# On Certain Subclasses of Meromorphic Functions with Positive and Fixed Second Coefficients Involving the Liu-Srivastava Linear Operator

Authors: N. Magesh; N. B. Gatti; S. Mayilvaganan
Journal: ISRN Mathematical Analysis
 (2012)

---

## Abstract

We introduce and study a subclassΣP(γ,k,λ,c) of meromorphic univalent functions defined by certain linear operator involving the generalized hypergeometric function. We obtain coefficient estimates, extreme points, growth and distortion inequalities, radii of meromorphic starlikeness, and convexity for the class ΣP(γ,k,λ,c) by fixing the second coefficient. Further, it is shown that the class ΣP(γ,k,λ) is closed under convex linear combination.


---

## Body

## 1. Introduction

LetΣ denote the class of functions of the form
(1.1)f(z)=z-1+∑n=1∞anzn,
which are analytic in the punctured unit disk(1.2)U*:={z:z∈C,0<|z|<1}=:U∖{0}.LetΣ𝒮,  Σ*(γ) and ΣK(γ),  (0≤γ<1) denote the subclasses of Σ that are meromorphically univalent, meromorphically starlike functions of order γ, and meromorphically convex functions of order γ, respectively. Analytically, f∈Σ*(γ) if and only if, f is of the form (1.1) and satisfies(1.3)-R{zf′(z)f(z)}>γ,z∈U,
similarly, f∈ΣK(γ), if and only if, f is of the form (1.1) and satisfies
(1.4)-R{1+zf′′(z)f′(z)}>γ,z∈U,
and similar other classes of meromorphically univalent functions have been extensively studied by Aouf and Darwish [1], Aouf and Joshi [2], Mogra et al. [3], Uralegaddi and Ganigi [4], Uralegaddi and Somanatha[5], and Owa and Pascu [6].Letf, g∈Σ, where f(z) is given by (1.1) and g(z) is defined by(1.5)g(z)=z-1+∑n=1∞bnzn.Then the Hadamard product (or convolution)f*g of the functions f(z) and g(z) is defined by(1.6)(f*g)(z):=z-1+∑n=1∞anbnzn=:(g*f)(z).
Let ΣP be the class of functions of the form
(1.7)f(z)=z-1+∑n=1∞anzn,an≥0,
that are analytic and univalent in 𝕌*.For complex parametersα1,…,αl and β1,…,βm(βj≠0,-1,…;j=1,2,…,m) the generalized hypergeometric function lFm(z) is defined by
(1.8)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 (θ)n is the Pochhammer symbol defined by(1.9)(θ)n=Γ(θ+n)Γ(θ)={1,n=0;θ∈C∖{0},θ(θ+1)(θ+2)⋅(θ+n-1),n∈N;θ∈C.Corresponding to a functionlFm(α1,…αl;β1,…,βm;z) defined by(1.10)F(α1,…αl;β1,…,βm;z):=z-1lFm(α1,…αl;β1,…,βm;z).Liu and Srivastava [7] considered a linear operator ℋ(α1,…αl;β1,…,βm):Σ→Σ defined by the following Hadamard product (or convolution):(1.11)H(α1,…αl;β1,…,βm)f(z)=F(α1,…αl;β1,…,βm;z)*f(z)=z-1+∑n=1∞|(α1)n+1…(αl)n+1(β1)n+1…(βm)n+1|anzn(n+1)!,
where, αi>0, (i=1,2,…l), βj>0, (j=1,2,…m),  l≤m+1;  l, m∈ℕ0=ℕ∪{0}. For notational simplicity, we use a shorter notations ℋml[α1] for  ℋ(α1,…αl;β1,…,βm) and
(1.12)Γn(α1)=(α1)n+1…(αl)n+1(β1)n+1…(βm)n+11(n+1)!,
unless otherwise stated in the sequel. We note that the linear operator Hml[α1] was earlier defined for multivalent functions by Dziok and Srivastava [8] and was investigated by Liu and Srivastava [7].Making use of the operatorℋml[α1], now we consider a subclass of functions in ΣP as follows.Definition 1.1.
For0≤γ<1 and k≥0 and 0≤λ<1/2, let Σ(γ,k,λ) denote a subclass of Σ consisting functions of the form (1.1) satisfying the condition that
(1.13)-R{z(H[α1]f(z))′+λz2(H[α1]f(z))′′(1-λ)H[α1]f(z)+λz(H[α1]f(z))′+γ}>k|z(H[α1]f(z))′+λz2(H[α1]f(z))′′(1-λ)H[α1]f(z)+λz(H[α1]f(z))′+1|,z∈U*,
where ℋ[α1]f(z) is given by (1.11). Furthermore, we say that a function f∈ΣP(γ,k,λ), whenever f(z) is of the form (1.7).
We observe that, by specializing the parametersl,  m,  α1,…,αl,  β1,…,βm,  k,  γ and λ the class leads to various subclasses. As for illustrations, we present some examples for the cases.Example 1.2.
Ifl=2 and m=1 with α1=1, α2=1, β1=1 and  f(z)  of the form (1.7), then we obtain the new subclass ℳP(γ,k,λ)   defined by
(1.14)-R{zf′(z)+λz2f′′(z)(1-λ)f(z)+λzf′(z)+γ}>k|zf′(z)+λz2f′′(z)(1-λ)f(z)+λzf′(z)+1|,z∈U*.Example 1.3.
Forl=2, m=1, α1=δ+1,  β1=α2=1, andf(z) of the form (1.7), then we get the new subclass 𝒟Pδ(γ,k,λ) defined by
(1.15)-R{z(Dδf(z))′+λz2(Dδf(z))′′(1-λ)Dδf(z)+λz(Dδf(z))′+γ}>k|z(Dδf(z))′+λz2(Dδf(z))′′(1-λ)Dδf(z)+λz(Dδf(z))′+1|,z∈U*,
where 𝒟δf(z)=1/(z(1-z)δ+1)*f(z)(δ>-1), is the differential operator which was introduced by Ganigi and Uralegaddi [9]. Also we note that the class 𝒟Pδ(γ,k,λ) was introduced by Atshan and Kulkarni [10].Example 1.4.
Forl=2, m=1,  α2=1, and f(z) of the form (1.7), then we obtain the new subclass ℒP(γ,k,λ) defined by
(1.16)-R{z(L[α1,β1]f(z))′+λz2(L[α1,β1]f(z))′′(1-λ)L[α1,β1]f(z)+λz(L[α1,β1]f(z))′+γ}>k|z(L[α1,β1]f(z))′+λz2(L[α1,β1]f(z))′′(1-λ)L[α1,β1]f(z)+λz(L[α1,β1]f(z))′+1|,z∈U*,
where the operator ℒ[α1;β1]f(z) was introduced and studied by Liu and Srivastava [11] (see also [12, 13]).
For the classΣP(γ,k,λ), the following characterization was given by Magesh et al. [14].Theorem 1.5.
Letf∈ΣP be given by (1.7). Then  f∈ΣP(γ,k,λ), if and only if,
(1.17)∑n=1∞(1+(n-1)λ)[n(k+1)+(k+γ)]Γn(α1)an≤(1-γ)(1-2λ),
where Γn(α1) is given by (1.12).For a function defined by (1.7) and in the class ΣP(γ,k,λ), Theorem 1.5, immediately yields(1.18)Γ1(α1)a1≤(1-γ)(1-2λ)(2k+γ+1).
Hence we may take(1.19)Γ1(α1)a1=(1-γ)(1-2λ)c(2k+γ+1),0<c<1.Motivated by Aouf and Darwish [1], Aouf and Joshi [2], Ghanim and Darus [12], and Uralegaddi [15], we now introduce the following class of functions and use the similar techniques to prove our results.LetΣP(γ,k,λ,c) be the subclass of ΣP(γ,k,λ) consisting functions in of the form(1.20)f(z)=1z+(1-γ)(1-2λ)c(2k+γ+1)z+∑n=2∞Γn(α1)anzn,0<c<1.In this paper, coefficient estimates, extreme points, growth and distortion bounds, radii of meromorphically starlikeness and convexity are discussed for the classΣP(γ,k,λ) by fixing the second coefficient. Further, it is shown that the class ΣP(γ,k,λ) is closed under convex linear combination.


## 2. Main Results

In our first theorem, we now find out the coefficient inequality for the classΣP(γ,k,λ,c).Theorem 2.1.
Let the functionf(z) defined by (1.20). Then  f∈ΣP(γ,k,λ,c), if and only if,
(2.1)∑n=2∞(1+(n-1)λ)[n(k+1)+(k+γ)]Γn(α1)an≤(1-γ)(1-2λ)(1-c).
The result is sharp.Proof.
By putting(2.2)Γ1(α1)a1=(1-γ)(1-2λ)c(2k+γ+1),0<c<1,
in (1.17), the result is easily derived. The result is sharp for the function
(2.3)f(z)=1z+(1-γ)(1-2λ)c(2k+γ+1)z+(1-γ)(1-2λ)(1-c)(1+(n-1)λ)[n(k+1)+(k+γ)]Γn(α1)zn,n≥2.Corollary 2.2.
If the functionf defined by (1.20) is in the class  ΣP(γ,k,λ,c), then
(2.4)an≤(1-γ)(1-2λ)(1-c)(1+(n-1)λ)[n(k+1)+(k+γ)]Γn(α1),n≥2.
The result is sharp for the function f(z) given by (2.3).Next we prove the following growth and distortion properties for the classΣP(γ,k,λ,c).Theorem 2.3.
If the functionf(z) defined by (1.20) is in the class  ΣP(γ,k,λ,c) for  0<|z|=r<1, then one has
(2.5)1r-(1-γ)(1-2λ)c(2k+γ+1)r-(1-γ)(1-2λ)(1-c)(1+λ)(3k+γ+2)r2≤|f(z)|≤1r+(1-γ)(1-2λ)c(2k+γ+1)r+(1-γ)(1-2λ)(1-c)(1+λ)(3k+γ+2)r2.
The result is sharp for the function f(z) given by
(2.6)f(z)=1z+(1-γ)(1-2λ)c(2k+γ+1)z+(1-γ)(1-2λ)(1-c)(1+λ)(3k+γ+2)z2.Proof.
SinceΣP(γ,k,λ,c), Theorem 2.1 yields
(2.7)Γn(α1)an≤(1-γ)(1-2λ)(1-c)(1+(n-1)λ)[n(k+1)+(k+γ)],n≥2.
Thus, for 0<|z|=r<1,
(2.8)|f(z)|≤1|z|+(1-γ)(1-2λ)c(2k+γ+1)|z|+∑n=2∞Γn(α1)an|z|n≤1r+(1-γ)(1-2λ)c(2k+γ+1)r+r2∑n=2∞Γn(α1)an≤1r+(1-γ)(1-2λ)c(2k+γ+1)r+(1-γ)(1-2λ)(1-c)(1+λ)(3k+γ+2)r2,|f(z)|≥1|z|-(1-γ)(1-2λ)c(2k+γ+1)|z|-∑n=2∞Γn(α1)an|z|n≥1r-(1-γ)(1-2λ)c(2k+γ+1)r-r2∑n=2∞Γn(α1)an≥1r-(1-γ)(1-2λ)c(2k+γ+1)r-(1-γ)(1-2λ)(1-c)(1+λ)(3k+γ+2)r2.
Thus the proof of the theorem is complete.Theorem 2.4.
If the functionf(z) defined by  (1.20) is in the class  ΣP(γ,k,λ,c) for  0<|z|=r<1, then one has
(2.9)1r2-(1-γ)(1-2λ)c(2k+γ+1)-(1-γ)(1-2λ)(1-c)(1+λ)(3k+γ+2)r≤|f′(z)|≤1r2+(1-γ)(1-2λ)c(2k+γ+1)+(1-γ)(1-2λ)(1-c)(1+λ)(3k+γ+2)r.
The result is sharp for the function f(z) given by
(2.10)f(z)=1z+(1-γ)(1-2λ)c(2k+γ+1)z+(1-γ)(1-2λ)(1-c)(1+λ)(3k+γ+2)z2.Proof.
In view of Theorem2.1, it follows that
(2.11)nΓn(α1)an≤n(1-γ)(1-2λ)(1-c)(1+(n-1)λ)[n(k+1)+(k+γ)],n≥2.
Thus, for 0<|z|=r<1 and making use of (2.11), we obtain
(2.12)|f′(z)|≤|-1z2|+(1-γ)(1-2λ)c(2k+γ+1)+∑n=2∞nΓn(α1)an|z|n-1≤1r2+(1-γ)(1-2λ)c(2k+γ+1)+r∑n=2∞nΓn(α1)an≤1r2+(1-γ)(1-2λ)c(2k+γ+1)+(1-γ)(1-2λ)(1-c)(1+λ)(3k+γ+2)r,|f′(z)|≥|-1z2|-(1-γ)(1-2λ)c(2k+γ+1)-∑n=2∞nΓn(α1)an|z|n-1≥1r2-(1-γ)(1-2λ)c(2k+γ+1)-r∑n=2∞nΓn(α1)an≥1r2-(1-γ)(1-2λ)c(2k+γ+1)-(1-γ)(1-2λ)(1-c)(1+λ)(3k+γ+2)r.Hence the result follows.Next, we will show that the classΣP(γ,k,λ,c) is closed under convex linear combination.Theorem 2.5.
If(2.13)f1(z)=1z+(1-γ)(1-2λ)c(2k+γ+1)z,
and for n≥2(2.14)fn(z)=1z+(1-γ)(1-2λ)c(2k+γ+1)z+∑n=2∞(1-γ)(1-2λ)(1-c)(1+(n-1)λ)[n(k+1)+(k+γ)]Γn(α1)zn.
Then f∈ΣP(γ,k,λ,c) if and only if it can expressed in the form
(2.15)f(z)=∑n=1∞μnfn(z),
where μn≥0 and ∑n=1∞μn≤1.Proof.
From (2.13), (2.14), and (2.15), we have
(2.16)f(z)=1z+(1-γ)(1-2λ)c(2k+γ+1)z+∑n=2∞(1-γ)(1-2λ)(1-c)μn(1+(n-1)λ)[n(k+1)+(k+γ)]Γn(α1)zn.
Since
(2.17)∑n=2∞(1-γ)(1-2λ)(1-c)μn(1+(n-1)λ)[n(k+1)+(k+γ)]Γn(α1)(1+(n-1)λ)[n(k+1)+(k+γ)]Γn(α1)(1-γ)(1-2λ)(1-c)=∑n=2∞μn=1-μ1≤1,
it follows from Theorem 1.5 that the function f∈ΣP(γ,k,λ,c). Conversely, suppose that f∈ΣP(γ,k,λ,c). Since
(2.18)an≤(1-γ)(1-2λ)(1-c)(1+(n-1)λ)[n(k+1)+(k+γ)]Γn(α1),n≥2.
Setting
(2.19)μn=(1+(n-1)λ)[n(k+1)+(k+γ)]Γn(α1)(1-γ)(1-2λ)(1-c)an,μ1=1-∑n=2∞μn.
it follows that
(2.20)f(z)=∑n=1∞μnfn(z).
This completes the proof of the theorem.Theorem 2.6.
The classΣP(γ,k,λ,c) is closed under linear combination.Proof.
Suppose that the functionf be given by (1.20), and let the function g be given by
(2.21)g(z)=1z+(1-γ)(1-2λ)c(2k+γ+1)z+∑n=2∞|bn|zn,n≥2.
Assuming that f and g are in the class ΣP(γ,k,λ,c), it is enough to prove that the function h defined by
(2.22)h(z)=μf(z)+(1-μ)g(z),0≤μ≤1,
is also in the class ΣP(γ,k,λ,c). Since
(2.23)h(z)=1z+(1-γ)(1-2λ)c(2k+γ+1)z+∑n=2∞|anμ+(1-μ)bn|zn,
we observe that
(2.24)∑n=2∞(1+(n-1)λ)[n(k+1)+(k+γ)]Γn(α1)|anμ+(1-μ)bn|≤(1-γ)(1-2λ)(1-c),
with the aid of Theorem 2.1. Thus, h∈ΣP(γ,k,λ,c).Next we determine the radii of meromorphically starlikeness and convexity of orderδ for functions in the class ΣP(γ,k,λ,c).Theorem 2.7.
Let the functionf(z) defined by (1.20) be in the class  ΣP(γ,k,λ,c), then (i)
f is meromorphically starlike of order δ(0≤δ<1) in the disk |z|<r1(γ,k,λ,c,δ) where r1(γ,k,λ,c,δ), is the largest value for which
(2.25)(3-δ)(1-γ)(1-2λ)c(2k+γ+1)r2+(n+2-δ)(1-γ)(1-2λ)(1-c)(1+(n-1)λ)[n(k+1)+(k+γ)]rn+1≤(1-δ),n≥2,(ii)
f is meromorphically convex of order δ(0≤δ<1) in the disk |z|<r2(γ,k,λ,c,δ) where r2(γ,k,λ,c,δ), is the largest value for which(2.26)(3-δ)(1-γ)(1-2λ)c(2k+γ+1)r2+n(n+2-δ)(1-γ)(1-2λ)(1-c)(1+(n-1)λ)[n(k+1)+(k+γ)]rn+1≤(1-δ),n≥2.
Each of these results is sharp for function f(z) given by (2.3).Proof.
It is enough to highlight that(2.27)|zf′(z)f(z)+1|≤1-δ,|z|<r1.
Thus, we have
(2.28)|zf′(z)f(z)+1|=|-1/z+((V)(E)c/(Q))z+∑n=2∞nΓn(α1)anzn+1/z+((V)(E)c/(Q))z+∑n=2∞Γn(α1)anzn1/z+((V)(E)c/(Q))z+∑n=2∞Γn(α1)anzn|,
where (Q) denotes (2k+γ+1),V denotes (1-γ), and (E) denotes (1-2λ).
Hence (2.28) holds true if
(2.29)2(1-γ)(1-2λ)c(2k+γ+1)r2+∑n=2∞(n+1)Γn(α1)anrn+1≤(1-δ)[1-(1-γ)(1-2λ)c(2k+γ+1)r2-∑n=2∞Γn(α1)anrn+1],
or,
(2.30)(3-δ)(1-γ)(1-2λ)c(2k+γ+1)r2+∑n=2∞(n+2-δ)Γn(α1)anrn+1≤(1-δ),
and it follows that, from (2.1), we may take
(2.31)an≤(1-γ)(1-2λ)(1-c)(1+(n-1)λ)[n(k+1)+(k+γ)]Γn(α1)μn,n≥2,
where μn≥0 and ∑n=2∞μn≤1.
For each fixedr, we choose the positive integer n0=n0(r) for which
(2.32)(n+2-δ)(1+(n-1)λ)[n(k+1)+(k+γ)]Γn(α1)rn+1
is maximal. Then it follows that
(2.33)∑n=2∞(n+2-δ)Γn(α1)anrn+1≤(n0+2-δ)(1-γ)(1-2λ)(1-c)(1+(n0-1)λ)[n0(k+1)+(k+γ)]rn0+1.
Then f is starlike of order δ in 0<|z|<r1(γ,k,λ,c,δ) provided that
(2.34)(3-δ)(1-γ)(1-2λ)c(2k+γ+1)r2+(n0+2-δ)(1-γ)(1-2λ)(1-c)(1+(n0-1)λ)[n0(k+1)+(k+γ)]rn0+1≤(1-δ).
We find the value r0=r0(k,c,δ,n) and the corresponding integer n0(r0) so that
(2.35)(3-δ)(1-γ)(1-2λ)c(2k+γ+1)r2+(n0+2-δ)(1-γ)(1-2λ)(1-c)(1+(n0-1)λ)[n0(k+1)+(k+γ)]rn0+1=(1-δ).
It is the value for which the function f(z) is starlike in 0<|z|<r0. (ii) In a similar manner, we can prove our result providing the radius of meromorphic convexity of order δ(0≤δ<1) for functions in the class ΣP(γ,k,λ,c), so we skip the details of the proof of (ii).



---

*Source: 698307-2012-05-08.xml*