On Some Integral Operators for Certain Classes of 𝑝-Valent Functions

Abstract

We study some generalized integral operators for the classes of p-valent functions with bounded radius and boundary rotation. Our work generalizes many previously known results. Many of our results are best possible.

1. Introduction

Let 𝐴_𝑝 denote the class of functions of the form 𝑓(𝑧)=𝑧^𝑝+^∞_𝑛=𝑝+1𝑎_𝑛𝑧^𝑛,𝑝∈𝑁={1,2,…}, (1.1) which are analytic in the open unit disc 𝑈={𝑧∶|𝑧|<1}.

Let 𝑓 and 𝑔 be analytic functions in 𝑈 we say that 𝑓 is subordinate to 𝑔, written as 𝑓≺𝑔; (1.2) if there exists a Schwarz function 𝑤(𝑧) in 𝑈, with 𝑤(0)=0 and |𝑤(𝑧)|<1(𝑧∈𝑈), such that 𝑓(𝑧)=𝑔(𝑤(𝑧)). (1.3) In particular, when 𝑔 is univalent, then the above subordination is equivalent to 𝑓(0)=0,𝑓(𝑈)⊆𝑔(𝑈). (1.4) For functions 𝑓,𝑔∈𝐴_𝑝, given by 𝑓(𝑧)=𝑧^𝑝+^∞_𝑛=𝑝+1𝑎_𝑛𝑧^𝑛,𝑔(𝑧)=𝑧^𝑝+^∞_𝑛=𝑝+1𝑏_𝑛𝑧^𝑛,𝑧∈𝑈, (1.5) we define the Hadamard product (or convolution) of 𝑓 and 𝑔 by 𝑓∗𝑔(𝑧)=𝑧^𝑝+^∞_𝑛=𝑝+1𝑎_𝑛𝑏_𝑛𝑧^𝑛,𝑧∈𝑈. (1.6) Janowski [1] defined the class 𝑃[𝐴,𝐵] as follows.

Let ℎ be a function, analytic in 𝑈, with ℎ(0)=1. Then ℎ is said to belong to the class 𝑃[𝐴,𝐵], −1≤𝐵<𝐴≤1, if and only if, for 𝑧∈𝑈, ℎ(𝑧)=1+𝐴𝑤(𝑧)(𝑧),1+𝐵𝑤where𝑤(𝑧)isaSchwarzfunction. (1.7) Or equivalently, we can say that ℎ∈𝑃[𝐴,𝐵], −1≤𝐵<𝐴≤1, if and only if, ℎ(𝑧)≺1+𝐴𝑧1+𝐵𝑧,𝑧∈𝑈. (1.8) Geometrically, ℎ(𝑧) is in the class 𝑃[𝐴,𝐵], if and only if, ℎ(0)=1 and the image of ℎ(𝑈) lies inside the open disc centered on the real axis with diameter end points, 𝐷₁=ℎ(−1)=1−𝐴1−𝐵,𝐷₂=ℎ(1)=1+𝐴1+𝐵,0<𝐷₁<1<𝐷₂. (1.9) Clearly 𝑃[𝐴,𝐵]⊂𝑃((1−𝐴)/(1−𝐵)).

In the recent paper, Noor [2] introduced the class 𝑃_𝑘(𝛼). We define it as follows. Let 𝑃_𝑘(𝛼),0≤𝛼<𝑝, be the class of functions 𝑝(𝑧) with 𝑝(0)=1 and satisfying the property 𝑝(𝑧)=12₀^2𝜋1+(1−2𝛼)𝑧𝑒^−𝑖𝑡1−𝑧𝑒^−𝑖𝑡𝑑𝜇(𝑡), (1.10) where 𝜇(𝑡) is a real-valued function of bounded variation on [0,2𝜋] and ∫₀^2𝜋𝑑𝜇(𝑡)=2 and ∫₀^2𝜋|𝑑𝜇(𝑡)|≤𝑘.

The classes 𝑉_𝑘(𝛼) and 𝑅_𝑘(𝛼) are related to the class 𝑃_𝑘(𝛼) and can be defined as 𝑓∈𝑉_𝑘(𝛼),iff𝑧𝑓^′(𝑧)^′𝑝𝑓^′(𝑧)∈𝑃_𝑘(𝛼),𝑧∈𝑈,𝑓∈𝑅_𝑘(𝛼),iff𝑧𝑓^′(𝑧)𝑝𝑓(𝑧)∈𝑃_𝑘(𝛼),𝑧∈𝑈. (1.11) We define a class 𝑃_𝑘[𝐴,𝐵] as follows.

Let 𝑃_𝑘[𝐴,𝐵], 𝑘≥2, −1≤𝐵<𝐴≤1, denote the class of 𝑝-valent analytic functions ℎ(𝑧) that are represented by ℎ(𝑧)=𝑘4+12ℎ₁(𝑧)−𝑘4−12ℎ₂(𝑧),𝑧∈𝑈, (1.12) where ℎ₁,ℎ₂∈𝑃[𝐴,𝐵]. For 𝐴=1−2𝛼(0≤𝛼<𝑝)and 𝐵=−1, it reduces to the class 𝑃_𝑘(𝛼) and 𝑃₂(𝛼)=𝑃(𝛼) is the class of 𝑝-valent analytic functions ℎ(𝑧)with Reℎ(𝑧)>𝛼, 𝑧∈𝑈. Taking 𝐴=1, 𝐵=−1, and 𝑝=1, we have 𝑃_𝑘[1,−1]=𝑃_𝑘(see [3]), and 𝑃₂[1,−1]=𝑃 is the class of functions with positive real part.

Definition 1.1. A function 𝑓, analytic in 𝑈, and given by (1.1) is said to be in the class 𝑅_𝑘[𝐴,𝐵]; −1≤𝐵<𝐴≤1, 𝑘≥2, if and only if, 𝑧𝑓^′(𝑧)𝑝𝑓(𝑧)∈𝑃_𝑘[𝐴,𝐵],𝑧∈𝑈. (1.13) For 𝑝=1, 𝑅_𝑘[𝐴,𝐵] is introduced and studied by Noor [4]. We note that 𝑅_𝑘[𝐴,𝐵]⊂𝑅_𝑘1−𝐴1−𝐵⊂𝑅_𝑘, (1.14) where 𝑅_𝑘 is the class of functions with bounded radius rotation (see [5]). For 𝑘=2, we have 𝑅₂[𝐴,𝐵]≡𝑆^∗_𝑝[𝐴,𝐵]⊂𝑆^∗_𝑝1−𝐴1−𝐵⊂𝑆^∗_𝑝, (1.15) where 𝑆^∗_𝑝 is the class of 𝑝-valent starlike functions. Similarly, we can define the class 𝑉_𝑘[𝐴,𝐵] as follows.

Definition 1.2. A function 𝑓, analytic in 𝑈, and given by (1.1) is said to be in the class 𝑉_𝑘[𝐴,𝐵]; −1≤𝐵<𝐴≤1, 𝑘≥2, if and only if, 𝑧𝑓^′(𝑧)^′𝑝𝑓^′(𝑧)∈𝑃_𝑘[𝐴,𝐵],𝑧∈𝑈. (1.16) It is clear that 𝑓∈𝑉_𝑘[𝐴,𝐵],iff𝑧𝑓^′(𝑧)𝑝∈𝑅_𝑘[𝐴,𝐵],𝑧∈𝑈. (1.17) For 𝑝=1, 𝑉_𝑘[𝐴,𝐵] is the class introduced and studied by Noor [4]. It is easy to see that, 𝑉_𝑘[𝐴,𝐵]⊂𝑉_𝑘1−𝐴1−𝐵⊂𝑉_𝑘, (1.18) where 𝑉_𝑘 is the class of functions with bounded boundary rotation see [5]. Also 𝑉₂[𝐴,𝐵]≡𝐶_𝑝[𝐴,𝐵]⊂𝐶_𝑝1−𝐴1−𝐵⊂𝐶_𝑝, (1.19) where 𝐶_𝑝 is the class of 𝑝-valent convex functions.

Very recently, Frasin [6], introduced the following general integral operators for 𝑝-valent functions, 𝐹_𝑝(𝑧)=^𝑧₀𝑝𝑡^𝑝−1𝑓₁(𝑡)𝑡^𝑝𝛼1⋯𝑓_𝑛(𝑡)𝑡^𝑝𝛼𝑛𝑑𝑡,𝐺_𝑝(𝑧)=^𝑧₀𝑝𝑡^𝑝−1𝑓^′₁(𝑡)𝑝𝑡^{𝑝−1𝛼1}⋯𝑓^′_𝑛(𝑡)𝑝𝑡^{𝑝−1𝛼𝑛}𝑑𝑡,where𝛼_𝑖∈ℂ,𝑧∈𝑈. (1.20)(1.21) Clearly, we may see that for 𝑝=1, these operators become the general integral operators 𝐹₁(𝑧)=𝐹_𝑛(𝑧),𝐺₁(𝑧)=𝐹_{𝛼1,𝛼2,…,𝛼𝑛}(𝑧), (1.22) introduced and studied by Breaz and Breaz [7] and Breaz et al. [8], (see also [9, 10]).

For 𝑝=𝑛=1, 𝛼₁=𝛼∈[0,1] in (1.20), we obtain the integral operator ∫^𝑧₀(𝑓(𝑡)/𝑡)^𝛼𝑑𝑡 studied in [11] and for 𝑝=𝑛=1, 𝛼₁=𝛿∈ℂ, |𝛿|<1/4in (1.21), we obtain the integral operator ∫^𝑧₀(𝑓^′(𝑡))^𝛿𝑑𝑡, studied in [12].

2. Main Results

Lemma 2.1. Let𝛽>0, 𝛽+𝛾>0, 𝛼∈[𝛼₀,1),with 𝛼₀=max{(𝛽−𝛾−1)/2𝛽,−𝛾/𝛽}. If 𝑝(𝑧)+𝑧𝑝^′(𝑧)(𝑧)≺()𝑧𝛽𝑝+𝛾1+1−2𝛼,1−𝑧 (2.1) then 𝑝(𝑧)(𝑧)≺()𝑧≺𝑄1+1−2𝛼,1−𝑧 (2.2) where 𝑄(𝑧)=1/𝛽𝐺(𝑧)−𝛾/𝛽, 𝐺(𝑧)=¹₀1−𝑧1−𝑡𝑧^{2𝛽(1−𝛼)}𝑡^{𝛽+𝛾−1}𝑑𝑡=₂𝐹₁()𝑧2𝛽1−𝛼,1,𝛽+𝛾+1;,1−𝑧𝜌=𝜌𝛼,𝛽,𝛾=𝛽+𝛾𝛽₂𝐹₁()−𝛾2𝛽1−𝛼,1,𝛽+𝛾+1;1/2𝛽, (2.3)(2.4)₂𝐹₁ denotes the Gauss hypergeometric function. From (2.2), we can deduce the sharp result 𝑝∈𝑃(𝜌), where 𝜌 is defined in (2.4). This result is a special case of one given in [11].

Proof. To prove this Lemma we use Theorem 3.2j of [11, page 97]. Take ℎ(𝑧)=(1+(1−2𝛼)𝑧)/(1−𝑧), 0≤𝛼<1 and 𝐻(𝑧)=𝛽ℎ(𝑧)+𝛾=𝑎+𝑏𝑧,1−𝑧 (2.5) where 𝑎=𝛽+𝛾 and 𝑏=𝛽(1−2𝛼)+𝛾.
Since 𝐻 is convex to apply Theorem 3.2j of [11, page 97] we only need to determine condition Re𝐻(𝑧)>0.
The range of |𝑧|≤1 under 𝐻(𝑧) is a half plane. In order to satisfy the required condition this half plane needs to lie in the right half plane. This requirement will be satisfied if Re𝐻(−1)=Re𝐻(𝑖) and Re𝐻(0)>Re𝐻(−1)≥0.Or we can write it as𝛽(1−𝛼)>0,𝛽𝛼+𝛾≥0. (2.6) When 𝛽>0, 𝛽+𝛾>0, these conditions imply that 𝛼∈[−𝛾/𝛽,1), and if 𝛽+𝛾>1, then 𝛼∈[(𝛽−𝛾−1)/2𝛽,1). Hence all the conditions of Theorem 3.2j of [11, page 97] are satisfied for 𝛼∈[𝛼₀,1), with 𝛼₀=max{(𝛽−𝛾−1)/2𝛽,−𝛾/𝛽}, thus we have the required result.

To show that the solution 𝑄(𝑧) can be represented in terms of hypergeometric functions we take 𝐴=1−2𝛼, 𝐵=−1, 𝑛=1 in Theorem 3.3d of [11, page 109].

Lemma 2.2. Let 𝑓∈𝑉_𝑘(𝛼), 0≤𝛼<𝑝, 𝑘≥2. Then 𝑓∈𝑅_𝑘(𝜌) in 𝑈, where =1𝜌=𝜌𝛼,𝑝₂𝐹₁().2𝑝1−𝛼,1,𝑝+1;1/2 (2.7) This result is sharp.

Proof. Let for 𝑘≥2, 𝑧∈𝑈, we have 𝑧𝑓^′(𝑧)(𝑧)(𝑧)=𝑘𝑝𝑓=ℎ4+12ℎ₁(𝑧)−𝑘4−12ℎ₂(𝑧), (2.8) where ℎ, ℎ_𝑖 are analytic in 𝑈 with ℎ(0)=1, ℎ_𝑖(0)=1, 𝑖=1,2.
We define𝜙_𝑝(𝑧)=𝑧+^∞_𝑛=11𝑝((𝑧1+𝑛−1))^𝑛,𝑧∈𝑈. (2.9) By using (2.8), with convolution technique, see [13], we have 𝜙_𝑝(𝑧)𝑧(𝑧)=𝑘∗ℎ4+12𝜙_𝑝(𝑧)𝑧∗ℎ₁(𝑧)−𝑘4−12𝜙_𝑝(𝑧)𝑧∗ℎ₂(𝑧). (2.10) This implies that, ℎ(𝑧)+𝑧ℎ^′(𝑧)(𝑧)=𝑘𝑝ℎ4+12ℎ₁(𝑧)+𝑧ℎ^′₁(𝑧)𝑝ℎ₁(𝑧)−𝑘4−12ℎ₂(𝑧)+𝑧ℎ^′₂(𝑧)𝑝ℎ₂(𝑧). (2.11) Logarithmic differentiation of (2.8) yields, 𝑧𝑓^′(𝑧)^′𝑝𝑓^′(𝑧)=ℎ(𝑧)+𝑧ℎ^′(𝑧)(𝑧)=𝑘𝑝ℎ4+12ℎ₁(𝑧)+𝑧ℎ^′₁(𝑧)𝑝ℎ₁(𝑧)−𝑘4−12ℎ₂(𝑧)+𝑧ℎ^′₂(𝑧)𝑝ℎ₂(𝑧). (2.12) Since (𝑧𝑓^′(𝑧))^′/𝑝𝑓^′(𝑧)∈𝑃_𝑘(𝛼), 0≤𝛼<𝑝, thus ℎ_𝑖(𝑧)+𝑧ℎ^′_𝑖(𝑧)𝑝ℎ_𝑖(𝑧)∈𝑃(𝛼),𝑖=1,2. (2.13) By using Lemma 2.1 (for 𝛽=𝑝and 𝛾=0), we deduce that ℎ_𝑖∈𝑃(𝜌), where 𝜌 is given in (2.7). This estimate is best possible because of the best dominant property of function 𝑄(𝑧), where 𝑄(𝑧)=₂1𝐹₁2𝑝(1−𝛼),1,𝑝+1;𝑧/(1−𝑧),𝑧∈𝑈. (2.14)

For 𝑝=1, we have the sharp result proved in [14].

We begin with the following theorem.

Theorem 2.3. (i) Let 𝛼_𝑖>0, 𝑓_𝑖∈𝑅_𝑘[𝐴,𝐵] for all 𝑖=1,2,…,𝑛, and, ∑^𝑛_𝑖=1𝛼_𝑖=1. Then the integral operator 𝐹_𝑝∈𝑉_𝑘[𝐴,𝐵] in 𝑈, where −1≤𝐵<𝐴≤1, 𝑘≥2.
(ii) Let 𝑓_𝑖∈𝑅_𝑘(𝛼), 𝛼_𝑖>0for all 𝑖=1,2,…,𝑛 with 𝛼=(1−𝐴)/(1−𝐵), 𝑘≥2. If ∑^𝑛_𝑖=1𝛼_𝑖=1, then the integral operator 𝐹_𝑝 defined by (1.20) also belongs to the class 𝑅_𝑘(𝜌) in 𝑈, where 𝜌=𝜌(𝛼,𝑝)is defined by (2.7). This result is sharp.

Proof (i). From (1.20), we can see that 𝐹_𝑝∈𝐴_𝑝 in 𝑈, and 𝐹^′_𝑝(𝑧)=𝑝𝑧^𝑝−1𝑓₁(𝑧)𝑧^𝑝𝛼1⋯𝑓_𝑛(𝑧)𝑧^𝑝𝛼𝑛. (2.15) Differentiating logarithmically and multiplying by 𝑧, we obtain, 𝑧𝐹_𝑝^′′(𝑧)𝐹^′_𝑝(𝑧)=𝑝−1+^𝑛_𝑖=1𝛼_𝑖𝑧𝑓^′_𝑖(𝑧)𝑓^′_𝑖(𝑧)−𝑝,𝑧∈𝑈. (2.16) Thus, we have 1+𝑧𝐹_𝑝^′′(𝑧)𝐹^′_𝑝(𝑧)=^𝑛_𝑖=1𝛼_𝑖𝑧𝑓^′_𝑖(𝑧)𝑓^′_𝑖(𝑧), (2.17) or 𝑧𝐹^′_𝑝(𝑧)^′𝑝𝐹^′_𝑝(𝑧)=^𝑛_𝑖=1𝛼_𝑖𝑧𝑓^′_𝑖(𝑧)𝑝𝑓^′_𝑖(𝑧)=𝑘4+12^𝑛_𝑖=1𝛼_𝑖𝑝_𝑖(𝑧)−𝑘4−12^𝑛_𝑖=1𝛼_𝑖ℎ_𝑖(𝑧), (2.18) where ℎ_𝑖,𝑝_𝑖∈𝑃[𝐴,𝐵], for all 𝑖=1,2,…,𝑛.
Since 𝑃[𝐴,𝐵] is a convex set, see [15], it follows that,𝑧𝐹^′_𝑝(𝑧)^′𝑝𝐹^′_𝑝(𝑧)=𝑘4+12𝐻₁(𝑧)−𝑘4−12𝐻₂(𝑧), (2.19) where 𝐻₁,𝐻₂∈𝑃[𝐴,𝐵] and therefore, 𝑧𝐹^′_𝑝(𝑧)^′𝑝𝐹^′_𝑝(𝑧)∈𝑃_𝑘[𝐴,𝐵],𝑧∈𝑈. (2.20) This proves the result.

Substituting 𝑝=1, in Theorem 2.3(i), we have the following corollary.

Corollary 2.4. Let 𝛼_𝑖>0, 𝑓_𝑖∈𝑅_𝑘[𝐴,𝐵] for all 𝑖=1,2,…,𝑛, −1≤𝐵<𝐴≤1, 𝑘≥2. Then the integral operator 𝐹_𝑝∈𝑉_𝑘[𝐴,𝐵] in 𝑈.

Remark 2.5. Letting 𝛼₁=𝛼, 𝛼₂=𝛽, and 𝑛=2 in Corollary 2.4, we obtain a result due to Noor [4].

For 𝑛=1, 𝛼₁=𝛼=1, and 𝑓₁=𝑓 in Theorem 2.3(i), we have the following.

Corollary 2.6. Let𝑓∈𝑅_𝑘[𝐴,𝐵] in 𝑈, −1≤𝐵<𝐴≤1, 𝑘≥2. Then the integral operator ∫^𝑧₀𝑝(𝑓(𝑡)/𝑡)𝑑𝑡∈𝑉_𝑘[𝐴,𝐵], 𝑧∈𝑈.

Proof (ii). Taking 𝐴=1−2𝛼, 𝐵=−1, with 𝛼=(1−𝐴)/(1−𝐵), we have for all 𝑖=1,2,…,𝑛, 𝑓_𝑖∈𝑅_𝑘[1−2𝛼,−1]=𝑅_𝑘(𝛼), (2.21) using part (i) of Theorem 2.3, we have 𝐹_𝑝∈𝑉_𝑘[1−2𝛼,−1]=𝑉_𝑘(𝛼)in𝑈. (2.22) Now using Lemma 2.2 for 𝐹_𝑝∈𝑉_𝑘(𝛼), 𝛼=(1−𝐴)/(1−𝐵) implies that 𝐹_𝑝∈𝑅_𝑘𝜌,where𝜌=𝜌𝛼,𝑝isdefinedin(2.7). (2.23) The sharpness of the result is clear from the function 𝑄(𝑧) defined by (2.14).

For 𝑝=1, we have the following corollary.

Corollary 2.7. Let 𝛼_𝑖>0, 𝑓_𝑖∈𝑅_𝑘(𝛼) for all 𝑖=1,2,…,𝑛, with 𝛼=(1−𝐴)/(1−𝐵) and 𝐴=1−2𝛼, 𝐵=−1. Then the integral operator 𝐹_𝑝 defined by (1.20) also belongs to the class 𝑅_𝑘(𝜌) in 𝑈, where 𝜌=𝜌(𝛼)=2𝛼−12−2^2(1−𝛼),if1𝛼≠21,2ln2if1𝛼=2. (2.24)

Remark 2.8. Letting 𝛼₁=𝜇, 𝛼₂=𝜂, and 𝑛=2 in Corollary 2.7, we have the sharp result proved in [14].

For 𝐴=1, 𝐵=−1, and 𝑝=1, we have 𝑓_𝑖∈𝑅_𝑘(0)impliesthat𝐹_𝑝∈𝑉_𝑘12in𝑈. (2.25)

Theorem 2.9. (i) Let 𝛼_𝑖>0,𝑓_𝑖∈𝑉_𝑘[𝐴,𝐵] for all 𝑖=1,2,…,𝑛. If ∑^𝑛_𝑖=1𝛼_𝑖=1, then the integral operator 𝐺_𝑝defined by (1.21), also belongs to the class 𝑉_𝑘[𝐴,𝐵] in 𝑈, where −1≤𝐵<𝐴≤1, 𝑘≥2.
(ii) Let for𝛼_𝑖>0, ∑^𝑛_𝑖=1𝛼_𝑖=1 and 𝑓_𝑖∈𝑉_𝑘(𝛼), for all 𝑖=1,2,…,𝑛 with 0≤𝛼<𝑝, 𝛼=(1−𝐴)/(1−𝐵), 𝑘≥2. Then the integral operator 𝐺_𝑝∈𝑅_𝑘(𝜌) in 𝑈, where 𝜌=𝜌(𝛼,𝑝)is defined by (2.7). This result is sharp.

Proof (i). From definition (1.20), we have 1+𝑧𝐺_𝑝^′′(𝑧)𝐺^′_𝑝(𝑧)=𝑝+^𝑛_𝑖=1𝛼_𝑖𝑧𝑓_𝑖^′′(𝑧)𝑓^′_𝑖(𝑧)=−𝑝+1_𝑛𝑖=1𝛼_𝑖𝑧𝑓^′_𝑖(𝑧)^′𝑓^′_𝑖(𝑧), (2.26) or 𝑧𝐺^′_𝑝(𝑧)^′𝑝𝐺^′_𝑝(𝑧)=^𝑛_𝑖=1𝛼_𝑖𝑧𝑓^′_𝑖(𝑧)^′𝑝𝑓^′_𝑖(𝑧)=𝑘4+12^𝑛_𝑖=1𝛼_𝑖𝑝_𝑖(𝑧)−𝑘4−12^𝑛_𝑖=1𝛼_𝑖ℎ_𝑖(𝑧), (2.27) where ℎ_𝑖,𝑝_𝑖∈𝑃[𝐴,𝐵], for all 𝑖=1,2,…,𝑛.
Since 𝑃[𝐴,𝐵] is a convex set, see [15], it follows that,𝑧𝐺^′_𝑝(𝑧)^′𝑝𝐺^′_𝑝(𝑧)=𝑘4+12𝐻₁(𝑧)−𝑘4−12𝐻₂(𝑧),𝑧∈𝑈, (2.28) where 𝐻₁,𝐻₂∈𝑃[𝐴,𝐵] and therefore, 𝑧𝐺^′_𝑝(𝑧)^′𝑝𝐺^′_𝑝(𝑧)∈𝑃_𝑘[𝐴,𝐵]in𝑈. (2.29) This implies that 𝐺_𝑝∈𝑉_𝑘[𝐴,𝐵].

Letting 𝑝=1 in Theorem 2.9(i), we have the following corollary.

Corollary 2.10. Let 𝛼_𝑖>0, 𝑓_𝑖∈𝑉_𝑘[𝐴,𝐵] for all 𝑖=1,2,…,𝑛 and −1≤𝐵<𝐴≤1, 𝑘≥2. If ∑^𝑛_𝑖=1𝛼_𝑖=1, then 𝐺_𝑝∈𝑉_𝑘[𝐴,𝐵] in 𝑈.

Proof (ii). Taking 𝐴=1−2𝛼, 𝐵=−1, we have for all 𝑖=1,2,…,𝑛𝑓_𝑖∈𝑉_𝑘[1−2𝛼,−1]=𝑉_𝑘(𝛼),where𝛼=1−𝐴.1−𝐵 (2.30) Now using part (i) of Theorem 2.9, we have 𝐺_𝑝∈𝑉_𝑘[1−2𝛼,−1]=𝑉_𝑘(𝛼)in𝑈. (2.31) Now using Lemma 2.2, for 𝛼=(1−𝐴)/(1−𝐵), we have 𝐺_𝑝∈𝑉_𝑘(𝛼)impliesthat𝐺_𝑝∈𝑅_𝑘𝜌,in𝑈,where𝜌=𝜌𝛼,𝑝isdefinedin(2.7). (2.32) The sharpness of the result is clear from the function 𝑄(𝑧) defined by (2.14).

For 𝑝=1, we have the following corollary.

Corollary 2.11. (i) Let 𝛼_𝑖>0, 𝑓_𝑖∈𝑉_𝑘(𝛼), 𝑖=1,2,…,𝑛, with 𝛼=(1−𝐴)/(1−𝐵) and𝐴=1−2𝛼, 𝐵=−1. Then 𝐺_𝑝∈𝑅_𝑘(𝜌) in 𝑈, where 𝜌=𝜌(𝛼) and defined in (2.24).
Also for 𝐴=1, 𝐵=−1, we have.
(ii) If𝑓_𝑖∈𝑉_𝑘(0) for all 𝑖=1,2,…,𝑛, then 𝐺_𝑝∈𝑅_𝑘(1/2) in 𝑈.

Acknowledgments

Authors are thankful to anonymous referees for their very constructive comments to improve this paper. Second author (P. Goswami) is also grateful to Professor S. P. Goyal for his guidance and constant encouragement.