On Certain Subclass of Analytic Functions with Fixed Point

Abstract

A new certain differential operator 𝑇^𝑚_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧) and a subclass 𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆) are introduced for functions of the form 𝑓(𝑧)=(𝑧−𝑤)−∑^∞_𝑛=2𝑎_𝑛(𝑧−𝑤)^𝑛 which are univalent in the unit disc 𝑈={𝑧∈ℂ∶|𝑧|<1}. In this paper, we obtain coefficient inequalities, distortion theorem, closure theorems, and class preserving integral operators of functions belonging to the class 𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆).

1. Introduction

Let 𝐻(𝑈) be the set of functions which are regular in the unit disc 𝑈={𝑧∈ℂ∶|𝑧|<1}, 𝐴={𝑓∈𝐻(𝑈)∶𝑓(0)=𝑓^′(0)−1=0} and 𝑆={𝑓∈𝐴∶𝑓 is univalent in 𝑈}.

We recall here the definitions of the well-known classes of starlike and convex functions: 𝑆^∗=𝑓∈𝐴∶Re𝑧𝑓^′(𝑧)𝑓(𝑧),𝑆>0,𝑧∈𝑈_𝑐=𝑓∈𝐴∶Re1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)>0,𝑧∈𝑈. (1)

Let 𝑤 be a fixed point in 𝑈 and 𝐴(𝑤)={𝑓∈𝐻(𝑈)∶𝑓(𝑤)=𝑓^′(𝑤)−1=0}.

In [1], Kanas and Rønning introduced the following classes: 𝑆_𝑤=𝑓∈𝐴(𝑤)∶𝑓isunivalentin𝑈,𝑆𝑇_𝑤=𝑆^∗_𝑤=𝑓∈𝑆_𝑤()𝑓∶Re𝑧−𝑤^′(𝑧)𝑓(𝑧)>0,𝑧∈𝑈,𝑆𝑉_𝑤=𝑆^𝑐_𝑤=𝑓∈𝑆_𝑤()𝑓∶1+Re𝑧−𝑤^′′(𝑧)𝑓^′(𝑧)>0,𝑧∈𝑈. (2) These classes are extensively studied by Acu and Owa [2].

The class 𝑆^∗_𝑤 is defined by geometric property that the image of any circular arc centered at 𝑤 is starlike with respect to 𝑓(𝑤) and the corresponding class 𝑆^𝑐_𝑤 is defined by the property that the image of any circular arc centered at 𝑤 is convex. We observe that the definitions are somewhat similar to the ones for uniformly starlike and convex functions introduced by Goodman in [3, 4], except that in this case the point 𝑤 is fixed.

In fact, several subclasses of 𝑆 have been introduced by a fixed geometric property of the image domain. It is interesting to note that these subclasses play an important role in other branches of mathematics. For example, starlike functions and convex functions play an important role in the solution of certain differential equations (see Robertson [5], Saitoh [6], Owa et al. [7], Reade and Silverman [8], and Sokół and Wiśniowska-Wajnryb [9]). One of the important problems in geometric function theory are the extremal problems, which pose an effective method for establishing the existence of analytic functions with certain natural properties. Extremal problems (supremum Re{𝑓}) play an important role in geometric function theory, for finding sharp estimates, coefficient bounds, and an extremal function. The results we obtained here may have potential application in other branches of mathematics, both pure and applied. For example the extremal problems are closely connected to Hele-Shaw flow of fluid mechanics [10], exterior inverse problems in potential theory, and many others.

Now let us begin with our definitions as follows.

The function 𝑓(𝑧) in 𝑆_𝑤 is said to be starlike functions of order 𝛾 if and only if ()𝑓Re𝑧−𝑤^′(𝑧)𝑓(𝑧)>𝛾(𝑧∈𝑈) (3) for some 𝛾(0≤𝛾<1). We denote by 𝑆𝑇_𝑤(𝛾) the class of all starlike functions of order 𝛾. Similarly, a function 𝑓(𝑧) in 𝑆_𝑤 is said to be convex of order 𝛾 if and only if ()𝑓Re1+𝑧−𝑤^′′(𝑧)𝑓^′(𝑧)>𝛾(𝑧∈𝑈) (4) for some 𝛾(0≤𝛾<1). We denote by 𝑆𝑉_𝑤(𝛾) the class of all convex functions of order 𝛾.

It is easy to see that 𝑆_𝑤 denoting the subclass of 𝐴(𝑤) has the series of expansion: 𝑓(𝑧)=(𝑧−𝑤)+^∞_𝑛=2𝑎_𝑛(𝑧−𝑤)^𝑛. (5)

For the function 𝑓(𝑧) in the class 𝑆_𝑤, we define the following new differential operator: 𝑇⁰𝑓(𝑧)(𝑧),𝑇=𝑓_{1𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)=()𝑓(𝑧)()𝑓1−𝛽𝜆−𝛼+𝛽𝜆−𝛼)(𝑧−𝑤^′(𝑧)+𝛿(𝑧−𝑤)²𝑓^′′(𝑧),𝑇²_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)=1−𝛽(𝜆−𝛼)𝑇¹_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)+𝛽(𝜆−𝛼)(𝑧−𝑤)𝑇¹_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)^′+𝛿(𝑧−𝑤)²𝑇¹_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)^′′, (6) and for 𝑚=1,2,3,…. 𝑇^𝑚_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)=1−𝛽(𝜆−𝛼)𝑇^𝑚−1_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧()𝑇)+𝛽𝜆−𝛼)(𝑧−𝑤^𝑚−1_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)^′+𝛿(𝑧−𝑤)²𝑇^𝑚−1_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)^′′=(𝑧−𝑤)+^∞_𝑛=21+(𝑛−1)𝛽(𝜆−𝛼)+𝑛𝛿^𝑚×𝑎_𝑛(𝑧−𝑤)^𝑛, (7) for 𝛼≥0, 𝛽≥0, 𝛿≥0, 𝜆≥0, and 𝑚∈ℕ₀=ℕ∪{0}.

It easily verified from (7) that 𝛽(𝜆−𝛼)(𝑧−𝑤)𝑇^𝑚_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)^′=𝑇^𝑚+1_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)−1−𝛽(𝜆−𝛼)𝑇^𝑚_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)−𝛿(𝑧−𝑤)²𝑇^𝑚_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)^′′. (8)

Remark 1. (i) When 𝑤=𝛿=0, we have the operator introduced and studied by Darus and Ibrahim (see [11]).
(ii) When 𝑤=𝛼=𝛿=0 and 𝛽=1, we have the operator introduced and studied by Al-Oboudi (see [12]).
(iii) When 𝛼=𝛿=0 and 𝜆=𝛽=1, we have the operator introduced and studied by Acu and Owa (see [2]).
(iv) And when 𝛼=𝛿=𝑤=0 and 𝜆=𝛽=1, we have the operator introduced and studied by Sǎlǎgean (see [13]).

With the help of the differential operator 𝑇^𝑚_{𝛼,𝛽,𝛿,𝜆}, we define the class 𝑆_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆) as follows.

Definition 2. A function 𝑓(𝑧)∈𝑆_𝑤 is said to be a member of the class 𝑆_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆) if it satisfies (𝑧−𝑤)𝑇^𝑚_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)^′𝑇^𝑚_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)<()𝑇−1𝑧−𝑤^𝑚_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)^′𝑇^𝑚_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)+1−2𝛾 (9) for some 0≤𝛾<1, 𝛼≥0, 𝛽≥0, 𝛿≥0, 𝜆≥0, and 𝑚∈ℕ₀ and for all 𝑧∈𝑈.

It is easy to check that 𝑆_0,0(0,1,𝛾,0,1) is the class of starlike functions of order 𝛾 and 𝑆_0,0(0,1,0,0,1) gives the all class of starlike functions.

Let 𝑇_𝑤 denote the subclass of 𝐴(𝑤) consisting of functions of the following form: 𝑓(𝑧)=(𝑧−𝑤)−^∞_𝑛=2𝑎_𝑛(𝑧−𝑤)^𝑛,𝑎_𝑛≥0. (10)

Further, we define the class 𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆) by 𝑆^∗_𝑚,𝑤𝛼,𝛽,𝛾,𝛿,𝜆=𝑆_𝑚,𝑤𝛼,𝛽,𝛾,𝛿,𝜆∩𝑇_𝑤. (11)

In this paper, coefficient inequalities, distortion theorem, and closure theorems of functions belonging to the class 𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆) are obtained. Finally, the class preserving integral operators of the form 𝐹(𝑧)=𝑐+1(𝑧−𝑤)^𝑐𝑧_𝑤(𝑡−𝑤)^𝑐−1𝑓(𝑡)𝑑𝑡(𝑐>−1) (12) for the class 𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆) is considered.

2. Coefficient Inequalities

Our first result provides a sufficient condition for a function, regular in 𝑈, to be in 𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆).

Theorem 3. A function 𝑓(𝑧) being defined by (10) is in 𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆) if and only if 𝑛−𝛾1+(^∞_𝑛=2𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚𝑎_𝑛≤1−𝛾𝑚∈ℕ₀, (13) where 0≤𝛾<1. The result (13) is sharp for functions of the following form: 𝑓_𝑛(𝑧)=()−𝑧−𝑤1−𝛾𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚(𝑧−𝑤)^𝑛𝑛≥2;𝑚∈ℕ₀. (14)

Proof. Suppose that (13) holds true for 0≤𝛾<1. consider the expression 𝑀𝑓,𝑓^′=(𝑧−𝑤)𝑇^𝑚_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)^′−𝑇^𝑚_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)−(𝑧−𝑤)𝑇^𝑚_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)^′+1−2𝛾𝑇^𝑚_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧). (15) Then for |𝑧−𝑤|=𝑟<1, we have 𝑀𝑓,𝑓^′=^∞_𝑛=2(𝑛−1)1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚×𝑎_𝑛(𝑧−𝑤)^𝑛−2)−1−𝛾(𝑧−𝑤_∞𝑛=2𝑛−2𝛾+11+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚×𝑎_𝑛(𝑧−𝑤)^𝑛,𝑀𝑓,𝑓^′≤^∞_𝑛=2()×()𝛽)𝑛−11+𝑛−1((𝜆−𝛼+𝑛𝛿)^𝑚𝑎_𝑛𝑟^𝑛−21−𝛾+^∞_𝑛=2×𝑛+2𝛾−11+𝜆𝑛+𝑝−2^𝑚𝑎_𝑛𝑟^𝑛≤^∞_𝑛=22𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚×𝑎_𝑛−21−𝛾≤0. (16)
So that 𝑓(𝑧)∈𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆).
 For the converse, assume that (𝑧−𝑤)𝑇^𝑚_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)^′𝑇^𝑚_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)/()𝑇−1𝑧−𝑤^𝑚_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)^′𝑇^𝑚_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧)=−+1−2𝛾_∞𝑛=2(𝑛−1)1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚×𝑎_𝑛(𝑧−𝑤)^𝑛×2()−1−𝛾𝑧−𝑤_∞𝑛=2𝑛−2𝛾+11+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚×𝑎_𝑛(𝑧−𝑤)^𝑛−1<1. (17) Since Re(𝑧−𝑤)≤|𝑧−𝑤| for all (𝑧−𝑤), it follows from (17) that Re^∞_𝑛=2(𝑛−1)1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚×𝑎_𝑛(𝑧−𝑤)^𝑛×2()−1−𝛾𝑧−𝑤_∞𝑛=2𝑛−2𝛾+11+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚×𝑎_𝑛(𝑧−𝑤)^𝑛−1<1. (18)
Choose values of (𝑧−𝑤) on the real axis so that (𝑧−𝑤)(𝑇^𝑚_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧))^′/𝑇^𝑚_{𝛼,𝛽,𝛿,𝜆}𝑓(𝑧) is real. Upon clearing the denominator in (18) and letting 𝑟→1⁻ through real values, we obtain ^∞_𝑛=2(𝑛−1)1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚𝑎_𝑛−<21−𝛾_∞𝑛=2𝑛−2𝛾+11+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚𝑎_𝑛. (19) This gives the required condition. Hence the theorem follows.

Corollary 4. Let the function 𝑓(𝑧) be defined by (10) and 𝑓(𝑧)∈𝑆_𝑤. If 𝑓∈𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆), then 𝑎_𝑛≤1−𝛾𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚,𝑛≥2. (20)
The result (20) is sharp for functions 𝑓_𝑛(𝑧) given by (14).

3. Distortion Theorem

A distortion property for functions in the class 𝑓∈𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆) is contained in

Theorem 5. Let the function 𝑓(𝑧) one has defined by (10) be in the class 𝑓∈𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆). Then for |𝑧−𝑤|=𝑟<1, we have 𝑟−1−𝛾2−𝛾1+𝛽((𝜆−𝛼)+2𝛿)^𝑚𝑟²≤𝑓(𝑧)≤𝑟+1−𝛾2−𝛾1+𝛽((𝜆−𝛼)+2𝛿)^𝑚𝑟²,21−1−𝛾2−𝛾1+𝛽((𝜆−𝛼)+2𝛿)^𝑚𝑟≤𝑓^′(𝑧2)≤1+1−𝛾2−𝛾1+𝛽((𝜆−𝛼)+2𝛿)^𝑚𝑟. (21)

Proof. Since 𝑓∈𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆), Theorem 3 readily yields the inequality ^∞_𝑛=2𝑎_𝑛≤1−𝛾2−𝛾1+𝛽((𝜆−𝛼)+2𝛿)^𝑚. (22) Thus, for |𝑧−𝑤|=𝑟<1, and making use of (22), we have 𝑓(𝑧)≤|𝑧−𝑤|+^∞_𝑛=2𝑎_𝑛|𝑧−𝑤|^𝑛≤𝑟+𝑟^2∞_𝑛=2𝑎_𝑛≤𝑟+1−𝛾2−𝛾1+𝛽((𝜆−𝛼)+2𝛿)^𝑚𝑟²,𝑓(𝑧)≥|𝑧−𝑤|−^∞_𝑛=2𝑎_𝑛|𝑧−𝑤|^𝑛≥𝑟−𝑟^2∞_𝑛=2𝑎_𝑛≥𝑟−1−𝛾2−𝛾1+𝛽((𝜆−𝛼)+2𝛿)^𝑚𝑟². (23) Also from Theorem 3, it follows that 2−𝛾1+𝛽((𝜆−𝛼)+2𝛿)^𝑚2^∞_𝑛=2𝑛𝑎_𝑛≤^∞_𝑛=2𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚𝑎_𝑛≤1−𝛾. (24) Hence 𝑓^′(𝑧)≤1+^∞_𝑛=2𝑛𝑎_𝑛|𝑧−𝑤|^𝑛−1≤1+𝑟^∞_𝑛=2𝑛𝑎_𝑛2≤1+1−𝛾2−𝛾1+𝛽((𝜆−𝛼)+2𝛿)^𝑚𝑓𝑟,_′(𝑧)≥1−^∞_𝑛=2𝑛𝑎_𝑛|𝑧−𝑤|^𝑛−1≥1−𝑟^∞_𝑛=2𝑛𝑎_𝑛2≥1−1−𝛾2−𝛾1+𝛽((𝜆−𝛼)+2𝛿)^𝑚𝑟. (25) This completes the proof of Theorem 5.

4. Radii of Starlikeness and Convexity

The radii of starlikeness and convexity for the class 𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆) is given by the following theorems.

Theorem 6. If the function 𝑓(𝑧) defined by (10) is in the class 𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆), then 𝑓(𝑧) is of order 𝜂(0≤𝜂<1) in |𝑧−𝑤|<𝑟₁, where 𝑟₁=inf_𝑛≥21−𝜂𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚𝑛−𝜂1−𝛾^1/𝑛. (26)
The result is sharp for the functions 𝑓_𝑛(𝑧) given by (14).

Proof. It suffices to prove that (𝑧−𝑤)𝑓^′(𝑧)𝑓(𝑧)−1≤1−𝜂, (27) for |𝑧−𝑤|<𝑟₁. We have (𝑧−𝑤)𝑓^′(𝑧)𝑓(𝑧)=−1−∑^∞_𝑛=2(𝑛−1)𝑎_𝑛(𝑧−𝑤)^𝑛(𝑧−𝑤)−∑^∞_𝑛=2𝑎_𝑛(𝑧−𝑤)^𝑛≤∑^∞_𝑛=2(𝑛−1)𝑎_𝑛|𝑧−𝑤|^𝑛1−∑^∞_𝑛=2𝑎_𝑛|𝑧−𝑤|^𝑛. (28) Hence (28) holds true if ^∞_𝑛=2(𝑛−1)𝑎_𝑛|𝑧−𝑤|^𝑛≤1−𝜂1−^∞_𝑛=2𝑎_𝑛|𝑧−𝑤|^𝑛 (29) or ^∞_𝑛=2𝑛−𝜂𝑎1−𝜂^𝑛|𝑧−𝑤|^𝑛≤1 (30) with the aid of (13), (30) is true if 𝑛−𝜂||1−𝜂𝑧−𝑤^𝑛≤𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚.1−𝛾 (31) Solving (31) for |𝑧−𝑤|, we obtain ||≤()𝛽)𝑧−𝑤1−𝜂𝑛−𝛾1+𝑛−1((𝜆−𝛼+𝑛𝛿)^𝑚𝑛−𝜂1−𝛾^1/𝑛(𝑛≥2). (32) This completes the proof of Theorem 6.

Theorem 7. If the function 𝑓(𝑧) defined by (10) is in the class 𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆), then 𝑓(𝑧) is convex of order 𝜂(0≤𝜂<1) in |𝑧−𝑤|<𝑟₂, where 𝑟₂=inf_𝑛≥2(1−𝜂)(𝑛−𝛾)1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚𝑛(𝑛+𝜂−2)(1−𝛾)^1/(𝑛−1)(𝑛≥2). (33)
The result is sharp for the functions 𝑓_𝑛(𝑧) given by (14).

Proof. By using the technique employed in the proof of Theorem 6, we can show that (𝑧−𝑤)𝑓^′′(𝑧)𝑓^′(𝑧)≤1−𝜂, (34) for |𝑧−𝑤|<𝑟₂, with the aid of Theorem 3. Thus we have the assertion of Theorem 7.

5. Closure Theorems

Let the functions 𝑓_𝑗(𝑧),𝑗=1,2,…,𝑙, be defined by 𝑓_𝑗(𝑧)=(𝑧−𝑤)−^∞_𝑛=2𝑎_𝑛,𝑗(𝑧−𝑤)^𝑛,𝑎_𝑛,𝑗≥0 (35) for 𝑧∈𝑈.

Theorem 8. Let the functions 𝑓_𝑗(𝑧) be defined by (35) in the class 𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆) for every 𝑗=1,2,…,𝑙. Then the function 𝐺(𝑧) defined by 𝐺(𝑧)=(𝑧−𝑤)−^∞_𝑛=2𝑏_𝑛(𝑧−𝑤)^𝑛,𝑏_𝑛≥0 (36) is a member of the class 𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆), where 𝑏_𝑛=1𝑙^𝑙_𝑗=1𝑎_𝑛,𝑗(𝑛≥2). (37)

Proof. Since 𝑓_𝑗(𝑧)∈𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆), it follows from Theorem 3 that 𝑛−𝛾1+(^∞_𝑛=2𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚𝑎_𝑛,𝑗≤1−𝛾 (38) for every 𝑗=1,2,…,𝑙. Hence, 𝑛−𝛾1+(^∞_𝑛=2𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚𝑏_𝑛=^∞_𝑛=2𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚×1𝑙^𝑙_𝑗=1𝑎_𝑛,𝑗=1𝑙^𝑙_𝑗=1^∞_𝑛=2×()𝛽)𝑛−𝛾1+𝑛−1((𝜆−𝛼+𝑛𝛿)^𝑚𝑎_𝑛,𝑗≤1𝑙^𝑙_𝑗=11−𝛾=1−𝛾 (39) which (in view of Theorem 5) implies that 𝐺(𝑧)∈𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆).

Theorem 9. The class 𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆) is closed under convex linear combination.

Proof. Suppose that the functions 𝑓_𝑗(𝑧)(𝑗=1,2) defined by (35) be in the class 𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆), it is sufficient to prove that the function 𝐻(𝑧)=𝜇𝑓₁(𝑧)+1−𝜇𝑓₂(𝑧)0≤𝜇≤1 (40) is also in the class 𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆). Since, for 0≤𝜇≤1, 𝐻(𝑧)=(𝑧−𝑤)−^∞_𝑛=2𝜇𝑎_𝑛,1+1−𝜇𝑎_𝑛,2(𝑧−𝑤)^𝑛, (41) we observe that 𝑛−𝛾1+(^∞_𝑛=2𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚×𝜇𝑎_𝑛,1+1−𝜇𝑎_𝑛,2=𝜇^∞_𝑛=2𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚𝑎_𝑛,1+1−𝜇^∞_𝑛=2×()𝛽)𝑛−𝛾1+𝑛−1((𝜆−𝛼+𝑛𝛿)^𝑚𝑎_𝑛,2≤𝜇1−𝛾+1−𝜇1−𝛾=1−𝛾. (42) Hence 𝐻(𝑧)∈𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆). This completes the proof of Theorem 9.

Theorem 10. Let 𝑓₀(𝑧)𝑓=𝑧−𝑤,_𝑛(𝑧)=()−𝑧−𝑤1−𝛾𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚(𝑧−𝑤)^𝑛𝑛≥2;𝑚∈ℕ₀. (43) Then 𝑓(𝑧)∈𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆) if and only if it can be expressed in the form 𝑓(𝑧)=^∞_𝑛=1𝜆_𝑛𝑓_𝑛(𝑧), (44) where 𝜆_𝑛≥0 and ∑^∞_𝑛=1𝜆_𝑛=1.

Proof. Suppose that 𝑓(𝑧)=^∞_𝑛=1𝜆_𝑛𝑓_𝑛(𝑧), (45) where 𝜆_𝑛≥0  (𝑛≥1) and ∑^∞_𝑛=1𝜆_𝑛=1. Then 𝑓(𝑧)=^∞_𝑛=1𝜆_𝑛𝑓_𝑛(𝑧)=𝜆₁𝑓₁(𝑧)+^∞_𝑛=2𝜆_𝑛𝑓_𝑛(𝑧)=()−𝑧−𝑤_∞𝑛=21−𝛾𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚×𝜆_𝑛(𝑧−𝑤)^𝑛. (46) Since 𝑛−𝛾1+(^∞_𝑛=2𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚⋅1−𝛾𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚𝜆_𝑛=1−𝛾^∞_𝑛=2𝜆_𝑛=1−𝛾1−𝜆₀≤1−𝛾, (47) it follows from Theorem 3 that the function 𝑓(𝑧)∈𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆). Conversely, let us suppose that 𝑓(𝑧)∈𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆). Since 𝑎_𝑛≤1−𝛾𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚𝑛≥2;𝑚∈ℕ₀. (48) Setting 𝜆_𝑛=𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚𝑎1−𝛾^𝑛,𝑛≥2;𝑚∈ℕ₀,𝜆₁=1−^∞_𝑛=2𝜆_𝑛, (49) it follows that 𝑓(𝑧)=𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆). This completes the proof of the theorem.

6. Extreme Points

Theorem 11. Let 𝑓₀(𝑧)𝑓=𝑧−𝑤,_𝑛(𝑧)=()−𝑧−𝑤1−𝛾𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚×(𝑧−𝑤)^𝑛𝑛≥2;𝑚∈ℕ₀. (50) Then 𝑓(𝑧)∈𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆) if and only if it can be expressed in the form 𝑓(𝑧)=^∞_𝑛=1𝜆_𝑛𝑓_𝑛(𝑧), (51) where 𝜆_𝑛≥0 and ∑^∞_𝑛=1𝜆_𝑛=1.

Proof. Suppose that 𝑓(𝑧)=^∞_𝑛=1𝜆_𝑛𝑓_𝑛(𝑧), (52) where 𝜆_𝑛≥0  (𝑛≥1) and ∑^∞_𝑛=1𝜆_𝑛=1. Then 𝑓(𝑧)=^∞_𝑛=1𝜆_𝑛𝑓_𝑛(𝑧)=𝜆₁𝑓₁(𝑧)+^∞_𝑛=2𝜆_𝑛𝑓_𝑛(𝑧)=()−𝑧−𝑤_∞𝑛=21−𝛾𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚×𝜆_𝑛(𝑧−𝑤)^𝑛. (53) Since 𝑛−𝛾1+(^∞_𝑛=2𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚⋅1−𝛾𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚𝜆_𝑛=1−𝛾^∞_𝑛=2𝜆_𝑛=1−𝛾1−𝜆₀≤1−𝛾, (54) it follows from Theorem 3 that the function 𝑓(𝑧)∈𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆). Conversely, let us suppose that 𝑓(𝑧)∈𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆). Since 𝑎_𝑛≤1−𝛾𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚𝑛≥2;𝑚∈ℕ₀. (55) Setting 𝜆_𝑛=𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚𝑎1−𝛾^𝑛,𝑛≥2;𝑚∈ℕ₀,𝜆₁=1−^∞_𝑛=2𝜆_𝑛, (56) it follows that 𝑓(𝑧)=𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆). This completes the proof of theorem.

7. Integral Operators

Theorem 12. If the function 𝑓(𝑧) defined by (10) is in the class 𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆), and let 𝑐 be a real number such that 𝑐>−1. Then the function 𝐹(𝑧) defined by 𝐹(𝑧)=𝑐+1(𝑧−𝑤)^𝑐𝑧_𝑤(𝑡−𝑤)^𝑐−1𝑓(𝑡)𝑑𝑡 (57) also belongs to the class 𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆).

Proof. From (57), it follows that 𝐹(𝑧)=(𝑧−𝑤)−∑^∞_𝑛=2𝑏_𝑛(𝑧−𝑤)^𝑛, where 𝑏_𝑛=((𝑐+1)/(𝑐+𝑛))𝑎_𝑛. Therefore ^∞_𝑛=2𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚𝑏_𝑛=^∞_𝑛=2𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚×𝑐+1𝑎𝑐+𝑛^𝑛≤^∞_𝑛=2𝑛−𝛾1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)^𝑚𝑎_𝑛≤1−𝛾, (58) since 𝑓(𝑧)∈𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆). Hence by Theorem 3, 𝐹(𝑧)∈𝑆^∗_𝑚,𝑤(𝛼,𝛽,𝛾,𝛿,𝜆).

Acknowledgment

The work here was fully supported by LRGS/TD/2011/UKM/ICT/03/02 and UKM-DLP-2011-050.