An integral operator on the classes S*(α) and CVH(β)

A N N A L E S

U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N – P O L O N I A

VOL. LXVII, NO. 2, 2013 SECTIO A 53–58

NICOLETA ULARU and NICOLETA BREAZ

An integral operator on the classes S

(α) and CVH(β)

Abstract. The purpose of this paper is to study some properties related to convexity order and coefficients estimation for a general integral operator. We find the convexity order for this operator, using the analytic functions from the class of starlike functions of orderα and from the class CVH(β) and also we estimate the first two coefficients for functions obtained by this operator applied on the classCVH(β).

1. Preliminary and definitions. We consider the class of analytic func- tions f(z), in the open unit disk, U = {z ∈ C : |z| < 1}, having the form:

(1.1) f(z) = z +

∞ j=2

a_jz^j, z∈ U.

This class is denoted by A. By S we denote the class of all functions from A which are univalent in U.

We denote byK(α) the class of all convex functions of order α (0 ≤ α < 1) that satisfy the inequality:

Re

zf^(z) f^(z) + 1



> α, z∈ U.

2000 Mathematics Subject Classification. 30C45.

Key words and phrases. Analytic function, integral operator, starlike function, convex function, coefficients estimation.

A function f ∈ A is in the class S^∗(α), of starlike functions of order α if Re

zf^(z) f(z)



> α, z∈ U.

These classes were introduced by Robertson in [4] and studied by many other authors.

We also consider the class CVH(β) which was introduced by Acu and Owa in [1]. An analytic function f is in the classCVH(β) with β > 0 if we have the following inequality:

(1.2) 

zf^(z)

f^(z) − 2β(√

2 − 1) + 1

 < Re√

2zf^(z) f^(z)



+ 2β(√

2 − 1) +√ 2, where z∈ U.

Remark 1. This class is well defined for Re√

2^zf_f^(z)^(z)



>2β(1−√ 2)−√

2.

For this class the following result was proved by Acu and Owa in [1].

Theorem 1.1. If f (z) = z +_∞

j=2a_jz^j belongs to the classCVH(β), β > 0, then

|a₂| ≤ 1 + 4β

2(1 + 2β), |a₃| ≤ (1 + 4β)(3 + 16β + 24β²) 12(1 + 2β)³ . For the analytic functions f_i and g_i we consider the operator

(1.3) K(z) =

z

0

n i=1

(g_i^(t))^ηi·

f_i(t) t

_γi dt,

for γ_i, η_i >0 with i = 1, n. This operator was studied by Pescar in [3] and Ularu in [5].

We study the properties of this operator on the classesCVH(β) and S^∗(α).

The idea of this paper was given by an open problem considered by N. Breaz, D. Breaz and Acu in [2].

2. Main results. Let φ= 1 −ⁿ

i=1

η_i− (2 −√ 2)ⁿ

i=1

η_iβ_i+ⁿ

i=1

γ_i(α_i− 1),

where β_i>0, α_i ∈ [0, 1) and η_i, γ_i>0 for all i = 1, n. For (2.1)

n i=1

η_i+ (2 −√ 2)ⁿ

i=1

η_iβ_i+ⁿ

i=1

γ_i(α_i− 1) ≤ 1 we have that0 ≤ φ < 1.

Theorem 2.1. If f_i ∈ S^∗(α_i) and g_i ∈ CVH(β_i), with β_i >0, 0 ≤ α_i <1 and η_i, γ_i >0 for all i = 1, n satisfying the condition (2.1), then the integral operator K(z) defined by (1.3) is in the class K(φ), 0 ≤ φ < 1 where

φ= 1 −ⁿ

i=1

η_i− (2 −√ 2)ⁿ

i=1

η_iβ_i+ⁿ

i=1

γ_i(α_i− 1).

Proof. From the definition of K(z) we obtain:

zK^(z) K^(z) =ⁿ

i=1



η_izg^_i(z) g_i^(z)

 +ⁿ

i=1

 γ_i

zf_i^(z) f_i(z) − 1

 . Further we have:

√2Re

zK^(z) K^(z) + 1



= Reⁿ

i=1

√2ηizg_i^(z) g_i^(z) +√

2 +√ 2Re

n i=1

γ_izf_i^(z) f_i(z) −√

2Re

n i=1

γ_i. We use the fact that f_i are starlike functions of order α_i and g_i∈ CVH(β_i) for i= 1, n:

√2Re

zK^(z) K^(z) + 1



>

n i=1

η_i

zg_i^(z)

g_i^(z) − 2β_i(√

2 − 1) + 1



−

n i=1

2ηiβ_i(√

2 − 1) + ηi√ 2

+√ 2

+√ 2

n i=1

γ_iα_i−√ 2

n i=1

γ_i

> −√ 2ⁿ

i=1

η_i− 2(√

2 − 1)ⁿ

i=1

η_iβ_i+√ 2

+√ 2ⁿ

i=1

γ_iα_i−√ 2ⁿ

i=1

γ_i.

From these inequalities we obtain that:

Re

zK^(z) K^(z) + 1



>1 −ⁿ

i=1

η_i− (2 −√ 2)ⁿ

i=1

η_iβ_i+ⁿ

i=1

γ_i(α_i− 1).

So we obtain the convexity order for the operator K(z) for functions in the classesS^∗(α_i) and CVH(β_i) for i = 1, n. 

For η₁ = η₂ = · · · = η_n= 1 and γ₁ = γ₂ = · · · = γ_n= 1 in the definition of K(z) given by (1.3) we obtain:

K₁(z) = z

0

n i=1

g_i^(t) · f_i(t) t dt for i= 1, n.

Corollary 2.2. If f_i ∈ S^∗(α_i) and g_i ∈ CVH(β_i), for β_i >0, 0 ≤ α_i <1 for all i= 1, n, then the integral operator

K₁(z) = z

0

n i=1

g_i^(t) · f_i(t) t dt is convex of order φ, where

φ= 1 − n − (2 −√ 2)ⁿ

i=1

β_i+ⁿ

i=1

(α_i− 1), for 0 ≤ φ < 1.

Next we will obtain the estimation for the coefficients of the operator K₁(z) defined above.

Theorem 2.3. Let f_i ∈ CVH(γ_i), g_i ∈ CVH(β_i), with β_i, γ_i >0 and g_i(z) = z+_∞

j=2a_i,jz^j, f_i(z) = z +_∞

j=2b_i,jz^j for all i = 1, n. If K1(z) = z +

_∞

j=2c_jz^j, then we obtain:

|c₂| ≤ 1 2

 _n



i=1

1 + 4γ_i

2(1 + 2γ_i)+ⁿ

i=1

1 + 4β_i 1 + 2β_i



and

|c₃| ≤ 1 3

 _n



i=1

(1 + 4γ_i)(3 + 16γ_i+ 24γ_i²) 12(1 + 2γi)³ +ⁿ⁻¹

k=1

 1 + 4γ_k 2(1 + 2γ_k)

n i=k+1

1 + 4γi

2(1 + 2γ_i)



+ⁿ

i=1

(1 + 4β_i)(3 + 16β_i+ 24β_i²) 12(1 + 2βi)³

+2 3

 2

n−1

k=1

 1 + 4βk

2(1 + 2β_k)

n i=k+1

1 + 4βi

2(1 + 4β_i)



+

 _n



i=1

1 + 4βi

2(1 + 2βi)

  _n



i=1

1 + 4γi

2(1 + 2γi)



.

Proof. From the definition of K₁(z) we obtain:

K₁^(z) =ⁿ

i=1

g^_i(z) ·f_i(z) z and further we get that:

1 +^∞

j=2

jc_jz^j−1=



1 +^∞

j=2

ja_1,jz^j−1

 . . .



1 +^∞

j=2

ja_n,jz^j−1



×



1 +^∞

j=2

b_1,jz^j−1

 . . .



1 +^∞

j=2

b_n,jz^j−1

 .

After some computation from the above relation we obtain:

(2.2) c₂ = 1

2

n i=1

b_i,2+

n i=1

a_i,2 and

(2.3)

c₃ = 1 3

n i=1

b_i,3+ⁿ

i=1

a_i,3+1 3

n−1

k=1

 b_k,2

n i=k+1

b_i,2



+ 4 3

n−1

k=1

 a_k,2

n i=k+1

a_i,2

 +2

3

 _n



i=1

a_i,2

  _n



i=1

b_i,2

 . From Theorem 1.1 we have the following inequalities for the coefficients:

|ai,2| ≤ 1 + 4βi

2(1 + 2β_i)

|a_i,3| ≤ (1 + 4β_i)(3 + 16β_i+ 24β_i²) 12(1 + 2β_i)³

and

|b_i,2| ≤ 1 + 4γ_i 2(1 + 2γi)

|b_i,3| ≤ (1 + 4γ_i)(3 + 16γ_i+ 24γ_i²) 12(1 + 2γ_i)³

for i = 1, n. Now we will use the inequalities in (2.2) and (2.3) and we obtain:

|c₂| ≤ 1 2

n i=1

|b_i,2| +ⁿ

i=1

|a_i,2|

≤ 1 2

 _n



i=1

1 + 4γi

2(1 + 2γi) +

n i=1

1 + 4βi

1 + 2βi



and

|c₃| ≤ 1 3

n i=1

|b_i,3| +ⁿ

i=1

|a_i,3| + 1 3

n−1

k=1



|b_k,2| ⁿ

i=k+1

|b_i,2|



+4 3

n−1

k=1



|a_k,2| ⁿ

i=k+1

|a_i,2|

 +2

3

 _n



i=1

|a_i,2|

  _n



i=1

|b_i,2|



≤ 1 3

 _n



i=1

(1 + 4γ_i)(3 + 16γ_i+ 24γ_i²) 12(1 + 2γ_i)³

+ⁿ⁻¹

k=1

 1 + 4γ_k 2(1 + 2γk)

n i=k+1

1 + 4γ_i 2(1 + 2γi)



+ⁿ

i=1

(1 + 4βi)(3 + 16βi+ 24β_i²) 12(1 + 2β_i)³

+2 3

 2ⁿ⁻¹

k=1

 1 + 4β_k 2(1 + 2βk)

n i=k+1

1 + 4β_i 2(1 + 4βi)



+

 _n



i=1

1 + 4β_i 2(1 + 2β_i)

  _n



i=1

1 + 4γ_i 2(1 + 2γ_i)



,

hence the proof is complete. 

References

[1] Acu, M., Owa, S., Convex functions associated with some hyperbola, J. Approx. Theory Appl.1 (1) (2005), 37–40.

[2] Breaz, N., Breaz, D., Acu, M., Some properties for an integral operator on theCVH(β)- class , IJOPCA2 (1) (2010), 53–58.

[3] Pescar, V., The univalence of an integral operator, Gen. Math.19 (4) (2011), 69–74.

[4] Robertson, M. S., Certain classes of starlike functions, Michigan Math. J. 76 (1) (1954), 755–758.

[5] Ularu, N., Convexity properties for an integral operator, Acta Univ. Apulensis Math.

Inform.27 (2011), 115–120.

Nicoleta Ularu Nicoleta Breaz

University of Pite¸sti “1 Decembrie 1918” University of Alba Iulia Tˆargul din Vale Str., No. 1 N. Iorga Str., No. 11–13

110040 Pite¸sti, Arge¸s 510009 Alba Iulia, Alba

Romania Romania

e-mail: nicoletaularu@yahoo.com e-mail:nbreaz@uab.ro Received September 7, 2012