# Connections between Certain Subclasses of Analytic Univalent Functions Based on Operators

Authors: R. Ezhilarasi; T. V. Sudharsan; Maisarah Haji Mohd; K. G. Subramanian
Journal: Journal of Complex Analysis
 (2017)

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## Abstract

In this paper, by applying the Hohlov linear operator, connections between the classSD(α),  α≥0, and two subclasses of the class A of normalized analytic functions are established. Also an integral operator related to hypergeometric function is considered.


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## Body

## 1. Introduction

LetA denote the family of functions f that are analytic in the open unit disk Δ={z:|z|<1} with the normalization(1)fz=z+∑n=2∞anzn. Let S denote the subclass of functions in A which are also univalent in ∆. A well-known subclass of S is the class ST (see, e.g., [1]) of starlike functions f of the form (1), satisfying Rezf′(z)/f(z)>0,z∈Δ. Another class UCD⁡(α),α≥0, introduced in [2], consists of functions f∈A satisfying Re(f′(z))≥α|zf′′(z)|,z∈Δ. Various properties of this class have been obtained in [2–4]. A related class SD(α) has been recently considered in [5], initially introduced in [6]. A function f of the form (1) is said to be in the class SD(α) if (2)Re⁡fzz≥αf′z-fzz,forα≥0.Theorem 1 (see [5]).
A functionf of the form (1) is in the class SD(α) if(3)∑n=2∞1+αn-1an≤1.Ponnusamy and Ronning [7] introduced and studied the class Rη(β)⊂S,  (0≤β<1) of functions f∈A for which there exists a number η∈(-π/2,π/2) such that Re⁡{eiη[f′(z)-β]}>0,z∈∆. If the function f of the form (1) belongs to the class Rη(β), then(4)an≤21-βcos⁡ηn,n∈N∖1. For complex numbers a,  b, and c(c≠0,-1,-2,…), the Gaussian hypergeometric function F12(z) is defined by(5)F21z=F21a,b,c;z=∑n=0∞anbncnznn!, where λn is the Pochhammer symbol given by(6)λn=1,n=0λλ+1λ+2⋯λ+n-1,n∈N=Γλ+nΓλ. It is known that(7)F12a,b,c;1=ΓcΓc-a-bΓc-aΓc-b,Re⁡c-a-b>0,c≠0,-1,-2,…. The Hadamard product (or convolution) of two functions f defined by (1) and g given by g(z)=z+∑n=2∞bnzn is defined by(8)f∗gz=z+∑n=2∞anbnzn. Hohlov [8] introduced a linear operator Ia,b,c:A→A, corresponding to the Gaussian hypergeometric function F12 which is defined by the convolution(9)Ia,b,cfz=zF12a,b,c,z∗fz,f∈A. For a function f of the form (1), we have(10)Ia,b,cfz=z+∑n=2∞an-1bn-1cn-11n-1anzn,z∈∆. The operator Ia,b,c is a natural extension of several operators such as Alexander, Libera, Bernardi, and Carlson-Shaffer operators denoted, respectively, by A,  L,  B, and L(a,c).Motivated by the work of Thulasiram et al. [9], in this paper, by applying the linear operator Ia,b,c, we establish some interesting connections between the class SD(α) and the classes ST and Rη(β),(β<1) consisting of functions f given by (1). Also we consider an integral operator related to the hypergeometric functions.


## 2. Main Results

In the sequel the functionf∈A is given by (1).Theorem 2.
Leta,b∈C∖{0}. Also let c be a real number such that c>|a|+|b|+2. If f∈ST and if the inequality (11)ΓcΓc-a-bΓc-aΓc-babc-a-b-1αa+1b+1c-a-b-2+2α+1+1≤2 is satisfied, then Ia,b,c(f)∈SD(α).Proof.
Letf∈ST. Applying the well-known estimate due to Nevanlinna [10] for the coefficients of the functions f∈ST, in view of Theorem 1, we need to prove that (12)∑n=2∞n1+αn-1an-1bn-1cn-11n-1≤1. By virtue of the relation |(d)n|≤(|d|)n, and on writing n+2=(n+1)+1 and (n+2)2=(n+1)2+2(n+1)+1 and using the fact that (a)n+k=(a)k(a+k)n, we have(13)∑n=2∞n1+αn-1an-1bn-1cn-11n-1≤∑n=2∞n1+αn-1an-1bn-1cn-11n-1=∑n=1∞n+1+αn+12-n+1anbncn1n=α∑n=0∞n+22an+1bn+1cn+11n+1+1-α∑n=0∞n+2an+1bn+1cn+11n+1=α∑n=0∞n+12an+1bn+1cn+11n+1+2α∑n=0∞n+1an+1bn+1cn+11n+1+1-α∑n=0∞n+1an+1bn+1cn+11n+1+∑n=0∞an+1bn+1cn+11n+1=α∑n=0∞an+2bn+2cn+21n+2α+1∑n=0∞an+1bn+1cn+11n+∑n=0∞anbncn1n-1=αa2b2c2∑n=0∞a+2nb+2nc+2n1n+2α+1abc∑n=0∞a+1nb+1nc+1n1n+∑n=0∞anbncn1n-1=αa2b2c2Γc+2Γc-a-b-2Γc-aΓc-b+2α+1abcΓc+1Γc-a-b-1Γc-aΓc-b+ΓcΓc-a-bΓc-aΓc-b-1=ΓcΓc-a-bΓc-aΓc-babc-a-b-1αa+1b+1c-a-b-2+2α+1+1-1≤1,which is satisfied by the hypothesis.On settingb=a¯, an improvement of the assertion of Theorem 2 is obtained as given in Theorem 3.Theorem 3.
Leta∈C∖{0}. Also let c be a real number such that c>max⁡{0,2Re⁡(a)+2}. If f∈ST, and if the inequality (14)ΓcΓc-2Re⁡aΓc-aΓc-a¯a2c-2Re⁡a-1αa+1a¯+1c-2Re⁡a-2+2α+1+1≤2 is satisfied, then Ia,a¯,c(f)∈SD(α).Theorem 4.
Leta,b∈C∖{0}. Also, let c be a real number such that c>|a|+|b|,|a|≠1,|b|≠1. If f∈Rη(β),(β<1) and the inequality (15)21-βcos⁡ηa-1b-1ΓcΓc-a-bΓc-aΓc-bc-a-b+αab+1-c-c+ab-a-b-αc-1≤1 is satisfied, then Ia,b,c(f)∈SD(α).Proof.
Letf be of the form (1) and let f∈Rη(β),(β<1). By virtue of Theorem 1 and in view of (10), it remains to show that (16)∑n=2∞1+αn-1an-1bn-1cn-11n-1an≤1. Using the inequality (4) and the relations (d)n=d(d+1)n-1 and |(d)n|≤(|d|)n, we obtain that(17)∑n=2∞1+αn-1an-1bn-1cn-11n-1an≤21-βcos⁡η∑n=2∞1+αn-1nan-1bn-1cn-11n-1=2α1-βcos⁡η∑n=2∞an-1bn-1cn-11n-1+21-α1-βcos⁡η∑n=2∞an-1bn-1cn-11n=2α1-βcos⁡η∑n=1∞anbncn1n+21-α1-βcos⁡η∑n=2∞a-1n/a-1b-1n/b-1c-1n/c-11n=2α1-βcos⁡η∑n=0∞anbncn1n-1+21-α1-βc-1cos⁡ηa-1b-1∑n=0∞a-1nb-1nc-1n1n-1-a-1b-1c-1=2α1-βcos⁡ηΓcΓc-a-bΓc-aΓc-b-1+21-α1-βc-1cos⁡ηa-1b-1Γc-1Γc-a-b+1Γc-aΓc-b-1-a-1b-1c-1=2α1-βcos⁡ηΓcΓc-a-bΓc-aΓc-b-1+21-α1-βcos⁡ηa-1b-1ΓcΓc-a-b·c-a-bΓc-aΓc-b-c-1-a-1b-1=21-βcos⁡ηΓcΓc-a-bΓc-aΓc-bα+1-αc-a-ba-1b-1+21-βcos⁡η-α-1-αc-1a-1b-1-1-α=21-βcos⁡ηa-1b-1ΓcΓc-a-bΓc-aΓc-bc-a-b+αab+1-c-c+ab-a-b-αc-1≤1.On takingb=a¯, an improvement of the assertion of Theorem 4 is obtained as given in Theorem 5.Theorem 5.
Leta∈C∖{0}. Further let c be a real number such that c>max⁡{0,2Re⁡(a)},  a≠1. If f∈Rη(β),(β<1) and the inequality (18)21-βcos⁡ηa-1a¯-1ΓcΓc-2Re⁡aΓc-aΓc-a¯c-2Re⁡a+αa2+1-c-c-a2-2Re⁡a-αc-1≤1is satisfied, then Ia,a¯,c(f)∈SD(α).


## 3. An Integral Operator

We now obtain results in connection with a particular integral operator [11] G(a,b;c;z) defined by(19)Ga,b;c;z=∫0zFa,b;c;tdt, where F(a,b;c;z)=F12(z) is given by (5).Theorem 6.
Leta,b∈C-{0}. Also let c be a real number such that c>|a|+|b|,|a|≠1,|b|≠1. Let G(a,b;c;z) be given by (19). If the hypergeometric inequality (20)ΓcΓc-a-bΓc-aΓc-bc1-α+αab+1-a+ba-1b-1-1-αc-1a-1b-1≤2is satisfied, then G(a,b,c,z)∈SD(α).Proof.
The functionG(a,b;c;z) has the series representation given by(21)z+∑n=2∞an-1bn-1cn-11nzn. In view of Theorem 1, it is enough to prove that(22)∑n=2∞1+αn-1an-1bn-1cn-11n≤1. Now(23)∑n=2∞1+αn-1an-1bn-1cn-11n≤1-α∑n=2∞an-1bn-1cn-11n+α∑n=2∞nan-1bn-1cn-11n=1-α∑n=2∞a-1nb-1nc-1a-1b-1c-1n1n+α∑n=2∞an-1bn-1cn-11n-1=1-αc-1a-1b-1∑n=0∞a-1nb-1nc-1n1n-1-a-1b-1c-1+α∑n=1∞anbncn1n=ΓcΓc-a-bΓc-aΓc-b1-αc-a-ba-1b-1+α-1-αc-1a-1b-1-1≤1,by hypothesis.A result analogous to Theorem6 can be stated for the class UCD(α) in Theorem 7.Theorem 7.
Leta,b∈C-{0}. Also let c be a real number such that c>|a|+|b|+1. Let f∈A and be of the form (1). If the hypergeometric inequality(24)ΓcΓc-a-b-1Γc-aΓc-bc-a-b+αab-1≤2 is satisfied, then G(a,b,c,z)∈UCD(α).



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