On an Integral Transform of a Class of Analytic Functions

Abstract

For 𝛼,𝛾≥0 and 𝛽<1, let 𝒲_𝛽(𝛼,𝛾) denote the class of all normalized analytic functions 𝑓 in the open unit disc 𝐸={𝑧∶|𝑧|<1} such that ℜ𝑒^𝑖𝜙((1−𝛼+2𝛾)(𝑓(𝑧)/𝑧)+(𝛼−2𝛾)𝑓^′(𝑧)+𝛾𝑧𝑓^′′(𝑧)−𝛽)>0, 𝑧∈𝐸 for some 𝜙∈ℝ. It is known (Noshiro (1934) and Warschawski (1935)) that functions in 𝒲_𝛽(1,0) are close-to-convex and hence univalent for 0≤𝛽<1. For 𝑓∈𝒲_𝛽(𝛼,𝛾), we consider the integral transform 𝐹(𝑧)=𝑉_𝜆(𝑓)(𝑧)∶=∫¹₀𝜆(𝑡)(𝑓(𝑡𝑧)/𝑡)𝑑𝑡, where 𝜆 is a nonnegative real-valued integrable function satisfying the condition ∫¹₀𝜆(𝑡)𝑑𝑡=1. The aim of present paper is, for given 𝛿<1, to find sharp values of 𝛽 such that (i) 𝑉_𝜆(𝑓)∈𝒲_𝛿(1,0) whenever 𝑓∈𝒲_𝛽(𝛼,𝛾) and (ii) 𝑉_𝜆(𝑓)∈𝒲_𝛿(𝛼,𝛾) whenever 𝑓∈𝒲_𝛽(𝛼,𝛾).

1. Introduction

Let 𝒜 denote the class of analytic functions 𝑓 defined in the open unit disc 𝐸={𝑧∶|𝑧|<1} with the normalizations 𝑓(0)=𝑓^′(0)−1=0, and let 𝑆 be the subclass of 𝒜 consisting of functions univalent in 𝐸. For any two functions 𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛 and 𝑔(𝑧)=𝑧+∑^∞_𝑛=2𝑏_𝑛𝑧^𝑛 in 𝒜, the Hadamard product (or convolution) of 𝑓 and 𝑔 is the function 𝑓∗𝑔 defined by 𝑓∗𝑔(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑏_𝑛𝑧^𝑛.(1.1) For 𝑓∈𝒜, Fournier and Ruscheweyh [1] introduced the integral operator 𝐹(𝑧)=𝑉_𝜆𝑓(𝑧)∶=¹₀𝜆(𝑡)𝑓(𝑡𝑧)𝑡𝑑𝑡,(1.2) where 𝜆 is a nonnegative real-valued integrable function satisfying the condition ∫¹₀𝜆(𝑡)𝑑𝑡=1. This operator contains some well-known operators such as Libera, Bernardi, and Komatu as its special cases. Fournier and Ruscheweyh [1] applied the famous duality theory to show that for a function 𝑓 in the class 𝒫𝛽∶=𝑓∈𝒜∶∃𝜙∈ℝ∣ℜ𝑒^𝑖𝜙𝑓^′(𝑧)−𝛽>0,𝑧∈𝐸,(1.3) the linear integral operator 𝑉_𝜆(𝑓) is univalent in 𝐸. Since then, this operator has been studied by a number of authors for various choices of 𝜆(𝑡). In another remarkable paper, Barnard et al. in [2] obtained conditions such that 𝑉_𝜆(𝑓)∈𝒫₁(𝛽) whenever 𝑓 is in the class 𝒫_𝛾𝛽∶=𝑓∈𝒜∶∃𝜙∈ℝ∣ℜ𝑒^𝑖𝜙𝑓(𝑧)1−𝛾𝑧+𝛾𝑓^′(𝑧)−𝛽>0,𝑧∈𝐸,(1.4) with 𝛽<1,𝛾≥0. Note that for 0≤𝛽<1, functions in 𝒫₁(𝛽)≡𝒫(𝛽) satisfy the condition ℜ𝑓^′(𝑧)>𝛽 in 𝐸 and thus are close-to-convex in 𝐸. A domain 𝐷 in ℂ is close-to-convex if its compliment in ℂ can be written as union of nonintersecting half lines.

In 2008, Ponnusamy and Rønning [3] discussed the univalence of 𝑉_𝜆(𝑓) for the functions in the class ℛ_𝛾𝛽∶=𝑓∈𝒜∶∃𝜙∈ℝ∣ℜ𝑒^𝑖𝜙𝑓^′(𝑧)+𝛾𝑧𝑓^′′(𝑧)−𝛽>0,𝑧∈𝐸.(1.5) In a very recent paper, Ali et al. [4] studied the class 𝒲_𝛽𝛼,𝛾∶=𝑓∈𝒜∶∃𝜙∈ℝ∣ℜ𝑒^𝑖𝜙𝑓(𝑧)1−𝛼+2𝛾𝑧+𝛼−2𝛾𝑓^′(𝑧)+𝛾𝑧𝑓^′′(𝑧)−𝛽>0,𝑧∈𝐸,(1.6) where 𝛼,𝛾≥0 and 𝛽<1. In this paper, they obtained sufficient conditions so that the integral transform 𝑉_𝜆(𝑓) maps normalized analytic functions 𝑓∈𝒲_𝛽(𝛼,𝛾) into the class of starlike functions. It is evident that 𝒲_𝛽(1,0)≡𝒫(𝛽), 𝒲_𝛽(𝛼,0)≡𝒫_𝛼(𝛽) and 𝒲_𝛽(1+2𝛾,𝛾)≡ℛ_𝛾(𝛽).

In the present paper, we shall mainly tackle the following problems. (1)For given 𝛿<1, find sharp values of 𝛽=𝛽(𝛿,𝛼) such that 𝑉_𝜆(𝑓)∈𝒲_𝛿(1,0) whenever 𝑓∈𝒲_𝛽(𝛼,𝛾). (2)For given 𝛿<1, find sharp values of 𝛽=𝛽(𝛿) such that 𝑉_𝜆(𝑓)∈𝒲_𝛿(𝛼,𝛾) whenever 𝑓∈𝒲_𝛽(𝛼,𝛾). To prove one of our results, we shall need the generalized hypergeometric function _𝑝𝐹_𝑞, so we define it here.

Let 𝛼_𝑗(𝑗=1,2,…,𝑝) and 𝛽_𝑗(𝑗=1,2,…,𝑞) be complex numbers with 𝛽_𝑗≠0,−1,−2,…(𝑗=1,2,…,𝑞). Then the generalized hypergeometric function _𝑝𝐹_𝑞 is defined by _𝑝𝐹_𝑞(𝑧)=_𝑝𝐹_𝑞𝛼₁,…,𝛼_𝑝;𝛽₁,…,𝛽_𝑞;𝑧=^∞_𝑛=0𝛼_1𝑛⋯𝛼_𝑝𝑛𝛽_1𝑛⋯𝛽_𝑞𝑛𝑧^𝑛,𝑛!𝑝≤𝑞+1(1.7) where (𝑎)_𝑛 is the Pochhammer symbol, defined in terms of the Gamma function, by (𝑎)_𝑛Γ()∶=𝑎+𝑛Γ(𝑎)=𝑎()⋯()1,𝑛=0,𝑎+1𝑎+𝑛−1,𝑛∈ℕ.(1.8) In particular, ₂𝐹₁ is called the Gaussian hypergeometric function. We note that the _𝑝𝐹_𝑞 series in (1.7) converges absolutely for |𝑧|<∞ if 𝑝<𝑞+1 and for 𝑧∈𝐸 if 𝑝=𝑞+1.

We shall also need the following lemma.

Lemma 1.1 (see [5]). Let 𝛽₁<1, 𝛽₂<1, and 𝜂∈ℝ. Then, for 𝑝,𝑞 analytic in 𝐸 with 𝑝(0)=𝑞(0)=1, the conditions ℜ𝑝(𝑧)>𝛽₁ and ℜ𝑒^𝑖𝜂(𝑞(𝑧)−𝛽₂)>0 imply ℜ𝑒^𝑖𝜂((𝑝∗𝑞)(𝑧)−𝛿)>0, where 1−𝛿=2(1−𝛽₁)(1−𝛽₂).

2. Main Results

We use the notations introduced in [4]. Let 𝜇≥0 and 𝜈≥0 satisfy 𝜇+𝜈=𝛼−𝛾,𝜇𝜈=𝛾.(2.1) When 𝛾=0, then 𝜇 is chosen to be 0, in which case, 𝜈=𝛼≥0. When 𝛼=1+2𝛾, (2.1) yields 𝜇+𝜈=1+𝛾=1+𝜇𝜈 or (𝜇−1)(1−𝜈)=0.(i)For 𝛾>0, then choosing 𝜇=1 gives 𝜈=𝛾.(ii)For 𝛾=0, then 𝜇=0 and 𝜈=𝛼=1.

Theorem 2.1. Let 𝜇≥0, 𝜈≥0 satisfy (2.1). Further, let 𝛿<1 be given, and define 𝛽=𝛽(𝛿,𝜇,𝜈) by 1−1−𝛿211−𝜈¹₀𝜆(𝑡)¹₀𝑑𝑠1+𝑡𝑠^𝜇1𝑑𝑡+𝜈−1¹₀𝜆(𝑡)¹₀𝑑𝜂𝑑𝜁1+𝑡𝜂^𝜈𝜁^𝜇𝑑𝑡⁻¹,𝛾≠0,1−1−𝛿211−𝛼¹₀𝜆(𝑡)11+𝑡𝑑𝑡+𝛼−1¹₀𝜆(𝑡)¹₀𝑑𝜂1+𝑡𝜂^𝛼𝑑𝑡⁻¹,𝛾=0𝜇=0,𝜈=𝛼>0.(2.2) If 𝑓∈𝒲_𝛽(𝛼,𝛾), then 𝐹=𝑉_𝜆(𝑓)∈𝒲_𝛿(1,0)⊂𝑆. The value of 𝛽 is sharp.

Proof. The case 𝛾=0(𝜇=0,𝜈=𝛼>0) corresponds to Theorem  1.5 in [2]. So we assume that 𝛾>0.
Define 𝑓(𝑧)1−𝛼+2𝛾𝑧+𝛼−2𝛾𝑓^′(𝑧)+𝛾𝑧𝑓^′′(𝑧)=𝐻(𝑧).(2.3) Writing 𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛, it follows that 𝐻(𝑧)=1+^∞_𝑛=1𝑎_𝑛+1(𝑛𝜈+1)𝑛𝜇+1𝑧^𝑛.(2.4) It is a simple exercise to see that𝑓^′(𝑧)=𝐻(𝑧)∗₃𝐹₂12,𝜈,1𝜇;1,1𝜈+1.𝜇+1;𝑧(2.5) Let 𝐹(𝑧)=𝑉_𝜆(𝑓)(𝑧), where 𝑉_𝜆(𝑓) is defined by (1.2). Then for 𝛾≠0, we can write 𝐹^′(𝑧)=𝑓^′(𝑧)∗¹₀𝜆(𝑡)(𝑧)∗1−𝑡𝑧𝑑𝑡=𝐻₃𝐹₂12,𝜈,1𝜇;1,1𝜈+1∗𝜇+1;𝑧¹₀𝜆(𝑡)(𝑧)∗1−𝑡𝑧𝑑𝑡=𝐻¹₀𝜆(𝑡)₃𝐹₂12,𝜈,1𝜇;1,1𝜈+1𝜇+1;𝑡𝑧𝑑𝑡.(2.6)
Since 𝑓∈𝒲_𝛽(𝛼,𝛾), it follows that ℜ{𝑒^𝑖𝜙(𝐻(𝑧)−𝛽)}>0 for some 𝜙∈ℝ. Now, for each 𝛾>0, we first claim that ℜ¹₀𝜆(𝑡)₃𝐹₂12,𝜈,1𝜇;1,1𝜈+1𝜇+1;𝑡𝑧𝑑𝑡>1−1−𝛿21−𝛽,𝑧∈𝐸,(2.7) which, by Lemma 1.1, implies that 𝐹∈𝒲_𝛿(1,0). Therefore, it suffices to verify the inequality (2.7). Using the identity (which can be checked by comparing the coefficients of 𝑧^𝑛 on both sides) ₃𝐹₂(2,𝑏,𝑐;𝑑,𝑒;𝑧)=(𝑑−1)₃𝐹₂(1,𝑏,𝑐;𝑑−1,𝑒;𝑧)−(𝑑−2)₃𝐹₂(1,𝑏,𝑐;𝑑,𝑒;𝑧),(2.8) it follows that ₃𝐹₂12,𝜈,1𝜇;1,1𝜈+1=1𝜇+1;𝑧𝜈¹₀𝑑𝑠1−𝑧𝑠^𝜇+11−𝜈¹₀𝑑𝜂𝑑𝜁1−𝑧𝜂^𝜈𝜁^𝜇.(2.9) Thus, ¹₀𝜆(𝑡)₃𝐹₂12,𝜈,1𝜇;1,1𝜈+1=𝜇+1;𝑡𝑧𝑑𝑡₁₀𝜆(𝑡1)𝜈¹₀𝑑𝑠1−𝑡𝑧𝑠^𝜇+11−𝜈¹₀𝑑𝜂𝑑𝜁1−𝑡𝑧𝜂^𝜈𝜁^𝜇𝑑𝑡.(2.10) Therefore, for 𝛾>0, we have ℜ¹₀𝜆(𝑡)₃𝐹₂12,𝜈,1𝜇;1,1𝜈+1>1𝜇+1;𝑡𝑧𝑑𝑡𝜈¹₀𝜆(𝑡)¹₀𝑑𝑠1+𝑡𝑠^𝜇1𝑑𝑡+𝜈−1¹₀𝜆(𝑡)¹₀𝑑𝜂𝑑𝜁1+𝑡𝜂^𝜈𝜁^𝜇𝑑𝑡=1−1−𝛿2,1−𝛽(2.11) in the view of .
To prove the sharpness, let 𝑓∈𝒲_𝛽(𝛼,𝛾) be the function determined by 𝑓(𝑧)1−𝛼+2𝛾𝑧+𝛼−2𝛾𝑓^′(𝑧)+𝛾𝑧𝑓^′′(𝑧)=𝛽+1−𝛽1+𝑧.1−𝑧(2.12) Using a series expansion, we see that we can write 𝑓(𝑧)=𝑧+^∞_𝑛=221−𝛽()𝑧𝑛𝜈+1−𝜈𝑛𝜇+1−𝜇^𝑛.(2.13) Then, 𝐹(𝑧)=𝑉_𝜆𝑓(𝑧)=𝑧+21−𝛽^∞_𝑛=2𝜓_𝑛()𝑧𝑛𝜈+1−𝜈𝑛𝜇+1−𝜇^𝑛,(2.14) where 𝜓_𝑛=∫¹₀𝜆(𝑡)𝑡^𝑛−1𝑑𝑡. Equation can be restated as 1=21−𝛽11−𝛿1−𝜈¹₀𝜆(𝑡)¹₀𝑑𝑠1+𝑡𝑠^𝜇1𝑑𝑡+𝜈−1¹₀𝜆(𝑡)¹₀𝑑𝜂𝑑𝜁1+𝑡𝜂^𝜈𝜁^𝜇=2𝑑𝑡1−𝛿1+¹₀𝜆(𝑡)−1𝜈¹₀𝑑𝑠1+𝑡𝑠^𝜇+1𝜈−1¹₀𝑑𝜂𝑑𝜁1+𝑡𝜂^𝜈𝜁^𝜇=2𝑑𝑡1−𝛿¹₀𝜆(𝑡)^∞_𝑛=2(−1)^𝑛−1𝑡^𝑛−1−1𝑛𝜇+1−𝜇𝜈+1𝜈1−1()2𝑛𝜈+1−𝜈𝑑𝑡=−1−𝛿^∞_𝑛=2(−1)^𝑛−1𝑛𝜓_𝑛(.𝑛𝜈+1−𝜈)𝑛𝜇+1−𝜇(2.15) Finally, 𝐹^′(𝑧)=1+21−𝛽^∞_𝑛=2𝑛𝜓_𝑛()𝑧𝑛𝜈+1−𝜈𝑛𝜇+1−𝜇^𝑛−1,(2.16) which for 𝑧=−1 takes the value 𝐹^′(−1)=1+21−𝛽^∞_𝑛=2(−1)^𝑛−1𝑛𝜓_𝑛(−()𝑛𝜈+1−𝜈)𝑛𝜇+1−𝜇=1+21−𝛽1−𝛿21−𝛽=𝛿.(2.17) This shows that the result is sharp.

Letting 𝛾=0 and 𝛼=1 in Theorem  1.1, we obtain the following result of Ruscheweyh [6].

Corollary 2.2. Let 𝛿<1, and define 𝛽=𝛽(𝛿,1)<1 by 𝛽(𝛿)=1−1−𝛿21−¹₀𝜆(𝑡)1+𝑡𝑑𝑡⁻¹.(2.18) If 𝑓∈𝒲_𝛽(1,0)≡𝒫₁(𝛽), then 𝐹=𝑉_𝜆(𝑓)∈𝒲_𝛿(1,0)⊂𝑆. The value of 𝛽 is sharp.

Theorem 2.3. Let 𝛿<1 and 𝛼,𝛾≥0, and define 𝛽=𝛽(𝛿)<1 by 𝛽1−𝛽=−¹₀𝜆(𝑡)(1−((1+𝛿)/(1−𝛿))𝑡)(1+𝑡)𝑑𝑡.(2.19) If 𝑓∈𝒲_𝛽(𝛼,𝛾), then 𝑉_𝜆(𝑓)∈𝒲_𝛿(𝛼,𝛾). The value of 𝛽 is sharp.

Proof. The idea of the proof is similar to the one used to prove Theorem  2 in [1].
Let 𝐹(𝑧)=𝑉_𝜆(𝑓)(𝑧)=∫¹₀𝜆(𝑡)(𝑓(𝑡𝑧)/𝑡)𝑑𝑡. Clearly, 𝐹^′(𝑧)=¹₀𝜆(𝑡)1−𝑡𝑧𝑑𝑡∗𝑓^′(𝑧).(2.20) Since, 𝑓∈𝒲_𝛽(𝛼,𝛾), so with 𝑔(𝑧)=1−𝛼+2𝛾𝑓(𝑧)/𝑧+𝛼−2𝛾𝑓^′(𝑧)+𝛾𝑧𝑓^′′(𝑧)−𝛽,1−𝛽(2.21) we have ℜ[𝑒^𝑖𝜙𝑔(𝑧)]>0, where 𝜙∈ℝ.
For 𝛾≠𝛼/2, 𝑓^′(𝑧)=1𝑔(𝑧)−𝛼−2𝛾𝛽+1−𝛽1−𝛼+2𝛾𝑓(𝑧)𝛼−2𝛾𝑧−𝛾𝛼−2𝛾𝑧𝑓^′′(𝑧).(2.22) Putting this value in (2.20), 𝐹^′(𝑧)=¹₀𝜆(𝑡)11−𝑡𝑧𝑑𝑡∗𝑔(𝑧)−𝛼−2𝛾𝛽+1−𝛽1−𝛼+2𝛾𝑓(𝑧)𝛼−2𝛾𝑧−𝛾𝛼−2𝛾𝑧𝑓^′′(𝑧).(2.23) Equivalently, 𝐹^′(𝑧)=1𝑔(𝑧)∗𝛼−2𝛾𝛽+1−𝛽¹₀𝜆(𝑡)−1−𝑡𝑧𝑑𝑡1−𝛼+2𝛾𝐹(𝑧)𝛼−2𝛾𝑧−𝛾𝛼−2𝛾𝑧𝐹^′′(𝑧).(2.24) Thus 1−𝛼+2𝛾(𝐹(𝑧)/𝑧)+𝛼−2𝛾𝐹^′(𝑧)+𝛾𝑧𝐹^′′(𝑧)=𝑔(𝑧)∗𝛽+1−𝛽¹₀𝜆(𝑡).1−𝑡𝑧𝑑𝑡(2.25) In the case when 𝛾=𝛼/2, 𝑔(𝑧)=𝑓(𝑧)/𝑧+𝛾𝑧𝑓^′′(𝑧)−𝛽.1−𝛽(2.26) Since 𝑓(𝑧)𝑧=𝛽+1−𝛽𝑔(𝑧)−𝛾𝑧𝑓^′′(𝑧),(2.27)This leads to,𝐹(𝑧)𝑧+𝛾𝑧𝐹^′′(𝑧)=𝑔(𝑧)∗𝛽+1−𝛽¹₀𝜆(𝑡),1−𝑡𝑧𝑑𝑡(2.28) which is clearly (2.25) with 𝛾=𝛼/2.
Further 𝐹∈𝒲_𝛿(𝛼,𝛾) if and only if 𝐺(𝑧)∶=(𝐹(𝑧)−𝛿𝑧)/(1−𝛿)∈𝒲₀(𝛼,𝛾). Now using (2.25), we obtain 𝐺(𝑧)1−𝛼+2𝛾𝑧+𝛼−𝛾𝐺^′(𝑧)+𝛾𝑧𝐺^′′(𝑧)=𝑔(𝑧)∗𝛽−𝛿+1−𝛿1−𝛽1−𝛿¹₀𝜆(𝑡).1−𝑡𝑧𝑑𝑡(2.29) Since ℜ𝑒^𝑖𝜙𝑔(𝑧)>0 for some 𝜙∈ℝ, it follows by duality principle [8, page 23] that 𝐺(𝑧)1−𝛼+2𝛾𝑧+𝛼−2𝛾𝐺^′(𝑧)+𝛾𝑧𝐺^′′(𝑧)≠0(2.30) if, and only if, ℜ𝛽−𝛿+1−𝛿1−𝛽1−𝛿¹₀𝜆(𝑡)>11−𝑡𝑧𝑑𝑡2.(2.31) Using ℜ(1/(1−𝑡𝑧))>1/(1+𝑡), we get ℜ𝛽−𝛿+1−𝛿1−𝛽1−𝛿¹₀𝜆(𝑡)>1−𝑡𝑧𝑑𝑡1−𝛽1−𝛿𝛽−𝛿1−𝛽+¹₀𝜆(𝑡).1+𝑡𝑑𝑡(2.32) By using (2.19), we have 𝛽−(1+𝛿)/21−𝛽=−¹₀𝜆(𝑡)(1+𝑡)𝑑𝑡.(2.33) Thus, 𝛽−𝛿1−𝛽+¹₀𝜆(𝑡)11+𝑡𝑑𝑡=21−𝛿,1−𝛽(2.34) which implies that ℜ𝛽−𝛿+1−𝛿1−𝛽1−𝛿¹₀𝜆(𝑡)>1−𝑡𝑧𝑑𝑡1−𝛽1−𝛿𝛽−𝛿1−𝛽+¹₀𝜆(𝑡)=11+𝑡𝑑𝑡2.(2.35) Thus, we deduce, using duality principle, that (1−𝛼+2𝛾)(𝐺(𝑧)/𝑧)+(𝛼−𝛾)𝐺^′(𝑧)+𝛾𝑧𝐺^′′(𝑧) is contained in a half plane not containing the origin. So, 𝐺∈𝒲₀(𝛼,𝛾) and hence 𝐹∈𝒲_𝛿(𝛼,𝛾).
To prove the sharpness, let 𝑓(𝑧)=𝑧+2(1−𝛽)∑^∞_𝑛=2(𝑧^𝑛/(𝑛𝜇+1−𝜇)(𝑛𝜈+1−𝜈)). 𝐹(𝑧)=𝑉_𝜆𝑓(𝑧)=𝑧+21−𝛽^∞_𝑛=2𝑧^𝑛𝜔_𝑛𝑛𝜇+1−𝜇(𝑛𝜈+1−𝜈),where𝜔_𝑛=¹₀𝜆(𝑡)𝑡^𝑛−1𝑑𝑡.(2.36) Further, 𝛽1−𝛽=−¹₀𝜆(𝑡)(1−((1+𝛿)/(1−𝛿))𝑡)(1+𝑡)𝑑𝑡(2.37) gives 𝛽1−𝛽=−1+¹₀𝜆(𝑡)(1+(1+𝛿)/(1−𝛿))(1+𝑡)𝑡𝑑𝑡,(2.38) or 1=21−𝛽1−𝛿¹₀𝑡𝜆(𝑡)21+𝑡𝑑𝑡=1−𝛿^∞_𝑛=2(−1)^𝑛𝜔_𝑛.(2.39) Further, assume that 𝐻(𝑧)=𝐹(𝑧)1−𝛼+2𝛾𝑧+𝛼−𝛾𝐹^′(𝑧)+𝛾𝑧𝐹^′′(𝑧).(2.40) Since 𝐹(𝑧)=𝑧+2(1−𝛽)∑^∞_𝑛=2(𝜔_𝑛𝑧^𝑛/(𝑛𝜇+1−𝜇)(𝑛𝜈+1−𝜈)),
so, 𝐻(𝑧)=1+21−𝛽^∞_𝑛=2𝜔_𝑛𝑧^𝑛−1.(2.41) Therefore, for 𝑧=−1, 𝐻(−1)=1−21−𝛽^∞_𝑛=2𝜔_𝑛(−1)^𝑛=1−21−𝛽1−𝛿21−𝛽=𝛿.(2.42) This shows that the result is sharp.

Letting 𝛾=0 in Theorem 2.3 above, we obtain the following result of Kim and Rønning [9].

Corollary 2.4. Let 𝛿<1 and 𝛼≥0, and define 𝛽=𝛽(𝛿) by 𝛽1−𝛽=−¹₀𝜆(𝑡)(1−((1+𝛿)/(1−𝛿))𝑡)(1+𝑡)𝑑𝑡.(2.43) If 𝑓∈𝒲_𝛽(𝛼,0)≡𝒫_𝛼(𝛽), then 𝑉_𝜆(𝑓)∈𝒲_𝛿(𝛼,0)≡𝒫_𝛼(𝛿). The value of 𝛽 is sharp.
Upon setting 𝜆(𝑡)=(1+𝑐)𝑡^𝑐 with −1<𝑐, we have the following corollary.

Corollary 2.5. Let 𝛿<1, 𝛼,𝛾≥0, and −1<𝑐≤0 be given, and let 𝐺(𝑧) be defined by 𝐺(𝑧)=(1+𝑐)𝑧^𝑐𝑧₀𝑢^𝑐−1𝑓(𝑢)𝑑𝑢.(2.44) Suppose that 𝑓∈𝒲_𝛽(𝛼,𝛾), then 𝐺∈𝒲_𝛿(𝛼,0), where 2()𝛽=1+𝑐₂𝐹₁(1,2+𝑐;3+𝑐,−1)−(2+𝑐)2(1+𝑐)₂𝐹₁().1,2+𝑐;3+𝑐,−1(2.45) The constant 𝛽 is sharp.

The special case of Corollary 2.5 (with 𝛾=0) has been obtained by Aghalary et al. [11].