# A Study on Becker's Univalence Criteria

Authors: Maslina Darus; Imran Faisal
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We study univalence properties for certain subclasses of univalent functions𝔎, 𝔎2, 𝔎3, and S(p), respectively. These subclasses are associated with a generalized integral operator. The extended Becker-typed univalence criteria will be studied for these subclasses.


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## Body

## 1. Introduction and Preliminaries

LetA denote the class of analytic functions f in the open unit disk 𝕌={z:|z|<1} normalized by f(0)=f′(0)-1=0. Thus, each f∈A has a Taylor series representation (1.1)f(z)=z+∑k=2∞akzk⋅
Let A2 be the subclass of A consisting of functions of the form (1.2)f(z)=z+∑k=3∞akzk⋅
Let 𝔎 be the univalent subclass of A which satisfies (1.3)|z2f′(z)(f(z))2-1|<1,z∈U.
Let 𝔎2 be the subclass of 𝔎 for which f′′(0)=0. Let 𝔎2,μ be the subclass of 𝔎2 consisting of functions of the form (1.2) which satisfy (1.4)|z2f′(z)(f(z))2-1|<μ,0<μ≤1,z∈U.
Next, we define a subclass S(p) of A consisting of all functions f(z) that satisfy (1.5)|(z(f(z)))′′|≤p,0<p≤2,p∈R,z∈U.
For functions f(z)=z+∑k=2∞akzk and g(z)=z+∑k=2∞bkzk, the Hadamard product (or convolution) f*g is defined as usual by (1.6)(f*g)(z)=z+∑k=2∞akbkzk⋅
Define the function φ(a,c;z) by (1.7)φ(a,c;z)=z2F(1,a,c;z)=∑k=0∞(a)k(c)kzk-1,c≠0,-1-,2,…,
where (a)kk is the famous Pochhammer symbol defined in terms of Gamma function. It is easily seen that φ(2-α,2;z) is a convex function, since zφ′(z)=φ(2-α,1;z)∈SV⋆(α)·Using the fractional derivative of orderα, Dzα [1], Owa and Srivastava [2] introduced the operator Ωα:A→A which is known as an extension of fractional derivative and fractional integral, as follows: (1.8)Ωαf(z)=Γ(2-α)zαDzαf(z),α≠2,3,4,…=z+∑k=2∞Γ(k+1)Γ(2-α)Γ(k+1-α)akzk=φ(2,2-α;z)*f(z)⋅
Note that Ω0f(z)=f(z)·For a functionf in A, we define Dλn,ν(α,β,μ)f(z):A→A, the linear fractional differential operator, as follows: (1.9)Iλ0,ν(α,β,μ)f(z)=f(z),Iλ1,ν(α,β,μ)f(z)=(ν-μ+β-λν+β)(Ωαf(z))+(μ+λν+β)z(Ωαf(z)),Iλ2,ν(α,β,μ)f(z)=Iλα(Iλ1,ν(α,β,μ)f(z))⋮Iλn,ν(α,β,μ)f(z)=Iλα(Iλn-1,ν(α,β,μ)f(z))⋅
If f is given by (1.1), then by (1.8) and (1.9), we see that (1.10)Iλn,ν(α,β,μ)f(z)=z+∑k=2∞((Γ(k+1)Γ(2-α)Γ(k+1-α))(ν+(μ+λ)(k-1)+βν+β))nakzk⋅
From (1.8) and (1.9), Dλn,ν(α,β,μ)f(z) can be written in terms of convolution as (1.11)Iλn,ν(α,β,μ)f(z)=[φ(2,2-α;z)*gβ,λμ,ν(z)⋯φ(2,2-α;z)*gβ,λμ,ν(z)]︸*f(z),
where (1.12)gβ,λμ,ν(z)=z-((ν-μ+β-λ)/(ν+β))z2(1-z)2=z-(ν-μ+β-λν+β)z2(1+2z+3z2+⋯)=z+(1+μ+λν+β)z2+(1+2μ+λν+β)z3⋯⋮gβ,λμ,ν(z)=z+∑k=2∞((ν+(μ+λ)(k-1)+βν+β))zk,(1.13)φ(2,2-α;z)*gβ,λμ,ν(z)⋯φ(2,2-α;z)*gβ,λμ,ν(z)=n-timesproduct,
which generalizes many operators. Indeed, if we choose suitably values of α, β, μ, and ν in (1.12), we have the following.(i)
β=1, μ=0, and α=0, we obtain Dα,λmf(z) given by Aouf et al. [3].(ii)
ν=1, β=0,  μ=0, and α=0, we obtain Dλmf(z) given by Al-Oboudi [4].(iii)
ν=1, β=0, μ=0, λ=1, and α=0, we obtain Dmf(z) given by Sălăgean [5].(iv)
ν=1, β=1, λ=1, μ=0, and α=0, we obtain Imf(z) given by Uralegaddi and Somanatha [6].(v)
β=1, λ=1, μ=0, and α=0, we obtain Im(ℓ)f(z) given by Cho and Srivastava [7] and Cho and Kim [8].(vi)
ν=1, β=0, μ=0, λ=0, and n=1, we obtain Owa and Srivastava differential operator [2].(vii)
ν=1, β=0, and μ=0, we obtain Dλn,αf(z) given by Al-Oboudi and Al-Amoudi [9, 10].(viii)
β=l,  μ=0, and α=p, we obtain Ipn(λ,l)f(z) given by Catas [11].(ix)
β=l, μ=0, α=p, and λ=1, we obtain Ipn(λ,l)f(z) given by Kumar et al. and Srivastava et al., respectively [12, 13].Next, we introduce a new family of integral operator by using generalized differential operator already defined above.Form∈ℕ∪{0} and γ1,γ2,γ3,…,γn, ρ∈ℂ∖{0,-1,-2,…}, we define a family of integral operators Υγi,η,λ(n,ρ,ν,α,β):Am→Am by (1.14)Υγi,η,λ(n,ρ,ν,α,β:z)={ρ∫0ztρ-1∏i=1m(Iλn,ν(α,β,μ,η)fi(t)t)1/γidt}1/ρ,fi∈A,
which generalize many integral operators. In fact, if we choose suitable values of parameters in this type of operator, we get the following interesting operators.(i)
ν=1, β=0, μ=0, α=0, γi=1/αi, and ρ=1, we obtain I(f1,…,fm) given by Bulut [14].(ii)
n=0, ν=1,  β=0,  μ=0, α=0, γi=1/(α-1), and ρ=n(α-1)+1, we obtain Fn,α(z) given by Breaz et al. [15].(iii)
n=0, ν=1, β=0, μ=0, α=0, γi=1/αi, and ρ=1, we obtain Fα(z) given by D. Breaz and N. Breaz [16].For our main result, we need the following lemmas.Lemma 1.1 (see [17, 18]).
Letc be a complex number, |c|≤1,  c≠-1. If f(z)=z+a2z2+⋯ is a regular function in 𝕌 and
(1.15)|c|z|2+(1-|z|2)zf′′(z)f′(z)|≤1,∀z∈U,
then the function f is regular and univalent in 𝕌.Lemma 1.2 (Schwarz Lemma).
Let the functionf(z) be regular in the disk 𝕌R={z∈ℂ:|z|<R} with |f(z)|<M. If f(z) has one zero with multiply ≥m for z=0, then
(1.16)|f(z)|≤MRm|z|m,∀z∈UR,
and equality holds only if f(z)=eiθ(M/Rm)|z|m, where θ is constant.Lemma 1.3 (see [19]).
Letδ be a complex number with Reδ>0 such that c∈ℂ,|c|≤1,c≠-1. If f∈A satisfies the condition
(1.17)|c|z|2δ+(1-|z|2δ)zf′′(z)δf′(z)|≤1,∀z∈U,
then the function
(1.18)Fδ(z)={δ∫0ztδ-1f′(t)dt}1/δ
is analytic and univalent in 𝕌.Lemma 1.4 (see [20]).
If a functionf∈S(p), then
(1.19)|z2f′(z)(f(z))2-1|≤p|z|2,∀z∈U.


## 2. Univalence Properties

In this section, we will discuss the univalence properties of the new family of integral operators mentioned above.Theorem 2.1.
Letc be a complex number, |Iλn,ν(α,β,μ,η)fi(z)|≤Mi, Mi≥1 for all i={1,2,3,…} and Iλn,ν(α,β,μ,η)fi(t)∈S(pi) for i={1,2,3,…} such that
(2.1)R(ρ)≥∑i=1∞((Mi-1)pi+2)Mi-1|γi|(Mi1),
where ρ, γi are complex numbers. If
(2.2)|c|≤1-1Re(ρ)∑i=1∞((Mi-1)pi+2)Mi-1|γi|(Mi-1),Mi≥1,
then the family Υγi,η,λ(n,ρ,ν,α,β:z) is univalent.Proof.
SinceIλn,ν(α,β,μ,η)fi(t)∈S(pi), so by Lemma 1.4, we have
(2.3)|z2(Iλn,ν(α,β,μ,η)fi(t))′(Iλn,ν(α,β,μ,η)fi(t))2-1|≤pi|z|2,∀z∈U.
Now, by using hypothesis, we have
(2.4)|Iλn,ν(α,β,μ,η)fi(z)|≤Mi,
so by Lemma 1.3, we get
(2.5)|Iλn,ν(α,β,μ,η)fi(z)|≤Miz,∵R=1.
Let
(2.6)Iλn,ν(α,β,μ)f(z)z=1+∑k=2∞((Γ(k+1)Γ(2-α)Γ(k+1-α))(ν+(μ+λ)(k-1)+βν+β))nakzk-1≠0=1ifz=0,
so
(2.7)∏i=1m(Iλn,ν(α,β,μ,η)fi(z)z)1/γi=(Iλn,ν(α,β,μ,η)f1(z)z)1/γ1⋯(Iλn,ν(α,β,μ,η)fm(z)z)1/γm=1.
Let
(2.8)F(z)=∫0z((Iλn,ν(α,β,μ,η)f1(t)t)1/γ1⋯(Iλn,ν(α,β,μ,η)fm(t)t)1/γm)dt,
which implies that
(2.9)F′(z)=((Iλn,ν(α,β,μ,η)f1(t)t)1/γ1⋯(Iλn,ν(α,β,μ,η)fm(t)t)1/γm)′zF′′(z)F′(z)=1γ1(z(Iλn,ν(α,β,μ,η)f1(z))′Iλn,ν(α,β,μ,η)f1(z)-1)+⋯+1γm(z(Iλn,ν(α,β,μ,η)fm(z))′Iλn,ν(α,β,μ,η)fm(z)-1),(2.10)zF′′(z)F′(z)=∑i=1m1γi(z(Iλn,ν(α,β,μ,η)fi(z))′Iλn,ν(α,β,μ,η)fi(z)-1).
This implies that
(2.11)|zF′′(z)F′(z)|≤∑i=1∞1|γi|(|z(Iλn,ν(α,β,μ,η)fi(z))′Iλn,ν(α,β,μ,η)fi(z)|+1),
or
(2.12)|zF′′(z)F′(z)|≤∑i=1∞1|γi|(|z(Iλn,ν(α,β,μ,η)fi(z))′(Iλn,ν(α,β,μ,η)fi(z))2||(Iλn,ν(α,β,μ,η)fi(z))(z)|+1)⋅
Using (2.5), we get
(2.13)|zF′′(z)F′(z)|≤∑i=1∞1|γi|(|z(Iλn,ν(α,β,μ,η)fi(z))′(Iλn,ν(α,β,μ,η)fi(z))2|Mi+1)⋅
This implies that
(2.14)|zF′′(z)F′(z)|≤∑i=1∞1|γi|(|z(Iλn,ν(α,β,μ,η)fi(z))′(Iλn,ν(α,β,μ,η)fi(z))2-1|Mi+Mi+1)⋅
By using (2.3), we get
(2.15)|zF′′(z)F′(z)|≤∑i=1∞1|γi|(pi|z|2Mi+Mi+1),
which implies that
(2.16)|zF′′(z)F′(z)|≤∑i=1∞1|γi|(piMi+(Mi+Mi2+Mi3+⋯)+1),
because Mi,Mi2,Mi3,…,≥1 implies that
(2.17)|zF′′(z)F′(z)|≤∑i=1∞1|γi|(piMi+(MiMi-1)+1)=∑i=1∞1|γi|(piMi+(2Mi-1Mi-1)),|zF′′(z)F′(z)|≤∑i=1∞1|γi|((piMi2-piMi+2Mi-1Mi-1))≤∑i=1∞1|γi|(((piMi-pi+2)Mi-1Mi-1)),|zF′′(z)F′(z)|≤∑i=1∞1|γi|((((Mi-1)pi+2)Mi-1Mi-1))⋅
Now, we calculate
(2.18)|c|z|2ρ+(1-|z|2ρ)zF′′(z)ρF′(z)|≤|c|+1|ρ||zF′′(z)F′(z)|≤|c|+1R(ρ)|zF′′(z)F′(z)|.
This implies that
(2.19)|c|z|2ρ+(1-|z|2ρ)zF′′(z)ρF′(z)|<|c|+1R(ρ)∑i=1∞1|γi|((((Mi-1)pi+2)Mi-1Mi-1))⋅
By using (2.46), we conclude that
(2.20)|c|z|2ρ+(1-|z|2ρ)zF′(z)ρF(z)|≤|c|+1|ρ||zF′′(z)F′(z)|≤1⋅
Hence, by Lemma 1.3, the family of integral operators Υγi,η,λ(n,ρ,ν,α,β:z) is univalent.Corollary 2.2.
Letc be a complex number, |Iλn,ν(α,β,μ,η)fi(z)|≤M, M≥1 for all i={1,2,3,…} and Iλn,ν(α,β,μ,η)fi(t)∈S(p),  Mi=M≥1, for all i={1,2,3,…} such that
(2.21)R(ρ)≥∑i=1∞((M-1)p+2)M-1|γi|(M-1),
where ρ, γi are complex numbers. If
(2.22)|c|≤1-1Re(ρ)∑i=1∞((M-1)pi+2)M-1|γi|(M-1),M≥1,
then the family Υγi,η,λ(n,ρ,ν,α,β:z) is univalent.Corollary 2.3.
Letc be a complex number, |Iλn,ν(α,β,μ,η)fi(z)|≤M, M≥1, for all i={1,2,3,…} and the family Iλn,ν(α,β,μ,η)fi(t)∈S(p), Mi=M≥1,  |γi|=|γ|, for all i={1,2,3,…} such that
(2.23)R(ρ)≥∑i=1∞((M-1)p+2)M-1|γ|(M-1),
where ρ, γi are complex numbers. If
(2.24)|c|≤1-1Re(ρ)∑i=1∞((M-1)pi+2)M-1|γ|(M-1),M≥1,
then the family Υγi,η,λ(n,ρ,ν,α,β:z) is univalent.Using the method given in the proof of Theorem2.1, one can prove the following results.Theorem 2.4.
Letc be a complex number, |Iλn,ν(α,β,μ,η)fi(z)|≤Mi, Mi≥1 for all i={1,2,3,…} and the family Iλn,ν(α,β,μ,η)fi(t)∈S(pi) for i={1,2,3,…} and c such that
(2.25)R(ρ)≥∑i=1∞(piMi-1)Mi+1|γi|piMi,
where ρ, γi are complex numbers. If
(2.26)|c|≤1-1Re(ρ)∑i=1∞(piMi-1)Mi+1|γi|(piMi),Mi≥1,
then the family Υγi,η,λ(n,ρ,ν,α,β:z) is univalent.Theorem 2.5.
Letc be a complex number, |Iλn,ν(α,β,μ,η)fi(z)|≤Mi, Mi≥1 for all i={1,2,3,…,n} and Iλn,ν(α,β,μ,η)fi(t)∈S(pi) for i={1,2,3,…,n} such that
(2.27)R(ρ)≥∑i=1n(pi(Mi-1)+Min-2)Mi+1|γi|(Mi-1),
where ρ, γi are complex numbers. If
(2.28)|c|≤1-1Re(ρ)∑i=1n(pi(Mi-1)+Min-2)Mi+1|γi|(Mi-1),Mi≥1,
then the family Υγi,η,λ(n,ρ,ν,α,β:z) is univalent.Theorem 2.6.
Letc be a complex number, |Iλn,ν(α,β,μ,η)fi(z)|≤Mi, Mi≥1 for all i={1,2,3,…,n} and Iλn,ν(α,β,μ,η)fi(t)∈S(pi) for i={1,2,3,…,n} such that
(2.29)R(ρ)≥∑i=1n(pi+(n(n+1)/2))Mi-1|γi|,
where ρ, γi are complex numbers. If
(2.30)|c|≤1-1Re(ρ)∑i=1n(pi+(n(n+1)/2))Mi-1|γi|,Mi≥1,
then the family Υγi,η,λ(n,ρ,ν,α,β:z) is univalent.Theorem 2.7.
Letc be a complex number, |Iλn,ν(α,β,μ,η)fi(z)|≤Mi, Mi≥1 for all i={1,2,3,…,n} and Iλn,ν(α,β,μ,η)fi(t)∈𝔎2,μi, for i={1,2,3,…,n} such that
(2.31)R(ρ)≥∑i=1n(μi+n(n+1))Mi|γi|,
where ρ, γi are complex numbers. If
(2.32)|c|≤1-1Re(ρ)∑i=1n(μi+n(n+1))Mi|γi|,Mi≥1,
then the family Υγi,η,λ(n,ρ,ν,α,β:z) is univalent.Proof.
Using the proof of Theorem2.1, we have
(2.33)|zF′′(z)F′(z)|≤∑i=1∞1|γi|(|z(Iλn,ν(α,β,μ,η)fi(z))′(Iλn,ν(α,β,μ,η)fi(z))2|Mi+1).
Since Iλn,ν(α,β,μ,η)fi(t)∈𝔎2,μi, so by using (1.4), we get
(2.34)|z2(Iλn,ν(α,β,μ,η)fi(z))′(Iλn,ν(α,β,μ,η)fi(z))2-1|<μi,0<μ≤1,z∈U.
So from (2.33), we get
(2.35)|zF′′(z)F′(z)|≤∑i=1∞1|γi|(|z(Iλn,ν(α,β,μ,η)fi(z))′(Iλn,ν(α,β,μ,η)fi(z))2-1|Mi+Mi+1),
or
(2.36)|zF′′(z)F′(z)|≤∑i=1∞1|γi|(μiMi+2Mi),Mi>1,|zF′′(z)F′(z)|≤∑i=1∞1|γi|(μiMi+2Mi+4Mi+⋯+n-times),Mi>1,|zF′′(z)F′(z)|≤∑i=1∞1|γi|(μiMi+n(n+1)Mi),Mi>1.
Now, we evaluate the expression
(2.37)|c|z|2ρ+(1-|z|2ρ)zF′′(z)ρF′(z)|≤|c|+1|ρ||zF′′(z)F′(z)|≤|c|+1R(ρ)|zF′′(z)F′(z)|,|c|z|2ρ+(1-|z|2ρ)zF′′(z)ρF′(z)|≤|c|+1R(ρ)∑i=1∞1|γi|(μiMi+n(n+1)Mi).
Using (2.45) and (2.46), we conclude that
(2.38)|c|z|2ρ+(1-|z|2ρ)zF′(z)ρF(z)|≤1.
Hence by using Lemma 1.3, the family Υγi,η,λ(n,ρ,ν,α,β:z) is univalent.Corollary 2.8.
Letc be a complex number, |Iλn,ν(α,β,μ,η)fi(z)|≤M, M≥1 for all i={1,2,3,…,n} and Iλn,ν(α,β,μ,η)fi(t)∈𝔎2,μi, for i={1,2,3,…,n} such that
(2.39)R(ρ)≥∑i=1n(μi+n(n+1))M|γi|,
where ρ, γi are complex numbers. If
(2.40)|c|≤1-1Re(ρ)∑i=1n(μi+n(n+1))M|γi|,M≥1,
then the family Υγi,η,λ(n,ρ,ν,α,β:z) is univalent.Corollary 2.9.
Letc be a complex number, |Iλn,ν(α,β,μ,η)fi(z)|≤M, M≥1, |γi|=|γ| for all i={1,2,3,…,n} and Iλn,ν(α,β,μ,η)fi(t)∈𝔎2,μi, for i={1,2,3,…,n} such that
(2.41)R(ρ)≥∑i=1n(μi+n(n+1))M|γ|,
where ρ, γi are complex numbers. If
(2.42)|c|≤1-1Re(ρ)∑i=1n(μi+n(n+1))M|γ|,M≥1,
then the family Υγi,η,λ(n,ρ,ν,α,β:z) is univalent.Using a similar method as in the proof of Theorem2.7, one can prove the following results.Theorem 2.10.
Letc be a complex number, |Iλn,ν(α,β,μ,η)fi(z)|≤Mi, Mi≥1 for all i={1,2,3,…,n} and Iλn,ν(α,β,μ,η)fi(t)∈𝔎2,μi, for i={1,2,3,…,n} such that
(2.43)R(ρ)≥∑i=1n(μiMi-1)Mi+MinMi|γi|(Mi-1),
where ρ, γi are complex numbers. If
(2.44)|c|≤1-1Re(ρ)∑i=1n(μiMi-1)Mi+MinMi|γi|(Mi-1),Mi≥1,
then the family Υγi,η,λ(n,ρ,ν,α,β:z) is univalent.Theorem 2.11.
Letc be a complex number, |Iλn,ν(α,β,μ,η)fi(z)|≤Mi, Mi≥1 for all i={1,2,3,…} and Iλn,ν(α,β,μ,η)fi(t)∈𝔎2,μi, for i={1,2,3,…} such that
(2.45)R(ρ)≥∑i=1n(μiMi-μi+2)Mi-1|γi|(Mi-1),
where ρ, γi are complex numbers. If
(2.46)|c|≤1-1Re(ρ)∑i=1n(μiMi-μi+2)Mi-1|γi|(Mi-1),Mi≥1,
then the family Υγi,η,λ(n,ρ,ν,α,β:z) is univalent.Note that some other related work involving integral operators regarding univalence criteria can also be found in [21–23].



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