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SOME PROPERTIES INVOLVING QUASI-CONVOLUTION PRODUCTS CONCERNING SOME SPECIAL CLASSES OF

UNIVALENT FUNCTIONS

Mugur Acu 1

Abstract. _{In this paper we give some sample properties involving} quasi-convolution products of functions with negative coefficients.

2000 Mathematics Subject Classification: 30C45

Keywords and phrases: Quasi-convolution product, uniform starlike func-tions with negative coefficients, uniform convex funcfunc-tions with negative co-efficients, Alexander integral operator, Bernardi integral operator

1. Introduction

Let _H(U) be the set of functions which are regular in the unit disc

U, A = _{f _{∈ H}(U) : f(0) = f′₍₀₎ ₋ _{1 = 0}_}_, _H_u(_U_{) =} _{_f _{∈ H}₍_U_{) :}

f is univalent in U_} and S =_{f _∈A:f is univalent in U_}. Let define the Alexander integral operator IA:A→A ,

f(z) = IAF(z) = z

Z

0

F(t)

t dt , z ∈U (1)

and the Bernardi integral operator Ia:A→A ,

f(z) =IaF(z) = 1 +a

za z

Z

0

F(t)_·ta−1

dt , a= 1,2,3, ... (2)

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It is easy to observe that for f(z) _∈ A, f(z) = z + ∞

X

j=2

ajzj, we have

IAf(z) = z+ ∞

X

j=2

aj

j z

j

and Iaf(z) =z+ ∞

X

j=2

a+ 1

a+jajz

j .

If we consider the functionsf(z), g(z)_∈A,f(z) =z₋

∞

X

j=2

aj ·zj , aj ≥0,

j = 2,3, ... z _∈U andg(z) =z₋

∞

X

j=2

bj·zj , bj ≥0 , j = 2,3, ... z ∈U, we define the quasi-convolution product of the functionsf and g by

(f_∗g)(z) =z₋

∞

X

j=2

aj ·bjzj.

Similarly, the quasi-convolution product of more than two functions can also be defined. The quasi-convolution product was used in previously papers by Kumar (see [4], [5]), Owa (see [8], [9]), Misra (see [7]) and many others.

The purpose of this note is to obtain some properties regarding certain subclasses of functions with negative coefficients, by using the integral oper-ators defined above and quasi-convolution products.

2. Preliminary results

Definition 1 [10] Let T be the set of all functions f _∈S having the form:

f(z) = z₋

∞

X

j=2

aj·zj , aj ≥0 , j = 2,3, ... z ∈U (3)

Theorem 1 [10] If f _∈A, having the form (3), then the next two assertions are equivalent:

(i)

∞

X

j=2

j_·aj ≤1

(ii) f _∈T

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Remark 1 Using the definitions of the Alexander and Bernardi integral op-erators and the previously theorem, we observe that for f _∈ T we have

IA(f) _∈ T and Ia(f) _∈ T. More, we observe that for f _∈ T, having the form (3), we have aj ∈

h

0,1 j

i

for all j _≥ 2. If we denote T_[0,1 j] = (

f(z) =z₋

∞

X

j=2

aj ·zj , aj ∈

0,1 j

, j = 2,3, ... z _∈U )

, we have T _⊂

T_[0_,1

j]. Also,it is easy to observe that f ∈ T[0,1j] we have IA(f) ∈ T[0,1j] and Ia(f)_∈T_[0,1

j].

Definition 2 [2] A functionf is said to be uniformly starlike in the unit disc

U iff is starlike and has the property that for every circular arc γ contained inU, with centerα also inU, the arcf(γ)is starlike with respect tof(α).We let U S∗ _{denote the class of all such functions. By taking} _{U T}∗ ₌_T _∩_{U S}∗ _we define the class of uniformly starlike functions with negative coefficients.

Remark 2 An arc f(γ) is starlike with respect to a point w0 = f(α) if

arg(f(z)₋w0) is nondecreasing as z traces γ in the positive direction. Theorem 2 [2] Let f _∈S, f(z) = z+P∞

n=2

an·zn, an∈C. If ∞

P

n=2

n_{· |}an| ≤ √

2 2 then f _∈U S∗_.

Definition 3 [3] A function f is uniformly convex in the disc U if f _∈SC (convex) and for every circular arc γ contained in U, with center α also in U , the arc f(γ) is also convex. We let U SC _{denote the class of all such} functions. By takingU TC ₌_T

∩U SC _{we define the class of uniformly convex} functions with negative coefficients.

Remark 3 The arc γ(t), a < t < b is convex if the argument of the tangent to γ(t) is nondecreasing with respect to t.

Theorem 3 [11] Let f _∈ T, f(z) = z ₋ P∞

n=2

an·zn. Then f ∈ U TC if and

only if :

∞

X

n=2

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Remark 4 In [6] the author showed that the Libera integral operator f :

A _→ A defined by f(z) = I1F(z) = 2 z

z

R

0

F(t)dt , z _∈ U preserve the class

U T∗_{. Also, in [1] is showed that the Alexander integral operator, defined} by (1), preserve the classes U T∗_, _{U T}C _{and the Bernardi integral operator,} defined by (2), preserve the class U TC_.

3. Main results

Remark 5 By using the series expansions of the functionsf1, f2, the Theo-rem 1 and the definition of the quasi-convolution product it is easy to observe that for f1 ∈T , f2 ∈T_[₀_,1

j], we have

f1∗f2 ∈T .

Theorem 4 If f1 ∈T , f2, f3 ∈T_[₀_,1

j], then f

1∗f2∗f3 ∈U TC.

Proof. Letf1(z) = z− ∞

X

j=2

a1_j _·zj, a1_j _≥0, j _≥2,f2(z) =z₋

∞

X

j=2

a2_j _·zj,a2 j ∈

h

0,1 j

i

, j _≥2,f3(z) =z− ∞

X

j=2

a3j ·zj, a 3 j ∈

0,1 j

, j _≥2, and (f1∗f2∗f3)(z) =z₋ P∞

j=2

cj·zj , where cj =a 1 j ·a

2 j ·a

3

j ≥0, j ≥2. From Remark 5 we have

f1∗f2∗f3 ∈T .

From Theorem 3 we obtain that is sufficient to prove that ∞

X

j=2

j(2j₋1)_·cj ≤1.

From a2 j, a

3 j ∈

h

0,1 j

i

, j _≥2 we have

j(2j ₋1)_·cj =j(2j−1)a 1 j ·a

2 j ·a

3 j ≤

2j₋1

j2 ·ja 1 j < ja

1

j. (4) Using Theorem 1 for the function f1 we have

∞

X

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From (4) and (5) we obtain ∞

X

j=2

j(2j ₋1)_·cj ≤1.This mean thatf1∗f2∗f3 ∈

U TC_{. Similarly, we obtain}

Corollary 1 If f1 ∈T , f2, . . . , f_p ∈T [0,1

j], p= 3,4, . . ., then f

1∗f2∗ · · · ∗

fp ∈U TC.

By using the Remark 4 and the Theorem 4 we obtain: Corollary 2 If f1 ∈ T , f2, f3 ∈ T_[₀_,1

j], then

IA(f1 ∗f2 ∗f3) ∈ U TC and

Ia(f1∗f2 ∗f3)∈U TC.

Theorem 5 If f1 ∈T , f2 ∈T_[₀_,1

j], then

IA(f1∗f2)∈U TC.

X

j=2

a1 j ·z

j_{, a}1

j ≥0, j ≥2,f2(z) =z− ∞

X

j=2

a2 j ·z

j_,_a2 j ∈

h

0,1 j

i

, j _≥ 2, and IA(f1∗f2)(z) = z− ∞

X

j=2

cj ·zj , where cj = 1 j ·a

1 j ·a

2 j ≥

0, j _≥2. From Remark 5 we havef1∗f2 ∈T, and from Remark 1 we obtain

IA(f1∗f2)∈T.

From Theorem 3 we obtain that is sufficient to prove that ∞

X

j=2

j(2j₋1)_·cj ≤1.

From a2 j ∈

h

0,1_ji , j _≥2 we have

j(2j₋1)_·cj =j(2j −1)· 1

j ·a

1 j ·a

2 j ≤

2j ₋1

j2 ·ja 1 j < ja

1

∞

X

j=2

ja1

j ≤1. (7)

X

j=2

j(2j ₋1)_·cj ≤1. This mean that IA(f1 ∗

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Corollary 3 If f1 ∈ T , f2, . . . , fp ∈ T_[0,1

j], p = 2,3, . . ., then IA(f

1∗f2 ∗

· · · ∗fp)_∈U TC_.

j], then

IA(f1)∗IA(f2)∈U TC.

X

j=2

a1_j _·zj, a1_j _≥0, j _≥2,f2(z) =z₋

∞

X

j=2

a2_j _·zj,a2 j ∈

h

0,1_ji , j _≥2, andIA(f1)∗IA(f2)(z) =z− ∞

X

j=2

cj ·zj , wherecj = 1 j2·a

1 j·a

2 j ≥

0, j _≥2. From Remarks 1 and 5 we obtainIA(f1)∗IA(f2)∈T. From Theorem 3 we obtain that is sufficient to prove that

∞

X

j=2

j(2j₋1)_·cj ≤1.

From a2

j ∈[0,1), j ≥2 we have

j(2j₋1)_·cj =j(2j−1)· 1

j2 ·a 1 j ·a

2 j ≤

2j₋1

j2 ·ja 1 j < ja

1

∞

X

j=2

ja1_j _≤1. (9)

X

j=2

j(2j₋1)_·cj ≤1. This mean that IA(f1)∗

IA(f2)∈U TC. Similarly, we obtain

Corollary 4 If f1 ∈ T , f2, . . . , fp ∈ T_[0,1 j],

p = 2,3, . . ., then IA(f1) ∗

IA(f2)∗ · · · ∗IA(fp)∈U TC. Theorem 7 If f1 ∈T , f2 ∈T_[₀_,1

j], then f

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Proof. Letf1(z) =z− ∞

X

j=2

aj ·zj, aj ≥0, j ≥2,f2(z) =z− ∞

X

j=2

bj ·zj, bj ∈

h

0,1 j

i

, j _≥2, and (f1∗f2)(z) =z− ∞

X

j=2

cj ·zj, cj =aj ·bj ≥0, j ≥2. From Remark 5 we havef1∗f2 ∈T. In this conditions, from Theorem 2, we obtain that is sufficient to prove that

∞

X

j=2

√

2_·j_·cj ≤1. Using f1 ∈T, from Theorem 1 we have

∞

X

j=2

j_·aj ≤1. (10)

Using bj ∈

h

0,1 j

i

, j _≥2, we have

√

2_·j_·cj =

√

2_·j_·aj ·bj ≤j·aj ·

√

2

j < j·aj. (11)

X

j=2

√

2_·j_·cj ≤1.This mean thatf1∗f2 ∈

U T∗_{.Similarly, we obtain}

Corollary 5 If f1 ∈T , f2, . . . , fp ∈T_[0,1 j],

p= 2,3, . . ., then f1∗f2∗ · · · ∗

fp ∈U T∗.

By using Remark 1 and Theorem 7 we obtain Theorem 8 If f1 ∈T , f2 ∈T_[₀_,1

j], then IA(f

1)∗IA(f2)∈U T∗ and Ia(f1)∗

Ia(f2)∈U T∗.

Corollary 6 If f1 ∈ T , f2, . . . , fp ∈ T_[0,1 j],

p = 2,3, . . ., then IA(f1) ∗

IA(f2)∗ · · · ∗IA(fp)∈U T∗ and Ia(f1)∗Ia(f2)∗ · · · ∗Ia(fp)∈U T∗. By using Remark 4 and Theorem 7 we obtain

j], then

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Corollary 7 If f1 ∈ T , f2, . . . , fp ∈ T_[0,1

j], p = 2,3, . . ., then IA(f

1∗f2 ∗

· · · ∗fp)_∈U T∗_.

References

[1] M. Acu,Some preserving properties of certain integral opeartors, Gen-eral Mathematics, Vol. 8(2000), no. 3-4, 23-30.

[2] A.W.Goodman,On uniformly Starlike Functions, Journal of Math.Anal. and Appl., 364-370, 155(1991).

[3] A.W.Goodman, On uniformly Convex Functions , Annales Polonici Matematici, LVIII, 86-92, 1991.

[4] V. Kumar, Hadamard product of certain starlike functions, J. Math. Anal. Appl., 113(1986), 230.

[5] V. Kumar, Hadamard product of certain starlike functions, J. Math. Anal. Appl., 126(1987), 70.

[6] I.Magda’s, An integral operator which preserve the class of uniformly starlike functions with negative coefficients, Mathematica Cluj, 41(64), 1(1999).

[7] A.K. Misra, Quasi-Hadamard product of analytic functions related to univalent functions, The Mathematics Student, 64(1995), 221.

[8] S. Owa,On the classes of univalent functions with negative coefficients, Math. Japan, 27(4)(1982), 409.

[9] S. Owa, On the Hadamard products of univalent functions, Tamkang J. Math., 14(1)(1983), 15.

[10] G.S.Sˇalˇagean,Geometria Planului Complex, Ed.Promedia Plus, Cluj-Napoca, 1997.

[11] K.G.Sumbramanian, G.Murugudusundaramoorthy, P.Balasubrahman-yam, H.Silvermen, Subclasses of uniformly convex and uniformly starlike functions , Math.Japonica, (3)1996, 517-522.

Author:

Mugur Acu

Department of Mathematics

University ”Lucian Blaga” of Sibiu Sibiu

Romania