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UNIVALENCE CRITERIA

Virgil Pescar

Abstract. _{In this paper we derive some criteria for univalence of analytic} functions and of an integral operator in the open unit disk.

2000 Mathematics Subject Classification: 30C45.

Key Words and Phrases: Univalence, Integral Operator.

1. Introduction

Let_A be the class of the functions of the form f(z) =z+

∞

X

n=2

anzn (1.1)

which are analytic in the open unit disk _U = _{z_∈C_:_|_z_|_<₁_}_{. Let} _S _{denote the}

subclass of _Aconsisting of all univalent functions f in_U.

2. Preliminary results

We need the following theorems.

Theorem 2.1._[1].If _f_{(z) =}_z₊_a₂_z2₊_{. . .} _{is analytic in} _U _and 1_{− |}z_|2

zf′′(z) f′(z)

≤

1 (2.1)

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Theorem 2.2._[4]. _Let _α _{be a complex number,}_{Re α >}₀ _and _f _{∈ A}_{. If} 1_{− |}z_|2Re α

Re α

zf′′(z) f′_(z)

≤

1 (2.2)

for all z_{∈ U}, then the function

Fα(z) =

α

Z z

0

uα−1f′(u)du 1

α

(2.3) is in the class _S.

Theorem 2.3._[5]Let _α _{be a complex number,} _{Re α >}₀ _and_f _{∈ A}_{. If} 1_{− |}z_|2Re α

Re α

zf′′(z) f′_(z)

≤

1

for all z_{∈ U}, then for any complex number β, Re β _≥Re α, the function

Fβ(z) =

β

Z z

0

uβ−1f′(u)du _β1

(2.4) is regular and univalent in _U.

Theorem 2.4._{(Schwarz)[2].Let} _f_(z) _{the function regular in the disk}

UR={z∈C:|z|< R}

with _|f(z)_|< M, M fixed. Iff(z) has in z= 0 one zero with multiply_≥m, then

|f(z)_{| ≤} M Rm|z|

m_{, z}

∈ UR (2.5)

the equality (in the inequality (2.5) for z₆= 0) can hold only if f(z) =eiθ M

Rmz m_,

where θ is constant.

Theorem 2.5._{[3]If the function} _g(z) _{is regular in} _U _and _|_g(z)_|_<₁ _in _U_{, then} for all ξ_{∈ U} andz_{∈ U} the following inequalities hold:

g(ξ)₋g(z) 1₋g(z)g(ξ)

≤

ξ₋z 1₋zξ

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and

|g′(z)_{| ≤} 1− |g(z)|

2

1_{− |}z_|2 (2.7)

the equalities hold only in the case g(z) = E₁₊(z+_uzu), where _|E|= 1 and _|u_|<1. Remark._[3]For _z_{= 0, from inequality (2.6) we have}

g(ξ)₋g(0) 1₋g(0)g(ξ)

<_|ξ_| (2.8)

and, hence

|g(ξ)_{| ≤} |ξ|+|g(0)|

1 +_|g(0)_||ξ_| (2.9)

Consideringg(0) =aand ξ=z,

|g(z)_{| ≤} |z|+|a|

1 +_|a_||z_| (2.10)

for all z_{∈ U}.

3. Main results

Theorem 3.1._{Let the function} _f _{∈ A}_{. If}

zf′′(z) f′_(z)

≤

3√3

2 (3.1)

for all z_{∈ U}, then the function f is in the class _S.

Proof. We consider the function g(z) = zf_f′′′₍(_zz₎), z ∈ U. We have g(0) = 0 and from (3.1) by Theorem 2.4 (Schwarz) we obtain

zf′′(z) f′_(z)

≤

3√3

2 |z| (3.2)

for all z_{∈ U}. From (3.2) we get (1_{− |}z_|2)

zf′′(z) f′_(z)

≤

3√3

2 (1− |z|

2₎_|_z_| _(3.3)

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Because

max

|z|<1[(1− |z|

2₎_|_z_|_{] =} 2

3√3, from (3.3) we obtain

(1_{− |}z_|2)

zf′′(z) f′_(z)

≤

1 (3.4)

for all z_{∈ U} and by Theorem 2.1 we obtain thatf is in the class_S.

Theorem 3.2._Let_α _{be a complex number,}_{Re α >}₀ _{and the function}_f _{∈ A}_{. If}

zf′′(z) f′_(z)

≤

(2Re α+ 1)2Re αRe α+1

2 (3.5)

for all z_{∈ U}, then the function

Fα(z) =

α

Z z

0

uα−1f′(u)du _α1

Proof. Let’s consider the function p(z) = zf_f′′′₍(_zz₎),z ∈ U. We have p(0) = 0 and from (3.5) by Theorem 2.4 (Schwarz) we get

zf′′(z) f′_(z)

≤

2 |z| (3.7)

for all z_{∈ U}. From (3.7) we obtain 1_{− |}z_|2Re α

Re α

zf′′(z) f′_(z)

≤

2 ·

1_{− |}z_|2Re α

Re α |z| (3.8) for all z_{∈ U}.

Since

max

|z|<1

1_{− |}z_|2Re α Re α |z|

= 2

(2Re α+ 1)2Re αRe α+1 from (3.8) we have

1_{− |}z_|2Re α Re α

zf′′(z) f′_(z)

≤

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for all z_{∈ U} and by Theorem 2.2 we obtain that the function Fα(z) is regular and

univalent in_U.

Remark 3.3._{From Theorem 3.2 for}α= 1 we obtain Theorem 3.1.

Theorem 3.4._Let_α _{be a complex number,}_{Re α >}₀ _{and the function}_f _{∈ A}_{. If}

zf′′(z) f′_(z)

≤

2 (3.10)

for all z_{∈ U}, then for any complex number β, Re β _≥Re α, the function

Fβ(z) =

β

Z z

0

uβ−1f′(u)du _β1

Proof. We consider the function ψ: (0,_∞)_→R_, _{ψ(x) =} 1−a2x

x , 0< a <1. The

function ψ(x) is the function decreasing for x _∈(0,1). If x1 =Re α ≤ x2 =Re β

and a=_|z_|,z_{∈ U} then

1_{− |}z_|2Re β Re β ≤

1_{− |}z_|2Re α

Re α (3.12)

for all z_{∈ U}. From (3.12) we obtain 1_{− |}z_|2Re β

Re β

zf′′(z) f′_(z)

≤

1_{− |}z_|2Re α Re α

zf′′(z) f′_(z)

(3.13) for all z_{∈ U}.

From (3.10) and Theorem 2.4 (Schwarz) we get

zf′′(z) f′_(z)

≤

2 |z|, z∈ U (3.14)

and, hence, we have 1_{− |}z_|2Re α

Re α

zf′′(z) f′_(z)

≤

2 ·

1_{− |}z_|2Re α

Re α |z| (3.15) for all z_{∈ U}.

Because

max

|z|<1

1_{− |}z_|2Re α

Re α |z|=

2

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1_{− |}z_|2Re β Re β

zf′′(z) f′_(z)

≤

1 (3.17)

for all z_{∈ U}.

From (3.17) and Theorem 2.2 we obtain that the function Fβ(z) is regular and

univalent in_U.

Theorem 3.5._Let _{α, β} _{complex numbers} _{Re β} _≥ _{Re α >} _{0, the function} _f _∈

A, f(z) =z+a2z2+. . . If

f′′(z) f′_(z)

<1 (3.18)

for all z_{∈ U} and

max

|z|<1

1_{− |}z_|2Re α

Re α |z|

|z_|+ 2_|a2|

1 + 2_|a2||z|

≤1 (3.19)

then the function Fβ(z) define by (2.4) is regular and univalent in U.

Proof. Let’s consider the regular functionp(z) = f_f′′′₍(_zz₎), z∈ U. We have|p(0)|= 2_|a2|and from (3.18) we obtain |p(z)|<1 for allz∈ U.

By Remark 2.6 we obtain

f′′(z) f′_(z)

≤

|z_|+ 2_|a2|

1 + 2_|a2||z|

(3.20) for all z_{∈ U}. From (3.20) we get

1_{− |}z_|2Re α

Re α

zf′′(z) f′_(z)

≤

1_{− |}z_|2Re α

Re α |z|

|z_|+ 2_|a2|

1 + 2_|a2||z|

(3.21) for all z_{∈ U}.

We consider the functionQ: [0,1]_→R

Q(x) = 1−x

2Re α

Re α x

x+ 2_|a2|

1 + 2_|a2|x

; x=_|z_|. BecauseQ 1₂>0 it results that

max

x∈(0,1)Q(x)>0.

Using this result and from (3.21) we conclude 1_{− |}z_|2Re α

Re α

zf′′(z) f′_(z)

≤

max

|z|<1

1_{− |}z_|2Re α Re α |z|

|z_|+ 2_|a2|

1 + 2_|a2||z|

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and hence, by (3.19) we obtain

1_{− |}z_|2Re α Re α

zf′′(z) f′_(z)

≤

1 (3.23)

for all z_{∈ U}. From (3.23) and by Theorem 2.3 we obtain that the functionFβ(z) is

in the class _S.

References

[1] Becker, J.,L¨ownersche Differentialgleichung und quasikonform fortsetzbare schlichte Functionen, J. Reine Angew. Math. 255 (1972), 23-43.

[2] Mayer, O.,The Functions Theory of One Variable Complex, Bucure¸sti, 1981. [3] Nehari, Z.,Conformal Mapping, Mc Graw-Hill Book Comp., New York, 1952 (Dover. Publ. Inc., 1975).

[4] Pascu, N. N., On a univalence criterion II, Itinerant Seminar on Functional Equations, Approximation and Convexity (Cluj-Napoca, 1985), 153-154.

[5] Pascu, N. N.,An improvement of Becker’s univalence criterion, Proceedings of the Commemorative Session Simion Stoilow, Bra¸sov, (1987), 43-48.

[6] Pescar, V., New Univalence Criteria, ”Transilvania” University of Bra¸sov, 2002.

[7] Pescar, V., Breaz, D. V., The Univalence of Integral Operators, Academic Publishing House, Sofia, 2008.

Virgil Pescar

Department of Mathematics

”Transilvania” University of Bra¸sov 500091 Bra¸sov, Romania