# Certain Subclasses of Starlike Functions of Complex Order Involving Generalized Hypergeometric Functions

Authors: G. Murugusundaramoorthy; N. Magesh
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

Making use of the generalized hypergeometric functions, we define a new subclass
of uniformly convex functions and a corresponding subclass of starlike functions with negative
coefficients and obtain coefficient estimates, extreme points, the radii of close-to-convexity, starlikeness
and convexity, and neighborhood results for the classTSml(α,β,γ). In particular, we
obtain integral means inequalities for the function f that belongs to the class TSml(α,β,γ) in the unit disc.


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## Body

## 1. Introduction

Let𝒜 denote the class of functions of the form(1.1)f(z)=z+∑n=2∞anzn
which are analytic in the open unit disc U={z:z∈ℂ,|z|<1} and normalized by f(0)=f′(0)-1=0. Also denote by T the subclass of 𝒜 consisting of functions of the form(1.2)f(z)=z-∑n=2∞|an|zn,
introduced and studied by Silverman [1]. For functions f∈𝒜 given by (1.1) and g∈𝒜 given by g(z)=z+∑n=2∞bnzn,  it is known [2] the definition of the Hadamard product (or Convolution) of f and g by(1.3)(f*g)(z)=z+∑n=2∞anbnzn,z∈U.For positive real values ofα1,…,αl and β1,…,βm(βj≠0,-1,…;j=1,2,…,m) the generalized hypergeometric functionFlm(z) is defined by(1.4)lFm(z)≡lFm(α1,…,αl;β1,…,βm;z):=∑n=0∞(α1)n⋯(αl)n(β1)n⋯(βm)nznn!(l≤m+1;l,m∈N0:=N∪{0};z∈U),
where N denotes the set of all positive integers and (λ)k is the Pochhammer symbol defined by(1.5)(λ)n={1,n=0,λ(λ+1)(λ+2)⋯(λ+n-1),n∈N.
The notation Flm is quite useful for representing many well-known functions such as the exponential, the Binomial, the Bessel, the Laguerre polynomial, and others; for example, see [3].LetH(α1,…,αl;β1,…,βm):𝒜→𝒜 be a linear operator defined by(1.6)H(α1,…,αl;β1,…,βm)f(z):=zlFm(α1,α2,…,αl;β1,β2,…,βm;z)*f(z)=z+∑n=2∞Γnanzn,
where(1.7)Γn=(α1)n-1⋯(αl)n-1(β1)n-1⋯(βm)n-11(n-1)!,
unless otherwise stated. For notational simplicity, we can use a shorter notation Hml[α1] for H(α1,…,αl;β1,…,βm) in the sequel. The linear operator Hml[α1] is called Dziok-Srivastava operator (see [3]), includ (as its special cases) various other linear operators introduced and studied by Carlson and Shaffer [4], Owa [5], Ruscheweyh [2], and Srivastava and Owa [6]. Motivated by Goodman [7, 8], Rønning [9, 10] introduced and studied the following subclasses of 𝒜. A function f∈𝒜 is said to be in the class Sp(α,β), uniformly β-starlike functions if it satisfies the condition(1.8)Re{zf′(z)f(z)-α}>β|zf′(z)f(z)-1|,(-1<α≤1;β≥0)z∈U
and is said to be in the class UCV(α,β),  uniformly β-convex functions if it satisfies the condition(1.9)Re{1+zf′′(z)f′(z)-α}>β|zf′′(z)f′(z)|,(-1<α≤1;β≥0)z∈U.
Indeed it follows from (1.8) and (1.9) that(1.10)f∈UCV(α,β)⇔zf′∈Sp(α,β).
The interesting geometric properties of these function classes were extensively studied by Kanas et al., in [11–14]. Motivated by Altintas et al. [15], Murugusundaramoorthy and Srivastava [16], and Murugusundaramoorthy and Magesh [17], now, we define a new subclass uniformly starlike functions of complex order.For-1≤α<1,β≥0, and γ∈ℂ∖{0}, we let TSml(α,β,γ) be the class of functions f satisfying (1.2) with the analytic criterion(1.11)Re{1+1γ(z(Hml[α1]f(z))′Hml[α1]f(z)-α)}>β|1+1γ(z(Hml[α1]f(z))′Hml[α1]f(z)-1)|,z∈U
where Hml[α1]f(z) is given by (1.6).By suitably specializing the values ofl,m,α1,α2,…,αl,β1,β2,…,βm,α, γ, and β the class TSml(α,β,γ), leads to various new subclasses of starlike functions of complex order. As for illustrations, we present some examples for the cases.Example 1.1.
Ifl=2,m=1 with α1=1,α2=1,β1=1, and f of the form of (1.2), then
(1.12)T𝕊(α,β,γ):={Re{1+1γ(zf′(z)f(z)-α)}>β|1+1γ(zf′(z)f(z)-1)|,z∈U}.Example 1.2.
Ifl=2,m=1 with α1=δ+1(δ>-1),α2=1,β1=1, and f of the form of (1.2), then
(1.13)TRδ(α,β,γ):={Re{1+1γ(z(Dδf(z))′Dδf(z)-α)}>β|1+1γ(z(Dδf(z))′Dδf(z)-1)|,z∈U},
where Dδ is called Ruscheweyh derivative of order δ(δ>-1) defined by
(1.14)Dδf(z):=z(1-z)δ+1*f(z)≡H12(δ+1,1;1)f(z).Example 1.3.
Ifl=2,m=1 with α1=μ+1(μ>-1), α2=1,β1=μ+2, and f of the form of (1.2), then
(1.15)TBμ(α,β,γ)={Re{1+1γ(z(Jμf(z))′Jμf(z)-α)}>β|1+1γ(z(Jμf(z))′Jμf(z)-1)|,z∈U},
where Jμ is a Bernardi operator [18] defined by
(1.16)Jμf(z):=μ+1zμ∫0ztμ-1f(t)dt≡H12(μ+1,1;μ+2)f(z).Example 1.4.
Ifl=2,m=1 with α1=a(a>0),α2=1,β1=c(c>0), and f of the form of (1.2), then
(1.17)TLca(α,β,γ):={Re{1+1γ(z(L(a,c)f(z))′L(a,c)f(z)-α)}>β|1+1γ(z(L(a,c)f(z))′L(a,c)f(z)-1)|,z∈U},
where L(a,c) is a well-known Carlson-Shaffer linear operator [4] defined by
(1.18)L(a,c)f(z):=(∑k=0∞(a)k(c)kzk+1)*f(z)≡H12(a,1;c)f(z).
Also TLca(α,β,γ) was studied by Murugusundaramoorthy and Magesh [19].The main object of this paper is to study some usual properties of the geometric function theory such as the coefficient bound, extreme points, radii of close to convexity, starlikeness, and convexity for the classTSml(α,β,γ). Further, we obtain neighborhood results and integral means inequalities for aforementioned class.


## 2. Basic Properties

First we obtain the necessary and sufficient condition for functionsf in the class TSml(α,β,γ).Theorem 2.1.
The necessary and sufficient condition forf of the form of (1.2) to be in the class TSml(α,β,γ) is
(2.1)∑n=2∞[(n+|γ|)(1-β)-(α-β)]Γn|an|≤(1-α)+|γ|(1-β),
where -1≤α<1,β≥0, and γ∈ℂ∖{0}.Proof.
Assume thatf∈TSml(α,β,γ), then
(2.2)Re{1+1γ(z(Hml[α1]f(z))′Hml[α1]f(z)-α)}>β|1+1γ(z(Hml[α1]f(z))′Hml[α1]f(z)-1)|,Re{1+1γ(z(1-α)-∑n=2∞(n-α)Γn|an|znz-∑n=2∞Γn|an|zn)}>β|1-1γ(∑n=2∞(n-1)Γn|an|znz-∑n=2∞Γn|an|zn)|.
Letting z→1- along the real axis, we have
(2.3){1+1|γ|((1-α)-∑n=2∞(n-α)Γn|an|1-∑n=2∞Γn|an|)}>β[1-1|γ|(∑n=2∞(n-1)Γn|an|1-∑n=2∞Γn|an|)].
Hence, by maximum modulus theorem, the simple computational leads the desired inequality
(2.4)∑n=2∞[(n+|γ|)(1-β)-(α-β)]Γn|an|≤(1-α)+|γ|(1-β).
Conversely, suppose that (2.1) is true for z∈U. Then
(2.5)Re{1+1γ(z(Hml[α1]f(z))′Hml[α1]f(z)-α)}-β|1+1γ(z(Hml[α1]f(z))′Hml[α1]f(z)-1)|>0
if
(2.6)1+1|γ|((1-α)-∑n=2∞(n-α)Γn|an||z|n-11-∑n=2∞Γn|an||z|n-1)-β[1-1|γ|(∑n=2∞(n-1)Γn|an||z|n-11-∑n=2∞Γn|an||z|n-1)]>0.
That is if
(2.7)∑n=2∞[(n+|γ|)(1-β)-(α-β)]Γn|an|≤(1-α)+|γ|(1-β),
which completes the proof.Corollary 2.2.
Let the functionf defined by (1.2) belong to TSml(α,β,γ). Then
(2.8)|an|≤[(1-α)+|γ|(1-β)][(n+|γ|)(1-β)-(α-β)]Γn,n≥2,-1≤α<1,β≥0 and, γ∈ℂ∖{0}, with equality for
(2.9)f(z)=z-[(1-α)+|γ|(1-β)][(n+|γ|)(1-β)-(α-β)]Γnzn.Next we state the following theorem on extreme points for the classTSml(α,β,γ) without proof.Theorem 2.3 (Extreme Points).
Let(2.10)f1(z)=z,fn(z)=z-[(1-α)+|γ|(1-β)][(n+|γ|)(1-β)-(α-β)]Γnznforn=2,3,4,….
Then f∈TSml(α,β,γ) if and only if f can be expressed in the form f(z)=∑n=1∞λnfn(z), where λn≥0 and ∑n=1∞λn=1.


## 3. Close-to-Convexity, Starlikeness, and Convexity

We determine the radii of close-to-convexity, starlikeness, and convexity results for functions in the classTSml(α,β,γ) in the following theorems.Theorem 3.1.
Letf∈TSml(α,β,γ). Then f is close-to-convex of order δ(0≤δ<1) in the disc |z|<r1, where
(3.1)r1=infn≥2[(1-δ)n[(n+|γ|)(1-β)-(α-β)][(1-α)+|γ|(1-β)]Γn]1/(n-1).Proof.
Letf belong to T. It is known [20] that f is close-to-convex of order δ, if it satisfies the condition
(3.2)|f′(z)-1|<1-δ.
For the left-hand side of (3.2) we have
(3.3)|f′(z)-1|≤∑n=2∞n|an||z|n-1.
The last expression is less than 1-δ if
(3.4)∑n=2∞n1-δ|an||z|n-1<1.
Using the fact that f∈TSml(α,β,γ) if and only if
(3.5)∑n=2∞[(n+|γ|)(1-β)-(α-β)](1-α)+|γ|(1-β)Γn|an|<1.
we can say that (3.2) is true if
(3.6)n1-δ|z|n-1≤[(n+|γ|)(1-β)-(α-β)](1-α)+|γ|(1-β)Γn.
Or, equivalently,
(3.7)|z|≤[(1-δ)[(n+|γ|)(1-β)-(α-β)]n[(1-α)+|γ|(1-β)]Γn]1/(n-1),
which completes the proof.Theorem 3.2.
Letf∈TSml(α,β,γ). Then the following are given.(1)
f is starlike of order δ(0≤δ<1) in the disc |z|<r2, where(3.8)r2=infn≥2{(1-δ)(n-δ)[(n+|γ|)(1-β)-(α-β)][(1-α)+|γ|(1-β)]Γn}1/(n-1).(2)
f is convex of order δ(0≤δ<1) in the unit disc |z|<r3, where(3.9)r3=infn≥2{(1-δ)n(n-δ)[(n+|γ|)(1-β)-(α-β)][(1-α)+|γ|(1-β)]Γn}1/(n-1).
Each of these results is sharp for the extremal function f given by (2.10).Proof.
Letf∈T. It is known [1] that f is starlike of order δ, if it satisfies the condition
(3.10)|zf′(z)f(z)-1|<1-δ.
For the left-hand side of (3.10) we have
(3.11)|zf′(z)f(z)-1|≤∑n=2∞(n-1)|an||z|n-11-∑n=2∞|an||z|n-1.
The last expression is less than 1-δ if
(3.12)∑n=2∞n-δ1-δ|an||z|n-1<1.
Using the fact that f∈TSml(α,β,γ) if and only if
(3.13)∑n=2∞[(n+|γ|)(1-β)-(α-β)](1-α)+|γ|(1-β)Γn|an|<1.
we can say that (3.10) is true if
(3.14)n-δ1-δ|z|n-1<[(n+|γ|)(1-β)-(α-β)](1-α)+|γ|(1-β)Γn.
Or, equivalently,
(3.15)|z|n-1<(1-δ)[(n+|γ|)(1-β)-(α-β)](n-δ)[(1-α)+|γ|(1-β)]Γn,
which yields the starlikeness of the family.
Using the fact thatf is convex if and only if zf′ is starlike, we can prove (2), on lines similar to those the proof of (1).


## 4. Integral Means

In order to find the integral means inequality and to verify the Silverman Conjuncture [21] for f∈TSml(α,β,γ) we need the following subordination result due to Littlewood [22].Lemma 4.1 (see [22]).
If the functionsf and g are analytic in U with g≺f, then
(4.1)∫02π|g(reiθ)|ηdθ≤∫02π|f(reiθ)|ηdθ,η>0,z=reiθ,0<r<1.Applying Theorem2.1 with the extremal function and Lemma 4.1, we prove the following theorem.Theorem 4.2.
Letη>0. If f∈TSml(α,β,γ) and {Φ(α,β,γ,n)}n=2∞ is nondecreasing sequence, then, for z=reiθ and 0<r<1, one has
(4.2)∫02π|f(reiθ)|ηdθ≤∫02π|f2(reiθ)|ηdθ,
where f2(z)=z-((1-α)+|γ|(1-β))z2/Φ(α,β,γ,2), and Φ(α,β,γ,n)=[(n+|γ|)(1-β)-(α-β)]Γn.Proof.
Letf of the form of (1.2) and f2(z)=z-((1-α)+|γ|(1-β))z2/Φ(α,β,γ,2), then we must show that
(4.3)∫02π|1-∑n=2∞anzn-1|ηdθ≤∫02π|1-(1-α)+|γ|(1-β)Φ(α,β,γ,2)z|ηdθ.
By Lemma 4.1, it suffices to show that
(4.4)1-∑n=2∞|an|zn-1≺1-(1-α)+|γ|(1-β)Φ(α,β,γ,2)z.
Setting
(4.5)1-∑n=2∞|an|zn-1=1-(1-α)+|γ|(1-β)Φ(α,β,γ,2)w(z),
from (4.5) and (2.1) we obtain
(4.6)|w(z)|=|∑n=2∞Φ(α,β,γ,n)(1-α)+|γ|(1-β)anzn-1|≤|z|∑n=2∞Φ(α,β,γ,n)(1-α)+|γ|(1-β)|an|≤|z|<1.
This completes the proof of Theorem 4.2.


## 5. Inclusion Relations InvolvingNδ(e)

To study about the inclusion relations involvingNδ(e) we need the following definitions due to Goodman [23] and Ruscheweyh [24]. The n,δ neighborhood of function f∈T is given by(5.1)Nδ(f)={g∈T:g(z)=z-∑n=2∞|bn|zn,∑n=2∞n|an-bn|≤δ}.
Particularly for the identity function e(z)=z, we have(5.2)Nδ(e)={g∈T:g(z)=z-∑n=2∞|bn|zn,∑n=2∞n|bn|≤δ}.Theorem 5.1.
Let(5.3)δ=2[(1-α)+|γ|(1-β)][(2+|γ|)(1-β)-(α-β)]Γ2,
where Γ2=(α1⋯α2)/(β1⋯β2). Then TSml(α,β,γ)⊂Nδ(e).Proof.
Forf∈TSml(α,β,γ), Theorem 2.1 yields
(5.4)[(2+|γ|)(1-β)-(α-β)]Γ2∑n=2∞|an|≤(1-α)+|γ|(1-β)
so that
(5.5)∑n=2∞|an|≤(1-α)+|γ|(1-β)[(2+|γ|)(1-β)-(α-β)]Γ2.
On the other hand, from (2.1) and (5.5) we have
(5.6)(1-β)Γ2∑n=2∞n|an|≤(1-α)+|γ|(1-β)+[(α-β)-|γ|(1-β)]Γ2∑n=2∞|an|≤(1-α)+|γ|(1-β)+[(α-β)-|γ|(1-β)]×Γ2(1-α)+|γ|(1-β)[(2+|γ|)(1-β)-(α-β)]Γ2≤[(1-α)+|γ|(1-β)]2(1-β)[(2+|γ|)(1-β)-(α-β)],∑n=2∞n|an|≤2[(1-α)+|γ|(1-β)][(2+|γ|)(1-β)-(α-β)]Γ2.Now we determine the neighborhood for each of the classTSml(α,β,γ) which we define as follows. A function f∈T is said to be in the class TSml(α,β,γ) if there exists a function g∈TSml(α,β,γ) such that(5.7)|f(z)g(z)-1|<1-η,(z∈U,0≤η<1).Theorem 5.2.
Ifg∈TSml(α,β,γ) and
(5.8)η=1-δ[(2+|γ|)(1-β)-(α-β)]Γ22[((2+|γ|)(1-β)-(α-β))Γ2-((1-α)+|γ|(1-β))],
where Γ2=(α1⋯α2)/(β1⋯β2), then Nδ(g)⊂TSml(α,β,γ,η).Proof.
Suppose thatf∈Nδ(g), then we find from (5.6) that
(5.9)∑n=2∞|an-bn|≤δ,
which implies the coefficient inequality
(5.10)∑n=2∞|an-bn|≤δ2.
Next, since g∈TSml(α,β,γ), we have
(5.11)∑n=2∞|bn|≤2[(1-α)+|γ|(1-β)][(2+|γ|)(1-β)-(α-β)]Γ2
so that
(5.12)|f(z)g(z)-1|<∑n=2∞|an-bn|1-∑n=2∞|bn|≤δ2×[(2+|γ|)(1-β)-(α-β)]Γ2[((2+|γ|)(1-β)-(α-β))Γ2-((1-α)+|γ|(1-β))]≤1-η,
provided that η is given precisely by (5.8). Thus by definition, f∈TSml(α,β,γ,η) for η given by (5.8), which completes the proof.


## Concluding Remarks

By suitably specializing the various parameters involved in Theorems 2.1 to 5.2, we can state the corresponding results for the new subclasses defined in Examples1.1 to 1.4 and also for many relatively more familiar function classes.



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