Quasi-Hadamard Product of Certain ω-Starlike and ω-Convex Functions with respect to Symmetric Points

Abstract

The main purpose of this paper is to derive some results associated with the quasi-Hadamard product of certain 𝜔-starlike and 𝜔-convex univalent analytic functions with respect to symmetric points.

1. Introduction and Definitions

Let 𝐴(𝜔) denote the class of functions of the form𝐹(𝑧)=(𝑧−𝜔)+^∞_𝑛=2𝑎_𝑛(𝑧−𝜔)^𝑛, (1) which are analytic and univalent in the open unit disk 𝑈={𝑧∶|𝑧|<1} and normalized with 𝐹(𝜔)=𝐹^′(𝜔)−1=0, where 𝜔 is a fixed point in 𝑈.

Throughout this paper, let the functions of the forms 𝑓(𝑧)=𝑎₁(𝑧−𝜔)+^∞_𝑛=2𝑎_𝑛(𝑧−𝜔)^𝑛𝑎₁>0,𝑎_𝑛,𝑓≥0_𝑖(𝑧)=𝑎_1,𝑖(𝑧−𝜔)+^∞_𝑛=2𝑎_𝑛,𝑖(𝑧−𝜔)^𝑛𝑎_1,𝑖>0,𝑎_𝑛,𝑖,𝑔(𝑧)≥0=𝑏₁(𝑧−𝜔)+^∞_𝑛=2𝑏_𝑛(𝑧−𝜔)^𝑛𝑏₁>0,𝑏_𝑛,𝑔≥0_𝑗(𝑧)=𝑏_1,𝑗(𝑧−𝜔)+^∞_𝑛=2𝑏_𝑛,𝑗(𝑧−𝜔)^𝑛𝑏_1,𝑗>0,𝑏_𝑛,𝑗≥0 (2)(3)(4)(5) be regular and univalent in the unit disk 𝑈.

Let 𝜔 be a fixed point in 𝑈, and 𝑆(𝜔)={𝐹∈𝐴(𝜔)∶𝐹isunivalentin𝑈}.

In [1], Kanas and Ronning defined the following classes of functions of 𝜔-starlike and 𝜔-convex, respectively, ST_𝜔=𝑆^∗_𝜔=(𝜔)()𝐹𝐹∈𝑆∶Re𝑧−𝜔^′(𝑧)𝐹(𝑧)>0,𝑧∈𝑈,CV_𝜔=𝑆^𝑐_𝜔=(𝜔)()𝐹𝐹∈𝑆∶1+Re𝑧−𝜔^′′(𝑧)𝐹^′(𝑧)>0,𝑧∈𝑈. (6) Recently, Acu and Owa [2], Oladipo [3], and Oladipo and Breaz [4] have studied the previous classes extensively.

Let 𝑆^∗_𝑠 be the subclass of 𝑆(𝜔) consisting of functions given by (1) with 𝜔=0 and satisfying the condition Re𝑧𝐹^′(𝑧)𝐹(𝑧)()().−𝐹−𝑧>0𝑧∈𝑈 (7) These functions are called starlike with respect to symmetric points and were introduced by Sakaguchi [5].

Motivated by the previous classes 𝑆^∗_𝑠, Oladipo [6] (see also [7]) defined the following classes of functions with respect to symmetric points.

Definition 1. (i) Let 𝑆^∗_𝑠(𝜔) be the subclass of 𝑆(𝜔) consisting of functions given by (1) satisfying the condition ()𝐹Re𝑧−𝜔^′(𝑧)𝐹(𝑧)()(),−𝐹−𝑧>0𝑧∈𝑈 (8) and 𝜔 is a fixed point in 𝑈. These functions are called 𝜔-starlike with respect to symmetric points.
(ii) Let 𝑆^𝑐_𝑠(𝜔) be the subclass of 𝑆(𝜔) consisting of functions given by (1) satisfying the condition )𝐹Re(𝑧−𝜔^′(𝑧)^′(𝐹(𝑧)−𝐹(−𝑧))^′>0(𝑧∈𝑈), (9) and 𝜔 is a fixed point in 𝑈. These functions are called 𝜔-convex with respect to symmetric points.
Suppose that 𝑓 and 𝑔 are two analytic functions in 𝑈. Then, we say that the function 𝑔 is subordinate to the function 𝑓, and we write 𝑔(𝑧)≺𝑓(𝑧)(𝑧∈𝑈), (10) if there exists a Schwarz function 𝜛(𝑧) with 𝜛(0)=0 and |𝜛(𝑧)|<1 such that 𝑔(𝑧)=𝑓(𝜛(𝑧))(𝑧∈𝑈). (11)
By applying the previous subordination definition, we define the following subclasses of 𝑆^∗_𝑠(𝜔) and 𝑆^𝑐_𝑠(𝜔).

Definition 2. (i) Let 𝑆^∗_𝑠(𝜔,𝐴,𝐵) be the subclass of 𝑆(𝜔) consisting of functions given by (1) satisfying the condition 2(𝑧−𝜔)𝐹^′(𝑧)𝐹(𝑧)()≺()−𝐹−𝑧1+𝐴𝑧−𝜔(),1+𝐵𝑧−𝜔 (12) where −1≤𝐵<𝐴≤1, 𝑧∈𝑈, and 𝜔 is a fixed point in 𝑈.
(ii) Let 𝑆^𝑐_𝑠(𝜔,𝐴,𝐵) be the subclass of 𝑆(𝜔) consisting of functions given by (1) satisfying the condition 2(𝑧−𝜔)𝐹^′(𝑧)^′(𝐹(𝑧)−𝐹(−𝑧))^′≺1+𝐴(𝑧−𝜔)(),1+𝐵𝑧−𝜔 (13) where −1≤𝐵<𝐴≤1, 𝑧∈𝑈, and 𝜔 is a fixed point in 𝑈.
For 𝜔=0, we have 𝑆^∗_𝑠(0,𝐴,𝐵)=𝑆^∗_𝑠(𝐴,𝐵),𝑆^𝑐_𝑠(0,𝐴,𝐵)=𝑆^𝑐_𝑠(𝐴,𝐵) (14) which were introduced by Goel and Mehrok [8] and Selvaraj and Vasanthi [9], respectively.
Using (12) and (13), we can easily obtain the characterization properties for the classes 𝑆^∗_𝑠(𝜔,𝐴,𝐵) and 𝑆^𝑐_𝑠(𝜔,𝐴,𝐵) as follows.

Lemma 3. A function 𝑓 defined by (2) belongs to the class 𝑆^∗_𝑠(𝜔,𝐴,𝐵), if it satisfies the condition 2𝑛−1−(^∞_𝑛=2−1)^𝑛+2𝐵𝑛+𝐴1−(−1)^𝑛𝑎_𝑛≤2(𝐴−𝐵)𝑎₁. (15)

Lemma 4. A function 𝑓 defined by (2) belongs to the class 𝑆^𝑐_𝑠(𝜔,𝐴,𝐵), if it satisfies the condition ^∞_𝑛=2𝑛2𝑛−1−(−1)^𝑛+2𝐵𝑛+𝐴1−(−1)^𝑛𝑎_𝑛≤2(𝐴−𝐵)𝑎₁. (16)

Now, we introduce the following class of analytic functions in 𝑈.

Definition 5. A function 𝑓 of the form (2), which is analytic in 𝑈, belongs to the class 𝑆^∗_𝑠,𝑘(𝜔,𝐴,𝐵), if it satisfies the condition ^∞_𝑛=2𝑛^𝑘2𝑛−1−(−1)^𝑛+2𝐵𝑛+𝐴1−(−1)^𝑛𝑎_𝑛≤2(𝐴−𝐵)𝑎₁, (17) where −1≤𝐵<𝐴≤1 and 𝑘 is any fixed nonnegative real number.
We note that, for any nonnegative real number 𝑘, the class 𝑆^∗_𝑠,𝑘(𝜔,𝐴,𝐵) is nonempty as the functions of the form 𝑓(𝑧)=𝑎₁()+𝑧−𝜔_∞𝑛=22(𝐴−𝐵)𝑎₁𝑛^𝑘2𝑛−1−(−1)^𝑛+2𝐵𝑛+𝐴1−(−1)^𝑛×𝜆_𝑛(𝑧−𝜔)^𝑛, (18) where 𝑎₁>0, 𝜆_𝑛≥0, and ∑^∞_𝑛=2𝜆_𝑛≤1, satisfy inequality (17).
Clearly, we have the following relationships:(i)𝑆^∗_𝑠,0(𝜔,𝐴,𝐵)≡𝑆^∗_𝑠(𝜔,𝐴,𝐵) and 𝑆^∗_𝑠,1(𝜔,𝐴,𝐵)≡𝑆^𝑐_𝑠(𝜔,𝐴,𝐵);(ii)𝑆^∗_𝑠,𝑘1(𝜔,𝐴,𝐵)⊂𝑆^∗_𝑠,𝑘2(𝜔,𝐴,𝐵)(𝑘₁>𝑘₂≥0);(iii)𝑆^∗_𝑠,𝑘(𝜔,𝐴,𝐵)⊂𝑆^∗_{𝑠,𝑘−1}(𝜔,𝐴,𝐵)⊂⋯⊂𝑆^𝑐_𝑠(𝜔,𝐴,𝐵)⊂𝑆^∗_𝑠(𝜔,𝐴,𝐵)(𝑘∈𝑁).
Let us define the quasi- Hadamard product of the functions 𝑓(𝑧) and 𝑔(𝑧) by 𝑓∗𝑔(𝑧)=𝑎₁𝑏₁(𝑧−𝜔)+^∞_𝑛=2𝑎_𝑛𝑏_𝑛(𝑧−𝜔)^𝑛. (19)

Similarly, we can define the quasi-Hadamard product of more than two functions; for example, 𝑓₁∗𝑓₂∗⋯∗𝑓_𝑚(𝑧)=^𝑚_𝑖=1𝑎_1,𝑖()+𝑧−𝜔_{∞𝑛=2𝑚𝑖=1}𝑎_𝑛,𝑖(𝑧−𝜔)^𝑛, (20) where the functions 𝑓_𝑖(𝑖=1,2,…,𝑚) are given by (3).

In this paper, we derive certain results associated with the quasi-Hadamard product of functions in the classes 𝑆^∗_𝑠,𝑘(𝜔,𝐴,𝐵), 𝑆^∗_𝑠(𝜔,𝐴,𝐵), and 𝑆^𝑐_𝑠(𝜔,𝐴,𝐵), which extend the results obtained by Kumar [10, 11], Darwish [12], and Aouf [13].

2. Main Results

Unless otherwise mentioned, we will assume throughout the following results that −1≤𝐵<𝐴≤1, 𝑧∈𝑈, 𝑘 is any fixed nonnegative real number, and 𝜔 is a fixed point in 𝑈.

Theorem 6. Let the functions 𝑓_𝑖(𝑧) defined by (3) be in the class 𝑆^𝑐_𝑠(𝜔,𝐴,𝐵) for every 𝑖=1,2,…,𝑚, and let the functions 𝑔_𝑗(𝑧) defined by (5) be in the class 𝑆^∗_𝑠,𝑘(𝜔,𝐴,𝐵) for every 𝑗=1,2,…,𝑞. Then, the quasi-Hadamard product 𝑓₁∗𝑓₂∗⋯∗𝑓_𝑚∗𝑔₁∗𝑔₂∗⋯∗𝑔_𝑞(𝑧) belongs to the class 𝑆^∗_{𝑠,2𝑚+(𝑘+1)𝑞−1}(𝜔,𝐴,𝐵).

Proof. Let 𝐺(𝑧)=𝑓₁∗𝑓₂∗⋯∗𝑓_𝑚∗𝑔₁∗𝑔₂∗⋯∗𝑔_𝑞(𝑧); then, 𝐺(𝑧)=^𝑚_𝑖=1𝑎^𝑞_1,𝑖𝑗=1𝑏_1,𝑗()+𝑧−𝜔_{∞𝑛=2𝑚𝑖=1}𝑎^𝑞_{𝑛,𝑖𝑗=1}𝑏_𝑛,𝑗(𝑧−𝜔)^𝑛. (21) It is sufficient to show that ^∞_𝑛=2𝑛^{2𝑚+(𝑘+1)𝑞−1}2𝑛−1−(−1)^𝑛+2𝐵𝑛+𝐴1−(−1)^𝑛×^𝑚_𝑖=1𝑎^𝑞_{𝑛,𝑖𝑗=1}𝑏_𝑛,𝑗≤2(𝐴−𝐵)^𝑚_𝑖=1𝑎^𝑞_1,𝑖𝑗=1𝑏_1,𝑗. (22) Since 𝑓_𝑖∈𝑆^𝑐_𝑠(𝜔,𝐴,𝐵), by Lemma 4, we have ^∞_𝑛=2𝑛2𝑛−1−(−1)^𝑛+2𝐵𝑛+𝐴1−(−1)^𝑛𝑎_𝑛,𝑖≤2(𝐴−𝐵)𝑎_1,𝑖, (23) for every 𝑖=1,2,…,𝑚. Thus, 𝑛2𝑛−1−(−1)^𝑛+2𝐵𝑛+𝐴1−(−1)^𝑛𝑎_𝑛,𝑖≤2(𝐴−𝐵)𝑎_1,𝑖, (24) or 𝑎_𝑛,𝑖≤2(𝐴−𝐵)𝑛2𝑛−1−(−1)^𝑛+2𝐵𝑛+𝐴1−(−1)^𝑛𝑎_1,𝑖, (25) for every 𝑖=1,2,…,𝑚. The right-hand expression of the last inequality is not greater than 𝑛⁻²𝑎_1,𝑖. Therefore, 𝑎_𝑛,𝑖≤𝑛⁻²𝑎_1,𝑖, (26) for every 𝑖=1,2,…,𝑚. Also, since 𝑔_𝑗∈𝑆^∗_𝑠,𝑘(𝜔,𝐴,𝐵), we find from (17) that ^∞_𝑛=2𝑛^𝑘2𝑛−1−(−1)^𝑛+2𝐵𝑛+𝐴1−(−1)^𝑛𝑏_𝑛,𝑗≤2(𝐴−𝐵)𝑏_1,𝑗, (27) for every 𝑗=1,2,…,𝑞. Hence, we obtain 𝑏_𝑛,𝑗≤𝑛^−(𝑘+1)𝑏_1,𝑗, (28) for every 𝑗=1,2,…,𝑞.
Using (26)–(28) for 𝑖=1,2,…,𝑚, 𝑗=𝑞, and 𝑗=1,2,…,𝑞−1, respectively, we have ^∞_𝑛=2𝑛^{2𝑚+(𝑘+1)𝑞−1}2𝑛−1−(−1)^𝑛+2𝐵𝑛+𝐴1−(−1)^𝑛×^𝑚_𝑖=1𝑎^𝑞_{𝑛,𝑖𝑗=1}𝑏_𝑛,𝑗≤^∞_𝑛=2𝑛^{2𝑚+(𝑘+1)𝑞−1}⋅𝑛^−2𝑚⋅𝑛^{−(𝑘+1)(𝑞−1)𝑚}_𝑖=1𝑎_1,𝑖^𝑞−1_𝑗=1𝑏_1,𝑗×2𝑛−1−(−1)^𝑛+2𝐵𝑛+𝐴1−(−1)^𝑛𝑏_𝑛,𝑞≤^𝑚_𝑖=1𝑎_1,𝑖^𝑞−1_𝑗=1𝑏_1,𝑗×^∞_𝑛=2𝑛^𝑘2𝑛−1−(−1)^𝑛+2𝐵𝑛+𝐴1−(−1)^𝑛𝑏_𝑛,𝑞≤2(𝐴−𝐵)^𝑚_𝑖=1𝑎^𝑞_1,𝑖𝑗=1𝑏_1,𝑗. (29) Thus, we have 𝐺(𝑧)∈𝑆^∗_{𝑠,2𝑚+(𝑘+1)𝑞−1}(𝜔,𝐴,𝐵). This completes the proof of Theorem 6.

Upon setting 𝑘=1 in Theorem 6, we obtain the following result.

Corollary 7. Let the functions 𝑓_𝑖(𝑧) defined by (3) and the functions 𝑔_𝑗(𝑧) defined by (5) belong to the class 𝑆^𝑐_𝑠(𝜔,𝐴,𝐵) for every 𝑖=1,2,…,𝑚 and 𝑗=1,2,…,𝑞. Then, the quasi-Hadamard product 𝑓₁∗𝑓₂∗⋯∗𝑓_𝑚∗𝑔₁∗𝑔₂∗⋯∗𝑔_𝑞(𝑧) belongs to the class 𝑆^∗_{𝑠,2𝑚+2𝑞−1}(𝜔,𝐴,𝐵).

Theorem 8. Let the functions 𝑓_𝑖(𝑧) defined by (3) be in the class 𝑆^∗_𝑠,𝑘(𝜔,𝐴,𝐵) for every 𝑖=1,2,…,𝑚, and let the functions 𝑔_𝑗(𝑧) defined by (5) be in the class 𝑆^∗_𝑠(𝜔,𝐴,𝐵) for every 𝑗=1,2,…,𝑞. Then, the quasi-Hadamard product 𝑓₁∗𝑓₂∗⋯∗𝑓_𝑚∗𝑔₁∗𝑔₂∗⋯∗𝑔_𝑞(𝑧) belongs to the class 𝑆^∗_{𝑠,(𝑘+1)𝑚+𝑞−1}(𝜔,𝐴,𝐵).

Proof. Suppose that 𝐺(𝑧) is defined as (21). To prove the theorem, we need to show that ^∞_𝑛=2𝑛^{(𝑘+1)𝑚+𝑞−1}2𝑛−1−(−1)^𝑛+2𝐵𝑛+𝐴1−(−1)^𝑛×^𝑚_𝑖=1𝑎^𝑞_{𝑛,𝑖𝑗=1}𝑏_𝑛,𝑗≤2(𝐴−𝐵)^𝑚_𝑖=1𝑎^𝑞_1,𝑖𝑗=1𝑏_1,𝑗. (30) Since 𝑓_𝑖∈𝑆^∗_𝑠,𝑘(𝜔,𝐴,𝐵), from (17), we have ^∞_𝑛=2𝑛^𝑘2𝑛−1−(−1)^𝑛+2𝐵𝑛+𝐴1−(−1)^𝑛𝑎_𝑛,𝑖≤2(𝐴−𝐵)𝑎_1,𝑖, (31) for every 𝑖=1,2,…,𝑚. Hence, we get 𝑎_𝑛,𝑖≤𝑛^−(𝑘+1)𝑎_1,𝑖, (32) for every 𝑖=1,2,…,𝑚. Further, since 𝑔_𝑗∈𝑆^∗_𝑠(𝜔,𝐴,𝐵), by Lemma 3, we have 2𝑛−1−(^∞_𝑛=2−1)^𝑛+2𝐵𝑛+𝐴1−(−1)^𝑛𝑏_𝑛,𝑗≤2(𝐴−𝐵)𝑏_1,𝑗, (33) for every 𝑗=1,2,…,𝑞. Whence, we obtain 𝑏_𝑛,𝑗≤𝑛⁻¹𝑏_1,𝑗, (34) for every 𝑗=1,2,…,𝑞.
Using (32)–(34) for 𝑖=𝑚, 𝑖=1,2,…,𝑚−1, and 𝑗=1,2,…,𝑞, respectively, we get ^∞_𝑛=2𝑛^{(𝑘+1)𝑚+𝑞−1}2𝑛−1−(−1)^𝑛+2𝐵𝑛+𝐴1−(−1)^𝑛×^𝑚_𝑖=1𝑎^𝑞_{𝑛,𝑖𝑗=1}𝑏_𝑛,𝑗≤^∞_𝑛=2𝑛^{(𝑘+1)𝑚+𝑞−1}⋅𝑛^{−(𝑘+1)(𝑚−1)}⋅𝑛^{−𝑞𝑚−1}_𝑖=1𝑎^𝑞_1,𝑖𝑗=1𝑏_1,𝑗×2𝑛−1−(−1)^𝑛+2𝐵𝑛+𝐴1−(−1)^𝑛𝑎_𝑛,𝑚≤^𝑚−1_𝑖=1𝑎^𝑞_1,𝑖𝑗=1𝑏_1,𝑗×^∞_𝑛=2𝑛^𝑘2𝑛−1−(−1)^𝑛+2𝐵𝑛+𝐴1−(−1)^𝑛𝑎_𝑛,𝑚≤2(𝐴−𝐵)^𝑚_𝑖=1𝑎^𝑞_1,𝑖𝑗=1𝑏_1,𝑗. (35) Therefore, we have 𝐺(𝑧)∈𝑆^∗_{𝑠,(𝑘+1)𝑚+𝑞−1}(𝜔,𝐴,𝐵). We complete the proof.

By taking 𝑘=0 in Theorem 8, we get the following result.

Corollary 9. Let the functions 𝑓_𝑖(𝑧) defined by (3) and the functions 𝑔_𝑗(𝑧) defined by (5) belong to the class 𝑆^∗_𝑠(𝜔,𝐴,𝐵) for every 𝑖=1,2,…,𝑚 and 𝑗=1,2,…,𝑞. Then, the quasi-Hadamard product 𝑓₁∗𝑓₂∗⋯∗𝑓_𝑚∗𝑔₁∗𝑔₂∗⋯∗𝑔_𝑞(𝑧) belongs to the class 𝑆^∗_{𝑠,𝑚+𝑞−1}(𝜔,𝐴,𝐵).

By putting 𝑘=0 in Theorem 6, or 𝑘=1 in Theorem 8, we obtain the following result.

Corollary 10. Let the functions 𝑓_𝑖(𝑧) defined by (3) be in the class 𝑆^𝑐_𝑠(𝜔,𝐴,𝐵) for every 𝑖=1,2,…,𝑚, and let the functions 𝑔_𝑗(𝑧) defined by (5) be in the class 𝑆^∗_𝑠(𝜔,𝐴,𝐵) for every 𝑗=1,2,…,𝑞. Then, the quasi-Hadamard product 𝑓₁∗𝑓₂∗⋯∗𝑓_𝑚∗𝑔₁∗𝑔₂∗⋯∗𝑔_𝑞(𝑧) belongs to the class 𝑆^∗_{𝑠,2𝑚+𝑞−1}(𝜔,𝐴,𝐵).

Next, we discuss some applications of Theorems 6 and 8.

Taking into account the quasi-Hadamard product of functions 𝑓₁(𝑧),𝑓₂(𝑧),…,𝑓_𝑚(𝑧) only, in the proof of Theorem 6, and using (23) and (26) for 𝑖=𝑚 and 𝑖=1,2,…,𝑚−1, respectively, we are led to the following.

Corollary 11. Let the functions 𝑓_𝑖(𝑧) defined by (3) belong to the class 𝑆^𝑐_𝑠(𝜔,𝐴,𝐵) for every 𝑖=1,2,…,𝑚. Then, the quasi- Hadamard product 𝑓₁∗𝑓₂∗⋯∗𝑓_𝑚(𝑧) belongs to the class 𝑆^∗_{𝑠,2𝑚−1}(𝜔,𝐴,𝐵).

Also, taking into account the quasi-Hadamard product of functions 𝑔₁(𝑧),𝑔₂(𝑧),…,𝑔_𝑞(𝑧) only, in the proof of Theorem 8, and using (33) and (34) for 𝑗=𝑞 and 𝑗=1,2,…,𝑞−1, respectively, we are led to the following.

Corollary 12. Let the functions 𝑔_𝑗(𝑧) defined by (5) belong to the class 𝑆^∗_𝑠(𝜔,𝐴,𝐵) for every 𝑗=1,2,…,𝑞. Then, the quasi-Hadamard product 𝑔₁∗𝑔₂∗⋯∗𝑔_𝑞(𝑧) belongs to the class 𝑆^∗_{𝑠,𝑞−1}(𝜔,𝐴,𝐵).

Remark 13. By taking 𝜔=0 in the previous results and making use of relationship (14), we obtain the corresponding results.

Acknowledgments

The present investigation is partly supported by the Natural Science Foundation of China under Grant 11271045, the Higher School Doctoral Foundation of China under Grant 20100003110004 and the Natural Science Foundation of Inner Mongolia of China under Grant 2010MS0117. The authors are thankful to the referees for their careful reading and making some helpful comments which have essentially improved the presentation of this paper.