# A New Class of Meromorphically Analytic Functions with Applications to the Generalized Hypergeometric Functions

Authors: Firas Ghanim; Maslina Darus
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We introduce a new subclass of meromorphically analytic functions, which is defined by means of a Hadamard product (or convolution). A characterization property such as the coefficient bound is obtained for this class. The other related properties, which are investigated in this paper, include the distortion and the radius of starlikeness. We also consider several applications of our main results to the generalized hypergeometric functions.


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## Body

## 1. Introduction

Let𝒜 be the class of functions f which are analytic in the open unit disk (1.1)U={z∈C:|z|<1}.
As usual, we denote by S the subclass of 𝒜, consisting of functions which are also univalent in U.Letw be a fixed point in U and A(w)={f∈H(D):f(w)=f′(w)-1=0}. In [1], Kanas and Ronning introduced the following classes(1.2)Sw={f∈A(w):fis univalent inU},STw={f∈A(w):Re((z-w)f′(z)f(z))>0,z∈U},CVw={f∈A:1+(Re(z-w)f′′(z)f′(z))>0,z∈U}.
Later, Acu and Owa [2] studied the classes extensively.The classSTw is defined by geometric property that the image of any circular arc centered at w is starlike with respect to f(w), and the corresponding class Swc is defined by the property that the image of any circular arc centered at w is convex. We observed that the definitions are somewhat similar to the ones introduced by Goodman in [3, 4] for uniformly starlike and convex functions except that, in this case, the point w is fixed.LetΣw denote the subclass of A(w) consisting of the function of the form(1.3)f(z)=1z-w+∑n=1∞an(z-w)n.
The functions f in Σw are said to be starlike functions of order β if and only if(1.4)Re{-(z-w)f′(z)f(z)}>β((z-w)∈U),
for some β(0≤β<1). We denote by Sw*(β) the class of all starlike functions of order β. Similarly, a function f in Sw is said to be convex of order β if and only if(1.5)Re(-1-(z-w)f′′(z)f′(z))>β((z-w)∈U),
for some β(0≤β<1). We denote by Cw(β) the class of all convex functions of order β.For the functionf∈Σw, we define (1.6)Iλ0f(z)=f(z),Iλ1f(z)=(z-w)f′(z)+2z-w,Iλ2f(z)=(z-w)(I1f(z))′+2z-w,
and, for k=1,2,3,…, we can write (1.7)Iλkf(z)=(z-w)(Ik-1f(z))′+2z-w=1z-w+∑n=1∞[1+λ(n-1)]kan(z-w)n,
where λ≥1,k≥0 and (z-w∈U).The differential operatorI1k is studied extensively by Ghanim and Darus [5, 6] and Ghanim et al. [7].The Hadamard product or convolution of the functionsf given by (1.3) with the function g and h given, respectively, by(1.8)g(z)=1z-w+∑n=1∞bn(z-w)n,h(z)=1z-w+∑n=1∞cn(z-w)n,
can be expressed as follows:(1.9)(f*g)(z)=1z-w+∑n=1∞anbn(z-w)n,(f*h)(z)=1z-w+∑n=1∞ancn(z-w)n.
Suppose that f and g are two analytic functions in the unit disk U. Then, we say that the function g is subordinate to the function f, and we write (1.10)g(z)≺f(z)(z∈U),
if there exists a Schwarz function ϖ(z) with ϖ(0)=0 and |ϖ(z)|<1 such that (1.11)g(z)=f(ϖ(z))(z∈U).
By applying the above subordination definition, we introduce here a new class Σw(A,B,k,α,λ) of meromorphically functions, which is defined as follows:Definition 1.1.
A functionf∈Σw of the form (1.3) is said to be in the class Σw(A,B,k,α,λ) if it satisfies the following subordination property:
(1.12)αIλk(f*g)(z)Iλk(f*h)(z)≺α-(A-B)(z-w)1+B(z-w)((z-w)∈U),
where -1≤B<A≤1,k≥0,α>0,λ≥1, with condition Iλk(f*h)(z)≠0.The purpose of this paper is to investigate the coefficient estimates, distortion properties, and the radius of starlikeness for the classΣw(A,B,k,α,λ). Some applications of the main results involving generalized hypergeometric functions are also considered.


## 2. Characterization and Other Related Properties

In this section, we begin by proving a characterization property which provides a necessary and sufficient condition for a functionf∈Σw of the form (1.3) to belong to the class Σw(A,B,k,α,λ) of meromorphically analytic functions.Theorem 2.1.
The functionf∈Σw is said to be a member of the class Σw(A,B,k,α,λ) if it satisfies
(2.1)∑n=1∞[1+λ(n-1)]k(αbn(1+B)-cn(α(1+B)+A-B))an≤A-B.
The equality is attained for the function fn(z) given by
(2.2)fn(z)=1z-w+(A-B)[1+λ(n-1)]k(αbn(1+B)-cn(α(1+B)+A-B))(z-w)n.Proof.
Letf∈Σw(A,B,k,α,λ), and suppose that
(2.3)αIλk(f*g)(z)Iλk(f*h)(z)=α-(A-B)(z-w)1+B(z-w).
Then, in view of (2.2), we have
(2.4)|α∑n=1∞[1+λ(n-1)]kan(bn-cn)(z-w)n+1(A-B)-∑n=1∞[1+λ(n-1)]kan(αBbn+{(A-B)-αB}cn)(z-w)n+1|≤α∑n=1∞[1+λ(n-1)]kan(bn-cn)|z-w|n+1(A-B)-∑n=1∞[1+λ(n-1)]kan(αBbn+{(A-B)-αB}cn)|z-w|n+1≤1.
Letting (z-w)→1, we get
(2.5)∑n=1∞[1+λ(n-1)]k(αbn(1+B)-cn(α(1+B)+A-B))an≤(A-B),
which is equivalent to our condition of the theorem, so that f∈Σw(A,B,k,α,λ). Hence we have the theorem.Theorem2.1 immediately yields the following result.Corollary 2.2.
If the functionf∈Σw belongs to the class Σw(A,B,k,α,λ), then
(2.6)an≤(A-B)[1+λ(n-1)]k(αbn(1+B)-cn(α(1+B)+A-B)),n≥1, where the equality holds true for the functions fn(z) given by (2.2).We now state the following growth and distortion properties for the classΣw(A,B,k,α,λ).Theorem 2.3.
If the functionf defined by (1.3) is in the class Σw(A,B,k,α,λ), then for 0<|z-w|=r<1, one has
(2.7)1r-(A-B)(αb1(1+B)-c1(α(1+B)+A-B))r≤|f(z)|≤1r+(A-B)(αb1(1+B)-c1(α(1+B)+A-B))r,1r2-(A-B)(αb1(1+B)-c1(α(1+B)+A-B))≤|f′(z)|≤1r2+(A-B)(αb1(1+B)-c1(α(1+B)+A-B)).Proof.
Sincef∈Σw(A,B,k,α,λ), Theorem 2.1 readily yields the inequality
(2.8)∑n=1∞an≤(A-B)(αb1(1+B)-c1(α(1+B)+A-B)).
Thus, for 0<|z-w|=r<1 and utilizing (2.8), we have
(2.9)|f(z)|≤1|z-w|+∑n=1man|(z-w)|n≤1r+r∑n=1man≤1r+(A-B)(αb1(1+B)-c1(α(1+B)+A-B))r,|f(z)|≥1|z-w|-∑n=1man|(z-w)|n≥1r-r∑n=1man≥1r-(A-B)(αb1(1+B)-c1(α(1+B)+A-B))r.
Also, from Theorem 2.1, we get
(2.10)∑n=1∞nan≤(A-B)(αb1(1+B)-c1(α(1+B)+A-B)).
Hence
(2.11)|f′(z)|≤1|z-w|2+∑n=1mnan|(z-w)|n-1≤1r+∑n=1mnan≤1r2+(A-B)(αb1(1+B)-c1(α(1+B)+A-B)),|f′(z)|≥1|z-w|2-∑n=1mnan|(z-w)|n-1≥1r2-∑n=1mnan≥1r2-(A-B)(αb1(1+B)-c1(α(1+B)+A-B)).
This completes the proof of Theorem 2.3.We next determine the radius of meromorphically starlikeness of the classΣw(A,B,k,α,λ), which is given by Theorem 2.4.Theorem 2.4.
If the functionf defined by (1.3) is in the class Σw(A,B,k,α,λ), then f is meromorphically starlike of order δ in the disk |z-w|<r1, where
(2.12)r1=inf⁡n≥1{(1-δ)(αbn(1+B)-cn(α(1+B)+A-B))(n+2-δ)(A-B)}1/(n+1).
The equality is attained for the function fn(z) given by (2.2).Proof.
It suffices to prove that(2.13)|(z-w)(Ikf(z))′Ikf(z)+1|≤1-δ.
For |z-w|<r1, we have
(2.14)|(z-w)(Ikf(z))′Ikf(z)+1|=|∑n=1∞(n+1)[1+λ(n-1)]kan(z-w)n1/(z-w)+∑n=1∞[1+λ(n-1)]kan(z-w)n|=|∑n=1∞(n+1)[1+λ(n-1)]kan(z-w)n+11+∑n=1∞[1+λ(n-1)]kan(z-w)n+1|≤∑n=1∞(n+1)[1+λ(n-1)]kan|z-w|n+11-∑n=1∞[1+λ(n-1)]kan|z-w|n+1.
Hence (2.14) holds true for
(2.15)∑n=1∞(n+1)[1+λ(n-1)]kan|z-w|n+1≤(1-δ)(1-∑n=1∞[1+λ(n-1)]kan|z-w|n+1)
or
(2.16)∑n=1∞(n+2-δ)[1+λ(n-1)]kan|z-w|n+1(1-δ)≤1.
With the aid of (2.1) and (2.16), it is true to say that for fixed n(2.17)(n+2-δ)[1+λ(n-1)]k|z-w|n+1(1-δ)≤[1+λ(n-1)]k(αbn(1+B)-cn(α(1+B)+A-B))(A-B)(n≥1).
Solving (2.17) for |z-w|, we obtain
(2.18)|z-w|<{(1-δ)(αbn(1+B)-cn(α(1+B)+A-B))(n+2-δ)(A-B)}1/(n+1).
This completes the proof of Theorem 2.4.


## 3. Applications Involving Generalized Hypergeometric Functions

Let us define the functionϕ̃(a,c;z) by(3.1)ϕ̃(a,c;z)=1z-w+∑n=0∞|(a)n+1(c)n+1|an(z-w)n,
for c≠0,-1,-2,…, and a∈ℂ/{0}, where (λ)n=λ(λ+1)n+1 is the Pochhammer symbol. We note that (3.2)ϕ̃(a,c;z)=1z-w2F1(1,a,c;z),
where (3.3)2F1(b,a,c;z)=∑n=0∞(b)n(a)n(c)n(z-w)nn!.
Corresponding to the function ϕ̃(a,c;z) and using the Hadamard product which was defined earlier in the introduction section for f(z)∈Σ, we define here a new linear operator L*(a,c) on Σ by (3.4)Lw*(a,c)f(z)=ϕ̃(a,c;z)*f(z)=1z-w+∑n=1∞|(a)n+1(c)n+1|an(z-w)n.
For a function f∈Lw*(a,c)f(z), we define (3.5)I0(Lw*(a,c)f(z))=Lw*(a,c)f(z),
and, for k=1,2,3,…,(3.6)Ik(Lw*(a,c)f(z))=z(Ik-1L*(a,c)f(z))′+2z-w=1z-w+∑n=1∞nk|(a)n+1(c)n+1|an(z-w)n.
We note Ik(Lw*(a,a)f(z)) studied by Ghanim and Darus [5, 6] and Ghanim et al. [7], and also, Ik(L0*(a,c)f(z)) studied by Ghanim and Darus [8, 9] and Ghanim et al. [10].The subordination relation (1.12) in conjunction with (3.4) and (3.6) takes the following form: (3.7)αIkLw*(a+1,c)f(z)IkLw*(a,c)f(z)=α-(A-B)(z-w)1+B(z-w)(0≤B<A≤1,k≥0,α>0).Definition 3.1.
A functionf∈Σw of the form (1.3) is said to be in the class Σw(A,B,k,α,a,c) if it satisfies the subordination relation (3.7) above.Theorem 3.2.
The functionf∈Σw is said to be a member of the class Σw(A,B,k,α,a,c) if it satisfies
(3.8)∑n=1∞nk(αbn(1+B)-cn(α(1+B)+A-B))|(a)n+1||(c)n+1|an≤(A-B).
The equality is attained for the function fn(z) given by
(3.9)fn(z)=1z-w+∑n=1∞(A-B)|(c)n+1|nk(αbn(1+B)-cn(α(1+B)+A-B))|(a)n+1|(z-w)n,n≥1.Proof.
By using the same technique employed in the proof of Theorem2.1 along with Definition 3.1, we can prove Theorem 3.2.The following consequences of Theorem3.2 can be deduced by applying (3.8) and (3.9) along with Definition 3.1.Corollary 3.3.
If the functionf∈Σw belongs to the class Σw(A,B,k,α,a,c), then
(3.10)an≤(A-B)|(c)n+1|nk(αbn(1+B)-cn(α(1+B)+A-B))|(a)n+1|,n≥1, where the equality holds true for the functions fn(z) given by (3.9).Corollary 3.4.
If the functionf defined by (1.3) is in the class Σw(A,B,k,α,a,c), then f is meromorphically starlike of order δ in the disk |z-w|<r3, where
(3.11)r3=inf⁡n≥1{(1-δ)(αbn(1+B)-cn(α(1+B)+A-B))|(c)n+1|(n+2-δ)(A-B)|(a)n+1|}1/(n+1).
The equality is attained for the function fn(z) given by (3.9).A slight background related to the formation of the present operator can be found in [11], and other work can be tackled using this type of operator. Also, the meromorphic functions with the generalized hypergeometric functions were considered recently by Dziok and Srivastava [12, 13], Liu [14], Liu and Srivastava [15], and Cho and Kim [16].



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