In this paper we define some subclass ofα - uniformly convex functions with respect to a convex domain included in right half planeD

21(2005), 49–54 www.emis.de/journals ISSN 1786-0091

SOME SUBCLASSES OF α-UNIFORMLY CONVEX FUNCTIONS

MUGUR ACU

Abstract. In this paper we define some subclass ofα - uniformly convex functions with respect to a convex domain included in right half planeD.

1. Introduction

LetH(U) be the set of functions which are regular in the unit discU, A={f ∈ H(U) :f(0) =f⁰(0)−1 = 0},

Hu(U) ={f ∈ H(U) :f is univalent inU} andS={f ∈A:f is univalent inU}.

Let consider the integral operatorL_a: A→Adefined as:

(1) f(z) =LaF(z) = 1 +a z^a

Zz

0

F(t)·t^a−1dt, a∈C, Re a≥0.

In the casea= 1,2,3, . . . this operator was introduced by S.D. Bernardi and it was studied by many authors in different general cases.

LetDⁿ be the S˘al˘agean differential operator (see [10]) defined as:

Dⁿ:A→A, n∈NandD⁰f(z) =f(z) D¹f(z) =Df(z) =zf⁰(z), Dⁿf(z) =D(Dⁿ⁻¹f(z)).

2. Preliminary results

Definition 2.1 ([4]). Let α ∈[0,1] andf ∈ A. We say thatf is α - uniformly convex function if:

Re

½

(1−α)zf⁰(z) f(z) +α

µ

1 +zf⁰⁰(z) f⁰(z)

¶¾

≥

¯¯

¯¯(1−α)

µzf⁰(z) f(z) −1

¶

+αzf⁰⁰(z) f⁰(z)

¯¯

¯¯ , z∈U.

We denote this class withU Mα.

2000Mathematics Subject Classification. 30C45.

Key words and phrases. Alexander integral operator, Briot-Bouquet differential subordination, Ruscheweyh operator.

49

u v

β >1

β = 1 0< β <1

γ β= 0

Figure 1

Remark 2.1. Geometric interpretation: f ∈U Mα if and only if J(α, f;z) = (1−α)zf⁰(z)

f(z) +α µ

1 + zf⁰⁰(z) f⁰(z)

¶

take all values in the parabolic region Ω ={w:|w−1| ≤Re w}={w=u+iv:v²≤ 2u−1}. We haveU M0 =SP, where the classSP was introduced by F. Ronning in [9] and U Mα ⊂Mα, whereMα is the well know class of α - convex functions introduced by P.T. Mocanu in [8].

Definition 2.2 ([1]). Letα∈[0,1] andn∈N. We say thatf ∈A is in the class U Dn,α(β, γ),β≥0,γ∈[−1,1), β+γ≥0 if

Re

·

(1−α)Dⁿ⁺¹f(z)

Dⁿf(z) +αDⁿ⁺²f(z) Dⁿ⁺¹f(z)

¸

≥β

¯¯

¯¯(1−α)Dⁿ⁺¹f(z)

Dⁿf(z) +αDⁿ⁺²f(z) Dⁿ⁺¹f(z)−1

¯¯

¯¯+γ.

Remark 2.2. Geometric interpretation: f ∈U Dn,α(β, γ) if and only if Jn(α, f;z) = (1−α)Dⁿ⁺¹f(z)

Dⁿf(z) +αDⁿ⁺²f(z) Dⁿ⁺¹f(z)

take all values in the convex domain included in right half planeD_β,γ, whereD_β,γ is an elliptic region forβ >1, a parabolic region forβ= 1, a hyperbolic region for 0< β <1, the half planeu > γ forβ = 0. (Figure 1.)

We haveU D0,α(1,0) =U Mα.

The next theorem is result of the so called “admissible functions method” intro- duced by P.T. Mocanu and S.S. Miller (see [5], [6], [7]).

Theorem 2.1. Let h convex in U and Re[βh(z) +δ] > 0, z ∈ U. If p ∈ H(U) with p(0) = h(0) and p satisfied the Briot-Bouquet differential subordina- tionp(z) + zp⁰(z)

βp(z) +δ ≺h(z), thenp(z)≺h(z).

Definition 2.3 ([3]). The function f ∈ A is n-starlike with respect to convex domain included in right half planeDif the differential expression Dⁿ⁺¹f(z)

Dⁿf(z) takes values in the domainD.

If we considerq(z) an univalent function withq(0) = 1, Re q(z)>0,q⁰(0)>0 which maps the unit discU into the convex domainD we have:

Dⁿ⁺¹f(z)

Dⁿf(z) ≺q(z).

We note byS_n^∗(q) the set of all these functions.

3. Main results

Letq(z) be an univalent function withq(0) = 1,q⁰(0)>0, which maps the unit discU into a convex domain included in right half planeD.

Definition 3.1. Let f ∈ A and α ∈ [0,1]. We say that f is α-uniform convex function with respect toD, if

J(α, f;z) = (1−α)zf⁰(z) f(z) +α

µ

1 + zf⁰⁰(z) f⁰(z)

¶

≺q(z).

We denote this class withU Mα(q).

Remark 3.1. Geometric interpretation: f ∈U Mα(q) if and only if J(α, f;z) take all values in the convex domain included in right half planD.

Remark 3.2. We haveU Mα(q)⊂Mα, whereMαis the well know class ofα-convex function. If we takeD= Ω (see Remark 2.1) we obtain the classU Mα.

Remark 3.3. From the above definition it easily results thatq1(z)≺q2(z) implies U Mα(q1)⊂U Mα(q2).

Theorem 3.1. For allα, α⁰ ∈[0,1]with α < α⁰ we have U Mα⁰(q)⊂U Mα(q).

Proof. Fromf ∈U Mα⁰(q) we have (2) J(α⁰, f;z) = (1−α⁰)zf⁰(z)

f(z) +α⁰ µ

1 + zf⁰⁰(z) f⁰(z)

¶

≺q(z),

where q(z) is univalent inU with q(0) = 1,q⁰(0) >0, and maps the unit discU into the convex domain included in right half planeD.

With notation zf⁰(z)

f(z) =p(z), wherep(z) = 1 +p₁z+. . . we have:

J(α⁰, f;z) =p(z) +α⁰·zp⁰(z) p(z) . From (2) we have p(z) +α⁰· zp⁰(z)

p(z) ≺q(z) with p(0) = q(0) and Re q(z)>0, z∈U.

In this conditions from Theorem 2.1, withδ= 0, we obtainp(z)≺q(z), orp(z) take all values inD.

If we consider the functiong: [0, α⁰]→C,g(u) =p(z) +u·zp⁰(z)

p(z) , withg(0) = p(z)∈D andg(α⁰) =J(α⁰, f;z)∈D. Since the geometric image ofg(α) is on the segment obtained by the union of the geometric image ofg(0) andg(α⁰), we have g(α)∈D orp(z) +αzp⁰(z)

p(z) ∈D.

Thus J(α, f;z) take all values in D, or J(α, f;z) ≺ q(z). This means f ∈

U Mα(q). ¤

Theorem 3.2. If F(z)∈U Mα(q) thenf(z) =La(F)(z)∈S₀^∗(q), where La is the integral operator defined by (1) andα∈[0,1].

Proof. From (1) we have

(1 +a)F(z) =af(z) +zf⁰(z).

With notation zf⁰(z)

f(z) =p(z), wherep(z) = 1 +p1z+. . . we have zF⁰(z)

F(z) =p(z) + zp⁰(z) p(z) +a. If we denote zF⁰(z)

F(z) =h(z), withh(0) = 1, we have fromF(z)∈U Mα(q) (see Definition 3.1):

h(z) +α·zh⁰(z)

h(z) ≺q(z),

where q(z) is univalent unU with q(0) = 1, q⁰(z)>0 and maps the unit disc U into the convex domain included in right half planeD.

From Theorem 2.1 we obtainh(z)≺q(z) orp(z) + zp⁰(z)

p(z) +a ≺q(z).

Using the hypothesis and the construction of the functionq(z) we obtain from Theorem 2.1 zf⁰(z)

f(z) =p(z)≺q(z) orf(z)∈S₀^∗(q)⊂S^∗. ¤ Definition 3.2. Letf ∈A,α∈[0,1] andn∈N. We say thatf isα−n-uniformly convex function with respect toD if

Jn(α, f;z) = (1−α)Dⁿ⁺¹f(z)

Dⁿf(z) +αDⁿ⁺²f(z)

Dⁿ⁺¹f(z) ≺q(z).

We denote this class withU Dn,α(q).

Remark 3.4. Geometric interpretation: f ∈U Dn,α(q) if and only ifJn(α, f;z) take all values in the convex domain included in right half planeD.

Remark 3.5. We haveU D0,α(q) =U Mα(q) and if in the above definition we con- siderD=Dβ,γ (see Remark 2.2) we obtain the class U Dn,α(β, γ).

Remark 3.6. It is easy to see thatq1(z)≺q2(z) impliesU Dn,α(q1)⊂U Dn,α(q2).

Theorem 3.3. For allα, α⁰ ∈[0,1]with α < α⁰ we have U Dn,α⁰(q)⊂U Dn,α(q).

Proof. Fromf ∈U Dn,α⁰(q) we have:

(3) Jn(α⁰, f;z) = (1−α⁰)Dⁿ⁺¹f(z)

Dⁿf(z) +α⁰Dⁿ⁺²f(z)

Dⁿ⁺¹f(z) ≺q(z),

where q(z) is univalent inU with q(0) = 1,q⁰(0) >0, and maps the unit discU into the convex domain included in right half planeD.

With notation Dⁿ⁺¹f(z)

Dⁿf(z) =p(z), wherep(z) = 1 +p1z+. . . we have Jn(α⁰, f;z) =p(z) +α⁰·zp⁰(z)

p(z) . From (3) we have p(z) +α⁰· zp⁰(z)

p(z) ≺q(z) with p(0) = q(0) and Re q(z)>0, z∈U. In this condition from Theorem 2.1 we obtain p(z)≺q(z), orp(z) take all values inD.

If we consider the function

g: [0, α⁰]→C, g(u) =p(z) +u·zp⁰(z) p(z) ,

withg(0) =p(z)∈D andg(α⁰) =Jn(α⁰, f;z)∈D, it easy to see that g(α) =p(z) +αzp⁰(z)

p(z) ∈D.

Thus we haveJn(α, f;z)≺q(z) orf ∈U Dn,α(q). ¤ Theorem 3.4. If F(z)∈ U Dn,α(q) then f(z) = La(F)(z) ∈S^∗_n(q), where La is the integral operator defined by (1).

Proof. From (1) we have (1 +a)F(z) =af(z) +zf⁰(z). By means of the application of the linear operatorDⁿ⁺¹ we obtain:

(1 +a)Dⁿ⁺¹F(z) =aDⁿ⁺¹f(z) +Dⁿ⁺¹(zf⁰(z)) or

(1 +a)Dⁿ⁺¹F(z) =aDⁿ⁺¹f(z) +Dⁿ⁺²f(z).

With notation Dⁿ⁺¹f(z)

Dⁿf(z) =p(z), wherep(z) = 1 +p1z+. . ., we have:

Dⁿ⁺¹F(z)

DⁿF(z) =p(z) + 1

p(z) +a·zp⁰(z).

If we denote Dⁿ⁺¹F(z)

DⁿF(z) =h(z), withh(0) = 1, we have from F∈U Dn,α(q):

h(z) +αzh⁰(z)

h(z) ≺q(z),

where q(z) is univalent inU with q(0) = 1,q⁰(0) >0, and maps the unit discU into the convex domain included in right half planeD.

From Theorem 2.1 we obtainh(z)≺q(z) orp(z) + zp⁰(z)

p(z) +a ≺q(z).

Using the hypothesis we obtain from Theorem 2.1 p(z)≺q(z) orf(z)∈S_n^∗(q).

¤

Remark 3.7. If we consider D=Dβ,γ in Theorem 3.3 and Theorem 3.4 we obtain the main results from [1] and if we takeD =Dβ,γ and α= 0 in Theorem 3.4 we obtain the Theorem 3.1 from [2].

References

[1] M. Acu. On a subclass ofα-uniform convex functions. submitted.

[2] M. Acu and D. Blezu. A preserving property of a Libera type operator.Filomat, (14):13–18, 2000.

[3] D. Blezu. On the n-uniformly close to convex functions with respect to a convex domain.

Gen. Math., 9(3-4):3–14, 2001.

[4] I. Magda¸s. Onα-uniformly convex functions.Mathematica, 43(66)(2):211–218 (2003), 2001.

[5] S. S. Miller and P. T. Mocanu. Differential subordinations and univalent functions.Michigan Math. J., 28(2):157–172, 1981.

[6] S. S. Miller and P. T. Mocanu. On some classes of first-order differential subordinations.

Michigan Math. J., 32(2):185–195, 1985.

[7] S. S. Miller and P. T. Mocanu. Univalent solutions of Briot-Bouquet differential equations.

J. Differential Equations, 56(3):297–309, 1985.

[8] P. T. Mocanu. Une propri´et´e de convexit´e g´en´eralis´ee dans la th´eorie de la repr´esentation conforme.Mathematica (Cluj), 11 (34):127–133, 1969.

[9] F. Rønning. On starlike functions associated with parabolic regions. Ann. Univ. Mariae Curie-SkÃlodowska Sect. A, 45:117–122 (1992), 1991.

[10] G. S¸. S˘al˘agean. On some classes of univalent functions. InSeminar of geometric function theory, volume 82 ofPreprint, pages 142–158. Univ. “Babe¸s-Bolyai”, Cluj, 1983.

Received March 24, 2004.

Department of Mathematics, University Lucian Blaga of Sibiu, Str. Dr. I. Rat.iu, No. 5-7, 550012 - Sibiu, Romania

E-mail address:acu [email protected]