Starlike and Convex Properties for Hypergeometric Functions

Abstract

The purpose of the present paper is to give some characterizations for a (Gaussian) hypergeometric function to be in various subclasses of starlike and convex functions. We also consider an integral operator related to the hypergeometric function.

1. Introduction

Let 𝒯 be the class consisting of functions of the form 𝑓(𝑧)=𝑧−^∞_𝑛=2𝑎_𝑛𝑧^𝑛,𝑎_𝑛≥0, (1.1) that are analytic and univalent in the open unit disk 𝕌={𝑧∶|𝑧|<1}. Let 𝒯^∗(𝛼) and 𝒞(𝛼) denote the subclasses of 𝒯 consisting of starlike and convex functions of order 𝛼(0≤𝛼<1), respectively [1].

Recently, Bharati et al. [2] introduced the following subclasses of starlike and convex functions.

Definition 1.1. A function 𝑓 of the form (1.1) is in 𝒮_𝑝𝒯(𝛼,𝛽) if it satisfies the condition Re{𝑧𝑓^′(𝑧)𝑓(𝑧)}≥𝛼|𝑧𝑓^′(𝑧)𝑓(𝑧)−1|+𝛽,𝛼≥0,0≤𝛽<1, (1.2) and 𝑓∈𝒰𝒞𝒯(𝛼,𝛽) if and only if 𝑧𝑓^′∈𝒮_𝑝𝒯(𝛼,𝛽).

Definition 1.2. A function 𝑓 of the form (1.1) is in 𝒫𝒯(𝛼) if it satisfies the condition Re{𝑧𝑓^′(𝑧)𝑓(𝑧)}+𝛼≥|𝑧𝑓^′(𝑧)𝑓(𝑧)−𝛼|,0<𝛼<∞, (1.3) and 𝑓∈𝒞𝒫𝒯(𝛼) if and only if 𝑧𝑓^′∈𝒫𝒯(𝛼).

Bharati et al. [2] showed that 𝒮_𝑝𝒯(𝛼,𝛽)=𝒯^∗((𝛼+𝛽)/(1+𝛼)),𝒰𝒞𝒯(𝛼,𝛽)=𝒞((𝛼+𝛽)/(1+𝛼)),𝒫𝒯(𝛼)=𝒯^∗(1−𝛼)(0<𝛼≤1), and 𝒞𝒫𝒯(𝛼)=𝒞(1−𝛼)(0<𝛼≤1). In particular, we note that 𝒰𝒞𝒯(1,0) is the class of uniformly convex functions given by Goodman [3] (also see [4–6]).

Let 𝐹(𝑎,𝑏;𝑐;𝑧) be the (Gaussian) hypergeometric function defined by 𝐹(𝑎,𝑏;𝑐;𝑧)=^∞_𝑛=0(𝑎)_𝑛(𝑏)_𝑛(𝑐)_𝑛(1)_𝑛𝑧^𝑛, (1.4) where 𝑐≠0,−1,−2,…, and (𝜆)_𝑛 is the Pochhammer symbol defined by (𝜆)_𝑛=1if𝑛=0,𝜆(𝜆+1)⋯(𝜆+𝑛−1)if𝑛∈ℕ={1,2,…}. (1.5) We note that 𝐹(𝑎,𝑏;𝑐;1) converges for Re(𝑐−𝑎−𝑏)>0 and is related to the Gamma function by 𝐹(𝑎,𝑏;𝑐;1)=Γ(𝑐)Γ(𝑐−𝑎−𝑏)Γ(𝑐−𝑎)Γ(𝑐−𝑏). (1.6) Silverman [7] gave necessary and sufficient conditions for 𝑧𝐹(𝑎,𝑏;𝑐;𝑧) to be in 𝒯^∗(𝛼) and 𝒞(𝛼), and also examined a linear operator acting on hypergeometric functions. For the other interesting developments for 𝑧𝐹(𝑎,𝑏;𝑐;𝑧) in connection with various subclasses of univalent functions, the readers can refer to the works of Carlson and Shaffer [8], Merkes and Scott [9], and Ruscheweyh and Singh [10].

In the present paper, we determine necessary and sufficient conditions for 𝑧𝐹(𝑎,𝑏;𝑐;𝑧) to be in 𝒮_𝑝𝒯(𝛼,𝛽),𝒰𝒞𝒯(𝛼,𝛽),𝒫𝒯(𝛼), and 𝒞𝒫𝒯(𝛼). Furthermore, we consider an integral operator related to the hypergeometric function.

2. Results

To establish our main results, we need the following lemmas due to Bharati et al. [2].

Lemma 2.1. (i) A function 𝑓 of the form (1.1) is in 𝒮_𝑝𝒯(𝛼,𝛽) if and only if it satisfies ^∞_𝑛=2(𝑛(1+𝛼)−(𝛼+𝛽))𝑎_𝑛≤1−𝛽. (2.1)
(ii) A function 𝑓 of the form (1.1) is in 𝒰𝒞𝒯(𝛼,𝛽) if and only if it satisfies ^∞_𝑛=2𝑛(𝑛(1+𝛼)−(𝛼+𝛽))𝑎_𝑛≤1−𝛽. (2.2)

Lemma 2.2. (i) A function 𝑓 of the form (1.1) is in 𝒫𝒯(𝛼) if and only if it satisfies ^∞_𝑛=2(𝑛−1+𝛼)𝑎_𝑛≤𝛼. (2.3)
(ii) A function 𝑓 of the form (1.1) is in 𝒞𝒫𝒯(𝛼) if and only if it satisfies ^∞_𝑛=2𝑛(𝑛−1+𝛼)𝑎_𝑛≤𝛼. (2.4)

Theorem 2.3. (i) If 𝑎,𝑏>−1,𝑐>0, and 𝑎𝑏<0, then 𝑧𝐹(𝑎,𝑏;𝑐;𝑧) is in 𝒮_𝑝𝒯(𝛼,𝛽) if and only if 𝑐≥𝑎+𝑏+1−(1+𝛼)𝑎𝑏(1−𝛽). (2.5)
(ii) If 𝑎,𝑏>0 and 𝑐>𝑎+𝑏+1, then 𝐹₁(𝑎,𝑏;𝑐;𝑧)=𝑧(2−𝐹(𝑎,𝑏;𝑐;𝑧)) is in 𝒮_𝑝𝒯(𝛼,𝛽) if and only if Γ(𝑐)Γ(𝑐−𝑎−𝑏)Γ(𝑐−𝑎)Γ(𝑐−𝑏)(1+(1+𝛼)𝑎𝑏(1−𝛽)(𝑐−𝑎−𝑏−1))≤2. (2.6)

Proof. (i) Since 𝑧𝐹(𝑎,𝑏;𝑐;𝑧)=𝑧+𝑎𝑏𝑐^∞_𝑛=2(𝑎+1)_𝑛−2(𝑏+1)_𝑛−2(𝑐+1)_𝑛−2(1)_𝑛−1𝑧^𝑛=𝑧−|𝑎𝑏𝑐|^∞_𝑛=2(𝑎+1)_𝑛−2(𝑏+1)_𝑛−2(𝑐+1)_𝑛−2(1)_𝑛−1𝑧^𝑛, (2.7) according to (i) of Lemma 2.1, we must show that ^∞_𝑛=2(𝑛(1+𝛼)−(𝛼+𝛽))(𝑎+1)_𝑛−2(𝑏+1)_𝑛−2(𝑐+1)_𝑛−2(1)_𝑛−1𝑐≤|𝑎𝑏|(1−𝛽). (2.8) Noting that (𝜆)_𝑛=𝜆(𝜆+1)_𝑛−1 and then applying (1.6), we have ^∞_𝑛=0((𝑛+2)(1+𝛼)−(𝛼+𝛽))(𝑎+1)_𝑛(𝑏+1)_𝑛(𝑐+1)_𝑛(1)_𝑛+1=(1+𝛼)^∞_𝑛=0(𝑎+1)_𝑛(𝑏+1)_𝑛(𝑐+1)_𝑛(1)_𝑛𝑐+(1−𝛽)𝑎𝑏^∞_𝑛=1(𝑎)_𝑛(𝑏)_𝑛(𝑐)_𝑛(1)_𝑛=(1+𝛼)Γ(𝑐+1)Γ(𝑐−𝑎−𝑏−1)𝑐Γ(𝑐−𝑎)Γ(𝑐−𝑏)+(1−𝛽)(𝑎𝑏Γ(𝑐)Γ(𝑐−𝑎−𝑏)Γ(𝑐−𝑎)Γ(𝑐−𝑏)−1). (2.9) Hence, (2.8) is equivalent to Γ(𝑐+1)Γ(𝑐−𝑎−𝑏−1)Γ(𝑐−𝑎)Γ(𝑐−𝑏)(1+𝛼+(1−𝛽)𝑐−𝑎−𝑏−1𝑐𝑎𝑏)≤(1−𝛽)(+𝑐|𝑎𝑏|𝑎𝑏)=0. (2.10) Thus, (2.10) is valid if and only if 1+𝛼+(1−𝛽)(𝑐−𝑎−𝑏−1)/(𝑎𝑏)≤0 or, equivalently, 𝑐≥𝑎+𝑏+1−(1+𝛼)𝑎𝑏/(1−𝛽).
(ii) Since 𝐹₁(𝑎,𝑏;𝑐;𝑧)=𝑧−^∞_𝑛=2(𝑎)_𝑛−1(𝑏)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1𝑧^𝑛, (2.11) by (i) of Lemma 2.1, we need only to show that ^∞_𝑛=2(𝑛(1+𝛼)−(𝛼+𝛽))(𝑎)_𝑛−1(𝑏)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1≤1−𝛽. (2.12) Now, ^∞_𝑛=2(𝑛(1+𝛼)−(𝛼+𝛽))(𝑎)_𝑛−1(𝑏)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1=(1+𝛼)^∞_𝑛=1𝑛(𝑎)_𝑛(𝑏)_𝑛(𝑐)_𝑛(1)_𝑛(1−𝛽)^∞_𝑛=1(𝑎)_𝑛(𝑏)_𝑛(𝑐)_𝑛(1)_𝑛=(1+𝛼)𝑎𝑏𝑐^∞_𝑛=1(𝑎+1)_𝑛−1(𝑏+1)_𝑛−1(𝑐+1)_𝑛−1(1)_𝑛−1+(1−𝛽)^∞_𝑛=1(𝑎)_𝑛(𝑏)_𝑛(𝑐)_𝑛(1)_𝑛=Γ(𝑐)Γ(𝑐−𝑎−𝑏)(Γ(𝑐−𝑎)Γ(𝑐−𝑏)(1+𝛼)𝑎𝑏𝑐−𝑎−𝑏−1+1−𝛽)−(1−𝛽). (2.13) But this last expression is bounded above by 1−𝛽 if and only if (2.6) holds.

Theorem 2.4. (i) If 𝑎,𝑏>−1,𝑎𝑏<0, and 𝑐>𝑎+𝑏+2, then 𝑧𝐹(𝑎,𝑏;𝑐;𝑧) is in 𝒰𝒞𝒯(𝛼,𝛽) if and only if (1+𝛼)(𝑎)₂(𝑏)₂+(3+2𝛼−𝛽)𝑎𝑏(𝑐−𝑎−𝑏−2)+(1−𝛽)(𝑐−𝑎−𝑏−2)₂≥0. (2.14)
(ii) If 𝑎,𝑏>0 and 𝑐>𝑎+𝑏+2, then 𝐹₁(𝑎,𝑏;𝑐;𝑧)=𝑧(2−𝐹(𝑎,𝑏;𝑐;𝑧)) is in 𝒰𝒞𝒯(𝛼,𝛽) if and only if Γ(𝑐)Γ(𝑐−𝑎−𝑏)(Γ(𝑐−𝑎)Γ(𝑐−𝑏)(1+𝛼)(𝑎)₂(𝑏)₂(1−𝛽)(𝑐−𝑎−𝑏−2)₂+(3+2𝛼−𝛽1−𝛽)(𝑎𝑏𝑐−𝑎−𝑏−1)+1)≤2. (2.15)

Proof. (i) Since 𝑧𝐹 has the form (2.7), we see from (ii) of Lemma 2.1 that our conclusion is equivalent to ^∞_𝑛=2𝑛(𝑛(1+𝛼)−(𝛼+𝛽))(𝑎+1)_𝑛−2(𝑏+1)_𝑛−2(𝑐+1)_𝑛−2(1)_𝑛−1≤𝑐|𝑎𝑏|(1−𝛽). (2.16) Writing (𝑛+2)((𝑛+2)(1+𝛼)−(𝛼+𝛽))=(1+𝛼)(𝑛+1)²+(2+𝛼−𝛽)(𝑛+1)+(1−𝛽), we see that ^∞_𝑛=0(𝑛+2)((𝑛+2)(1+𝛼)−(𝛼+𝛽))(𝑎+1)_𝑛(𝑏+1)_𝑛(𝑐+1)_𝑛(1)_𝑛+1=(1+𝛼)^∞_𝑛=0(𝑛+1)(𝑎+1)_𝑛(𝑏+1)_𝑛(𝑐+1)_𝑛(1)_𝑛+(2+𝛼−𝛽)^∞_𝑛=0(𝑎+1)_𝑛(𝑏+1)_𝑛(𝑐+1)_𝑛(1)_𝑛+(1−𝛽)^∞_𝑛=0(𝑎+1)_𝑛(𝑏+1)_𝑛(𝑐+1)_𝑛(1)_𝑛+1=(1+𝛼)(𝑎+1)(𝑏+1)𝑐+1^∞_𝑛=0(𝑎+2)_𝑛(𝑏+2)_𝑛(𝑐+2)_𝑛(1)_𝑛+(3+2𝛼−𝛽)^∞_𝑛=0(𝑎+1)_𝑛(𝑏+1)_𝑛(𝑐+1)_𝑛(1)_𝑛𝑐+(1−𝛽)𝑎𝑏^∞_𝑛=1(𝑎)_𝑛(𝑏)_𝑛(𝑐)_𝑛(1)_𝑛=Γ(𝑐+1)Γ(𝑐−𝑎−𝑏−2)+Γ(𝑐−𝑎)Γ(𝑐−𝑏)((1+𝛼)(𝑎+1)(𝑏+1)+(3+2𝛼−𝛽)(𝑐−𝑎−𝑏−2)1−𝛽𝑎𝑏(𝑐−𝑎−𝑏−2)₂)−(1−𝛽)𝑐.𝑎𝑏 (2.17) This last expression is bounded above by (1−𝛽)𝑐/|𝑎𝑏| if and only if (1+𝛼)(𝑎+1)(𝑏+1)+(3+2𝛼−𝛽)(𝑐−𝑎−𝑏−2)+1−𝛽𝑎𝑏(𝑐−𝑎−𝑏−2)₂≤0, (2.18) which is equivalent to (2.14).
(ii) In view of (ii) of Lemma 2.1, we need only to show that ^∞_𝑛=2𝑛(𝑛(1+𝛼)−(𝛼+𝛽))(𝑎)_𝑛−1(𝑏)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1≤1−𝛽. (2.19) Now, ^∞_𝑛=0(𝑛+2)((𝑛+2)(1+𝛼)−(𝛼+𝛽))(𝑎)_𝑛+1(𝑏)_𝑛+1(𝑐)_𝑛+1(1)_𝑛+1=(1+𝛼)^∞_𝑛=0(𝑛+2)²(𝑎)_𝑛+1(𝑏)_𝑛+1(𝑐)_𝑛+1(1)_𝑛+1−(𝛼+𝛽)^∞_𝑛=0(𝑛+2)(𝑎)_𝑛+1(𝑏)_𝑛+1(𝑐)_𝑛+1(1)_𝑛+1. (2.20) Writing 𝑛+2=(𝑛+1)+1, we have ^∞_𝑛=0(𝑛+2)(𝑎)_𝑛+1(𝑏)_𝑛+1(𝑐)_𝑛+1(1)_𝑛+1=^∞_𝑛=0(𝑎)_𝑛+1(𝑏)_𝑛+1(𝑐)_𝑛+1(1)_𝑛+^∞_𝑛=0(𝑎)_𝑛+1(𝑏)_𝑛+1(𝑐)_𝑛+1(1)_{𝑛+1∞𝑛=0},(𝑛+2)²(𝑎)_𝑛+1(𝑏)_𝑛+1(𝑐)_𝑛+1(1)^∞_𝑛+1=_𝑛=0(𝑛+1)(𝑎)_𝑛+1(𝑏)_𝑛+1(𝑐)_𝑛+1(1)_𝑛+2^∞_𝑛=0(𝑎)_𝑛+1(𝑏)_𝑛+1(𝑐)_𝑛+1(1)_𝑛+^∞_𝑛=0(𝑎)_𝑛+1(𝑏)_𝑛+1(𝑐)_𝑛+1(1)_𝑛+1=^∞_𝑛=1(𝑎)_𝑛+1(𝑏)_𝑛+1(𝑐)_𝑛+1(1)_𝑛−1+3^∞_𝑛=0(𝑎)_𝑛+1(𝑏)_𝑛+1(𝑐)_𝑛+1(1)_𝑛+^∞_𝑛=1(𝑎)_𝑛(𝑏)_𝑛(𝑐)_𝑛(1)_𝑛. (2.21)(1) Substituting (2.21) into the right-hand side of (2.20), we obtain (1+𝛼)^∞_𝑛=0(𝑎)_𝑛+2(𝑏)_𝑛+2(𝑐)_𝑛+2(1)_𝑛+(3+2𝛼−𝛽)^∞_𝑛=0(𝑎)_𝑛+1(𝑏)_𝑛+1(𝑐)_𝑛+1(1)_𝑛+(1−𝛽)^∞_𝑛=0(𝑎)_𝑛+1(𝑏)_𝑛+1(𝑐)_𝑛+1(1)_𝑛+1. (2.22) Since (𝑎)_𝑛+𝑘=(𝑎)_𝑘(𝑎+𝑘)_𝑛, we write (2.22) as (1+𝛼)(𝑎)₂(𝑏)₂(𝑐)₂Γ(𝑐+2)Γ(𝑐−𝑎−𝑏−2)Γ(𝑐−𝑎)Γ(𝑐−𝑏)+(3+2𝛼−𝛽)𝑎𝑏𝑐Γ(𝑐+1)Γ(𝑐−𝑎−𝑏−1)Γ(𝑐−𝑎)Γ(𝑐−𝑏)+(1−𝛽)(Γ(𝑐)Γ(𝑐−𝑎−𝑏)Γ(𝑐−𝑎)Γ(𝑐−𝑏)−1). (2.23) By simplification, we see that the last expression is bounded above by 1−𝛽 if and only if (2.15) holds.

Theorem 2.5. (i) If 𝑎,𝑏>−1,𝑐>0, and 𝑎𝑏<0, then 𝑧𝐹(𝑎,𝑏;𝑐;𝑧) is in 𝒫𝒯(𝛼) if and only if 𝑐≥𝑎+𝑏+1−𝑎𝑏𝛼. (2.24)
(ii) If 𝑎,𝑏>0 and 𝑐>𝑎+𝑏+1, then 𝐹₁(𝑎,𝑏;𝑐;𝑧)=𝑧(2−𝐹(𝑎,𝑏;𝑐;𝑧)) is in 𝒫𝒯(𝛼) if and only if Γ(𝑐)Γ(𝑐−𝑎−𝑏)Γ(𝑐−𝑎)Γ(𝑐−𝑏)(1+𝑎𝑏𝛼(𝑐−𝑎−𝑏−1))≤2. (2.25)

Proof. (i) Since 𝑧𝐹(𝑎,𝑏;𝑐;𝑧)=𝑧+𝑎𝑏𝑐^∞_𝑛=2(𝑎+1)_𝑛−2(𝑏+1)_𝑛−2(𝑐+1)_𝑛−2(1)_𝑛−1𝑧^𝑛=𝑧−|𝑎𝑏|𝑐^∞_𝑛=2(𝑎+1)_𝑛−2(𝑏+1)_𝑛−2(𝑐+1)_𝑛−2(1)_𝑛−1𝑧^𝑛, (2.26) according to (i) of Lemma 2.2, we must show that ^∞_𝑛=2(𝑛−1+𝛼)(𝑎+1)_𝑛−2(𝑏+1)_𝑛−2(𝑐+1)_𝑛−2(1)_𝑛−1≤𝑐|𝑎𝑏|𝛼. (2.27) Noting that (𝜆)_𝑛=𝜆(𝜆+1)_𝑛−1 and then applying (1.6), we have ^∞_𝑛=0(𝑛+1+𝛼)(𝑎+1)_𝑛(𝑏+1)_𝑛(𝑐+1)_𝑛(1)_𝑛+1=^∞_𝑛=0(𝑎+1)_𝑛(𝑏+1)_𝑛(𝑐+1)_𝑛(1)_𝑛𝑐+𝛼𝑎𝑏^∞_𝑛=1(𝑎)_𝑛(𝑏)_𝑛(𝑐)_𝑛(1)_𝑛=Γ(𝑐+1)Γ(𝑐−𝑎−𝑏−1)𝑐Γ(𝑐−𝑎)Γ(𝑐−𝑏)+𝛼(𝑎𝑏Γ(𝑐)Γ(𝑐−𝑎−𝑏)Γ(𝑐−𝑎)Γ(𝑐−𝑏)−1). (2.28) Hence, (2.27) is equivalent to Γ(𝑐+1)Γ(𝑐−𝑎−𝑏−1)Γ(𝑐−𝑎)Γ(𝑐−𝑏)(1+𝛼𝑐−𝑎−𝑏−1𝑐𝑎𝑏)≤𝛼(−𝑐|𝑎𝑏|𝑎𝑏)=0. (2.29) Thus, (2.29) is valid if and only if 1+𝛼(𝑐−𝑎−𝑏−1)/𝑎𝑏≤0 or, equivalently, 𝑐≥𝑎+𝑏+1−𝑎𝑏/𝛼.
(ii) Since 𝐹₁(𝑎,𝑏;𝑐;𝑧)=𝑧−^∞_𝑛=2(𝑎)_𝑛−1(𝑏)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1𝑧^𝑛, (2.30) by (i) of Lemma 2.2, we need only to show that ^∞_𝑛=2(𝑛−1+𝛼)(𝑎)_𝑛−1(𝑏)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1≤𝛼. (2.31) Now, ^∞_𝑛=2(𝑛−1+𝛼)(𝑎)_𝑛−1(𝑏)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1=^∞_𝑛=1𝑛(𝑎)_𝑛(𝑏)_𝑛(𝑐)_𝑛(1)_𝑛+𝛼^∞_𝑛=1(𝑎)_𝑛(𝑏)_𝑛(𝑐)_𝑛(1)_𝑛=𝑎𝑏𝑐^∞_𝑛=1(𝑎+1)_𝑛−1(𝑏+1)_𝑛−1(𝑐+1)_𝑛−1(1)_𝑛−1+𝛼^∞_𝑛=1(𝑎)_𝑛(𝑏)_𝑛(𝑐)_𝑛(1)_𝑛=Γ(𝑐)Γ(𝑐−𝑎−𝑏)(Γ(𝑐−𝑎)Γ(𝑐−𝑏)𝑎𝑏𝑐−𝑎−𝑏−1+𝛼)−𝛼. (2.32) But this last expression is bounded above by 𝛼 if and only if (2.25) holds.

Theorem 2.6. (i) If 𝑎,𝑏>−1,𝑎𝑏<0, and 𝑐>𝑎+𝑏+2, then 𝑧𝐹(𝑎,𝑏;𝑐;𝑧) is in 𝒞𝒫𝒯(𝛼) if and only if (𝑎)₂(𝑏)₂+(2+𝛼)𝑎𝑏(𝑐−𝑎−𝑏−2)+𝛼(𝑐−𝑎−𝑏−2)₂≥0. (2.33)
(ii) If 𝑎,𝑏>0 and 𝑐>𝑎+𝑏+2, then 𝐹₁(𝑎,𝑏;𝑐;𝑧)=𝑧(2−𝐹(𝑎,𝑏;𝑐;𝑧)) is in 𝒞𝒫𝒯(𝛼) if and only if Γ(𝑐)Γ(𝑐−𝑎−𝑏)(Γ(𝑐−𝑎)Γ(𝑐−𝑏)(𝑎)₂(𝑏)₂𝛼(𝑐−𝑎−𝑏−2)₂+(2+𝛼𝛼)(𝑎𝑏𝑐−𝑎−𝑏−1)+1)≤2. (2.34)

Proof. (i) Since 𝑧𝐹 has the form (2.26), we see from (ii) of Lemma 2.2 that our conclusion is equivalent to ^∞_𝑛=2𝑛(𝑛−1+𝛼)(𝑎+1)_𝑛−2(𝑏+1)_𝑛−2(𝑐+1)_𝑛−2(1)_𝑛−1≤𝑐|𝑎𝑏|𝛼. (2.35) Writing (𝑛+2)(𝑛+1+𝛼)=(𝑛+1)²+(1+𝛼)(𝑛+1)+𝛼, we see that ^∞∑_𝑛=0(𝑛+2)(𝑛+1+𝛼)(𝑎+1)_𝑛(𝑏+1)_𝑛(𝑐+1)_𝑛(1)_𝑛+1=∑^∞_𝑛=0(𝑛+1)(𝑎+1)_𝑛(𝑏+1)_𝑛(𝑐+1)_𝑛(1)_𝑛+(1+𝛼)∑^∞_𝑛=0(𝑎+1)_𝑛(𝑏+1)_𝑛(𝑐+1)_𝑛(1)_𝑛+𝛼∑^∞_𝑛=0(𝑎+1)_𝑛(𝑏+1)_𝑛(𝑐+1)_𝑛(1)_𝑛+1=(𝑎+1)(𝑏+1)𝑐+1^∞_𝑛∑₌₀(𝑎+2)_𝑛(𝑏+2)_𝑛(𝑐+2)_𝑛(1)_𝑛+(2+𝛼)∑^∞_𝑛=0(𝑎+1)_𝑛(𝑏+1)_𝑛(𝑐+1)_𝑛(1)_𝑛𝑐+𝛼𝑎𝑏^∞_𝑛∑₌₁(𝑎)_𝑛(𝑏)_𝑛(𝑐)_𝑛(1)_𝑛=Γ(𝑐+1)Γ(𝑐−𝑎−𝑏−2)𝛼Γ(𝑐−𝑎)Γ(𝑐−𝑏)((𝑎+1)(𝑏+1)+(2+𝛼)(𝑐−𝑎−𝑏−2)+𝑎𝑏(𝑐−𝑎−𝑏)₂)−𝛼𝑐.𝑎𝑏 (2.36) This last expression is bounded above by 𝛼𝑐/|𝑎𝑏| if and only if (𝑎+1)(𝑏+1)+(2+𝛼)(𝑐−𝑎−𝑏−2)+(𝛼/𝑎𝑏)(𝑐−𝑎−𝑏−1)₂≤0, which is equivalent to (2.33).
(ii) In view of (ii) of Lemma 2.2, we need only to show that ^∞_𝑛=2𝑛(𝑛−1+𝛼)(𝑎)_𝑛−1(𝑏)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1≤𝛼. (2.37) Now, ^∞_𝑛=0(𝑛+2)(𝑛+2−(1−𝛼))(𝑎)_𝑛+1(𝑏)_𝑛+1(𝑐)_𝑛+1(1)_𝑛+1=^∞_𝑛=0(𝑛+2)²(𝑎)_𝑛+1(𝑏)_𝑛+1(𝑐)_𝑛+1(1)_𝑛+1−(1−𝛼)^∞_𝑛=0(𝑛+2)(𝑎)_𝑛+1(𝑏)_𝑛+1(𝑐)_𝑛+1(1)_𝑛+1. (2.38) Substituting (2.21) into the right-hand side of (2.38), we obtain ^∞_𝑛=0(𝑎)_𝑛+2(𝑏)_𝑛+2(𝑐)_𝑛+2(1)_𝑛+(2+𝛼)^∞_𝑛=0(𝑎)_𝑛+1(𝑏)_𝑛+1(𝑐)_𝑛+1(1)_𝑛+𝛼^∞_𝑛=1(𝑎)_𝑛(𝑏)_𝑛(𝑐)_𝑛(1)_𝑛. (2.39) Since (𝑎)_𝑛+𝑘=(𝑎)_𝑘(𝑎+𝑘)_𝑛, we may write (2.39) as (𝑎)₂(𝑏)₂(𝑐)₂Γ(𝑐+2)Γ(𝑐−𝑎−𝑏−2)Γ(𝑐−𝑎)Γ(𝑐−𝑏)+(2+𝛼)𝑎𝑏𝑐Γ(𝑐+1)Γ(𝑐−𝑎−𝑏−1)Γ(𝑐−𝑎)Γ(𝑐−𝑏)+𝛼(Γ(𝑐)Γ(𝑐−𝑎−𝑏)Γ(𝑐−𝑎)Γ(𝑐−𝑏)−1). (2.40) By simplification, we see that the last expression is bounded above by 𝛼 if and only if (2.34) holds.

3. An Integral Operator

In the next theorems, we obtain similar-type results in connection with a particular integral operator 𝐺(𝑎,𝑏;𝑐;𝑧) acting on 𝐹(𝑎,𝑏;𝑐;𝑧) as follows: 𝐺(𝑎,𝑏;𝑐;𝑧)=^𝑧₀𝐹(𝑎,𝑏;𝑐;𝑡)𝑑𝑡. (3.1)

Theorem 3.1. Let 𝑎,𝑏>−1,𝑎𝑏<0, and 𝑐>max{0,𝑎+𝑏}. Then,
(i)𝐺(𝑎,𝑏;𝑐;𝑧) defined by (3.1) is in 𝒮_𝑝𝒯(𝛼,𝛽) if and only if Γ(𝑐+1)Γ(𝑐−𝑎−𝑏)(Γ(𝑐−𝑎)Γ(𝑐−𝑏)(1+𝛼)−𝑎𝑏(𝛼+𝛽)(𝑐−𝑎−𝑏)(𝑎−1)₂(𝑏−1)₂)+(𝛼+𝛽)(𝑐−1)₂(𝑎−1)₂(𝑏−1)₂≤0; (3.2)(ii)𝐺(𝑎,𝑏;𝑐;𝑧) defined by (3.1) is in 𝒫𝒯(𝛼) if and only if Γ(𝑐+1)Γ(𝑐−𝑎−𝑏)(1Γ(𝑐−𝑎)Γ(𝑐−𝑏)+𝑎𝑏(𝛼−1)(𝑐−𝑎−𝑏)(𝑎−1)₂(𝑏−1)₂)−(𝛼−1)(𝑐−𝑎)₂(𝑎−1)₂(𝑏−1)₂≤0. (3.3)

Proof. (i) Since 𝐺(𝑎,𝑏;𝑐;𝑧)=𝑧−|𝑎𝑏|𝑐^∞_𝑛=2(𝑎+1)_𝑛−2(𝑏+1)_𝑛−2(𝑐+1)_𝑛−2(1)_𝑛𝑧^𝑛, (3.4) by (i) of Lemma 2.1, we need only to show that ^∞_𝑛=2(𝑛(1+𝛼)−(𝛼+𝛽))(𝑎+1)_𝑛−2(𝑏+1)_𝑛−2(𝑐+1)_𝑛−2(1)_𝑛𝑐≤(1−𝛽)|𝑎𝑏|. (3.5) Now, ^∞_𝑛=0((𝑛+2)(1+𝛼)−(𝛼+𝛽))(𝑎+1)_𝑛(𝑏+1)_𝑛(𝑐+1)_𝑛(1)_𝑛+2=(1+𝛼)^∞_𝑛=0(𝑎+1)_𝑛(𝑏+1)_𝑛(𝑐+1)_𝑛(1)_𝑛+1𝑐−(𝛼+𝛽)𝑎𝑏^∞_𝑛=1(𝑎)_𝑛(𝑏)_𝑛(𝑐)_𝑛(1)_𝑛+1=Γ(𝑐+1)Γ(𝑐−𝑎−𝑏)(Γ(𝑐−𝑎)Γ(𝑐−𝑏)1+𝛼−𝑎𝑏(𝛼+𝛽)(𝑐−𝑎−𝑏)(𝑎−1)₂(𝑏−1)₂)+(𝛼+𝛽)(𝑐−1)₂(𝑎−1)₂(𝑏−1)₂𝑐−(1−𝛽)𝑐𝑎𝑏≤(1−𝛽),|𝑎𝑏| (3.6) which is equivalent to (3.2).
(ii) According to (i) of Lemma 2.2, it is sufficient to show that ^∞_𝑛=2(𝑛−1+𝛼)(𝑎+1)_𝑛−2(𝑏+1)_𝑛−2(𝑐+1)_𝑛−2(1)_𝑛𝑐≤𝛼|𝑎𝑏|. (3.7) Now, ^∞_𝑛=0(𝑛+1+𝛼)(𝑎+1)_𝑛(𝑏+1)_𝑛(𝑐+1)_𝑛(1)_𝑛+2=^∞_𝑛=0(𝑎+1)_𝑛(𝑏+1)_𝑛(𝑐+1)_𝑛(1)_𝑛+1+(𝛼−1)^∞_𝑛=0(𝑎+1)_𝑛(𝑏+1)_𝑛(𝑐+1)_𝑛(1)_𝑛+2=𝑐(𝑎𝑏Γ(𝑐)Γ(𝑐−𝑎−𝑏)𝑐Γ(𝑐−𝑎)Γ(𝑐−𝑏)−1)+(𝛼−1)(𝑎𝑏(𝑐−1)((𝑎−1)(𝑏−1)Γ(𝑐−1)Γ(𝑐−𝑎−𝑏+1)=Γ(𝑐−𝑎)Γ(𝑐−𝑏)−1)−1)Γ(𝑐+1)Γ(𝑐−𝑎−𝑏)(1Γ(𝑐−𝑎)Γ(𝑐−𝑏)+𝑎𝑏(𝛼−1)(𝑐−𝑎−𝑏)(𝑎−1)₂(𝑏−1)₂)−(𝛼−1)(𝑐−1)₂(𝑎−1)₂(𝑏−1)₂𝑐−𝛼𝑐𝑎𝑏≤𝛼,|𝑎𝑏| (3.8) which is equivalent to (3.3).
Now, we observe that 𝐺(𝑎,𝑏;𝑐;𝑧)∈𝒰𝒞𝒯(𝛼,𝛽)(𝒞𝒫𝒯(𝛼)) if and only if 𝑧𝐹(𝑎,𝑏;𝑐;𝑧)∈𝒮_𝑝𝒯(𝛼,𝛽)(𝒫𝒯(𝛼)). Thus, any result of functions belonging to the class 𝒮_𝑝𝒯(𝛼,𝛽)(𝒫𝒯(𝛼)) about 𝑧𝐹 leads to that of functions belonging to the class 𝒰𝒞𝒯(𝛼,𝛽)(𝒞𝒫𝒯(𝛼)). Hence, we obtain the following analogues to Theorems 2.3 and 2.5.

Theorem 3.2. Let 𝑎,𝑏>−1,𝑎𝑏<0, and 𝑐>𝑎+𝑏+2. Then,
(i)𝐺(𝑎,𝑏;𝑐;𝑧) defined by (3.1) is in 𝒰𝒞𝒯(𝛼,𝛽) if and only if 𝑐≥𝑎+𝑏+1−(1+𝛼)𝑎𝑏(1−𝛽); (3.9)(ii)𝐺(𝑎,𝑏;𝑐;𝑧) defined by (3.1) is in 𝒞𝒫𝒯(𝛼) if and only if 𝑐≥𝑎+𝑏+1−𝑎𝑏𝛼. (3.10)