B) denote the class of functions f ∈ Awhose coefficients satisfy the conditions (17). We note that Hem

λ(g, δ;A, B)⊆ Hλm(g, δ;A, B). Theorem 3. Let the function f(z) defined by (1) be in the class Hem

λ(g, δ;A, B) where −1 ≤ B < A ≤ 1. Also let C denote the familiar class of functions f ∈ A

which are also univalent and convex in U. Then

σ2(δ, λ, m, A, B)

2(A−B)|δ|+σ2(δ, λ, m, A, B)

(f∗g)(z)≺g(z) (z∈ U;g∈ C), (21) and

Re(f(z))>−(A−B)|δ|+σ2(δ, λ, m, A, B)

σ2(δ, λ, m, A, B) (z∈ U). (22) The constant σ2(δ, λ, m, A, B)

2(A−B)|δ|+σ2(δ, λ, m, A, B)

is the best estimate. Proof. Let f(z)∈Hem

λ(δ;A, B) and let g(z) =z+ P∞

k=2bkzk∈ C. Then

σ2(δ, λ, m, A, B)

2(A−B)|δ|+σ2(δ, λ, m, A, B)

(f∗g)(z)

= σ2(δ, λ, m, A, B)

2(A−B)|δ|+σ2(δ, λ, m, A, B)

z+ ∞ X

k=2 akbkzk

.

Thus, by Definition 1, the assertion of the theorem will hold if the sequence

σ2(δ, λ, m, A, B)

2(A−B)|δ|+σ2(δ, λ, m, A, B) ak

k=1

is a subordinating factor sequence, with a1 = 1. In view of Lemma 1, this will be

true if and only if

Re

1 + 2

∞ X

k=1

σ2(δ, λ, m, A, B)

2(A−B)|δ|+σ2(δ, λ, m, A, B)

akzk

(10)

Now

Re

1+ σ2(δ, λ, m, A, B) (A−B)|δ|+σ2(δ, λ, m, A, B)

∞ X

k=1 akzk

=Re

1+ σ2(δ, λ, m, A, B)

(A−B)|δ|+σ2(δ, λ, m, A, B)a1z

+ 1

(A−B)|δ|+σ2(δ, λ, m, A, B)

∞ X

k=2

σ2(δ, λ, m, A, B)akzk

≥1− σ2(δ, λ, m, A, B)

(A−B)|δ|+σ2(δ, λ, m, A, B) r

+ 1

|(A−B)|δ|+σ2(δ, λ, m, A, B)|

∞ X

k=2

σk(δ, λ, m, A, B)|ak|rk

.

Since σk(δ, λ, m, A, B) is a real increasing function ofk (k≥2) 1− σ2(δ, λ, m, A, B)

(A−B)|δ|+σ2(δ, λ, m, A, B)

r+

1

|(A−B)|δ|+σ2(δ, λ, m, A, B)|

∞ X

k=2

σk(δ, λ, m, A, B)|ak|rk

>1−

σ2(δ, λ, m, A, B)

(A−B)|δ|+σ2(δ, λ, m, A, B)

r+ (A−B)|δ|

(A−B)|δ|+σ2(δ, λ, m, A, B) r

= 1−r >0.

Thus (23) holds true in U. This proves the inequality (21). The inequality (22) follows by taking the convex function g(z) = 1−zz = z +P∞k=2zk in (21). To prove the sharpness of the constant σ2(δ, λ, m, A,B)

2(A−B)|δ|+σ2(δ, λ, m, A,B)

, we consider f0(z) ∈

e

Hm

λ(g, δ;A, B) given by

f0(z) =z−

(A−B)|δ| σ2(δ, λ, m, A, B)

z2 (−1≤B < A≤1).

Thus from (21), we have

σ2(δ, λ, m, A, B)

2(A−B)|δ|+σ2(b, λ, m, A, B)

f0(z)≺ z

(11)

It can be easily verified that min

Re

σ2(δ, λ, m, A, B)

2(A−B)|δ|+σ2(δ, λ, m, A, B)f0(z)

=−1

2 (z∈ U). This shows that the constant σ2(δ, λ, m, A,B)

2(A−B)|δ|+σ2(δ, λ, m, A,B) is best possible.

Remark 3.By specializing the parameters, the above result reduces to various other results obtained by several authors.

References

[1] F. M. Al-Oboudi, On univalent functions defined by a generalized S˘al˘agean operator, Int. J. Math. Math. Sci. 2004, no. 25-28, 1429–1436.

[2] M. K. Aouf, S. Owa and M. Obradovi´c, Certain classes of analytic functions of complex order and type beta, Rend. Mat. Appl. (7) 11(1991), no. 4, 691–714.

[3] M. K. Aouf, H. E. Darwish and A. A. Attiya,On a class of certain analytic functions of complex order, Indian J. Pure Appl. Math. 32 (2001), no. 10, 1443– 1452.

[4] A. A. Attiya,On a generalization class of bounded starlike functions of com-plex order, Appl. Math. Comput. 187 (2007), no. 1, 62–67.

[5] B. C. Carlson and D. B. Shaffer,Starlike and prestarlike hypergeometric func-tions, SIAM J. Math. Anal. 15(1984), no. 4, 737–745.

[6] J. Dziok and H. M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput., 103(1999), 1-13.

[7] J. Dziok and H. M. Srivastava, Some sublasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeometric function, Adv. Stud. Contemp. Math., 5(2002), 115-125.

[8] J. Dziok and H. M. Srivastava,Certain sublasses of analytic functions associ-ated with the generalized hypergeometric function, Integral Transforms Spec. Funct., 14(2003), 7-18.

[9] P.L.Duren,Univalent functions, Springer - verlag, Berlin, 1983.

[10] Yu. E. Hohlov,Operators and operations in the class of univalent functions, Izv. Vyss. Ucebn. Zaved. Matematika, 10(1978), 83-89.

[11] G. Murugusundaramoorthy and N. Magesh, Starlike and convex functions of complex order involving the Dziok-Srivastava operator, Integral Transforms Spec. Funct. 18 (2007), no. 5-6, 419–425.

[12.] M. A. Nasr and M. K. Aouf, Starlike function of complex order, J. Natur. Sci. Math. 25(1985), no. 1, 1–12.

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[14.] G. S¸. S˘al˘agean, Subclasses of univalent functions, in Complex analysis— fifth Romanian-Finnish seminar, Part 1 (Bucharest, 1981), 362–372, Lecture Notes in Math., 1013, Springer, Berlin.

[15.] N. S. Sohi and L. P. Singh,A class of bounded starlike functions of complex order, Indian J. Math. 33(1991), no. 1, 29–35.

[16.] C.Selvaraj and K.R.Karthikeyan,Differential sandwich theorems for cer-tain subclasses of analytic functions, Math. Commun. 12(2008), no. 2.

[17.] C.Selvaraj and K.R.Karthikeyan, Certain classes of analytic functions of complex order involving a family of generalized differential operators, J. Math. Inequ. Appl. 2 (2008), no. 4, 449-458.

[18.] H. S. Wilf, Subordinating factor sequences for convex maps of the unit circle, Proc. Amer. Math. Soc. 12 (1961), 689–693.

C.Selvaraj

Department of Mathematics, Presidency College,

email: pamc9439@yahoo.co.in C. Santhosh Moni

R.M.K.Engineering College R.S.M.Nagar,

Kavaraipettai-601206,

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