On Certain Sufficient Condition Involving Gaussian Hypergeometric Functions

Abstract

The authors define a new subclass of 𝒜 of functions involving complex order in the open unit disk 𝕌. For this new class, we obtain certain inclusion properties involving the Gaussian hypergeometric functions.

1. Introduction and Motivation

Let 𝒜 be the class of functions 𝑓 normalized by𝑓(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛, (1.1) which are analytic in the open unit disk 𝕌={𝑧∶𝑧∈ℂ,|𝑧|<1}. (1.2) As usual, we denote by 𝒮 the subclass of 𝒜 consisting of functions which are also univalent in 𝕌. A function 𝑓∈𝒜 is said to be starlike of order 𝛼 in 𝕌(0≤𝛼<1), if and only if ℜ(𝑧𝑓^′(𝑧)𝑓(𝑧))>𝛼(𝑧∈𝕌;0≤𝛼<1). (1.3)

This function class is denoted by 𝒮^∗(𝛼). We also write 𝒮^∗(0)=∶𝒮^∗, where 𝒮^∗ denotes the class of functions 𝑓∈𝒜 that are starlike in 𝕌 with respect to the origin.

A function 𝑓∈𝒜 is said to be convex of order 𝛼 in 𝕌(0≤𝛼<1) if and only if ℜ(1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧))>𝛼(𝑧∈𝕌;0≤𝛼<1). (1.4)

The class of convex functions is denoted by the class 𝒦(𝛼). Further, 𝒦=𝒦(0), the well-known standard class of convex functions. It is an established fact that 𝑓∈𝒦(𝛼)⇔𝑧𝑓^′∈𝒮^∗(𝛼). (1.5)

A function 𝑓∈𝒜 is said to be in the class 𝑈𝐶𝑉 of uniformly convex functions in 𝕌 if 𝑓 is a normalized convex function in 𝕌 and has the property that, for every circular arc 𝛿 contained in the unit disk 𝕌, with center 𝜁 also in U, the image curve 𝑓(𝛿) is a convex arc. The function class 𝑈𝐶𝑉 was introduced by Goodman [1].

For functions 𝑓∈𝒜 given by (1.1) and 𝑔∈𝒜 given by 𝑔(𝑧)=𝑧+∑^∞_𝑛=2𝑏_𝑛𝑧^𝑛, we define the Hadamard product (or Convolution) of 𝑓 and 𝑔 by (𝑓∗𝑔)(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑏_𝑛𝑧^𝑛,𝑧∈𝕌. (1.6)

Furthermore, we denote by 𝑘−𝑈𝐶𝑉 and 𝑘−𝑆𝑇 two interesting subclasses of 𝒮 consisting, respectively, of functions which are 𝑘-uniformly convex and 𝑘-starlike in 𝕌. Thus, we have 𝑘−𝑈𝐶𝑉∶={𝑓∈𝒮∶ℜ(1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧))>𝑘|𝑧𝑓^′′(𝑧)𝑓^′(𝑧)|(𝑧∈𝕌;0≤𝑘<∞)},𝑘−𝑆𝑇∶={𝑓∈𝒮∶ℜ(𝑧𝑓^′(𝑧)𝑓(𝑧))>𝑘|𝑧𝑓^′(𝑧)𝑓(𝑧)−1|(𝑧∈𝕌;0≤𝑘<∞)}. (1.7) The class 𝑘−𝑈𝐶𝑉 was introduced by Kanas and Wiśniowska [2], where its geometric definition and connections with the conic domains were considered. The class 𝑘−𝑆𝑇 was investigated in [3]. In fact, it is related to the class 𝑘−𝑈𝐶𝑉 by means of the well-known Alexander equivalence between the usual classes of convex and starlike functions; see also the work of Kanas and Srivastava [4] for further developments involving each of the classes 𝑘−𝑈𝐶𝑉 and 𝑘−𝑆𝑇. In particular, when 𝑘=1, we obtain 1−𝑈𝐶𝑉≡𝑈𝐶𝑉,1−𝑆𝑇=𝑆𝑃, (1.8) where 𝑈𝐶𝑉 and 𝑆𝑃 are the familiar classes of uniformly convex functions and parabolic starlike functions in 𝕌, respectively (see for details, [1, 5]). In fact, by making use of a certain fractional calculus operator, Srivastava and Mishra [6] presented a systematic and unified study of the classes 𝑈𝐶𝑉 and 𝑆𝑃.

A function 𝑓∈𝒜 is said to be in the class 𝑃^𝜏_𝛾(𝐴,𝐵)⊂𝒜 if it satisfies the inequality |𝑓^′(𝑧)+𝛾𝑧𝑓^′′(𝑧)−1(𝐴−𝐵)𝜏−𝐵[𝑓^′(𝑧)+𝛾𝑧𝑓^′′(𝑧)−1]|<1(𝑧∈𝕌;𝜏∈ℂ⧵{0},−1≤𝐵<𝐴≤1,0≤𝛾<1). (1.9)

The class 𝑃^𝜏₀(𝐴,𝐵) was introduced earlier by Dixit and Pal [7]. Two of the many interesting subclasses of the class 𝑃^𝜏_𝛾(𝐴,𝐵) are worthy of mention here. First of all, by setting 𝛾=0,𝜏=𝑒^𝑖𝜂𝜋cos𝜂(−2𝜋<𝜂<2),𝐴=1−2𝛽(0≤𝛽<1),𝐵=−1, (1.10) the class 𝑃^𝜏_𝛾(𝐴,𝐵) reduces essentially to the class 𝑅_𝜂(𝛽) introduced and studied by Ponnusamy and Rønning [8], where 𝑅_𝜂(𝛽)={𝑓∈𝒜∶ℜ(𝑒^𝑖𝜂(𝑓^′𝜋(𝑧)−𝛽))>0(𝑧∈𝕌;−2𝜋<𝜂<2,0≤𝛽<1)}. (1.11) Secondly, if we put 𝛾=0,𝜏=1,𝐴=𝛽,𝐵=−𝛽(0<𝛽≤1), (1.12) we obtain the class of functions 𝑓∈𝒜 satisfying the inequality |𝑓^′(𝑧)−1𝑓^′(𝑧)+1|<𝛽(𝑧∈𝕌;0<𝛽≤1) (1.13) which was studied by (among others) Padmanabhan [9] and Caplinger and Causey [10].

Finally, many of the authors have also studied the class 𝑃¹_𝛾(𝐴,𝐵). For details of these works one can refer to the works of Ding Gong [11], R. Singh and S. Singh [12], Owa and Wu [13], and also the references cited by them. Although, many mapping properties of the class 𝑃¹_𝛾(𝐴,𝐵) have been studied by these authors, they did not study any mapping properties involving the hypergeometric functions.

The Gaussian hypergeometric function 𝐹(𝑎,𝑏;𝑐,𝑧), 𝑧∈𝕌 is given by 𝐹(𝑎,𝑏;𝑐;𝑧)=^∞_𝑛=0(𝑎)_𝑛(𝑏)_𝑛(𝑐)_𝑛(1)_𝑛𝑧^𝑛, (1.14) is the solution of the homogeneous hypergeometric differential equation 𝑧(1−𝑧)𝑤^′′(𝑧)+[𝑐−(𝑎+𝑏+1)𝑧]𝑤^′(𝑧)−𝑎𝑏𝑤(𝑧)=0 (1.15) and has rich applications in various fields such as conformal mappings, quasiconformal theory, and continued fractions.

Here, 𝑎, 𝑏, 𝑐 are complex numbers such that 𝑐≠0,−1,−2,−3,…, (𝑎)₀=1 for 𝑎≠0, and for each positive integer 𝑛, (𝑎)_𝑛=𝑎(𝑎+1)(𝑎+2)⋯(𝑎+𝑛−1) is the Pochhammer symbol. In the case of 𝑐=−𝑘, 𝑘=0,1,2,…, 𝐹(𝑎,𝑏;𝑐;𝑧) is defined if 𝑎=−𝑗 or 𝑏=−𝑗, where 𝑗≤𝑘. In this situation, 𝐹(𝑎,𝑏;𝑐;𝑧) becomes a polynomial of degree 𝑗 in 𝑧. Results regarding 𝐹(𝑎,𝑏;𝑐;𝑧) when ℜ(𝑐−𝑎−𝑏) is positive, zero, or negative are abundant in the literature. In particular when ℜ(𝑐−𝑎−𝑏)>0, the function is bounded. This and the zero balanced case ℜ(𝑐−𝑎−𝑏)=0 are discussed in detail by many authors (see [14, 15]). The hypergeometric function 𝐹(𝑎,𝑏;𝑐;𝑧) has been studied extensively by various authors and it plays an important role in Geometric Function Theory. It is useful in unifying various functions by giving appropriate values to the parameters 𝑎, 𝑏, and 𝑐. We refer to [8, 16–19] and references therein for some important results.

In particular, the close-to-convexity (in turn the univalency), convexity, starlikeness, (for details on these technical terms we refer to [5]), and various other properties of these hypergeometric functions were examined based on the conditions on 𝑎, 𝑏, and 𝑐 in [8]. For more interesting properties of hypergeometric functions, one can also refer to [20, 21].

Let 𝑓(𝑧) and 𝑔(𝑧) be analytic in 𝕌 and 𝑔(𝑧) univalent. Then we say that 𝑓(𝑧) is subordinate to 𝑔(𝑧) written as 𝑓(𝑧)≺𝑔(𝑧) if 𝑓(0)=𝑔(0) and 𝑓(𝕌)⊂𝑔(𝕌).

For 𝑓∈𝒜, we recall that the operator 𝐼_{𝑎,𝑏,𝑐}(𝑓) of Hohlov [22] which maps 𝒜 into itself defined by 𝐼_{𝑎,𝑏,𝑐}(𝑓)(𝑧)=𝑧𝐹(𝑎,𝑏;𝑐;𝑧)∗𝑓(𝑧), (1.16) where ∗ denotes usual Hadamard product of power series. Therefore, for a function 𝑓 defined by (1.1), we have 𝐼_{𝑎,𝑏,𝑐}(𝑓)(𝑧)=𝑧+^∞_𝑛=2(𝑎)_𝑛−1(𝑏)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1𝑎_𝑛𝑧^𝑛. (1.17)

Using the integral representation, 𝐹(𝑎,𝑏;𝑐;𝑧)=Γ(𝑐)Γ(𝑏)Γ(𝑐−𝑏)¹₀𝑡^𝑏−1(1−𝑡)^{𝑐−𝑏−1}𝑑𝑡(1−𝑡𝑧)^𝑎,ℜ(𝑐)>𝑅(𝑏)>0, (1.18) we can write [𝐼_{𝑎,𝑏,𝑐}(𝑓)](𝑧)=Γ(𝑐)Γ(𝑏)Γ(𝑐−𝑏)¹₀𝑡^𝑏−1(1−𝑡)^{𝑐−𝑏−1}𝑓(𝑡𝑧)𝑡𝑧𝑑𝑡∗(1−𝑡𝑧)^𝑎. (1.19)

When 𝑓(𝑧) equals the convex function 𝑧/(1−𝑧), then the operator 𝐼_{𝑎,𝑏,𝑐}(𝑓) in this case becomes 𝑧𝐹(𝑎,𝑏;𝑐;𝑧). For 𝑎=1, 𝑏=1+𝛿, 𝑐=2+𝛿 with ℜ(𝛿)>−1 then the convolution operator 𝐼_{𝑎,𝑏,𝑐}(𝑓) turns into Bernardi operator 𝐵_𝑓(𝑧)=[𝐼_{𝑎,𝑏,𝑐}(𝑓)](𝑧)=1+𝛿𝑧^𝛿1₀𝑡^𝛿−1𝑓(𝑡)𝑑𝑡. (1.20) Indeed, 𝐼_1,1,2(𝑓) and 𝐼_1,2,3(𝑓) are known as Alexander and Libera operators, respectively.

Let 0≤𝑘<∞, and let 𝑓∈𝒜 be of the form (1.1). If 𝑓∈𝑘−𝑈𝐶𝑉, then the following coefficient inequalities hold true (cf. [2]): |𝑎_𝑛|≤(𝑃₁)_𝑛−1(1)_𝑛,𝑛∈ℕ⧵{1}, (1.21) where 𝑃₁=𝑃₁(𝑘) is the coefficient of 𝑧 in the function 𝑝_𝑘(𝑧)=1+^∞_𝑛=1𝑃_𝑛(𝑘)𝑧^𝑛, (1.22) which is the extremal function for the class 𝑃(𝑝_𝑘) related to the class 𝑘−𝑈𝐶𝑉 by the range of the expression 1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)(𝑧∈𝕌), (1.23) where 𝑃₁=𝑃₁(𝑘) is given, as above, by (1.22).

Similarly, if 𝑓 of the form (1.1) belong to the class 𝑘−𝑆𝑇, then (cf. [3]) |𝑎_𝑛|≤(𝑃₁)_𝑛−1(1)_𝑛−1,𝑛∈ℕ⧵{1}, (1.24) where 𝑃₁=𝑃₁(𝑘) is given, as above by (1.22).

2. Properties of 𝑃^𝜏_𝛾(𝐴,𝐵)

Theorem 2.1. Let 𝑓∈𝒮 and be of the form (1.1). If 𝑓∈𝑃^𝜏_𝛾(𝐴,𝐵), then |𝑎_𝑛|≤(𝐴−𝐵)|𝜏|.𝑛(1+𝛾(𝑛−1)) (2.1) The estimate is sharp.

Proof. Since 𝑓∈𝑃^𝜏_𝛾(𝐴,𝐵), we have 11+𝜏[𝑓^′(𝑧)+𝛾𝑧𝑓^′′(𝑧)−1]=1+𝐴𝑤(𝑧),1+𝐵𝑤(𝑧) (2.2) where 𝑤(𝑧) is analytic in 𝕌 and satisfies the condition 𝑤(0)=0 and |𝑤(𝑧)|<1 for 𝑧∈𝕌. Hence, we have 1𝜏[𝑓^′(𝑧)+𝛾𝑧𝑓^′′𝐵(𝑧)−1]=𝑤(𝑧)[(𝐴−𝐵)−𝜏(𝑓^′(𝑧)+𝛾𝑧𝑓^′′(𝑧)−1)]. (2.3) Using 𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛 and 𝑤(𝑧)=∑^∞_𝑛=1𝑏_𝑛𝑧^𝑛, we have 𝐵{(𝐴−𝐵)−𝜏^∞_𝑛=2(1+𝛾(𝑛−1))𝑛𝑎_𝑛𝑧^𝑛−1}[^∞_𝑛=1𝑏_𝑛𝑧^𝑛1]=𝜏[^∞_𝑛=2(1+𝛾(𝑛−1))𝑛𝑎_𝑛𝑧^𝑛−1]. (2.4) By equating the coefficients, we observe that the coefficient 𝑎_𝑛 in the right-hand side depends only on 𝑎₂,𝑎₃,…,𝑎_𝑛−1 on the left-hand side of the above expression. This gives 𝐵[(𝐴−𝐵)−𝜏(^𝑘−1_𝑛=2[1+𝛾(𝑛−1)]𝑛𝑎_𝑛𝑧^𝑛−1=1)]𝑤(𝑧)𝜏(^𝑘_𝑛=2[1+𝛾(𝑛−1)]𝑛𝑎_𝑛𝑧^𝑛−1)+^∞_𝑛=𝑘+1𝑑_𝑛𝑧^𝑛−1. (2.5) By using |𝑤(𝑧)|<1, we get 𝐵|(𝐴−𝐵)−𝜏(^𝑘−1_𝑛=2[1+𝛾(𝑛−1)]𝑛𝑎_𝑛𝑧^𝑛−11)|≥|𝜏(^𝑘_𝑛=2[1+𝛾(𝑛−1)]𝑛𝑎_𝑛𝑧^𝑛−1)+^∞_𝑛=𝑘+1𝑑_𝑛𝑧^𝑛−1|. (2.6) Squaring both sides of (2.6) and integrating around |𝑧|=𝑟, 0<𝑟<1, we obtain (𝐴−𝐵)²+𝐵²|𝜏|²(^𝑘−1_𝑛=2[1+𝛾(𝑛−1)]²𝑛²|𝑎_𝑛|²𝑟^2𝑛−2)≥1|𝜏|²(^𝑘_𝑛=2[1+𝛾(𝑛−1)]²𝑛²|𝑎_𝑛|²𝑟^2𝑛−2)+^∞_𝑛=𝑘+1|𝑑_𝑛|²𝑟^2𝑛−2. (2.7) By letting 𝑟→1, we conclude that (𝐴−𝐵)²+𝐵²|𝜏|²(^𝑘−1_𝑛=2[1+𝛾(𝑛−1)]²𝑛²|𝑎_𝑛|²1)≥|𝜏|²(^𝑘_𝑛=2[1+𝛾(𝑛−1)]²𝑛²|𝑎_𝑛|²) (2.8) or (^𝑘_𝑛=2[1+𝛾(𝑛−1)]²𝑛²|𝑎_𝑛|²)≤(𝐴−𝐵)²|𝜏|²+𝐵²(^𝑘−1_𝑛=2[1+𝛾(𝑛−1)]²𝑛²|𝑎_𝑛|²). (2.9) By making use of the fact that −1≤𝐵<1, we get [1+𝛾(𝑛−1)]²𝑛²|𝑎_𝑛|²≤(𝐴−𝐵)²|𝜏|². (2.10) This gives |𝑎_𝑛|≤(𝐴−𝐵)|𝜏|𝑛(1+𝛾(𝑛−1)),𝑛=2,3,…. (2.11) The result is sharp for the function 𝑧𝑓(𝑧)=𝜏∬¹₀𝑢^1/𝛾−11+𝐴𝜏(𝑢𝑣𝑧)^𝑛−1+𝐵(1−𝜏)(𝑢𝑣𝑧)^𝑛−11+𝐵(𝑢𝑣𝑧)^𝑛−1𝑧𝑑𝑢𝑑𝑣,𝛾>0,₁₀1+𝐴𝜏(𝑢𝑧)^𝑛−1+𝐵(1−𝜏)(𝑢𝑧)^𝑛−11+𝐵(𝑢𝑧)^𝑛−1𝑑𝑢,𝛾=0. (2.12)

Theorem 2.2. Let 𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛. Then a sufficient condition for 𝑓∈𝑃^𝜏_𝛾(𝐴,𝐵) is ^∞_𝑛=2𝑛(1+|𝐵|)[1+𝛾(𝑛−1)]|𝑎_𝑛|≤(𝐴−𝐵)|𝜏|. (2.13) The result is sharp for the function 𝑓(𝑧)=𝑧+(𝐴−𝐵)|𝜏|𝑧𝑛(1+|𝐵|)[1+𝛾(𝑛−1)]^𝑛,𝑛≥2. (2.14)

Proof. In view of (2.13), |1𝜏(𝑓^′(𝑧)+𝛾𝑧𝑓^′′𝐵(𝑧)−1)|−|(𝐴−𝐵)−𝜏(𝑓^′(𝑧)+𝛾𝑧𝑓^′′1(𝑧)−1)|=|𝜏(1+^∞_𝑛=2𝑛𝑎_𝑛𝑧^𝑛−1+𝛾^∞_𝑛=2𝑛(𝑛−1)𝑎_𝑛𝑧^𝑛−1𝐵−1)|−|(𝐴−𝐵)−𝜏(1+^∞_𝑛=2𝑛𝑎_𝑛𝑧^𝑛−1+𝛾^∞_𝑛=2𝑛(𝑛−1)𝑎_𝑛𝑧^𝑛−1≤1−1)||𝜏|^∞_𝑛=2𝑛(1+𝛾(𝑛−1))|𝑎_𝑛||𝑧|^𝑛−1−{(𝐴−𝐵)−|𝐵|(|𝜏|^∞_𝑛=2𝑛(1+𝛾(𝑛−1)))|𝑎_𝑛||𝑧|^𝑛−1} (2.15) which is clearly less than or equal to zero for all |𝑧|=𝑟, 0<𝑟<1. Letting 𝑟→1, we get |𝑓^′(𝑧)+𝛾𝑧𝑓^′′(𝑧)−1(𝐴−𝐵)𝜏−𝐵[𝑓^′(𝑧)+𝛾𝑧𝑓^′′(𝑧)−1]|<1. (2.16) Thus, 𝑓∈𝑃^𝜏_𝛾(𝐴,𝐵).

3. Results Involving Gaussian Hypergeometric Function

Theorem 3.1. Let 𝑎,𝑏∈ℂ⧵{0}. Also, let c be a real number such that 𝑐>|𝑎|+|𝑏|+2. Then a sufficient condition for the function 𝑧𝐹(𝑎,𝑏;𝑐;𝑧) to be in the class 𝑃^𝜏_𝛾(𝐴,𝐵) is that 𝑆≤(𝐴−𝐵)|𝜏|1+|𝐵|+1, (3.1) where 𝑆=Γ(𝑐)Γ(𝑐−|𝑎|−|𝑏|−2)Γ(𝑐−|𝑎|)Γ(𝑐−|𝑏|)×[𝛾|𝑎𝑏|(1+|𝑎|)(1+|𝑏|)+(1+2𝛾)|𝑎𝑏|(𝑐−|𝑎|−|𝑏|−2)+(𝑐−|𝑎|−|𝑏|−2)(𝑐−|𝑎|−|𝑏|−1)]. (3.2)

Proof. 𝑧𝐹(𝑎,𝑏;𝑐;𝑧) has the series representation given by 𝑧𝐹(𝑎,𝑏;𝑐;𝑧)=𝑧+^∞_𝑛=2(𝑎)_𝑛−1(𝑏)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1𝑧^𝑛. (3.3) In view of Theorem 2.2, it suffices to show that 𝑆(𝑎,𝑏,𝑐,𝛾)∶=^∞_𝑛=2𝑛(1+|𝐵|)(1+𝛾(𝑛−1))|(𝑎)_𝑛−1(𝑏)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1|≤(𝐴−𝐵)|𝜏|. (3.4)
From the fact that |(𝑎)_𝑛|≤(|𝑎|)_𝑛, we observe that 𝑐 is real and positive, under the hypothesis 𝑆(𝑎,𝑏,𝑐,𝛾)≤^∞_𝑛=2𝑛[1+𝛾(𝑛−1)](|𝑎|)_𝑛−1(|𝑏|)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1. (3.5) By writing 𝑛[1+𝛾(𝑛−1)] as, 𝛾(𝑛−1)(𝑛−2)+(𝑛−1)(1+2𝛾)+1, we get 𝑆(𝑎,𝑏,𝑐,𝛾)≤𝛾^∞_𝑛=2(𝑛−1)(𝑛−2)(|𝑎|)_𝑛−1(|𝑏|)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1+(1+2𝛾)^∞_𝑛=2(𝑛−1)(|𝑎|)_𝑛−1(|𝑏|)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1+^∞_𝑛=2(|𝑎|)_𝑛−1(|𝑏|)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1=𝛾^∞_𝑛=3(|𝑎|)_𝑛−1(|𝑏|)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−3+(1+2𝛾)^∞_𝑛=2(|𝑎|)_𝑛−1(|𝑏|)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−2+^∞_𝑛=2(|𝑎|)_𝑛−1(|𝑏|)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1. (3.6) Using the fact that (𝑎)_𝑛=𝑎(𝑎+1)_𝑛−1, (3.7) it is easy to see that 𝑆(𝑎,𝑏,𝑐,𝛾)≤𝛾|𝑎𝑏|(1+|𝑎|)(1+|𝑏|)𝑐(1+𝑐)^∞_𝑛=3(2+|𝑎|)_𝑛−3(2+|𝑏|)_𝑛−3(2+𝑐)_𝑛−3(1)_𝑛−3+(1+2𝛾)|𝑎𝑏|𝑐^∞_𝑛=2(1+|𝑎|)_𝑛−2(1+|𝑏|)_𝑛−2(1+𝑐)_𝑛−2(1)_𝑛−2+^∞_𝑛=2(|𝑎|)_𝑛−1(|𝑏|)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1. (3.8) From (1.14), 𝑆(𝑎,𝑏,𝑐,𝛾)≤𝛾|𝑎𝑏|(1+|𝑎|)(1+|𝑏|)𝑐(1+𝑐)𝐹(2+|𝑎|,2+|𝑏|;2+𝑐;1)+(1+2𝛾)|𝑎𝑏|𝑐𝐹(1+|𝑎|,1+|𝑏|;1+𝑐;1)+𝐹(|𝑎|,|𝑏|;𝑐;1)−1. (3.9) By using the Gauss summation theorem 𝐹(𝑎,𝑏;𝑐;1)=Γ(𝑐−𝑎−𝑏)Γ(𝑐),Γ(𝑐−𝑎)Γ(𝑐−𝑏) (3.10) we get 𝑆(𝑎,𝑏,𝑐,𝛾)≤𝛾|𝑎𝑏|(1+|𝑎|)(1+|𝑏|)𝑐(1+𝑐)Γ(𝑐−|𝑎|−|𝑏|−2)Γ(𝑐+2)Γ(𝑐−|𝑎|)Γ(𝑐−|𝑏|)+(1+2𝛾)|𝑎𝑏|𝑐Γ(𝑐−|𝑎|−|𝑏|−1)Γ(𝑐+1)+Γ(𝑐−|𝑎|)Γ(𝑐−|𝑏|)Γ(𝑐−|𝑎|−|𝑏|)Γ(𝑐)Γ(𝑐−|𝑎|)Γ(𝑐−|𝑏|)−1. (3.11) Equation (3.4) now follows by an application of (3.1) and (3.2).

Theorem 3.2. Let 𝑎,𝑏∈ℂ⧵{0}. Also, let 𝑐 be a real number such that 𝑐>|𝑎|+|𝑏|. If 𝑓∈𝑃^𝜏_𝛾(𝐴,𝐵), and if the inequality Γ(𝑐)Γ(𝑐−|𝑎|−|𝑏|)≤1Γ(𝑐−|𝑎|)Γ(𝑐−|𝑏|)1+|𝐵|+1 (3.12) is satisfied, then 𝑧𝐹(𝑎,𝑏;𝑐;𝑧^𝑘)∗𝑓(𝑧)∈𝑃^𝜏_𝛾(𝐴,𝐵), where 𝑘∈ℕ.

Proof. Let 𝑓 be of the form (1.1) belong to the class 𝑃^𝜏_𝛾(𝐴,𝐵). By virtue of Theorem 2.2, it suffices to show that 𝑆₀∶=^∞_𝑛=2(𝑘(𝑛−1)+1)(1+|𝐵|)(1+𝛾𝑘(𝑛−1))|(𝑎)_𝑛−1(𝑏)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1𝑎_{𝑘(𝑛−1)+1}|≤(𝐴−𝐵)|𝜏|. (3.13) Taking into account inequality (2.1) and the relation |(𝑎)_𝑛−1|≤(|𝑎|)_𝑛−1, we deduce that 𝑆₀≤(1+|𝐵|)^∞_𝑛=2(𝑘(𝑛−1)+1)(1+𝛾𝑘(𝑛−1))(𝐴−𝐵)|𝜏||(𝑘(𝑛−1)+1)(1+𝛾𝑘(𝑛−1))(𝑎)_𝑛−1(𝑏)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1|≤(1+|𝐵|)(𝐴−𝐵)|𝜏|^∞_𝑛=2(|𝑎|)_𝑛−1(|𝑏|)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1=(1+|𝐵|)(𝐴−𝐵)|𝜏|(𝐹(|𝑎|,|𝑏|;𝑐;1)−1) (3.14) which is bounded previously by (𝐴−𝐵)|𝜏|, in view of inequality (3.12).

Repeating the previous reasoning for 𝑏=𝑎, we can improve the assertion of Theorem 3.2 as follows.

Theorem 3.3. Let 𝑎∈ℂ⧵{0}. Also, let 𝑐 be a real number such that 𝑐>max{0,2ℜ(𝑎)}. If 𝑓∈𝑃^𝜏_𝛾(𝐴,𝐵), and if the inequality Γ(𝑐)Γ(𝑐−2ℜ(𝑎))Γ(𝑐−|𝑎|)Γ(𝑐−|≤1𝑎|)1+|𝐵|+1 (3.15) is satisfied, then 𝑧𝐹(𝑎,𝑎;𝑐;𝑧^𝑘)∗𝑓(𝑧)∈𝑃^𝜏_𝛾(𝐴,𝐵), where 𝑘∈ℕ.

In the special case when 𝑏=1, Theorem 3.2 immediately yields the following new result.

Theorem 3.4. Let 𝑎∈ℂ⧵{0}. Also, let 𝑐 be a real number such that 𝑐>|𝑎|+1. If 𝑓∈𝑃^𝜏_𝛾(𝐴,𝐵), and if the inequality 𝑐−1≤1𝑐−|𝑎|−11+|𝐵|+1 (3.16) is satisfied, then 𝑧𝐹(𝑎,1;𝑐;𝑧^𝑘)∗𝑓(𝑧)∈𝑃^𝜏_𝛾(𝐴,𝐵), where 𝑘∈ℕ.

Theorem 3.5. Let 𝑎,𝑏∈ℂ⧵{0}. Also, let 𝑐 be a real number such that 𝑐>|𝑎|+|𝑏|+3. If 𝑓∈𝒮, and if the inequality |𝑎||𝑏|Γ(𝑐)Γ(𝑐−|𝑎|−|𝑏|−1)Γ(𝑐−|𝑎|)Γ(𝑐−|𝑏|)[4+(1+|𝑎|)(1+|𝑏|)𝑐−|𝑎|−|𝑏|−2(5+(2+|𝑎|)(2+|𝑏|)+𝑐−|𝑎|−|𝑏|−3)]Γ(𝑐)Γ(𝑐−|𝑎|−|𝑏|)Γ(𝑐−|𝑎|)Γ(𝑐−|𝑏|)[1+3|𝑎||𝑏|+𝑐−|𝑎|−|𝑏|−1|𝑎||𝑏|(1+|𝑎|)(1+|𝑏|)]≤𝑐(1+𝑐)(𝐴−𝐵)|𝜏|1+|𝐵|+1 (3.17) is satisfied, then 𝐼_{𝑎,𝑏,𝑐}(𝑓)∈𝑃^𝜏_𝛾(𝐴,𝐵).

Proof. Let 𝑓∈𝒮. Applying the well-known estimate for the coefficients of the functions 𝑓∈𝒮, due to de Branges [23], we need to show that ^∞_𝑛=2𝑛²[1+𝛾(𝑛−1)]|(𝑎)_𝑛−1(𝑏)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1|≤(𝐴−𝐵)|𝜏|.1+|𝐵| (3.18) The left-hand side of (3.18) can be written as ^∞_𝑛=2𝑛²|(𝑎)_𝑛−1(𝑏)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1|+𝛾^∞_𝑛=2𝑛²(𝑛−1)|(𝑎)_𝑛−1(𝑏)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1|. (3.19) The second expression of (3.19), by virtue of the triangle inequality for the pochhammer symbol |(𝑎)_𝑛−1|≤(|𝑎|)_𝑛−1, is less than or equal to 𝛾^∞_𝑛=2𝑛²(𝑛−1)(|𝑎|)_𝑛−1(|𝑏|)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1=𝛾^∞_𝑛=1(𝑛+1)²(|𝑎|)_𝑛(|𝑏|)_𝑛(𝑐)_𝑛(1)_𝑛−1=∶𝑆₁. (3.20) Now, making use of the relation (3.7), we get 𝑆₁=𝛾|𝑎𝑏|𝑐^∞_𝑛=1(𝑛+1)²(|𝑎|+1)_𝑛−1(|𝑏|+1)_𝑛−1(𝑐+1)_𝑛−1(1)_𝑛−1=𝛾|𝑎𝑏|𝑐^∞_𝑛=0(𝑛+2)²(|𝑎|+1)_𝑛(|𝑏|+1)_𝑛(𝑐+1)_𝑛(1)_𝑛=𝛾|𝑎𝑏|𝑐^∞_𝑛=0𝑛(𝑛−1)(|𝑎|+1)_𝑛(|𝑏|+1)_𝑛(𝑐+1)_𝑛(1)_𝑛+5𝛾|𝑎𝑏|𝑐^∞_𝑛=0𝑛(|𝑎|+1)_𝑛(|𝑏|+1)_𝑛(𝑐+1)_𝑛(1)_𝑛+4𝛾|𝑎𝑏|𝑐^∞_𝑛=0(|𝑎|+1)_𝑛(|𝑏|+1)_𝑛(𝑐+1)_𝑛(1)_𝑛, (3.21) where we are writing (𝑛+2)²=𝑛(𝑛−1)+5𝑛+4. By repeating the use of (3.7) and the Gauss summation formula, we have 𝑆₁≤|𝑎𝑏|(|𝑎|+1)(|𝑏|+1)(|𝑎|+2)(|𝑏|+2)Γ(𝑐)Γ(𝑐−|𝑎|−|𝑏|−3)+Γ(𝑐−|𝑎|)Γ(𝑐−|𝑏|)5|𝑎𝑏|(|𝑎|+1)(|𝑏|+1)Γ(𝑐)Γ(𝑐−|𝑎|−|𝑏|−2)+Γ(𝑐−|𝑎|)Γ(𝑐−|𝑏|)4|𝑎𝑏|Γ(𝑐)Γ(𝑐−|𝑎|−|𝑏|−1)=Γ(𝑐−|𝑎|)Γ(𝑐−|𝑏|)|𝑎𝑏|Γ(𝑐)Γ(𝑐−|𝑎|−|𝑏|)Γ(𝑐−|𝑎|)Γ(𝑐−|𝑏|)[4+(1+|𝑎|)(1+|𝑏|)𝑐−|𝑎|−|𝑏|−2(5+(2+|𝑎|)(2+|𝑏|)𝑐−|𝑎|−|𝑏|−3)]. (3.22) As a next step, we consider the first expression of equation. By making use of the triangle inequality for the pochhammer symbol as stated in evaluating 𝑆₁, we get ^∞_𝑛=2𝑛²|(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+2)²(|𝑎|)_𝑛+1(|𝑏|)_𝑛+1(𝑐)_𝑛+1(1)_𝑛+1∶=𝑆₂. (3.23) Now making use of relation (3.7), we obtain 𝑆₂=^∞_𝑛=0(𝑛+1)²(|𝑎|)_𝑛+1(|𝑏|)_𝑛+1(𝑐)_𝑛+1(1)_𝑛+1+2^∞_𝑛=0(𝑛+1)(|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+1, (3.24) where we write (𝑛+2)²=(𝑛+1)²+2(𝑛+1)+1. By repeating the use of (3.7) and the Gauss summation formula, we have 𝑆₂≤Γ(𝑐)Γ(𝑐−|𝑎|−|𝑏|)[Γ(𝑐−|𝑎|)Γ(𝑐−|𝑏|)|𝑎𝑏|(|𝑎|+1)(|𝑏|+1)+𝑐(1+𝑐)3|𝑎𝑏|𝑐−|𝑎|−|𝑏|+1]. (3.25) The proof of Theorem 3.5 now follows by an application of the inequalities of the terms dealing with 𝑆₁,𝑆₂ and inequality (3.17).

Repeating the previous reasoning for 𝑏=𝑎, we can improve the assertion of Theorem 3.5 as follows.

Theorem 3.6. Let 𝑎,𝑏∈ℂ⧵{0}. Also, let 𝑐 be a real number such that 𝑐>max{0,2ℜ(𝑎)+3)}. If 𝑓∈𝒮, and if the inequality |𝑎|²Γ(𝑐)Γ(𝑐−2ℜ(𝑎)−1)Γ(𝑐−|𝑎|)Γ(𝑐−|𝑎|)[4+(1+|𝑎|)(1+|𝑎|)𝑐−2ℜ(𝑎)−2(5+(2+|𝑎|)(2+|𝑎|)+𝑐−2ℜ(𝑎)−3)]Γ(𝑐)Γ(𝑐−2ℜ(𝑎))Γ(𝑐−|𝑎|)Γ(𝑐−|𝑎|)[1+3|𝑎|²+𝑐−2ℜ(𝑎)−1|𝑎|²(1+|𝑎|)(1+|𝑎|)]≤𝑐(1+𝑐)(𝐴−𝐵)|𝜏|1+|𝐵|+1 (3.26) is satisfied, then 𝐼_{𝑎,𝑎,𝑐}(𝑓)∈𝑃^𝜏_𝛾(𝐴,𝐵).

In the special case when 𝑏=1, Theorem 3.2 immediately yields a result concerning the Carlson-Shaffer operator ℒ(𝑎,𝑐).

Theorem 3.7. Let 𝑎∈ℂ⧵{0}. Also, let 𝑐 be a real number such that 𝑐>|𝑎|+4. If 𝑓∈𝒮, and if the inequality |𝑎|(𝑐−1)(𝑐−|𝑎|−1)(𝑐−|𝑎|−2)[4+2(1+|𝑎|)𝑐−|𝑎|−3(5+3(2+|𝑎|)+𝑐𝑐−|𝑎|−4)](𝑐−|𝑎|−1)[1+3|𝑎|+𝑐−|𝑎|−22|𝑎|(1+|𝑎|)]≤𝑐(1+𝑐)(𝐴−𝐵)|𝜏|1+|𝐵|+1 (3.27) is satisfied, then ℒ(𝑎,𝑐)(𝑓)∈𝑃^𝜏_𝛾(𝐴,𝐵).

Theorem 3.8. Let 𝑎,𝑏∈ℂ⧵{0}. Also, let 𝑐 be a real number such that 𝑐>|𝑎|+|𝑏|+𝑃₁, where 𝑃₁=𝑃₁(𝑘) is given with (1.22). If, for some 𝑘 (0≤𝑘<∞), 𝑓∈𝑘−𝑈𝐶𝑉, and the inequality ₃𝐹₂(|𝑎|,|𝑏|,𝑃₁𝑃;𝑐,1;1)+¹𝛾|𝑎𝑏|𝑐³𝐹₂(|𝑎|+1,|𝑏|+1,𝑃₁+1;𝑐+1,2;1)≤(𝐴−𝐵)|𝜏|1+|𝐵|+1 (3.28) is satisfied, then 𝐼_{𝑎,𝑏,𝑐}(𝑓)∈𝑃^𝜏_𝛾(𝐴,𝐵).

Proof. By means of (1.17) and (2.13), the following inequality must be satisfied: ^∞_𝑛=2𝑛[1+𝛾(𝑛−1)]|(𝑎)_𝑛−1(𝑏)_𝑛−1(𝑐)_𝑛−1(1)_𝑛−1𝑎_𝑛|≤(𝐴−𝐵)|𝜏|.1+|𝐵| (3.29)
Applying the estimates for the coefficients given by (1.21), and making use of the relations (3.7) and |(𝑑)_𝑛|≤(|𝑑|)_𝑛, condition (3.29) will be satisfied if (1+|𝐵|)[₃𝐹₂(|𝑎|,|𝑏|,𝑃₁𝑃;𝑐,1;1)−1]+(1+|𝐵|)¹𝛾|𝑎𝑏|𝑐³𝐹₂(|𝑎|+1,|𝑏|+1,𝑃₁+1;𝑐+1,2;1)≤(𝐴−𝐵)|𝜏| (3.30) provided 𝑐>|𝑎|+|𝑏|+𝑃₁. The proof of the Theorem 3.8 is now completed by virtue of hypothesis (3.28).

Theorem 3.9. Let 𝑎,𝑏∈ℂ⧵{0}. Also, let 𝑐 be a real number such that 𝑐>|𝑎|+|𝑏|+𝑃₁, where 𝑃₁=𝑃₁(𝑘) is given with (1.22). If, for some 𝑘(0≤𝑘<∞),𝑓∈𝑘−𝑆𝑇, and the inequality ₃𝐹₂(|𝑎|,|𝑏|,𝑃₁𝑃;𝑐,1;1)+¹(1+𝛾)|𝑎𝑏|𝑐³𝐹₂(|𝑎|+1,|𝑏|+1,𝑃₁+𝑃+1;𝑐+1,2;1)₁𝛾|𝑎𝑏|𝑐³𝐹₂(|𝑎|+1,|𝑏|+1,𝑃₁+1;𝑐+1,1;1)≤(𝐴−𝐵)|𝜏|1+|𝐵|+1 (3.31) is satisfied, then 𝐼_{𝑎,𝑏,𝑐}(𝑓)∈𝑃^𝜏_𝛾(𝐴,𝐵).

Proof. Proceeding as in the proof of Theorem 3.8, and applying the estimates for the coefficients given by (1.24) instead of (1.21), and making use of relations (3.7) and |(𝑑)_𝑛|≤(|𝑑|)_𝑛, the proof of the theorem by virtue of hypothesis (3.31) is complete.

Acknowledgment

The authors sincerely thank the referees for their suggestions.