A CLASS OF UNIVALENT FUNCTIONS WHICH EXTENDS THE CLASS OF MOCANU FUNCTIONS

(1)

A CLASS OF UNIVALENT FUNCTIONS WHICH EXTENDS THE CLASS OF MOCANU FUNCTIONS

GH. OROS and GEORGIA IRINA OROS

We introduce a subclass of starlike functions which extends the well-known class of alpha-convex functions (Mocanu functions) and the class of convex functions.

AMS 2000 Subject Classification: Primary 30C45; Secondary 30C55.

Key words: holomorphic function, starlike function, starlike function of orderλ.

1. INTRODUCTION AND PRELIMINARIES

LetU be the unit disc U ={z ∈C:|z|<1} of the complex plane. Let H(U) be the space of holomorphic functions in U and H[a, n] ={f ∈ H(U), f(z) = a+anzⁿ+an+1zⁿ⁺¹ +· · ·, z ∈ U}, A_n = {f ∈ H(U), f(z) = z+a_n+1zⁿ⁺¹+a_n+2zⁿ⁺²+· · · , z ∈U} withA₁ =A,

S^∗ =

f ∈A, Rezf⁰(z)

f(z) >0, z ∈U

, the class of starlike functions in U, and

S^∗(α) =

f ∈A, Rezf⁰(z)

f(z) > α, z∈U

, the class of starlike functions of order α, 0≤α <1.

We also let Mα=

f ∈A, Re

(1−α)zf⁰(z) f(z) +α

zf⁰⁰(z) f⁰(z) + 1

>0, z ∈U

the class of α-convex functions (or Mocanu functions) and K=

f ∈A, Rezf⁰⁰(z)

f⁰(z) + 1>0, z ∈U

, the class of normalized convex function in U.

MATH. REPORTS10(60),2 (2008), 165–168

(2)

166 Gh. Oros and Georgia Irina Oros 2

Theorem 1 ([3]). If α ≥0 and f ∈ A, then f ∈ Mα if and only if the function

F(z) =f(z)

zf⁰(z) f(z)

α

is starlike in U.

It is well-known that the function f ∈ A is convex if and only if the function F(z) =zf⁰(z), z∈U, is starlike inU.

In order to prove the new results we shall use

Lemma A ([1]). Let ψ :C² → C, satisfy the condition Reψ(is, σ) ≤0, z∈U,for s, σ ∈R, σ ≤ −(1 +s²)/2.

Ifp(z) = 1+p₁z+p₂z²+· · · satisfiesReψ[p(z), zp⁰(z)]>0thenRep(z)>

0, z∈U.

More general forms of this lemma can be found in [1].

2. MAIN RESULT

Definition 1. Let α, β ∈R and f ∈A with ^f^(z)f_z⁰^(z) 6= 0, for z∈ U. We say that the function f belongs to the class Mα,β if the function F :U →C defined by

(1) F(z) =z

f(z) z

β zf⁰(z)

f(z) α

is a starlike function in U.

Theorem 2. Let α∈R andβ ≥1. Then M_α,β⊂S^∗. Proof. Letf ∈A,f(z) =z+a2z²+· · ·,z∈U, and

(2) zf⁰(z)

f(z) =p(z) = 1 +p1z¹+p2z²+· · ·, z∈U.

Iff ∈Mα,β then

(3) RezF⁰(z)

F(z) >0, z∈U.

By differentiating (1) with respect to zand using (2), we obtain

(4) zF⁰(z)

F(z) = 1−β+βp(z) +αzp⁰(z)

p(z) , z∈U.

Let

(5) ψ(p(z), zp⁰(z)) = 1−β+βp(z) +αzp⁰(z)

p(z) , z∈U.

(3)

3 A class of univalent functions which extends the class of Mocanu functions 167

Then inequality (3) becomes

(6) Reψ(p(z), zp⁰(z))>0, z∈U.

In order to prove Theorem 2 we shall use Lemma A. For this we calculate Reψ(is, σ) = Re

h

1−β+βis+ασ is i

= Re

1−β+βis−ασi s

= 1−β≤0, for any s, σ∈R,σ≤ −(1 +s²)/2. Lemma A implies that Rep(z)>0,z∈U, which is equivalent to

Rezf⁰(z)

f(z) >0, z∈U.

Hence f is starlike inU, i.e., M_α,β ⊂S^∗.

Remark1. If β = 1, α≥0 then F(z) =f(z)h_zf0(z) f(z)

iα

and M_α,1 =M_α, i.e., the class Mα,1 coincides with the class of α-convex functions (Mocanu functions).

Remark2. Forα=β = 1, we haveF(z) =zf⁰(z), z∈U and M_1,1=K, i.e., the class M1,1 coincides with the class of convex functions.

Theorem 3. Let α, β be real numbers with α ≥ 0, β ≥ 1 and let λ∈ (0, λ1) where

(7) λ₁ = 2β−α−2 +p

(2β−α−2)²+ 8αβ

4β , λ₁ ∈[0,1).

If f ∈Mα,β then f ∈S^∗(λ).

Proof. By differentiating (1) with respect toz and letting

(8) zf⁰(z)

f(z) = (1−λ)p(z) +λ, z∈U, we obtain

zF⁰(z)

F(z) = 1−β+β[(1−λ)p(z) +λ] +α (1−λ)zp⁰(z)

(1−λ)p(z) +λ, z∈U.

Let

(9) ψ(p(z), zp⁰(z)) = 1−β+β[(1−λ)p(z) +λ] + α(1−λ)zp⁰(z)

(1−λ)p(z) +λ, z∈U.

Since f ∈M_α,β we have

RezF⁰(z)

F(z) >0, z∈U, which is equivalent to

(10) Reψ(p(z), zp⁰(z))>0, z∈U.

(4)

168 Gh. Oros and Georgia Irina Oros 4

We have

(11) Reψ(is, σ) = Re

1−β+β[(1−λ)is+λ] + α(1−λ)σ (1−λ)is+λ

=

= 1−β+λβ+α (1−λ)λσ

λ²+ (1−λ)²s² ≤1−β+λβ−α (1−λ)λ(1 +s²) 2[λ²+ (1−λ)²s²] ≤

≤ 2(1−β+λβ)[λ²+ (1−λ)²s²]−α(1−λ)λ(1 +s²) 2[λ²+ (1−λ)²s²] ≤

≤ s²[2(1−β+λβ)(1−λ)²−αλ(1−λ)] + 2λ²(1−β+λβ)−αλ(1−λ)

2[s²(1−λ)²+λ²] ≤0,

for any α > 0, β ≥ 1, λ ∈ (0, λ1) where λ1 is given by (7). Since (11) contradicts (10), by Lemma A we have Rep(z)>0,z∈U. From (8) we have

p(z) = 1 1−λ

zf⁰(z) f(z) −λ

and

Rep(z) = 1 1−λ

Rezf⁰(z) f(z) −λ

>0, z∈U.

Hence Re hzf⁰(z)

f(z) −λ i

>0, which is equivalent to Rezf⁰(z)

f(z) > λ, z∈U, i.e., f ∈S^∗(λ).

Remark 3. If β = 1 then λ₁ = ^−α+

√ α²+8α

4 = ϕ(α) and we deduce that Mα ⊂S^∗(ϕ(α)). This result was obtained in [2].

REFERENCES

[1] S.S. Miller and P.T. Mocanu,Second order differential inequalities in the complex plane.

J. Math. Appl.65(1978),2, 289–305.

[2] S.S. Miller, P.T. Mocanu and M.O. Reade,On generalized convexity in conformal map- pings, II. Rev. Roumaine Math. Pures Appl.21(1976),2, 219–225.

[3] P.T. Mocanu,Une propriété de convexité generalisée dans la théorie de representation.

Mathematica (Cluj)11(34) (1969), 127–133.

Received 18 June 2007 University of Oradea

Department of Mathematics Str. Armatei Romˆane nr. 5 410087 Oradea, Romania georgia oros ro@yahoo.co.uk