Generalized k-Uniformly Close-to-Convex Functions Associated with Conic Regions

Abstract

We define and study some subclasses of analytic functions by using a certain multiplier transformation. These functions map the open unit disc onto the domains formed by parabolic and hyperbolic regions and extend the concept of uniformly close-to-convexity. Some interesting properties of these classes, which include inclusion results, coefficient problems, and invariance under certain integral operators, are discussed. The results are shown to be the best possible.

1. Introduction

Let 𝐴 denote the class of analytic functions 𝑓 defined in the unit disc 𝐸={𝑧∶|𝑧|<1} and satisfying the condition 𝑓(0)=0,𝑓′(0)=1. Let 𝑆,𝑆^∗(𝛾),𝐶(𝛾) and 𝐾(𝛾) be the subclasses of 𝐴 consisting of functions which are univalent, starlike of order 𝛾, convex of order 𝛾, and close-to-convex of order 𝛾, respectively, 0≤𝛾<1. Let 𝑆^∗(0)=𝑆^∗,𝐶(0)=𝐶 and 𝐾(0)=𝐾.

For analytic functions 𝑓(𝑧)=∑^∞_𝑛=0𝑎_𝑛𝑧^𝑛 and 𝑔(𝑧)=∑^∞_𝑛=0𝑏_𝑛𝑧^𝑛, by 𝑓∗𝑔 we denote the convolution (Hadamard product) of 𝑓 and 𝑔, defined by𝑓∗𝑔(𝑧)=^∞_𝑛=0𝑎_𝑛𝑏_𝑛𝑧^𝑛. (1.1)

We say that a function𝑓∈𝐴 is subordinate to a function 𝐹∈𝐴 and write 𝑓(𝑧)≺𝐹(𝑧) if and only if there exists an analytic function 𝑤(𝑧),𝑤(0)=0,|𝑤(𝑧)|<1 for 𝑧∈𝐸 such that 𝑓(𝑧)=𝐹(𝑤(𝑧)),𝑧∈𝐸.

If 𝐹 is univalent in 𝐸, then𝑓(𝑧)≺𝐹(𝑧)⟺𝑓(0)=𝐹(0),𝑓(𝐸)⊂𝐹(𝐸). (1.2)

For 𝑘∈[0,1], define the domain Ω_𝑘 as follows, see [1]:Ω_𝑘=𝑢+𝑖𝑣∶𝑢>𝑘(𝑢−1)²+𝑣². (1.3) For fixed 𝑘,Ω_𝑘 represents the conic region bounded, successively, by the imaginary axis (𝑘=0), the right branch of hyperbola (0<𝑘<1), a parabola (𝑘=1).

Related with Ω_𝑘, the domain Ω_𝑘,𝛾 is defined in [2] as follows:Ω_𝑘,𝛾=1−𝛾Ω_𝑘+𝛾,0≤𝛾<1. (1.4)

The functions which play the role of extremal functions for the conic regions Ω_𝑘,𝛾 are denoted by 𝑝_𝑘,𝛾(𝑧) with 𝑝_𝑘,𝛾(0)=1, and 𝑝_𝑘,𝛾′(0)>0 are univalent, map 𝐸 onto Ω_𝑘,𝛾, and are given as𝑝_𝑘,𝛾(𝑧)=1+1−2𝛾𝑧()21−𝑧,𝑘=0,1+1−𝛾𝜋²√log1+𝑧1−√𝑧²2,𝑘=1,1+1−𝛾1−𝑘²sinh²2𝜋arccos𝑘arctanh√𝑧,(0<𝑘<1). (1.5)

It has been shown [3, 4] that 𝑝_𝑘,𝛾(𝑧) is continuous as regards to 𝑘 and has real coefficients for all 𝑘∈[0,1].

Let 𝑃(𝑝_𝑘,𝛾) be the class of functions 𝑝(𝑧) which are analytic in 𝐸 with 𝑝(0)=1 such that 𝑝(𝑧)≺𝑝_𝑘,𝛾(𝑧) for 𝑧∈𝐸. It can easily be seen that 𝑃(𝑝_𝑘,𝛾)⊂𝑃, where 𝑃 is the class of Caratheodory functions of positive real part.

The class 𝑃_𝑚(𝑝_𝑘,𝛾) is defined in [5] as follows.

Let 𝑝(𝑧) be analytic in 𝐸 with 𝑝(0)=1. Then 𝑝∈𝑃_𝑚(𝑝_𝑘,𝛾) if and only if, for 𝑚≥2,0≤𝛾<1,𝑘∈[0,1],𝑧∈𝐸,𝑝(𝑧)=𝑚4+12𝑝₁(𝑧)−𝑚4−12𝑝₂(𝑧),𝑝₁,𝑝₂∈𝑃𝑝_𝑘,𝛾. (1.6)

For 𝑘=0,𝛾=0, the class 𝑃_𝑚(𝑝_0,0) coincides with the class 𝑃_𝑚 introduced by Pinchuk in [6]. Also 𝑃₂=𝑃.

The generalized Harwitz-Lerch Zeta function [7] 𝜙(𝑧,𝜆,𝜇) is given as𝜙𝑧,𝜆,𝜇=^∞_𝑛=0𝑧^𝑛𝜇+𝑛^𝜆,𝜆∈ℂ,𝜇∈ℂ⧵𝑍⁻={−1,−2,…}. (1.7)

Using (1.7), the following family of linear operators, see [7–9], is defined in terms of the Hadamard product as𝐽_𝜆,𝜇𝑓(𝑧)=𝐻_𝜆,𝜇(𝑧)∗𝑓(𝑧), (1.8) where 𝑓∈𝐴,𝐻_𝜆,𝜇(𝑧)=1+𝜇^𝜆𝜙𝑧,𝜆,𝜇−𝜇^−𝜆,(𝑧∈𝐸), (1.9)

and 𝜙(𝑧,𝜆,𝜇) is given by (1.7).

From (1.7) and (1.8), we can write𝐽_𝜆,𝜇𝑓(𝑧)=𝑧+^∞_𝑛=21+𝜇𝑛+𝜇^𝜆𝑎_𝑛𝑧^𝑛. (1.10)

For the different permissible values of parameters 𝜆 and 𝜇, the operator 𝐽_𝜆,𝜇 has been studied in [3, 4, 7, 10–12].

We observe some special cases of the operator (1.10) as given below

(i) 𝐽_0,𝜇𝑓(𝑧)=𝑓(𝑧),

(ii) 𝐽_1,0𝑓(𝑧)=𝑧+∑^∞_𝑛=2(𝑎_𝑛/𝑛)𝑧^𝑛=∫^𝑧₀(𝑓(𝑡)/𝑡)𝑑𝑡,

(iii) 𝐽_1,𝜇𝑓(𝑧)=𝑧+∑^∞_𝑛=2((1+𝜇)/(𝑛+𝜇))𝑎_𝑛𝑧^𝑛=((1+𝜇)/𝑧^𝜇)∫^𝑧₀𝑡^𝜇−1𝑓(𝑡)𝑑𝑡,(𝜇>−1).

We remark that 𝐽_1,1𝑓(𝑧) is the well-known Libera operator and 𝐽_1,𝜇𝑓(𝑧) is the generalized Bernardi operator, see [13, 14]. Also 𝐽_𝜆,1𝑓(𝑧)=𝐿_𝜆𝑓(𝑧) represents the operator closely related to the multiplier transformation studied by Flett [3].

We define the operator 𝐼_𝜆,𝜇∶𝐴→𝐴 as𝐼_𝜆,𝜇𝑓(𝑧)∗𝐽_𝜆,𝜇𝑓(𝑧)=𝑧(),1−𝑧𝜆real,𝜇>−1, (1.11)see [15]. This gives us𝐼_𝜆,𝜇𝑓(𝑧)=𝑧+^∞_𝑛=2𝑛+𝜇1+𝜇^𝜆𝑎_𝑛𝑧^𝑛,𝜆real,𝜇>−1. (1.12)

From (1.12), the following identity can easily be verified𝑧𝐼_𝜆,𝜇𝑓(𝑧)^′=𝜇+1𝐼_𝜆+1,𝜇𝑓(𝑧)−𝜇𝐼_𝜆,𝜇𝑓(𝑧). (1.13)

Remark 1.1. (i)For 𝑘∈(0,1), we note that the domain Ω_𝑘 given by (1.3) represents the following hyperbolic region:𝑘𝑢+²1−𝑘²²−𝑘²1−𝑘²𝑣²>𝑘1−𝑘²²,𝑘𝑢>.𝑘+1 (1.14)
The extremal function 𝑝_𝑘,𝛾(𝑧), for 0<𝑘<1, can be written as𝑝_𝑘,𝛾(𝑧),=1−𝛾𝑝_𝑘(𝑧)+𝛾, (1.15)where 𝑝_𝑘(𝑧), in a simplified form, is given below 𝑝_𝑘(𝑧)1=1+2sin²𝜎1+√𝑧1−√𝑧^2𝜎/𝜋+1−√𝑧1+√𝑧^2𝜎/𝜋,1−2=1+2sin²𝜎^∞_𝑛=1^2𝑛_𝑗=0(−1)^𝑗𝑗𝑧2𝜎/𝜋−2𝜎/𝜋2𝑛−𝑗^𝑛,𝜎=1+8𝜋sin𝜎²𝑧+⋯,(𝑧∈𝐸,𝜎=arccos𝑘), (1.16) and the branch of √𝑧 is chosen such that Im√𝑧≥0.
It is easy to see that, for ℎ∈𝑃(𝑝_𝑘,𝛾),Reℎ(𝑧)>(𝑘+𝛾)/(𝑘+1),𝑘∈(0,1). That is𝑃𝑝_𝑘,𝛾⊂𝑃𝑘+𝛾1+𝑘, (1.17)and the order (𝑘+𝛾)/(1+𝑘) is sharp with the extremal function 𝑝(𝑧)=(1−𝛾)𝑝_𝑘(𝑧)+𝛾, where 𝑝_𝑘(𝑧) is given by (1.16).(ii) For 𝑘=1, the extremal function𝑝₁(𝑧)2=1+𝜋²√log1+𝑧1−√𝑧²,8=1+𝜋^2∞_𝑛=11𝑛^𝑛−1_𝑗=01𝑧2𝑗+1^𝑛,8=1+𝜋²2𝑧+3𝑧²+23𝑧45³+44𝑧105⁴+⋯,(𝑧∈𝐸), (1.18) maps 𝐸 conformally onto the parabolic region Ω₁={𝑢+𝑖𝑣∶𝑢>(𝑣²+1)/2}.
It can easily be verified that Re𝑝₁(𝑧)>1/2 and, in this case, the order 1/2 is sharp.

We now define the following.

Definition 1.2. Let 𝑓∈𝐴 and let the operator 𝐼_𝜆,𝜇𝑓 be defined by (1.12). Then 𝑓∈𝑘−∪𝑅^𝛾_𝑚(𝜆,𝜇) for 𝑚≥2,𝑘∈[0,1] and 𝛾∈[0,1) if and only if 𝑧𝐼_𝜆,𝜇𝑓(𝑧)′𝐼_𝜆,𝜇𝑔(𝑧)∈𝑃_𝑚𝑝_𝑘,𝛾,𝑧∈𝐸. (1.19)
We note the following.
(i)For 𝑚=2,𝑘=0, and 𝜆=0, the class 𝑘−∪𝑅^𝛾_𝑚(𝜆,𝜇) reduces to 𝑆^∗(𝛾), and 𝜆=0 gives us the class 𝑘−∪𝑆𝑇 of uniformly starlike functions, see [2, 16].(ii)0−∪𝑅⁰_𝑚(0,𝜇)=𝑅_𝑚 is the class of functions of bounded radius rotation, see [13, 14].(iii)We denote 𝑘−∪𝑅^𝛾_𝑚(0,𝜇) as 𝑘−∪𝑅^𝛾_𝑚, see [5].(iv)Let 𝑚=2. Then 𝑓∈𝑘−∪𝑅^𝛾₂(𝜆,𝜇) implies that𝑧𝐼Re^𝜆,𝜇𝑓(𝑧)′𝐼_𝜆,𝜇𝑓(𝑧)𝑧𝐼>𝑘^𝜆,𝜇𝑓(𝑧)′𝐼_𝜆,𝜇𝑓(𝑧)−1+𝛾, (1.20) and we note that, for 0≤𝑘₂<𝑘₁, 𝑘₁−∪𝑅^𝛾₂(𝜆,𝜇)⊂𝑘₂−∪𝑅^𝛾₂(𝜆,𝜇).

Definition 1.3. Let 𝑓∈𝐴. Then 𝑓∈𝑘−∪𝑇^𝛾_𝑚(𝜆,𝜇) if and only if there exists 𝑔∈𝑘−∪𝑅^𝛾₂(𝜆,𝜇) such that

𝑧𝐼_𝜆,𝜇𝑓(𝑧)′𝐼_𝜆,𝜇𝑔(𝑧)∈𝑃_𝑚𝑝_𝑘,𝛾in𝐸. (1.21)

Special Cases.
(i)0−∪𝑇^𝛾₂(0,𝜇)=𝐾(𝛾). (ii)For 𝑘=𝛾=𝜆=0, we obtain the class 𝑇_𝑚 introduced and discussed in [17].(iii)When we take 𝑚=2 and 𝜆=0, then 𝑘−∪𝑇^𝛾₂(0,𝜇)=𝑘−∪𝐾_𝛾, the class of uniformly close-to-convex functions, see [2].

2. Preliminary Results

We need the following results in our investigation.

Lemma 2.1 (see [18]). Let 𝑞(𝑧) be convex in 𝐸 and 𝑗∶𝐸→ℂ with Re[𝑗(𝑧)]>0,𝑧∈𝐸. If 𝑝(𝑧), analytic in 𝐸 with 𝑝(0)=1, satisfies 𝑝(𝑧)+𝑗(𝑧)𝑧𝑝^′(𝑧)≺𝑞(𝑧), (2.1) then 𝑝(𝑧)≺𝑞(𝑧). (2.2)

In the following, one gives an easy extension of a result proved in [1].

Lemma 2.2 (see [5]). Let 𝑘≥0 and let 𝛽,𝛿 be any complex numbers with 𝛽≠0 and Re((𝛽𝑘/(𝑘+1))+𝛿)>𝛾. If ℎ(𝑧) is analytic in 𝐸,ℎ(0)=1 and satisfies ℎ(𝑧)+𝑧ℎ^′(𝑧)𝛽ℎ(𝑧)+𝛿≺𝑝_𝑘,𝛾(𝑧), (2.3) and 𝑞_𝑘,𝛾(𝑧) is an analytic solution of 𝑞_𝑘,𝛾(𝑧)+𝑧𝑞_𝑘,𝛾′(𝑧)𝛽𝑞_𝑘,𝛾(𝑧)+𝛿=𝑝_𝑘,𝛾(𝑧), (2.4)then 𝑞_𝑘,𝛾(𝑧) is univalent, ℎ(𝑧)≺𝑞_𝑘,𝛾(𝑧)≺𝑝_𝑘,𝛾(𝑧), (2.5) and 𝑞_𝑘,𝛾(𝑧) is the best dominant of (2.3).

Lemma 2.3 (see [19]). If 𝑓∈ℂ,𝑔∈𝑆^∗, then for each ℎ analytic in 𝐸 with ℎ(0)=1, 𝑓∗ℎ𝑔(𝐸)(𝐸)⊂𝑓∗𝑔Coℎ(𝐸), (2.6)where Coℎ(𝐸) denotes the convex hull of ℎ(𝐸).

Lemma 2.4 (see [18]). Let 𝑢=𝑢₁+𝑖𝑢₂,𝑣=𝑣₁+𝑖𝑣₂ and let 𝜓(𝑢,𝑣) be a complex-valued function satisfying the conditions:(i)𝜓(𝑢,𝑣) is continuous in a domain 𝐷⊂ℂ²,(ii)(1,0)∈𝐷 and Re𝜓(1,0)>0,(iii)Re𝜓(𝑖𝑢₂,𝑣₁)≤0, whenever (𝑖𝑢₂,𝑣₁)∈𝐷 and 𝑣₁≤−(1/2)(1+𝑢²₂).If ℎ(𝑧)=1+𝑐₁𝑧+𝑐₂𝑧²+⋯ is a function analytic in 𝐸 such that (ℎ(𝑧),𝑧ℎ′(𝑧))∈𝐷 and Re𝜓(ℎ(𝑧),𝑧ℎ′(𝑧))>0 for 𝑧∈𝐸, then Reℎ(𝑧)>0 in 𝐸.

Lemma 2.5 (see [20]). Let ℎ∈𝑃_𝑚(𝜌),0≤𝜌<1. Then, with =𝑟𝑒^𝑖𝜃,𝑧∈𝐸, one has(i)(1/2𝜋)∫₀^2𝜋|ℎ(𝑟𝑒^𝑖𝜃)|²𝑑𝜃≤(1−[𝑚²(1−𝜌)²−1]𝑟²)/(1−𝑟²), (ii)(1/2𝜋)∫₀^2𝜋|ℎ′(𝑟𝑒^𝑖𝜃)|𝑑𝜃≤𝑚(1−𝜌)/(1−𝑟²).

Lemma 2.6 (see [5]). Let 𝑓∈𝑘−∪𝑅^𝛾_𝑚. Then there exist 𝑠₁,𝑠₂∈𝑘−∪𝑅^𝛾₂ such that

𝑓(𝑧)=𝑠₁(𝑧)^(𝑚+2)/4𝑠₂(𝑧)^(𝑚−2)/4,𝑘≥0,𝑚≥2,𝑧∈𝐸. (2.7)

Lemma 2.7. Let 𝑝∈𝑃_𝑚(𝑝_𝑘,𝛾) and 𝑝(𝑧)=1+∑^∞_𝑛=1𝑐_𝑛𝑧^𝑛. Then 𝑐_𝑛≤𝑚2𝛿_𝑘,𝛾,𝑛≥1, (2.8) where 𝛿_𝑘,𝛾=81−𝛾cos⁻¹𝑘²𝜋²1−𝑘²8,0≤𝑘<1,1−𝛾𝜋²,𝑘=1. (2.9)

Proof. Let 𝑝(𝑧)=(𝑚/4+1/2)𝑝₁(𝑧)−(𝑚/4−1/2)𝑝₂(𝑧). Then, 𝑝_𝑖(𝑧)≺𝑝_𝑘,𝛾(𝑧)=1+𝛿_𝑘,𝛾𝑧+⋯,𝑖=1,2. (2.10)
Now the proof follows immediately by using the well-known Rogosinski’s result, see [21].

3. Main Results

We shall assume throughout, unless stated otherwise, that 𝑘∈[0,1],𝑚≥0,0≤𝛾<1,𝜆∈ℂ,𝜇>−1 and 𝑧∈𝐸.

Theorem 3.1. Let 𝑓∈𝑘−∪𝑅^𝛾_𝑚(𝜆,𝜇). Let, for 𝛼,𝛽>0, 𝐹(𝑧)=1+𝛽𝑧^−𝛽𝑧₀𝑡^𝛽−1′𝑓^𝛼(𝑡)𝑑𝑡^1/𝛼. (3.1) Then, 𝐹∈𝑘−∪𝑅^𝛾_𝑚(𝜆,𝜇) in 𝐸.

Proof. Set 𝑧𝐼_𝜆,𝜇𝐹(𝑧)′𝐼_𝜆,𝜇𝐹(𝑧)(𝑧)=𝑚=𝐻4+12𝐻₁(𝑧)−𝑚4−12𝐻₂(𝑧). (3.2) We note 𝐻(𝑧) is analytic in 𝐸 with 𝐻(0)=1.
From (3.1), we have𝑧^𝛽𝐼_𝜆,𝜇𝐹(𝑧)^𝛼′=𝑧^𝛽−1𝐼_𝜆,𝜇𝑓(𝑧)^𝛼. (3.3)
That is𝐼_𝜆,𝜇𝐹(𝑧)^𝛼𝛽+𝛼𝐻(𝑧)=𝐼_𝜆,𝜇𝑓(𝑧)^𝛼. (3.4)
Logarithmic differentiation of (3.4) and simple computations give us𝐻(𝑧)+𝑧𝐻^′(𝑧)(𝑧)=𝑧𝐼𝛼𝐻+𝛽^𝜆,𝜇𝑓(𝑧)^′𝐼_𝜆,𝜇𝑓(𝑧)∈𝑃_𝑚𝑝_𝑘,𝛾. (3.5)
Define𝜙_𝑎,𝑏(𝑧)=1𝑧1+𝑏(1−𝑧)^𝑎+1+𝑏𝑧𝑏+1(1−𝑧)^𝑎+2,(𝑎>0,𝑏≥0), (3.6) then, with 𝑎=1/𝛼,𝑏=𝛽/𝛼, we have 𝐻(𝑧)∗𝜙_𝑎,𝑏(𝑧)𝑧=𝐻(𝑧)+𝑧𝐻^′(𝑧)𝛼𝐻(𝑧)+𝛽. (3.7)
From (3.2), (3.5), and (3.7), it follows that𝐻_𝑖(𝑧)+𝑧𝐻_𝑖^′(𝑧)𝛼𝐻_𝑖(𝑧)𝑝+𝛽∈𝑃_𝑘,𝛾,𝑧∈𝐸,𝑖=1,2. (3.8)
On applying Lemma 2.2, we obtain𝐻_𝑖(𝑧)≺𝑞_𝑘,𝛾(𝑧)≺𝑝_𝑘,𝛾(𝑧)in𝐸, (3.9)where 𝑞_𝑘,𝛾(𝑧) is the best dominant and is given as 𝑞_𝑘,𝛾(𝑧)=𝛼¹₀𝑡^{𝛽+𝛼−1}exp_𝑧^𝑡𝑧𝑝_𝑘,𝛾(𝑢)−1𝑢𝑑𝑢^𝛼𝑑𝑡⁻¹−𝛽𝛼. (3.10)
Consequently it follows, from (3.2), that 𝐻∈𝑃_𝑚(𝑝_𝑘,𝛾) and 𝐹∈𝑘−∪𝑇^𝛾_𝑚(𝜆,𝜇) in 𝐸.

For 𝑘=0,𝛾=0, we have the following special case.

Corollary 3.2. Let 𝑓∈0−∪𝑅⁰_𝑚(𝜆,𝜇)=𝑅_𝑚(𝜆,𝜇) and let 𝐹(𝑧) be defined by (3.1). Then, 𝐹∈𝑅^𝛾1_𝑚(𝜆,𝜇), where

𝛾₁=21+2𝛽+1+2𝛽²+8𝛼. (3.11)

Proof. We write 𝑧𝐼_𝜆,𝜇𝐹(𝑧)′𝐼_𝜆,𝜇𝐹(𝑧)=1−𝛾₁𝐻(𝑧)+𝛾₁=𝑚4+121−𝛾₁𝐻₁(𝑧)+𝛾₁−𝑚4−121−𝛾₁𝐻₂(𝑧)+𝛾₁, (3.12) and proceeding as in Theorem 3.1, we obtain 𝑧𝐼_𝜆,𝜇𝑓(𝑧)′𝐼_𝜆,𝜇𝑓(𝑧)=𝑚4+121−𝛾₁𝐻₁(𝑧)+𝛼𝑧𝐻^′₁(𝑧)𝛼𝐻₁(𝑧)+𝛼𝛾₁+𝛽/1−𝛾₁+𝛾₁−𝑚4−121−𝛾₁𝐻₂(𝑧)+𝛼𝑧𝐻^′₂(𝑧)𝛼𝐻₂(𝑧)+𝛼𝛾₁+𝛽/1−𝛾₁+𝛾₁. (3.13)
We construct the functional 𝜓(𝑢,𝑣) by taking 𝑢=𝐻_𝑖(𝑧),  𝑣=𝑧𝐻_𝑖^′(𝑧), as𝜓()𝑣𝑢,𝑣=𝑢+𝛼𝑢+𝛼𝛾₁+𝛽/1−𝛾₁+𝛾₁1−𝛾₁. (3.14)
The first two conditions of Lemma 2.4 can easily be verified. For condition (iii), we proceed as follows:Re𝜓𝑖𝑢₂,𝑣₁=𝛾₁1−𝛾₁+Re1−𝛾₁𝑣₁𝛼𝛾₁+𝛽+𝑖𝛼1−𝛾₁𝑢₂,=11−𝛾₁𝛾₁+1−𝛾₁𝛼𝛾₁+𝛽𝑣₁𝛼𝛾₁+𝛽²+𝛼²1−𝛾₁²𝑢²₂,≤11−𝛾₁𝛾₁−1−𝛾₁𝛼𝛾₁+𝛽1+𝑢²₂2𝛼𝛾₁+𝛽²+𝛼²1−𝛾₁²𝑢²₂,=11−𝛾₁𝐴+𝐵𝑢²₂,𝑣2𝐶¹≤−1+𝑢²₂2, (3.15) where 𝐴=2𝛾₁(𝛼𝛾₁+𝛽)²−(1−𝛾₁)(𝛼𝛾₁+𝛽),𝐵=2𝛾₁𝛼²(1−𝛾₁)²−(1−𝛾₁)(𝛼𝛾₁+𝛽),𝐶={(𝛼𝛾₁+𝛽)²+𝛼²(1−𝛾₁)²𝑢²₂}>0.
The right-hand side of (3.15) is less than equal to zero when 𝐴≤0 and 𝐵≤0. From 𝐴≤0, we obtain 𝛾₁ as given by (3.11), and 𝐵≤0 ensures that 𝛾₁∈[0,1).
This shows that all the conditions of Lemma 2.4 are satisfied and therefore Re𝐻_𝑖(𝑧)>0. This implies 𝐻∈𝑃_𝑚 and consequently 𝐼_𝜆,𝜇𝐹∈𝑅^𝛾1_𝑚. That is 𝐹∈𝑅^𝛾1_𝑚(𝜆,𝜇) as required.

By taking 𝛼=1,𝛽=0,𝜆=0, and 𝑚=2, we obtain a well-known result that every convex function is starlike of order 1/2. Also, for 𝛽=1,𝜆=0,𝛼=1, and 𝑚=2, we obtain from (3.1) the Libera operator and in this case we obtain a known result with 𝛾₁=2/(3+√17) for starlike functions, see [18].

Assigning permissible values to different parameters, we obtain several new and known results from Theorem 3.1 and Corollary 3.2.

Theorem 3.3. Let 𝑓∈𝑘−∪𝑇^𝛾_𝑚(𝜆,𝜇) and let 𝐹(𝑧) be defined by (3.1). Then 𝐹∈𝑘−∪𝑇^𝛾_𝑚(𝜆,𝜇).

Proof. We can write (3.1) as 𝐼_𝜆,𝜇𝐹(𝑧)=1+𝛽𝑧^−𝛽𝑧₀𝑡^𝛽−1′𝐼_𝜆,𝜇𝑓(𝑡)^𝛼𝑑𝑡^1/𝛼,𝑓∈𝑘−∪𝑇^𝛾_𝑚,=𝐼𝜆,𝜇_𝜆,𝜇𝑓(𝑧)𝑧^𝛼∗ℎ_𝛼,𝛽(𝑧)𝑧^1/𝛼, (3.16)where ℎ_𝛼,𝛽(𝑧)=^∞_𝑛=1𝑧^𝑛𝑛+𝛼+𝛽, (3.17) is convex in 𝐸.
Let 𝑓∈𝑘−∪𝑇^𝛾_𝑚(𝜆,𝜇). Then there exists some 𝑔∈𝑘−∪𝑅^𝛾₂(𝜆,𝜇) such that𝑧𝐼_𝜆,𝜇𝑓(𝑧)′𝐼_𝜆,𝜇𝑔(𝑧)∈𝑃_𝑚𝑝_𝑘,𝛾. (3.18)𝐺(𝑧)=𝛽+1𝑧^−𝛽𝑧₀𝑡^𝛽−1𝑔^𝛼(𝑡)𝑑𝑡^1/𝛼. (3.19)
From Theorem 3.1, it follows that 𝐺∈𝑘−∪𝑅^𝛾₂(𝜆,𝜇). We can write (3.19) as𝐼_𝜆,𝜇𝐺(𝑧)𝐼=𝑧^𝜆,𝜇𝑔(𝑧)𝑧∗ℎ_𝛼,𝛽(𝑧)𝑧^1/𝛼, (3.20) where ℎ_𝛼,𝛽(𝑧), given by (3.17), is convex in 𝐸.
Since 𝑔∈𝑘−∪𝑅^𝛾₂(𝜆,𝜇), so 𝐼_𝜆,𝜇𝑔∈𝑆^∗((𝑘+𝛾)/(𝑘+1))⊂𝑆^∗. It can easily be shown that 𝑧(𝐼_𝜆,𝜇𝑔/𝑧)^𝛼 and 𝑧(𝐼_𝜆,𝜇𝐺/𝑧)^𝛼 are in the class 𝑆^∗.
Now, from (3.1), we have𝑧𝐼_𝜆,𝜇𝐹(𝑧)^′𝐼_𝜆,𝜇𝐹(𝑧)^𝛼−1𝐼_𝜆,𝜇𝐺(𝑧)^𝛼=ℎ_𝛼,𝛽(𝑧)∗𝑧𝐼_𝜆,𝜇𝑔(𝑧)/𝑧^𝛼𝑧𝐼_𝜆,𝜇𝑓(𝑧)^′𝐼_𝜆,𝜇𝑓(𝑧)^𝛼−1/𝐼_𝜆,𝜇𝑔(𝑧)^𝛼ℎ_𝛼,𝛽(𝑧)∗𝑧𝐼_𝜆,𝜇𝑔(𝑧)/𝑧^𝛼=𝑚4+12ℎ_𝛼,𝛽(𝑧)∗𝑧𝐼_𝜆,𝜇𝑔(𝑧)/𝑧^𝛼ℎ₁(𝑧)ℎ_𝛼,𝛽(𝑧)∗𝑧𝐼_𝜆,𝜇𝑔(𝑧)/𝑧^𝛼−𝑚4−12ℎ_𝛼,𝛽(𝑧)∗𝑧𝐼_𝜆,𝜇𝑔(𝑧)/𝑧^𝛼ℎ₂(𝑧)ℎ_𝛼,𝛽(𝑧)∗𝑧𝐼_𝜆,𝜇𝑔(𝑧)/𝑧^𝛼. (3.21)
We use Lemma 2.3 with ℎ_𝑖≺𝑝_𝑘,𝛾,𝑖=1,2, to haveℎ_𝛼,𝛽(𝑧)∗𝑧𝐼_𝜆,𝜇𝑔(𝑧)/𝑧^𝛼ℎ_𝑖(𝑧)ℎ_𝛼,𝛽(𝑧)∗𝑧𝐼_𝜆,𝜇𝑔(𝑧)/𝑧^𝛼≺𝑝_𝑘,𝛾(𝑧)in𝐸. (3.22)
Thus from (3.19), (3.21), and (3.22) we obtain the required result that 𝐹∈𝑘−∪𝑇^𝛾_𝑚(𝜆,𝜇). This completes the proof.

As a special case we note that, for 𝜆=0=𝑘, the subclass 𝑇^𝛾_𝑚⊂𝑇_𝑚 is invariant under the integral operator defined by (3.1).

Theorem 3.4. One has𝑘−∪𝑅^𝛾_𝑚𝜆+1,𝜇⊂𝑘−∪𝑅^𝛾_𝑚𝜆,𝜇. (3.23)

Proof. Let 𝑓∈𝑘−∪𝑅^𝛾_𝑚(𝜆+1,𝜇) and let 𝑧𝐼_𝜆,𝜇𝑓(𝑧)′𝐼_𝜆,𝜇𝑓(𝑧)=𝐻(𝑧), (3.24) where 𝐻(𝑧) is analytic in 𝐸 and is defined by (3.2).
Then, from (1.13), we have𝑧𝐼_𝜆+1,𝜇𝑓(𝑧)′𝐼_𝜆+1,𝜇𝑓(𝑧)=𝐻(𝑧)+𝑧𝐻^′(𝑧)𝐻(𝑧)+𝜇∈𝑃_𝑚𝑝_𝑘,𝛾. (3.25)
Applying similar technique used before, we have from (3.2) and (3.7) for 𝑖=1,2𝐻_𝑖(𝑧)+𝑧𝐻_𝑖^′(𝑧)𝐻_𝑖(𝑧)+𝜇≺𝑝_𝑘,𝛾. (3.26)
Thus, using Lemma 2.2, it follows that 𝐻_𝑖≺𝑝_𝑘,𝛾,𝑖=1,2 and 𝑧∈𝐸, consequently 𝐻∈𝑃_𝑚(𝑝_𝑘,𝛾) in 𝐸 and this completes the proof.

As special cases, we have the following.(i)Let 𝑚=2,𝜆≥0. Then, from Theorem 3.4, it easily follows that𝑘−∪𝑅^𝛾₂𝜆,𝜇⊂𝑘−∪𝑅^𝛾₂0,𝜇⊂𝑆^∗𝑘+𝛾1+𝑘⊂𝑆^∗. (3.27)(ii)Let 𝑘=0 and 𝜆≥0. Then 𝑓∈0−∪𝑅^𝛾_𝑚(𝜆,𝜇) implies 𝑓∈𝑅^𝛾_𝑚⊂𝑅_𝑚, that is, 𝑓(𝑧) is a function of bounded radius rotation in 𝐸.

Theorem 3.5. One has𝑘−∪𝑇^𝛾_𝑚𝜆+1,𝜇⊂𝑘−∪𝑇^𝛾_𝑚𝜆,𝜇,𝜇,𝜆≥0. (3.28)

Proof. Let 𝑓∈𝑘−∪𝑇^𝛾_𝑚(𝜆+1,𝜇). Then, for 𝑧∈𝐸, 𝑧𝐼_𝜆+1,𝜇𝑓(𝑧)′𝐼_𝜆+1,𝜇𝑔(𝑧)∈𝑃_𝑚𝑝_𝑘,𝛾, (3.29)for some 𝑔∈𝑘−∪𝑅^𝛾₂(𝜆+1,𝜇).
We define an analytic function ℎ(𝑧) in 𝐸 such that𝑧𝐼_𝜆,𝜇𝑓(𝑧)′𝐼_𝜆,𝜇𝑔(𝑧)(𝑧)=𝑚=ℎ4+12ℎ₁(𝑧)−𝑚4−12ℎ₂(𝑧), (3.30) where ℎ(0)=1. We shall show that ℎ∈𝑃_𝑚(𝑝_𝑘,𝛾) in 𝐸.
Since 𝑔∈𝑘−∪𝑅^𝛾₂(𝜆+1,𝜇) and 𝑘−∪𝑅^𝛾₂(𝜆+1,𝜇)⊂𝑘−∪𝑅^𝛾₂(𝜆,𝜇), we have𝑧𝐼_𝜆,𝜇𝑔(𝑧)′𝐼_𝜆,𝜇𝑔(𝑧)=ℎ₀(𝑧),ℎ₀∈𝑃𝑝_𝑘,𝛾,𝑧∈𝐸. (3.31)
Now, on using (1.13), we have𝑧𝐼_𝜆+1,𝜇𝑓(𝑧)′𝐼_𝜆+1,𝜇𝑔(𝑧)=1/𝜇+1𝑧𝐼_𝜆,𝜇𝑧𝑓^′(𝑧)′+𝜇/𝜇+1𝐼_𝜆,𝜇𝑧𝑓^′(𝑧)1/𝜇+1𝑧𝐼_𝜆,𝜇𝑔(𝑧)′+𝜇/𝜇+1𝐼_𝜆,𝜇𝑔(𝑧),=𝑧𝐼_𝜆,𝜇𝑧𝑓^′(𝑧)′/𝐼_𝜆,𝜇𝑔(𝑧)+𝜇ℎ(𝑧)ℎ₀(𝑧).+𝜇 (3.32)
Differentiation of (3.30) gives us𝑧𝑧𝐼_𝜆,𝜇𝑓(𝑧)′′𝐼_𝜆,𝜇𝑔(𝑧)=𝑧ℎ^′(𝑧)+(ℎ(𝑧))ℎ₀(𝑧), (3.33) and using (3.33) in (3.32), we obtain 𝑧𝐼_𝜆+1,𝜇𝑓(𝑧)′𝐼_𝜆+1,𝜇𝑔(𝑧)=ℎ(𝑧)+𝑧ℎ^′(𝑧)ℎ₀(𝑧)=𝑚+𝜇4+12ℎ₁(𝑧)+𝑧ℎ^′₁(𝑧)ℎ₀(𝑧)−𝑚+𝜇4−12ℎ₂(𝑧)+𝑧ℎ^′₂(𝑧)ℎ₀(𝑧).+𝜇 (3.34)
Since 𝑓∈𝑘−∪𝑇^𝛾_𝑚(𝜆+1,𝜇), we have with 1/𝐻₀(𝑧)={ℎ₀(𝑧)+𝜇}∈𝑃,ℎ_𝑖(𝑧)+𝐻₀(𝑧)𝑧ℎ^′_𝑖(𝑧)≺𝑝_𝑘,𝛾(𝑧)in𝐸, (3.35) and thus, applying Lemma 2.1, we have ℎ_𝑖(𝑧)≺𝑝_𝑘,𝛾(𝑧) in 𝐸. This shows ℎ∈𝑃_𝑚(𝑝_𝑘,𝛾) in 𝐸 and consequently 𝑓∈𝑘−∪𝑇^𝛾_𝑚(𝜆,𝜇).

As a special case, we note that 𝑓∈𝑘−∪𝑇^𝛾₂(𝜆,𝜇) is a close-to-convex function for 𝑧∈𝐸.

Theorem 3.6. Let 𝑓∈𝑘−∪𝑅^𝛾₂(𝜆,𝜇) and let 𝜙(𝑧) be convex in 𝐸. Then (𝑓∗𝜙)∈𝑘−∪𝑅^𝛾₂(𝜆,𝜇) for 𝑧∈𝐸.

Proof. We have 𝑧𝐼_𝜆,𝜇𝜙∗𝑓′𝐼_𝜆,𝜇=𝐼𝜙∗𝑓𝜙∗𝑧_𝜆,𝜇𝑓′𝜙∗𝐼_𝜆,𝜇𝑓,=𝜙∗𝑧𝐼_𝜆,𝜇𝑓′/𝐼_𝜆,𝜇𝑓𝐼_𝜆,𝜇𝑓𝜙∗𝐼_𝜆,𝜇𝑓. (3.36)
Now 𝑘−∪𝑅^𝛾₂(𝜆,𝜇)⊂𝑆^∗((𝑘+𝛾)/(1+𝑘))⊂𝑆^∗, and 𝜙 is a convex in 𝐸, we use Lemma 2.3 to (3.36) and conclude that (𝜙∗𝑓)∈𝑘−∪𝑅^𝛾₂(𝜆,𝜇) for 𝑧∈𝐸. This completes the proof.

Remark 3.7. Following the similar technique, we can easily extend Theorem 3.6 to the class 𝑘−∪𝑇^𝛾_𝑚(𝜆,𝜇), that is, 𝑘−∪𝑇^𝛾_𝑚(𝜆,𝜇), is invariant under convolution with convex function.

3.1. Applications of Theorem 3.6

The classes 𝑘−∪𝑅^𝛾₂(𝜆,𝜇) and 𝑘−∪𝑇^𝛾_𝑚(𝜆,𝜇) are preserved under the following integral operators:

(1) 𝑓₁(𝑧)=∫^𝑧₀(𝑓(𝑡)/𝑡)𝑑𝑡=(𝜙₁∗𝑓)(𝑧), where 𝜙₁(𝑧)=−log(1−𝑧),

(2) 𝑓₂(𝑧)=(2/𝑧)∫^𝑧₀𝑓(𝑡)𝑑𝑡=(𝜙₂∗𝑓)(𝑧), where 𝜙₂(𝑧)=−2[𝑧−log(1−𝑧)]/𝑧,

(3) 𝑓₃(𝑧)=∫^𝑧₀((𝑓(𝑡)−𝑓(𝑡𝑥))/(𝑡−𝑡𝑥))𝑑𝑡=(𝜙₃∗𝑓)(𝑧),|𝑥|≤1,𝑥≠1, where 𝜙₃(𝑧)=(1/(1−𝑥))log((1−𝑥𝑧)/(1−𝑧)),|𝑥|≤1,𝑥≠1,

(4) 𝑓₄(𝑧)=((1+𝑐)/𝑧^𝑐)∫^𝑧₀𝑡^𝑐−1𝑓(𝑡)𝑑𝑡=(𝜙₄∗𝑓)(𝑧),Re𝑐>0, where 𝜙₄(𝑧)=∑^∞_𝑛=1((1+𝑐)/(𝑛+𝑐))𝑧^𝑛,Re𝑐>0,

The proof is immediate since 𝜙_𝑖(𝑧) is convex in 𝐸 for 𝑖=1,2,3,4.

With essentially the same method together with Lemma 2.7, we can easily prove the following sharp coefficient results.

Theorem 3.8. Let 𝑓∈𝑘−∪𝑅^𝛾_𝑚(𝜆,𝑚) and let it be given by 𝑓(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛. (3.37) Then 𝑎_𝑛≤𝑚2[(𝑛−1)!]1+𝜇𝑛+𝜇^𝜆𝛿_{𝑘,𝛾𝑛−1},(𝑛≥2), (3.38) where (𝜌)_𝑛 is Pochhamer symbol defined, in terms of Gamma function Γ, by 𝜌_𝑛=Γ𝑛+𝜌Γ𝜌=𝜌⋯1,𝑛=0,𝜌+1𝜌+2𝜌+𝑛−1,𝑛∈𝑁, (3.39) and 𝛿_𝑘,𝛾 is as given by (2.9).
As special case, one notes that
(i)𝜆=0,𝑚=2, then one has𝑎_𝑛≤𝛿_{𝑘,𝛾𝑛−1}(𝑛−1)!, (3.40) see [2].(ii)Let 𝜆=0,𝑛=2. Then,𝑎_𝑛≤𝑚2𝛿_𝑘,𝛾. (3.41)
This coefficient bound is well known for 𝑚=2, see [2].

Using Theorem 3.8 with 𝑚=2, the following result can easily be proved.

Theorem 3.9. Let 𝑓∶𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛∈𝑘−∪𝑇^𝛾_𝑚(𝜆,𝜇). Then, for 𝑛≥2𝑎_𝑛≤1+𝜇𝑛+𝜇^𝜆𝛿_{𝑘,𝛾𝑛−1}+𝑚𝛿𝑛!2𝑛^{𝑘,𝛾𝑛−1}_𝑗=1𝛿_{𝑘,𝛾𝑗−1}𝑗−1!, (3.42)where 𝛿_𝑘,𝛾 is as given by (2.9).

By assigning different permissible values to the parameters, we obtain several known results, see [2, 22].

We now prove the following.

Theorem 3.10. Let 𝑓∶𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛∈𝑘−∪𝑅^𝛾_𝑚(𝜆,𝜇) with (𝑚+2)(1−𝛾)/2(1+𝑘)>1. Then, for 𝑛≥1, 𝑎_𝑛+1−𝑎_𝑛≤𝑐₁𝑚,𝛾,𝑘,𝜆,𝜇𝑛^{[{((𝑚/2)+1)((1−𝛾)/(1+𝑘))}−(2+𝜆)]}, (3.43) where 𝑐₁(𝑚,𝛾,𝑘,𝜆,𝜇) is a constant.

Proof. Let 𝐹(𝑧)=𝐼_𝜆,𝜇𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝐴_𝑛𝑧^𝑛,𝐴_𝑛=((𝑛+𝜇)/(1+𝜇))^𝜆𝑎_𝑛. Since 𝑓∈𝑘−∪𝑅^𝛾_𝑚(𝜆,𝜇), we can write 𝑧𝐹^′(𝑧)=𝐹(𝑧)ℎ(𝑧),ℎ∈𝑃_𝑚𝑝_𝑘,𝛾. (3.44)
That is𝑧𝑧𝐹^′(𝑧)′=𝐹(𝑧)ℎ²(𝑧)+𝑧ℎ^′(𝑧). (3.45)
Now, 𝐹∈𝑘−∪𝑅^𝛾_𝑚⊂𝑅_𝑚((𝑘+𝛾)/(1+𝑘)), and it follows from a result proved in [5] that there exist 𝑠₁,𝑠₂∈𝑆^∗((𝑘+𝛾)/(𝑘+1)) such that𝐹(𝑧)=𝑠₁(𝑧)^(𝑚+2)/4𝑠₂(𝑧)^(𝑚−2)/4=𝑔,𝑚≥2,₁(𝑧)^{(1−(𝑘+𝛾)/(𝑘+1))(𝑚+2)/4}𝑔₂(𝑧)^{(1−(𝑘+𝛾)/(𝑘+1))(𝑚−2)/4},𝑔₁,𝑔₂∈𝑆^∗,=𝑔₁(𝑧)^{((1−𝛾)/(𝑘+1))((𝑚+2)/4)}𝑔₂(𝑧)^{((1−𝛾)/(𝑘+1))((𝑚−2)/4)}. (3.46)Thus, for 𝜉∈𝐸 and 𝑛≥1, (𝑛+1)²𝜉𝐴_𝑛+1−𝑛²𝐴_𝑛≤12𝜋𝑟^𝑛+2₀^2𝜋𝑧−𝜉𝑧𝑧𝐹^′(𝑧)′=1𝑑𝜃2𝜋𝑟^𝑛+2₀^2𝜋𝑧−𝜉|𝐹(𝑧)|ℎ²(𝑧)+𝑧ℎ^′(𝑧=1)𝑑𝜃2𝜋𝑟^𝑛+2₀^2𝜋𝑔𝑧−𝜉¹(𝑧)^(𝑚/4+1/2)𝑔₂(𝑧)^{(𝑚/4−1/2)((1−𝛾)/(𝑘+1))}ℎ²(𝑧)+𝑧ℎ^′(𝑧)𝑑𝜃, (3.47) where 𝑔₁,𝑔₂∈𝑆^∗ and ℎ∈𝑃_𝑚(𝑝_𝑘,𝛾) in 𝐸.
Let 0<𝑟<1. Then, by a result [23], there exists a 𝜉 with |𝜉|=𝑟 such that, for 𝑧,|𝑧|=𝑟𝑧−𝜉𝑔₁(𝑧)≤2𝑟²1−𝑟². (3.48) We now use (3.48), distortion theorems for starlike functions 𝑔₁,𝑔₂ for ((𝑚/2)+1)((1−𝛾)/(1+𝑘))>1, and Lemma 2.5 with 𝜌=(𝑘+𝜌)/(𝑘+1),𝑟=(1−1/𝑛),𝑛→∞ and obtain from (3.47), (𝑛+1)²𝜉𝑛+𝜇+11+𝜇^𝜆𝑎_𝑛+1−𝑛²𝑛+𝜇1+𝜇^𝜆𝑎_𝑛≤𝐶𝑚,𝛾,𝑘𝑛^{{((𝑚/2)+1)((1−𝛾)/(1+𝑘))}}. (3.49) From (3.48), we easily obtain the required result given by (3.43), (𝑛→∞).
This completes the proof.

Using the similar technique, we can easily prove the following.

Theorem 3.11. Let 𝑓∶𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛∈𝑘−∪𝑇^𝛾_𝑚(𝜆,𝜇). Then 𝑎_𝑛=𝑂(1)𝑛^{{2((1−𝛾)/(1+𝑘))−𝜆−1}},(𝑛⟶∞), (3.50) where 𝑂(1) is a constant depending on 𝛾,𝑘,𝑚,𝜇, and 𝜆 only. The exponent {2((1−𝛾)/(1+𝑘))−𝜆−1} is best possible.

Acknowledgment

The authors are grateful to Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Islamabad, Pakistan, for providing excellent research facilities and environment.