# Certain Properties of a Class of Close-to-Convex Functions Related to Conic Domains

Authors: Wasim Ul-Haq; Shahid Mahmood
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We aim to define a new class of close-to-convex functions which is related to conic domains. Many interesting properties such as sufficiency criteria, inclusion results, and integral preserving properties are investigated here. Some interesting consequences of our results are also observed.


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## Body

## 1. Introduction

Let𝒜 be the class of functions f(1)f(z)=z+∑n=2∞anzn,
which are analytic in the open unit disc E={z∈ℂ:|z|<1}. Let f and g be analytic in E, and we say that f is subordinate to g, written as f(z)≺g(z) if there exists a Schwarz function w, which is analytic in E with w(0)=0 and |w(z)|<1  (z∈E), such that f(z)=g(w(z)). In particular, when g is univalent, then the above subordination is equivalent to f(0)=g(0) and f(E)⊆g(E); see [1].Kanas and Wisniowska [2, 3] introduced and studied the classes of k-uniformly convex functions denoted by k-𝒰𝒞𝒱 and the corresponding class k-𝒮𝒯 related by the Alexander-type relation. Later, the class k-uniformly close-to-convex functions denoted by k-𝒰𝒦 defined as
(2)k-𝒰𝒦={f∈𝒜:Re(zf′(z)g(z))Re(zf′(z)g(z))>k|zf′(z)g(z)-1|,Re(zf′(z)g(z))g(z)∈k-𝒮𝒯,z∈E}
was considered by Acu [4]; for study details on these classes, we refer to [5–7]. All these above mentioned classes were generalized to the classes 𝒮𝒟(k,δ), 𝒞𝒟(k,δ), and 𝒦𝒟(k,β,δ) by Shams et al. [8] and Srivastava et al. [9], respectively. The classes 𝒮𝒟(k,δ) and 𝒞𝒟(k,δ) are defined as
(3)𝒮𝒟(k,δ)={f∈𝒜:Re(zf′(z)f(z))Re(zf′(z)f(z))>k|zf′(z)f(z)-1|Re(zf′(z)f(z))+δ,z∈E},𝒞𝒟(k,δ)={f∈𝒜:Re(1+zf′′(z)f′(z))Re(1+zf′′(z)f′(z))>k|zf′′(z)f′(z)|Re(1+zf′′(z)f′(z))+δ,z∈E},
where k≥0, 0≤β, δ<1. The class 𝒦𝒟(k,β,δ) known as k-uniformly close-to-convex functions of order β type δ is the class of all those functions f∈𝒜 which satisfies the following condition:
(4)Re(f′(z)g′(z))≥k|f′(z)g′(z)-1|+β,(k≥0,0≤β,δ<1;z∈E),
for some g∈𝒞𝒟(k,δ).Motivated by the work of Noor et al. [10–13], we define the following.Definition 1.
Letf∈𝒜. Then, f is in the class 𝒯𝒟(k,α,β,γ,δ) if and only if, for α≥0, 0≤β,γ,δ<1,
(5)Re(H(α,β,γ;f,g))≥k|H(α,β,γ;f,g)-1|,(k≥0;z∈E),
for some g∈𝒞𝒟(k,δ), where
(6)H(α,β,γ;f,g)=1-α1-β[f′(z)g′(z)-β]=+α1-γ[(zf′(z))′g′(z)-γ].Special Cases
(i)
(
k
,
0
,
β
,
γ
,
δ
)
=
𝒦
𝒟
(
k
,
β
,
δ
); see [9].
(ii)
𝒯
𝒟
(
1,0
,
β
,
γ
,
0
)
=
𝒰
𝒦
(
β
) and 𝒯𝒟(1,1,β,γ,0)=𝒰𝒬(γ), the classes of uniformly close-to-convex and quasiconvex functions introduced and investigated in [14].
(iii)
𝒯
𝒟
(
0
,
α
,
0,0
,
0
)
=
𝒬
α, the class of alpha quasiconvex functions, introduced and studied in [11].
(iv)
𝒯
𝒟
(
0,0
β
,
γ
,
δ
)
=
𝒦
(
β
,
δ
), the class of close-to-convex functions of order β type δ, [15].
(v)
𝒯
𝒟
(
0,1
,
β
,
γ
,
δ
)
=
C
*
(
γ
,
δ
), the class of quasiconvex functions of order γ type δ, [16].The conditions k≥0, α≥0, 0≤β,γ,δ<1 on the parameters are assumed throughout the entire paper unless otherwise mentioned.Geometric Interpretation. A function f∈𝒜 is in the class 𝒯𝒟(k,α,β,γ,δ) if and only if the functional H(α,β,γ,δ;f,g) takes all the values in the conic domain Ωk defined as follows:
(7)Ωk={u+ιv;u>k(u-1)2+v2}.
Extremal functions for these conic regions are denoted by pk(z), which are analytic in E and map E onto Ωk such that pk(0)=1 and pk′(0)>1. These functions are given as:
(8)pk(z)={1+z1-z,k=0,1+2π2(log1+z1-z)2,k=1,1+21-k2sinh2[(2πarccosk)arctanhz],0<k<1,1+1k2-1sin(π2R(t)∫0u(z)/t11-x21-(tx)2dx)+1k2-1,k>1,
where u(z)=(z-t)/(1-tz),t∈(0,1),z∈E, and z is chosen such that k=cosh(πR′(t)/4R(t)), where R(t) is Legendre's complete elliptic integral of the first kind and R′(t) is complementary integral of R(t); see [2, 3].


## 2. Preliminaries Result

We require the following results which are essential in our investigations.Lemma 2 (see [17, page 70]).
Leth be convex function in E and q:E↦C  with Req(z)>0,  z∈E. If p is analytic in E with p(0)=h(0), then
(9)p(z)+q(z)zp′(z)≺h(z)impliesp(z)≺h(z).Lemma 3 (see [17, page 195]).
Leth be convex function in E with h(0)=0 and A≥0. Suppose that j≥4/h′(0)  and that B(z),  C(z), and D(z) are analytic in E and satisfy
(10)ReB(z)≥A+|C(z)-1|-Re(C(z)-1)+jD(z),
for z∈E. If p is analytic in E with p(z)=a1z+a2z2+a3z3+⋯ and the following subordination relation holds:
(11)Az2p′′(z)+B(z)zp′(z)+C(z)p(z)+D(z)≺h(z),
then
(12)p(z)≺h(z).Lemma 4 (see [12]).
Iff(z)≺h(z) and g(z)≺h(z), then for α∈[0,1],
(13)(1-α)f(z)+αg(z)≺h(z).


## 3. Main Results

First, we prove the following sufficiency criteria for the functions in the class𝒯𝒟(k,α,β,γ,δ).Theorem 5.
A functionf∈𝒜 is said to be in the class 𝒯𝒟(k,α,β,γ,δ), if
(14)∑n=2∞Φn(k;α,β,γ,δ)<(1-β)(1-γ),
where
(15)Φn(k;α,β,γ,δ)=(k+1)[(1-α)(1-γ)n+α(1-β)n2]|an|=+[(k+2-β)(1-γ)+α(k+1)(γ-β)n|bn|].Proof.
Let us assume that relation (6) holds. Now, it is sufficient to show that
(16)k|H(α,β,γ,δ;f,g)-1|-Re[H(α,β,γ,δ;f,g)-1]<1.
First, we consider(17)|H(α,β,γ,δ;f,g)-1|=|1-α1-β[f′(z)g′(z)-β]+α1-γ[(zf′(z))′g′(z)-γ]-1|=|1-α1-βf′(z)g′(z)+α1-γ(zf′(z))′g′(z)-(1-α)β1-β-αγ1-γ-11-α1-βf′(z)g′(z)+α1-γ(zf′(z))′g′(z)|=|(1-α)(1-γ)f′(z)(1-β)(1-γ)g′(z)+α(1-β)(zf′(z))′(1-β)(1-γ)g′(z)-[(1-γ)+α(γ-β)]g′(z)(1-β)(1-γ)g′(z)|.
Using (1) and the series g(z)=z+∑n=2∞bnzn in (17), we have
(18)|H(α,β,γ,δ;f,g)-1|=|(1-α)(1-γ)(1+∑n=2∞nanzn-1)(1-β)(1-γ)(1+∑n=2∞nbnzn-1)≤≤+α(1-β)(1+∑n=2∞n2anzn-1)(1-β)(1-γ)(1+∑n=2∞nbnzn-1)≤≤-[(1-γ)+α(γ-β)](1+∑n=2∞nbnzn-1)(1-β)(1-γ)(1+∑n=2∞nbnzn-1)|=|(1-α)(1-γ)(∑n=2∞nanzn-1)(1-β)(1-γ)(1+∑n=2∞nbnzn-1)≤≤+α(1-β)(∑n=2∞n2anzn-1)(1-β)(1-γ)(1+∑n=2∞nbnzn-1)≤≤-[(1-γ)+α(γ-β)](∑n=2∞nbnzn-1)(1-β)(1-γ)(1+∑n=2∞nbnzn-1)|≤|(1-α)(1-γ)(∑n=2∞n|an||z|n-1)(1-β)(1-γ)(1-∑n=2∞n|bn||z|n-1)≤≤+α(1-β)(∑n=2∞n2|an||z|n-1)(1-β)(1-γ)(1-∑n=2∞n|bn||z|n-1)≤≤+[(1-γ)+α(γ-β)](∑n=2∞n|bn||z|n-1)(1-β)(1-γ)(1-∑n=2∞n|bn||z|n-1)|≤|(1-α)(1-γ)(∑n=2∞n|an|)(1-β)(1-γ)(1-∑n=2∞n|bn|)≤≤+α(1-β)(∑n=2∞n2|an|)(1-β)(1-γ)(1-∑n=2∞n|bn|)≤≤+[(1-γ)+α(γ-β)](∑n=2∞n|bn|)(1-β)(1-γ)(1-∑n=2∞n|bn|)|.
Now,
(19)k|H(α,β,γ,δ;f,g)-1|-Re[H(α,β,γ,δ;f,g)-1]≤(k+1)|H(α,β,γ,δ;f,g)-1|≤(k+1)[(1-α)(1-γ)(∑n=2∞n|an|)(1-β)(1-γ)(1-∑n=2∞n|bn|)≤(k+1)+α(1-β)(∑n=2∞n2|an|)(1-β)(1-γ)(1-∑n=2∞n|bn|)≤(k+1)+[(1-γ)+α(γ-β)](∑n=2∞n|bn|)(1-β)(1-γ)(1-∑n=2∞n|bn|)].
The last inequality is bounded above by 1, if
(20)(k+1)[(1-α)(1-γ)(∑n=2∞‍n|an|)+α(1-β)(∑n=2∞‍n2|an|)(k+1)+[(1-γ)+α(γ-β)](∑n=2∞‍n|bn|)]≤(1-β)(1-γ)(1-∑n=2∞‍n|bn|).
Hence,
(21)∑n=2∞Φn(k;α,β,γ,δ)≤(1-β)(1-γ),
where Φn(k;α,β,γ,δ) is given by (15). This completes the proof.When we takeα=0, k=1, and g(z)=z in the above theorem, we obtain the following sufficient condition for the functions to be in the class 𝒰𝒦(β) which is proved in [14].Corollary 6 (see [14]).
A functionf∈A is said to be in the class 𝒰𝒦(β) if
(22)∑n=2∞‍n|an|≤(1-β)2.Corollary 7 (see [14]).
A functionf∈A is said to be in the class 𝒰𝒬(γ) if
(23)∑n=2∞‍n2|an|≤(1-γ)2.The above corollary is obtained when we takeα=1, k=1,  and   g(z)=z in Theorem 5.Theorem 8.
Letf∈𝒯𝒟(k,α,β,γ,δ) and α≥4(β/(1+3β)). Then, f∈𝒦𝒟(k,0,δ).Proof.
Let(24)f′(z)g′(z)=p(z),
where p(z) is analytic and p(0)=1. Now differentiating (24), we have
(25)(zf′(z))′g′(z)=zp′(z)+p(z)ψ(z),
where ψ(z)=(zg′(z))′/g′(z). Using (24) and (25) in relation (6), we obtain
(26)H(α,β,γ,δ;f(z))=1-α1-β[p(z)-β]+α1-γ[zp′(z)+p(z)ψ(z)-γ]=α1-γzp′(z)+[(1-α)(1-γ)+α(1-β)ψ(z)(1-β)(1-γ)]p(z)=-β(1-α)(1-γ)+αγ(1-β)(1-β)(1-γ)=B(z)zp′(z)+C(z)p(z)+D(z),
where
(27)B(z)=α1-γ,C(z)=(1-α)(1-γ)+α(1-β)ψ(z)(1-β)(1-γ),D(z)=-β(1-α)(1-γ)+αγ(1-β)(1-β)(1-γ).
Now, sincef∈𝒯𝒟(k,α,β,γ,δ), we have
(28)B(z)zp′(z)+C(z)p(z)+D(z)≺pk(z).
Replacingp(z) by p*(z)=p(z)-1 and pk(z) by pk*(z)=pk(z)-1, the above subordination is equivalent to
(29)B(z)zp*′(z)+C(z)p*(z)+D*(z)≺pk*(z),
where D*(z)=C(z)+D(z)-1. Using Lemma 3 with A=0, we obtain
(30)p*(z)≺pk*(z).
This implies that
(31)f′(z)g′(z)=p(z)≺pk(z).
Hence, f∈𝒦𝒟(k,0,δ). This completes the proof.Corollary 9.
Letf∈𝒯𝒟(k,α,0,0,0)=𝒬α. Then, f∈𝒦𝒟(0,0,0)=𝒦. That is, 𝒬α⊂𝒦,  α≥0.The above result is well-known inclusion proved in [11].Forf∈A, consider the following integral operator defined by
(32)F(z)=Im[f]=m+1zm∫0ztm-1f(t)dt,m=1,2,3….
This operator was given by Bernardi [18] in 1969. In particular, the operator I1 was considered by Libera [19]. Now let us prove the following.Theorem 10.
Letf∈𝒯𝒟(k,α,β,γ,δ). Then, Im[f]∈𝒦𝒟(k,0,δ).Proof.
Let the functiong be such that (6) is satisfied. It can easily be seen that according to [4], the function G=Im[f]∈𝒞𝒟(k,δ), and from (32), we deduce
(33)(1+m)f′(z)=(1+m)F′(z)+zF′′(z),(34)(1+m)g′(z)=(1+m)G′(z)+zG′′(z).
If we let p(z)=F′(z)/G′(z) and q(z)=1/(m+1+(zG′′(z)/G′)), then simple computations yield us
(35)f′(z)g′(z)=(1+m)F′(z)+zF′′(z)(1+m)G′(z)+zG′′(z)=zp(z)g′(z)+zp(z)g′′(z)+zg′(z)p′(z)(1+m)G′(z)+zG′′(z)=p(z)+q(z)zp′(z).
Let
(36)f′(z)g′(z)=p(z)+zp′(z)q(z)=h(z),
where h(z) is analytic and p(0)=1. From (36), we have
(37)(zf′(z))′g′(z)=ψ(z)h(z)+zh′(z),
where ψ(z)=(zg′(z))′/g′(z). Using (36) and (37) in (6), we have
(38)H(α,β,γ,δ;f(z))=1-α1-β[h(z)-β]+α1-γ[zh′(z)+h(z)ψ(z)-γ]=α1-γzh′(z)=+[(1-α)(1-γ)+α(1-β)ψ(z)(1-β)(1-γ)]h(z)=-β(1-α)(1-γ)+αγ(1-β)(1-β)(1-γ)=B(z)zh′(z)+C(z)h(z)+D(z),
where
(39)B(z)=α1-γ,C(z)=(1-α)(1-γ)+α(1-β)ψ(z)(1-β)(1-γ),D(z)=-β(1-α)(1-γ)+αγ(1-β)(1-β)(1-γ).
Now proceeding in the similar manner as in the proof of Theorem 5 and using Lemma 3 with A=0, we obtain
(40)f′(z)g′(z)=h(z)≺pk(z).
From (36), it implies that
(41)p(z)+zp′(z)q(z)≺pk(z).
By employing Lemma 2, we immediately obtain the desired result.Theorem 11.
Forα>α1≥0,
(42)𝒯𝒟(k,α,β,γ,δ)⊆𝒯𝒟(k,α1,β,γ,δ).Proof.
Letf∈𝒯𝒟(k,α,β,γ,δ). Then, consider
(43)H(α1,β,γ,δ;f,g)=1-α11-β[f′(z)g′(z)-β]=+α11-γ[(zf′(z))′g′(z)-γ].
After some simple computations, we have
(44)H(α1,β,γ,δ;f,g)=11-β(1-α1α)[f′(z)g′(z)-β]=+α1α[1-α1-β(f′(z)g′(z)-β)+α1-γ((zf′(z))′g′(z)-γ)]=(1-α1α)H(0,β,γ,δ;f,g)+α1αH(α,β,γ,δ;f,g).
Now, since f∈𝒯𝒟(k,α,β,γ,δ), we have H(α,β,γ,δ;f,g)≺pk(z). Also, Theorem 8 gives us that H(0,β,γ,δ;f,g)≺pk(z). The use of Lemma 4 leads us to the required relation; that is, H(α1,β,γ,δ;f,g)≺pk(z). This completes the proof.



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