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volume 7, issue 5, article 177, 2006.

Received 12 September, 2006;

accepted 18 October, 2006.

Communicated by:G. Kohr

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Journal of Inequalities in Pure and Applied Mathematics

SOME CONVEXITY PROPERTIES FOR A GENERAL INTEGRAL OPERATOR

DANIEL BREAZ AND NICOLETA BREAZ

Department of Mathematics

" 1 Decembrie 1918 " University Alba Iulia

Romania.

EMail:dbreaz@uab.ro EMail:nbreaz@uab.ro

c

2000Victoria University ISSN (electronic): 1443-5756 250-06

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Some Convexity Properties for a General Integral Operator Daniel Breaz and Nicoleta Breaz

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J. Ineq. Pure and Appl. Math. 7(5) Art. 177, 2006

http://jipam.vu.edu.au

Abstract

In this paper we consider the classes of starlike functions, starlike functions of orderα, convex functions, convex functions of orderαand the classes of the univalent functions denoted bySH(β),SP andSP(α, β). On these classes we study the convexity andα- order convexity for a general integral operator.

2000 Mathematics Subject Classification:30C45.

Key words: Univalent function, Integral operator, Convex function, Analytic function, Starlike function.

1. Introduction

LetU ={z ∈C,|z|<1}be the unit disc of the complex plane and denote by H(U), the class of the holomorphic functions inU.Consider

A=

f ∈H(U), f(z) =z+a₂z²+a₃z³+· · · , z ∈U

the class of analytic functions in U and S = {f ∈A:f is univalent in U}.

We denote byS^∗ the class of starlike functions that are defined as holomorphic functions in the unit disc with the propertiesf(0) =f⁰(0)−1 = 0and

Rezf⁰(z)

f(z) >0, z∈U.

A functionf ∈ Ais a starlike function by the orderα,0≤ α <1iff satisfies the inequality

Rezf⁰(z)

f(z) > α, z ∈U.

We denote this class by S^∗(α). Also, we denote by K the class of convex functions that are defined as holomorphic functions in the unit disc with the propertiesf(0) =f⁰(0)−1 = 0and

Re

zf⁰⁰(z) f⁰(z) + 1

>0, z ∈U.

A functionf ∈ Ais a convex function by the orderα,0≤ α < 1iff verifies the inequality

Re

zf⁰⁰(z) f⁰(z) + 1

> α, z ∈U.

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We denote this class byK(α).

In the paper [5] J. Stankiewicz and A. Wisniowska introduced the class of univalent functions,SH(β),β >0defined by:

(1.1)

zf⁰(z)

f(z) −2β√

2−1

<Re √

2zf⁰(z) f(z)

+ 2β√

2−1

, f ∈S, for allz ∈U.

Also, in the paper [3] F. Ronning introduced the class of univalent functions, SP, defined by

(1.2) Rezf⁰(z)

f(z) >

zf⁰(z) f(z) −1

, f ∈S,

for allz ∈U. The geometric interpretation of the relation (1.2) is that the class SP is the class of all functionsf ∈ S for which the expressionzf⁰(z)/f(z), z ∈U takes all values in the parabolic region

Ω = {ω:|ω−1| ≤Reω}=

ω=u+iv :v² ≤2u−1 .

In the paper [3] F. Ronning introduced the class of univalent functionsSP(α, β), α >0, β ∈[0,1), as the class of all functionsf ∈Swhich have the property:

(1.3)

zf⁰(z)

f(z) −(α+β)

≤Rezf⁰(z)

f(z) +α−β,

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for allz ∈U. Geometric interpretation:f ∈SP(α, β)if and only ifzf⁰(z)/f(z), z ∈U takes all values in the parabolic region

Ω_α,β ={ω :|ω−(α+β)| ≤Reω+α−β}

=

ω =u+iv :v² ≤4α(u−β) .

We consider the integral operatorF_n, defined by:

(1.4) F_n(z) =

Z z

0

f₁(t) t

α1

· · · · ·

f_n(t) t

αn

dt

and we study its properties.

Remark 1. We observe that for n = 1 and α₁ = 1 we obtain the integral operator of Alexander,F (z) =Rz

0 f(t)

t dt.

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2. Main Results

Theorem 2.1. Letα_i, i∈ {1, . . . , n}be real numbers with the propertiesα_i >0 fori∈ {1, . . . , n}and

n

X

i=1

α_i ≤n+ 1.

We suppose that the functions f_i, i = {1, . . . , n}are the starlike functions by order _α¹

i, i ∈ {1, . . . , n}, that isf_i ∈ S^∗

1 αi

for alli ∈ {1, . . . , n}.In these conditions the integral operator defined in (1.4) is convex.

Proof. We calculate forFn the derivatives of the first and second order. From (1.4) we obtain:

F_n⁰ (z) =

f1(z) z

α1

· · · · ·

fn(z) z

αn

and

F_n⁰⁰(z) =

n

X

i=1

α_i

f_i(z) z

αi−1

zf_i⁰(z)−f_i(z) zfi(z)

ⁿ Y

j=1

j6=i

f_j(z) z

αj

.

F_n⁰⁰(z) F_n⁰ (z) =α₁

zf₁⁰(z)−f₁(z) zf₁(z)

+· · ·+α_n

zf_n⁰ (z)−f_n(z) zf_n(z)

,

(2.1) F_n⁰⁰(z) F_n⁰ (z) =α₁

f₁⁰ (z) f₁(z) − 1

z

+· · ·+α_n

f_n⁰ (z) f_n(z) − 1

z

.

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By multiplying the relation (2.1) withzwe obtain:

(2.2) zF_n⁰⁰(z) F_n⁰ (z) =

n

X

i=1

α_i

zf_i⁰(z) f_i(z) −1

=

n

X

i=1

α_izf_i⁰(z)

f_i(z) −α₁− · · · −α_n. The relation (2.2) is equivalent with

(2.3) zF_n⁰⁰(z)

F_n⁰ (z) + 1 =α₁zf₁⁰(z)

f₁(z) +· · ·+α_nzf_n⁰ (z)

f_n(z) −α₁− · · · −α_n+ 1.

From (2.3) we obtain that:

(2.4) Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

=α₁Rezf₁⁰ (z)

f₁(z) +· · ·+α_nRezf_n⁰ (z)

f_n(z) −α₁− · · · −α_n+ 1.

But f_i ∈ S^∗

1 αi

, for all i ∈ {1, . . . , n}, so Re^zf_fⁱ⁰^(z)

i(z) > _α¹

i, for all i ∈ {1, . . . , n}.We apply this affirmation in the equality (2.4) and obtain:

Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

> α₁ 1 α1

+· · ·+α_n 1 αn

−α₁− · · · −α_n+ 1 (2.5)

=n+ 1−

n

X

i=1

α_i.

But, in accordance with the hypothesis, we obtain:

Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

>0 so,Fnis a convex function.

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Theorem 2.2. Letα_i, i∈ {1, . . . , n}, be real numbers with the propertiesα_i >

0fori∈ {1, . . . , n}and

n

X

i=1

α_i ≤1.

We suppose that the functions fi, i = {1, . . . , n}, are the starlike functions.

Then the integral operator defined in (1.4) is convex by order,1−Pn i=1α_i. Proof. Following the same steps as in Theorem2.1, we obtain:

(2.6) zF_n⁰⁰(z) F_n⁰ (z) =

n

X

i=1

α_i

zf_i⁰(z) fi(z) −1

=

n

X

i=1

α_izf_i⁰(z)

fi(z) −α₁− · · · −α_n. The relation (2.6) is equivalent with

(2.7) zF_n⁰⁰(z)

F_n⁰ (z) + 1 =α1

zf₁⁰(z)

f₁(z) +· · ·+αn

zf_n⁰ (z)

f_n(z) −α1− · · · −αn+ 1.

From (2.7) we obtain that:

(2.8) Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

=α₁Rezf₁⁰ (z)

f₁(z) +· · ·+α_nRezf_n⁰ (z)

f_n(z) −α₁− · · · −α_n+ 1.

Butf_i ∈ S^∗ for alli∈ {1, . . . , n},soRe^zf_fⁱ⁰^(z)

i(z) >0for alli∈ {1, . . . , n}.We apply this affirmation in the equality (2.8) and obtain that:

(2.9) Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

> α₁·0+· · ·+α_n·0−α₁−· · ·−α_n+1 = 1−

n

X

i=1

α_i.

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But in accordance with the inequality (2.9), obtain that Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

>1−

n

X

i=1

αi

so,F_nis a convex function by order1−Pn i=1α_i.

Theorem 2.3. Letα_i, i∈ {1, . . . , n}, be real numbers with the propertiesα_i >

0, fori∈ {1, . . . , n}and

(2.10)

n

X

i=1

α_i ≤

√2 2β √

2−1 +√

2.

We suppose thatf_i ∈ SH(β), for i = {1, . . . , n}andβ > 0. In these condi- tions, the integral operator defined in (1.4) is convex.

Proof. Following the same steps as in Theorem2.1, we obtain that:

(2.11) zF_n⁰⁰(z) F_n⁰ (z) + 1 =

n

X

i=1

α_izf_i⁰(z) f_i(z) −

n

X

i=1

α_i+ 1.

We multiply the relation (2.11) with√

2and obtain:

(2.12) √

2

zF_n⁰⁰(z) F_n⁰ (z) + 1

=

n

X

i=1

√

2α_izf_i⁰(z) f_i(z) −√

2

n

X

i=1

α_i+√ 2.

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The equality (2.12) is equivalent with:

√2

zF_n⁰⁰(z) F_n⁰ (z) + 1

=

n

X

i=1

α_i√

2zf_i⁰(z)

fi(z) + 2α_iβ√

2−1

−

n

X

i=1

2α_iβ√

2−1

−√ 2

n

X

i=1

α_i+√ 2.

We calculate the real part from both terms of the above equality and obtain:

√ 2 Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

=

n

X

i=1

α_i

Re

√

2zf_i⁰(z) f_i(z)

+ 2β√

2−1

−

n

X

i=1

2α_iβ√

2−1

−√ 2

n

X

i=1

α_i+√ 2.

Because f_i ∈ SH(β) for i = {1, . . . , n}, we apply in the above relation the inequality (1.1) and obtain:

√ 2 Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

>

n

X

i=1

α_i

zf_i⁰(z)

f_i(z) −2β√

2−1

−

n

X

i=1

2α_iβ√

2−1

−√ 2

n

X

i=1

α_i+√ 2.

Becauseα_i

zf_i⁰(z)

fi(z) −2β √

2−1

>0, for alli∈ {1, . . . , n}, we obtain that (2.13) √

2 Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

>−

n

X

i=1

2α_iβ√ 2−1

−√ 2

n

X

i=1

α_i+√ 2.

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Using the hypothesis (2.10), we have:

(2.14) Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

>0,

so,F_nis a convex function.

Corollary 2.4. Letαbe real numbers with the properties0< α≤

√ 2 2β(^√²⁻¹)⁺^√², β >0. We suppose that the functionsf ∈SH(β). In these conditions the inte- gral operator,F (z) =Rz

0

f(t) t

α

dtis convex.

Proof. In Theorem2.3, we considern= 1,α₁ =αandf₁ =f.

Theorem 2.5. Letα_i, i∈ {1, . . . , n}be real numbers with the propertiesα_i >0 fori∈ {1, . . . , n},

(2.15)

n

X

i=1

αi <1

and 1− Pn

i=1α_i ∈ [0,1). We consider the functions f_i, f_i ∈ SP for i = {1, . . . , n}. In these conditions, the integral operator defined in (1.4) is convex by1−Pn

i=1α_iorder.

Proof. Following the same steps as in Theorem2.1, we have:

(2.16) zF_n⁰⁰(z) F_n⁰ (z) + 1 =

n

X

i=1

αi

zf_i⁰(z) f_i(z) −

n

X

i=1

αi+ 1.

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We calculate the real part from both terms of the above equality and obtain:

(2.17) Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

=

n

X

i=1

αiRe

zf_i⁰(z) f_i(z)

−

n

X

i=1

αi+ 1.

Becausef_i ∈SP fori={1, . . . , n}we apply in the above relation the inequality (1.2) and obtain:

(2.18) Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

>

n

X

i=1

α_i

−

n

X

i=1

α_i+ 1.

Becauseα_i

>0, for alli∈ {1, . . . , n}, we get

(2.19) Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

>1−

n

X

i=1

α_i.

Using the hypothesis, we obtain that F_n is a convex function by1−Pn i=1α_i order.

Remark 2. IfPn

i=1α_i = 1then

(2.20) Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

>0, so,F_nis a convex function.

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Corollary 2.6. Letγbe a real number with the property0< γ <1. We suppose thatf ∈SP. In these conditions the integral operatorF (z) = Rz

0

f(t) t

γ

dtis convex of1−γorder.

Proof. In Theorem2.5, we considern= 1,α1 =γandf1 =f.

Theorem 2.7. We suppose thatf ∈SP. In this condition, the integral operator of Alexander, defined by

(2.21) F₁(z) =

Z z

0

f(t) t dt,

is convex.

Proof. We have:

(2.22) Re

zF₁⁰⁰(z) F₁⁰(z) + 1

= Rezf⁰(z) f(z) >

zf⁰(z) f(z) −1

>0.

So, the relation (2.22) implies that the Alexander operator is convex.

Remark 3. Theorem2.7can be obtained from Corollary2.6, forγ = 1.

Theorem 2.8. Letα_i, i∈ {1, . . . , n}be real numbers with the propertiesα_i >0 fori∈ {1, . . . , n},

(2.23)

n

X

i=1

α_i < 1

α−β+ 1, α >0, β ∈[0,1)

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and (β−α−1)Pn

i=1α_i + 1 ∈ (0,1). We suppose that f_i ∈ SP (α, β), for i = {1, . . . , n}. In these conditions, the integral operator defined in (1.4) is convex by(β−α−1)Pn

i=1α_i+ 1order.

Proof. Following the same steps as in Theorem2.1, we obtain that:

(2.24) zF_n⁰⁰(z) F_n⁰ (z) =

n

X

i=1

αi

zf_i⁰(z)

f_i(z) +α−β

+ (β−α−1)

n

X

i=1

αi.

and

(2.25) zF_n⁰⁰(z)

F_n⁰ (z) + 1 =

n

X

i=1

α_i

zf_i⁰(z)

f_i(z) +α−β

+ (β−α−1)

n

X

i=1

α_i+ 1.

We calculate the real part from both terms of the above equality and get:

(2.26) Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

= Re ( _n

X

i=1

αi

zf_i⁰(z)

f_i(z) +α−β )

+ (β−α−1)

n

X

i=1

α_i+ 1.

Becausef_i ∈ SP (α, β)fori = {1, . . . , n}we apply in the above relation the inequality (1.3) and obtain:

(2.27) Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

≥

n

X

i=1

α_i

zf_i⁰(z)

f_i(z) −(α+β)

+ (β−α−1)

n

X

i=1

α_i+ 1.

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Since αi

zf_i⁰(z)

fi(z) −(α+β)

> 0, for all i ∈ {1, . . . , n}, using the inequality (1.3), we have

(2.28) Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

≥(β−α−1)

n

X

i=1

α_i+ 1 >0.

From (2.28), since(β−α−1)Pn

i=1αi+ 1∈(0,1), we obtain that the integral operator defined in (1.4) is convex by(β−α−1)Pn

i=1α_i+ 1order.

Corollary 2.9. Let γ be a real number with the property 0 < γ < _α−β+1¹ , α > 0, β ∈ [0,1). We suppose thatf ∈ SP (α, β). In these conditions, the integral operatorF(z) = Rz

0

f(t) t

γ

dtis convex.

Proof. In Theorem2.8, we considern= 1,α₁ =γandf₁ =f.

Forα = β ∈(0,1)we obtain the class S(α, α)that is characterized by the property

(2.29)

zf⁰(z) f(z) −2α

≤Rezf⁰(z) f(z) .

Corollary 2.10. Let α_i, i ∈ {1, . . . , n} be real numbers with the properties α_i >0fori∈ {1, . . . , n}and

(2.30) 1−

n

X

i=1

α_i ∈[0,1).

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We consider the functions f_i, f_i ∈ SP (α, α), i = {1, . . . , n}, α ∈ (0,1). In these conditions, the integral operator defined in (1.4) is convex by1−Pn

i=1αi

order.

Proof. From (1.4) we obtain

(2.31) zF_n⁰⁰(z)

F_n⁰ (z) =

n

X

i=1

α_izf_i⁰(z) f_i(z) −

n

X

i=1

α_i,

which is equivalent with

(2.32) Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

=

n

X

i=1

α_iRezf_i⁰(z) f_i(z) −

n

X

i=1

α_i+ 1.

From (2.31) and (2.32), we have:

(2.33) Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

>

n

X

i=1

α_i

zf_i⁰(z) f_i(z) −2α

+ 1−

n

X

i=1

α_i.

SincePn i=1α_i

zf_i⁰(z) fi(z) −2α

>0, for alli∈ {1, . . . , n}, from (2.33), we get:

(2.34) Re

zF_n⁰⁰(z) F_n⁰ (z) + 1

>1−

n

X

i=1

α_i.

Now, from (2.34) we obtain that the operator defined in (1.4) is convex by1− Pn

i=1αiorder.

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References

[1] M. ACU, Close to convex functions associated with some hyperbola, Lib- ertas Mathematica, XXV (2005), 109–114.

[2] D. BREAZ AND N. BREAZ, Two integral operators, Studia Universitatis Babe¸s -Bolyai, Mathematica, Cluj Napoca, (3) (2002), 13–21.

[3] F. RONNING, Integral reprezentations of bounded starlike functions, Ann.

Polon. Math., LX(3) (1995), 289–297.

[4] F. RONNING, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118(1) (1993), 190–196.

[5] J. STANKIEWICZ AND A. WISNIOWSKA, Starlike functions associated with some hyperbola, Folia Scientiarum Universitatis Tehnicae Resoviensis 147, Matematyka, 19 (1996), 117–126.