A New Subclass of Harmonic Univalent Functions Associated with Dziok-Srivastava Operator

Abstract

The purpose of the present paper is to study a certain subclass of harmonic univalent functions associated with Dziok-Srivastava operator. We obtain coefficient conditions, distortion bounds, and extreme points for the above class of harmonic univalent functions belonging to this class and discuss a class preserving integral operator. We also show that class studied in this paper is closed under convolution and convex combination. The results obtained for the class reduced to the corresponding results for several known classes in the literature are briefly indicated.

1. Introduction

A continuous complex-valued function 𝑓=𝑢+𝑖𝑣 defined in a simply connected domain 𝐷, is said to be harmonic in 𝐷 if both 𝑢 and 𝑣 are harmonic in 𝐷. In any simply connected domain 𝐷 we can write 𝑓=ℎ+𝑔, where ℎ and 𝑔 are analytic in 𝐷. A necessary and sufficient condition for 𝑓 to be locally univalent and sense preserving in 𝐷 is that |ℎ^′(𝑧)|>|𝑔^′(𝑧)|,𝑧∈𝐷. See Clunie and Sheil-Small [1].

Denote by 𝑆_𝐻 the class of functions 𝑓=ℎ+𝑔 that are harmonic univalent and sense preserving in the unit disk 𝑈={𝑧:|𝑧|<1} for which 𝑓(0)=𝑓_𝑧(0)−1=0. Then for 𝑓=ℎ+𝑔∈𝑆_𝐻, we may express the analytic function ℎ and 𝑔 as ℎ(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑧^𝑘,𝑔(𝑧)=^∞_𝑘=1𝑏_𝑘𝑧^𝑘,𝑏₁<1. (1.1)

Note that 𝑆_𝐻, is reduced to  𝑆  the class of normalized analytic univalent functions if the coanalytic part of 𝑓=ℎ+𝑔 is identically zero.

For more basic results on harmonic univalent functions, one may refer to the following introductory text book by Duren [2] (see also [3–5]).

For 𝛼_𝑗∈𝐶,(𝑗=1,2,3,…,𝑞) and 𝛽_𝑗∈𝐶−{0,−1,−2,−3,…},  (𝑗=1,2,3,…,𝑠), the generalized hypergeometric functions are defined by _𝑞𝐹_𝑠𝛼₁,𝛼₂,…,𝛼_𝑞;𝛽₁,𝛽₂,…,𝛽_𝑠;𝑧=^∞_𝑘=0𝛼_1𝑘𝛼_2𝑘⋯𝛼_𝑞𝑘𝑧^𝑘𝛽_1𝑘𝛽_2𝑘⋯𝛽_𝑠𝑘,𝑘!𝑞≤𝑠+1;𝑞,𝑠∈𝑁₀={0,1,2,…}, (1.2) where (𝛼)_𝑘 is the Pochhammer symbol defined by (𝛼)_𝑘=Γ(𝛼+𝑘)Γ(𝛼)=𝛼(𝛼+1)⋯(𝛼+𝑘−1),if𝑘=1,2,3,…,1,if𝑘=0. (1.3)

Corresponding to the function ℎ𝛼₁,𝛼₂,…,𝛼_𝑞;𝛽₁,𝛽₂,…,𝛽_𝑠;𝑧=𝑧_𝑞𝐹_𝑠𝛼₁,𝛼₂,…,𝛼_𝑞;𝛽₁,𝛽₂,…,𝛽_𝑠;𝑧. (1.4)

The Dziok-Srivastava operator [6, 7] 𝐻_𝑞,𝑠(𝛼₁,𝛼₂,…,𝛼_𝑞;𝛽₁,𝛽₂,…,𝛽_𝑠) is defined for 𝑓∈𝑆 by 𝐻_𝑞,𝑠(𝛼₁,𝛼₂,…,𝛼_𝑞;𝛽₁,𝛽₂,…,𝛽_𝑠)𝑓(𝑧)=ℎ(𝛼₁,𝛼₂,…,𝛼_𝑞;𝛽₁,𝛽₂,…,𝛽_𝑠;𝑧)∗𝑓(𝑧)=𝑧+^∞_𝑘=2𝛼_1𝑘−1𝛼_2𝑘−1…𝛼_𝑞𝑘−1𝛽_1𝑘−1𝛽_2𝑘−1…𝛽_𝑠𝑘−1𝑎_𝑘𝑧^𝑘()!,𝑘−1 (1.5)

where * stands for convolution of two power series.

To make the notation simple, we write 𝐻_𝑞,𝑠𝛼₁𝑓(𝑧)=𝐻_𝑞,𝑠𝛼₁,𝛼₂,…,𝛼_𝑞;𝛽₁,𝛽₂,…,𝛽_𝑠𝑓(𝑧). (1.6)

Special cases of the Dziok-Srivastava operator includes the Hohlov operator [8], the Carlson-Shaffer operator 𝐿(𝑎,𝑐) [9], the Ruscheweyh derivative operator 𝐷^𝑛 [10], and the Srivastava-Owa fractional derivative operators ([11–13]).

We define the Dziok-Srivastava operator of the harmonic functions 𝑓=ℎ+𝑔 given by (1.1) as 𝐻_𝑞,𝑠𝛼₁𝑓(𝑧)=𝐻_𝑞,𝑠𝛼₁ℎ(𝑧)+𝐻_𝑞,𝑠𝛼₁𝑔(𝑧). (1.7)

Recently, Porwal [14, Chapter 5] defined the subclass 𝑀_𝐻(𝛽)⊂𝑆_𝐻 consisting of harmonic univalent functions 𝑓(𝑧) satisfying the following condition:𝑀_𝐻𝛽=𝑓∈𝑆_𝐻∶Re𝑧ℎ^′(𝑧)−𝑧𝑔^′(𝑧)ℎ(𝑧)+𝑔(𝑧),4<𝛽1<𝛽≤3,𝑧∈𝑈. (1.8)

He proved that if 𝑓=ℎ+𝑔 is given by (1.1) and if ^∞_𝑘=2𝑘−𝛽𝑎𝛽−1^𝑘+^∞_𝑘=1𝑘+𝛽𝑏𝛽−1^𝑘4≤1,1<𝛽≤3. (1.9)

For 𝑔≡0 the class of 𝑀_𝐻(𝛽) is reduced to the class 𝑀(𝛽) studied by Uralegaddi et al. [15].

Generalizing the class 𝑀_𝐻(𝛽), we let 𝑀_𝐻(𝛼₁,𝛽) denote the family of functions 𝑓=ℎ+𝑔 of form (1.1) which satisfy the condition 𝑧𝐻Re^𝑞,𝑠𝛼₁ℎ(𝑧)^′−𝑧𝐻_𝑞,𝑠𝛼₁𝑔(𝑧)^′𝐻_𝑞,𝑠𝛼₁ℎ(𝑧)+𝐻_𝑞,𝑠𝛼₁𝑔(𝑧)<𝛽, (1.10) where (1<𝛽≤4/3) and Γ(𝛼₁,𝑘)=|(𝛼₁)_𝑘−1(𝛼₂)_𝑘−1⋯(𝛼_𝑞)_𝑘−1/(𝛽₁)_𝑘−1(𝛽₂)_𝑘−1⋯(𝛽_𝑠)_𝑘−1|.

Further, let 𝑀_𝐻(𝛼₁,𝛽) be the subclass of 𝑀_𝐻(𝛼₁,𝛽) consisting of functions of the form 𝑓(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑧^𝑘−^∞_𝑘=1𝑏_𝑘𝑧^𝑘. (1.11)

In this paper, we give a sufficient condition for 𝑓=ℎ+𝑔, given by (1.1) to be in 𝑀_𝐻(𝛼₁,𝛽), and it is shown that this condition is also necessary for functions in 𝑀_𝐻(𝛼₁,𝛽). We then obtain distortion theorem, extreme points, convolution conditions, and convex combinations and discuss a class preserving integral operator for functions in 𝑀_𝐻(𝛼₁,𝛽).

2. Main Results

First, we give a sufficient coefficient bound for the class 𝑀_𝐻(𝛼₁,𝛽).

Theorem 2.1. If 𝑓=ℎ+𝑔∈𝑆_𝐻 is given by (1.1) and if ^∞_𝑘=2𝑘−𝛽Γ𝛼𝛽−1¹,𝑘()!𝑎𝑘−1^𝑘+^∞_𝑘=1𝑘+𝛽Γ𝛼𝛽−1¹,𝑘()!𝑏𝑘−1^𝑘≤1, (2.1) then 𝑓∈𝑀_𝐻(𝛼₁,𝛽).

Proof. Let ^∞_𝑘=2𝑘−𝛽Γ𝛼𝛽−1¹,𝑘()!𝑎𝑘−1^𝑘+^∞_𝑘=1𝑘+𝛽Γ𝛼𝛽−1¹,𝑘()!𝑏𝑘−1^𝑘≤1. (2.2)
It suffices to show that 𝑧𝐻_𝑞,𝑠𝛼₁ℎ(𝑧)^′−𝑧𝐻_𝑞,𝑠𝛼₁𝑔(𝑧)^′/𝐻_𝑞,𝑠𝛼₁ℎ(𝑧)+𝐻_𝑞,𝑠𝛼₁𝑔(𝑧)−1𝑧𝐻_𝑞,𝑠𝛼₁ℎ(𝑧)^′−𝑧𝐻_𝑞,𝑠𝛼₁𝑔(𝑧)^′/𝐻_𝑞,𝑠𝛼₁ℎ(𝑧)+𝐻_𝑞,𝑠𝛼₁𝑔(𝑧)−2𝛽−1<1,𝑧∈𝑈. (2.3)
We have 𝑧𝐻_𝑞,𝑠𝛼₁ℎ(𝑧)^′−𝑧𝐻_𝑞,𝑠𝛼₁𝑔(𝑧)^′/𝐻_𝑞,𝑠𝛼₁ℎ(𝑧)+𝐻_𝑞,𝑠𝛼₁𝑔(𝑧)−1𝑧𝐻_𝑞,𝑠𝛼₁ℎ(𝑧)^′−𝑧𝐻_𝑞,𝑠𝛼₁𝑔(𝑧)^′/𝐻_𝑞,𝑠𝛼₁ℎ(𝑧)+𝐻_𝑞,𝑠𝛼₁𝑔(𝑧)−≤∑2𝛽−1_∞𝑘=2(𝑘−1)𝒜𝑎_𝑘|𝑧|^𝑘−1+∑^∞_𝑘=1(𝑘+1)𝒜𝑏_𝑘|𝑧|^𝑘−12𝛽−1−∑^∞_𝑘=2𝑘−2𝛽+1𝒜𝑎_𝑘|𝑧|^𝑘−1−∑^∞_𝑘=1𝑘+2𝛽−1𝒜𝑏_𝑘|𝑧|^𝑘−1≤∑^∞_𝑘=2(𝑘−1)𝒜𝑎_𝑘+∑^∞_𝑘=1(𝑘+1)𝒜𝑏_𝑘2𝛽−1−∑^∞_𝑘=2𝑘−2𝛽+1𝒜𝑎_𝑘−∑^∞_𝑘=1𝑘+2𝛽−1𝒜𝑏_𝑘, (2.4) where 𝒜 denotes (Γ(𝛼₁,𝑘)/(𝑘−1)!).
The last expression is bounded above by 1, if ^∞_𝑘=2()Γ𝛼𝑘−1¹,𝑘()!𝑎𝑘−1^𝑘+^∞_𝑘=1()Γ𝛼𝑘+1¹,𝑘()!𝑏𝑘−1^𝑘≤2𝛽−1−^∞_𝑘=2Γ𝛼𝑘−2𝛽+1¹,𝑘()!𝑎𝑘−1^𝑘−^∞_𝑘=1Γ𝛼𝑘+2𝛽−1¹,𝑘()!𝑏𝑘−1^𝑘, (2.5) which is equivalent to ^∞_𝑘=2𝑘−𝛽Γ𝛼𝛽−1¹,𝑘()!𝑎𝑘−1^𝑘+^∞_𝑘=1𝑘+𝛽Γ𝛼𝛽−1¹,𝑘()!𝑏𝑘−1^𝑘≤1. (2.6)
But (2.6) is true by hypothesis.
Hence 𝑧𝐻_𝑞,𝑠𝛼₁ℎ(𝑧)^′−𝑧𝐻_𝑞,𝑠𝛼₁𝑔(𝑧)^′/𝐻_𝑞,𝑠𝛼₁ℎ(𝑧)+𝐻_𝑞,𝑠𝛼₁𝑔(𝑧)−1𝑧𝐻_𝑞,𝑠𝛼₁ℎ(𝑧)^′−𝑧𝐻_𝑞,𝑠𝛼₁𝑔(𝑧)^′/𝐻_𝑞,𝑠𝛼₁ℎ(𝑧)+𝐻_𝑞,𝑠𝛼₁𝑔(𝑧)−2𝛽−1<1,𝑧∈𝑈, (2.7) and the theorem is proved.
In the following theorem, it is proved that the condition (2.1) is also necessary for functions 𝑓=ℎ+𝑔∈𝑀_𝐻(𝛼₁,𝛽) that are given by (1.11).

Theorem 2.2. A function 𝑓(𝑧)=𝑧+∑^∞_𝑘=2|𝑎_𝑘|𝑧^𝑘−∑^∞_𝑘=1|𝑏_𝑘|𝑧^𝑘  is in 𝑀_𝐻(𝛼₁,𝛽), if and only if ^∞_𝑘=2Γ𝛼𝑘−𝛽¹,𝑘()!𝑎𝑘−1^𝑘+^∞_𝑘=1Γ𝛼𝑘+𝛽¹,𝑘()!𝑏𝑘−1^𝑘≤𝛽−1. (2.8)

Proof. Since 𝑀_𝐻(𝛼₁,𝛽)⊂𝑀_𝐻(𝛼₁,𝛽), we only need to prove the “only if” part of the theorem. For this we show that 𝑓∉𝑀_𝐻(𝛼₁,𝛽) if the condition (2.8) does not hold.
Note that a necessary and sufficient condition for 𝑓=ℎ+𝑔 given by (1.9) is in 𝑀_𝐻(𝛼₁,𝛽) if 𝑧𝐻Re^𝑞,𝑠𝛼₁ℎ(𝑧)^′−𝑧𝐻_𝑞,𝑠𝛼₁𝑔(𝑧)^′𝐻_𝑞,𝑠𝛼₁ℎ(𝑧)+𝐻_𝑞,𝑠𝛼₁𝑔(𝑧)<𝛽 (2.9) is equivalent to Re𝛽−1𝑧−∑^∞_𝑘=2𝑘−𝛽Γ𝛼₁,𝑘/(𝑘−1)!𝑎_𝑘𝑧^𝑘−∑^∞_𝑘=1𝑘+𝛽Γ𝛼₁,𝑘/(𝑘−1)!𝑏_𝑘𝑧^𝑘𝑧+∑^∞_𝑘=2Γ𝛼₁,𝑘/(𝑘−1)!𝑎_𝑘𝑧^𝑘−∑^∞_𝑘=1Γ𝛼₁,𝑘/(𝑘−1)!𝑏_𝑘𝑧^𝑘≥0. (2.10)
The above condition must hold for all values of 𝑧, |𝑧|=𝑟<1. Upon choosing the values of 𝑧 on the positive real axis where 0≤𝑧=𝑟<1, we must have 𝛽−1−∑^∞_𝑘=2𝑘−𝛽Γ𝛼₁,𝑘/(𝑘−1)!𝑎_𝑘𝑟^𝑘−1−∑^∞_𝑘=1𝑘+𝛽Γ𝛼₁,𝑘/(𝑘−1)!𝑏_𝑘𝑟^𝑘−11+∑^∞_𝑘=2Γ𝛼₁,𝑘/(𝑘−1)!𝑎_𝑘𝑟^𝑘−1−∑^∞_𝑘=1Γ𝛼₁,𝑘/(𝑘−1)!𝑏_𝑘𝑟^𝑘−1≥0. (2.11)
If the condition (2.8) does not hold then the numerator of (2.11) is negative for 𝑟 and sufficiently close to 1. Thus there exists a 𝑧₀=𝑟₀ in (0,1) for which the quotient in (2.11) is negative. This contradicts the required condition for 𝑓∈𝑀_𝐻(𝛼₁,𝛽) and so the proof is complete.
Next, we determine the extreme points of the closed convex hulls of 𝑀_𝐻(𝛼₁,𝛽), denoted by clco𝑀_𝐻(𝛼₁,𝛽).

Theorem 2.3. 𝑓∈clco𝑀_𝐻(𝛼₁,𝛽), if and only if 𝑓(𝑧)=^∞_𝑘=1𝑥_𝑘ℎ_𝑘(𝑧)+𝑦_𝑘𝑔_𝑘(𝑧), (2.12) where ℎ₁(𝑧)ℎ=𝑧,_𝑘(𝑧)=𝑧+𝛽−1()!𝑘−𝛽𝑘−1Γ𝛼₁𝑧,𝑘^𝑘(),𝑔,𝑘=2,3,…_𝑘(𝑧)=𝑧−𝛽−1()!𝑘+𝛽𝑘−1Γ𝛼₁,𝑘𝑧^𝑘,𝑘=(1,2,3,…),^∞_𝑘=1𝑥_𝑘+𝑦_𝑘=1, (2.13)𝑥_𝑘≥0, and 𝑦_𝑘≥0. In particular the extreme points of 𝑀_𝐻(𝛼₁,𝛽) are {ℎ_𝑘} and {𝑔_𝑘}.

Proof. For functions 𝑓 of the form (2.12), we have 𝑓(𝑧)=^∞_𝑘=1𝑥_𝑘ℎ_𝑘(𝑧)+𝑦_𝑘𝑔_𝑘(𝑧)=𝑧+^∞_𝑘=2𝛽−1()!𝑘−𝛽𝑘−1Γ𝛼₁𝑥,𝑘^𝑘𝑧^𝑘−^∞_𝑘=1𝛽−1()!𝑘+𝛽𝑘−1Γ𝛼₁𝑦,𝑘^𝑘𝑧^𝑘. (2.14)
Then ^∞_𝑘=2𝑘−𝛽Γ𝛼𝛽−1¹,𝑘(𝑘−1)!𝛽−1()!𝑘−𝛽𝑘−1Γ𝛼₁𝑥,𝑘^𝑘+^∞_𝑘=1𝑘+𝛽Γ𝛼𝛽−1¹,𝑘(𝑘−1)!𝛽−1()!𝑘+𝛽𝑘−1Γ𝛼₁𝑦,𝑘^𝑘=^∞_𝑘=2𝑥_𝑘+^∞_𝑘=1𝑦_𝑘=1−𝑥₁≤1, (2.15)
and so  𝑓∈clco𝑀_𝐻(𝛼₁,𝛽).
Conversely, suppose that 𝑓∈clco𝑀_𝐻(𝛼₁,𝛽). Set 𝑥_𝑘=((𝑘−𝛽)/(𝛽−1))((𝑘−1)!/Γ(𝛼₁,𝑘))|𝑎_𝑘|,  (𝑘=2,3,4,…) and 𝑦_𝑘=((𝑘+𝛽)/(𝛽−1))((𝑘−1)!/Γ(𝛼₁,𝑘))|𝑏_𝑘|,  (𝑘=1,2,3,…).
Then note that by Theorem 2.2, 0≤𝑥_𝑘≤1,(𝑘=2,3,4,…) and 0≤𝑦_𝑘≤1,(𝑘=1,2,3,…). We define 𝑥₁=1−∑^∞_𝑘=2𝑥_𝑘−∑^∞_𝑘=1𝑦_𝑘, and by Theorem 2.2, 𝑥₁≥0.
Consequently, we obtain 𝑓(𝑧)=∑^∞_𝑘=1(𝑥_𝑘ℎ_𝑘(𝑧)+𝑦_𝑘𝑔_𝑘(𝑧))  as required.

Theorem 2.4. If 𝑓∈𝑀_𝐻(𝛼₁,𝛽), then 𝑓(𝑧)≤1+𝑏₁1𝑟+Γ𝛼₁,2𝛽−1−2−𝛽𝛽+1𝑏2−𝛽¹𝑟²,|𝑧|𝑓(𝑧≥𝑏=𝑟<1,)1−₁1𝑟−Γ𝛼₁,2𝛽−1−2−𝛽𝛽+1𝑏2−𝛽¹𝑟²,|𝑧|=𝑟<1. (2.16)

Proof. We only prove the right hand inequality. The proof for left hand inequality is similar and will be omitted. Let 𝑓∈𝑀_𝐻(𝛼₁,𝛽). Taking the absolute value of 𝑓, we have 𝑓(𝑧)≤1+𝑏₁𝑟+^∞_𝑘=2𝑎_𝑘+𝑏_𝑘𝑟^𝑘≤1+𝑏₁𝑟+^∞_𝑘=2𝑎_𝑘+𝑏_𝑘𝑟²=1+𝑏₁𝑟+𝛽−112−𝛽Γ𝛼₁,2^∞_𝑘=22−𝛽Γ𝛼₁,2𝑎𝛽−1^𝑘+2−𝛽Γ𝛼₁,2𝑏𝛽−1^𝑘𝑟²≤1+𝑏₁𝑟+𝛽−112−𝛽Γ𝛼₁,2^∞_𝑘=2𝑘−𝛽Γ𝛼𝛽−1¹,𝑘()!𝑎𝑘−1^𝑘+𝑘+𝛽Γ𝛼𝛽−1¹,𝑘()!𝑏𝑘−1^𝑘𝑟²≤1+𝑏₁𝑟+𝛽−112−𝛽Γ𝛼₁,21−𝛽+1𝑏𝛽−1¹𝑟²=1+𝑏₁1𝑟+Γ𝛼₁,2𝛽−1−2−𝛽𝛽+1𝑏2−𝛽¹𝑟². (2.17)
For our next theorem, we need to define the convolution of two harmonic functions. For harmonic functions of the form 𝑓(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑧^𝑘−^∞_𝑘=1𝑏_𝑘𝑧^𝑘,𝐹(𝑧)=𝑧+^∞_𝑘=2𝐴_𝑘𝑧^𝑘−^∞_𝑘=1𝐵_𝑘𝑧^𝑘. (2.18) We define the convolution of two harmonic functions 𝑓 and 𝐹 as 𝑓∗𝐹(𝑧)=𝑓(𝑧)∗𝐹(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝐴_𝑘𝑧^𝑘−^∞_𝑘=1𝑏_𝑘𝐵_𝑘𝑧^𝑘. (2.19)
Using this definition, we show that the class 𝑀_𝐻(𝛼₁,𝛽) is closed under convolution.

Theorem 2.5. For 1<𝛾≤𝛽≤4/3, let 𝑓∈𝑀_𝐻(𝛼₁,𝛾) and 𝐹∈𝑀_𝐻(𝛼₁,𝛽). Then 𝑓∗𝐹∈𝑀_𝐻𝛼₁,𝛾⊆𝑀_𝐻𝛼₁,𝛽. (2.20)

Proof. Let 𝑓(𝑧)=𝑧+∑^∞_𝑘=2|𝑎_𝑘|𝑧^𝑘−∑^∞_𝑘=1|𝑏_𝑘|𝑧^𝑘 be in 𝑀_𝐻(𝛼₁,𝛾), and 𝐹(𝑧)=𝑧+∑^∞_𝑘=2|𝐴_𝑘|𝑧^𝑘−∑^∞_𝑘=1|𝐵_𝑘|𝑧^𝑘 be in 𝑀_𝐻(𝛼₁,𝛽).
Then the convolution 𝑓∗𝐹 is given by (2.19). We wish to show that the coefficients of 𝑓∗𝐹 satisfy the required condition given in Theorem 2.2. For 𝐹(𝑧)∈𝑀_𝐻(𝛼₁,𝛽) we note that |𝐴_𝑘|≤1 and |𝐵_𝑘|≤1. Now, for the convolution function 𝑓∗𝐹, we obtain ^∞_𝑘=2(𝑘−𝛼)()Γ𝛼𝛼−1¹,𝑘()!𝑎𝑘−1^𝑘𝐴_𝑘+^∞_𝑘=1(𝑘+𝛼)()Γ𝛼𝛼−1¹,𝑘()!𝑏𝑘−1^𝑘𝐵_𝑘≤^∞_𝑘=2(𝑘−𝛼)()Γ𝛼𝛼−1¹,𝑘()!𝑎𝑘−1^𝑘+^∞_𝑘=1(𝑘+𝛼)()Γ𝛼𝛼−1¹,𝑘()!𝑏𝑘−1^𝑘≤1,since𝑓∈𝑀_𝐻𝛼₁,𝛾. (2.21)
Therefore 𝑓∗𝐹∈𝑀_𝐻(𝛼₁,𝛾)⊆𝑀_𝐻(𝛼₁,𝛽).

Theorem 2.6. The class 𝑀_𝐻(𝛼₁,𝛽) is closed under convex combination.

Proof. For 𝑖=1,2,3,…, let 𝑓_𝑖(𝑧)∈𝑀_𝐻(𝛼₁,𝛽), where 𝑓_𝑖(𝑧) is given by 𝑓_𝑖(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑖𝑧^𝑘−^∞_𝑘=1𝑏_𝑘𝑖𝑧^𝑘. (2.22)
Then by Theorem 2.2, ^∞_𝑘=2𝑘−𝛽Γ𝛼𝛽−1¹,𝑘()!𝑎𝑘−1^𝑘𝑖+^∞_𝑘=1𝑘+𝛽Γ𝛼𝛽−1¹,𝑘()!𝑏𝑘−1^𝑘𝑖≤1. (2.23)
For ∑^∞_𝑖−1𝑡_𝑖=1, 0≤𝑡_𝑖≤1, the convex combination of 𝑓_𝑖 may be written as ^∞_𝑖−1𝑡_𝑖𝑓_𝑖(𝑧)=𝑧+^∞_𝑘=2^∞_𝑖−1𝑡_𝑖𝑎_𝑘𝑖𝑧^𝑘−^∞_𝑘=1^∞_𝑖−1𝑡_𝑖𝑏_𝑘𝑖𝑧^𝑘. (2.24)
Then by (2.23), ^∞_𝑘=2𝑘−𝛽Γ𝛼𝛽−1¹,𝑘(𝑘−1)!^∞_𝑖−1𝑡_𝑖𝑎_𝑘𝑖+^∞_𝑘=1𝑘+𝛽Γ𝛼𝛽−1¹,𝑘(𝑘−1)!^∞_𝑖−1𝑡_𝑖𝑏_𝑘𝑖=^∞_𝑖−1𝑡_𝑖^∞_𝑘=2𝑘−𝛽Γ𝛼𝛽−1¹,𝑘()!𝑎𝑘−1^𝑘𝑖+^∞_𝑘=1𝑘+𝛽Γ𝛼𝛽−1¹,𝑘()!𝑏𝑘−1^𝑘𝑖≤^∞_𝑖−1𝑡_𝑖=1. (2.25)
This is the condition required by Theorem 2.2 and so ∑^∞_𝑖−1𝑡_𝑖𝑓_𝑖(𝑧)∈𝑀_𝐻(𝛼₁,𝛽).

3. A Family of Class Preserving Integral Operator

Let 𝑓(𝑧)=ℎ(𝑧)+𝑔(𝑧) be defined by (1.1); then 𝐹(𝑧) defined by the relation 𝐹(𝑧)=𝑐+1𝑧^𝑐𝑧₀𝑡^𝑐−1ℎ(𝑡)𝑑𝑡+𝑐+1𝑧^𝑐𝑧₀𝑡^𝑐−1𝑔(𝑡)𝑑𝑡,(𝑐>−1). (3.1)

Theorem 3.1. Let 𝑓(𝑧)=ℎ(𝑧)+𝑔(𝑧)∈𝑆_𝐻 be given by (1.11) and 𝑓∈𝑀_𝐻(𝛼₁,𝛽), then 𝐹(𝑧) is defined by (3.1) also belong to 𝑀_𝐻(𝛼₁,𝛽).

Proof. Let 𝑓(𝑧)=𝑧+∑^∞_𝑘=2|𝑎_𝑘|𝑧^𝑘−∑^∞_𝑘=1|𝑏_𝑘|𝑧^𝑘 be in 𝑀_𝐻(𝛼₁,𝛽), then by Theorem 2.2, we have ^∞_𝑘=2𝑘−𝛽Γ𝛼𝛽−1¹,𝑘()!𝑎𝑘−1^𝑘+^∞_𝑘=1𝑘+𝛽Γ𝛼𝛽−1¹,𝑘()!𝑏𝑘−1^𝑘≤1. (3.2)
By definition of 𝐹(𝑧), we have 𝐹(𝑧)=𝑧+^∞_𝑘=2(𝑐+1)()𝑎𝑐+𝑘^𝑘𝑧^𝑘−^∞_𝑘=1(𝑐+1)()𝑏𝑐+𝑘^𝑘𝑧^𝑘. (3.3)
Now we have ^∞_𝑘=2𝑘−𝛽Γ𝛼𝛽−1¹,𝑘()!()𝑘−1𝑐+1()𝑎𝑐+𝑘^𝑘+^∞_𝑘=1𝑘+𝛽Γ𝛼𝛽−1¹,𝑘()!()𝑘−1𝑐+1()𝑏𝑐+𝑘^𝑘≤^∞_𝑘=2𝑘−𝛽Γ𝛼𝛽−1¹,𝑘()!𝑎𝑘−1^𝑘+^∞_𝑘=1𝑘+𝛽Γ𝛼𝛽−1¹,𝑘()!𝑏𝑘−1^𝑘≤1. (3.4)
Thus 𝐹(𝑧)∈𝑀_𝐻(𝛼₁,𝛽).

Acknowledgments

The first author is thankful to University Grants Commission, New Delhi (Government of India) for supporting this paper financially under research project (no. 8-2 (223)/2011 (MRP/Sc/NRCB)). The authors are also thankful to the reviewers for giving fruitful suggestions and comments.