We introduce a subclass of k-uniformly convex functions of order α with negative coefficients by using the multiplier transformations in the open unit disk U={z∈ℂ:|z|<1}. We obtain coefficient estimates, radii of convexity and close-to-convexity, extreme points, and integral means inequalities for the function f that belongs to the class 𝒩mℓ(α,β,k,υ).
1. Introduction
Let 𝒩 denote the class of functions of the form:f(z)β=zβ+∑n=2∞βanzβ+n-1,β>0,
which are analytic and univalent in the open unit disk U={z∈ℂ:|z|<1} (see [1]). Also denote by ℳ the subclass of 𝒩 consisting of functions of the form:f(z)β=zβ-∑n=2∞βanzβ+n-1,(an≥0,β>0).
For any integer m, we define the multiplier transformations Imℓ (see [2, 3]) of functions f∈𝒩(n) byImlf(z)β=zβ-∑n=2∞β(β+lβ+l+n-1)manzβ+n-1=zβ-∑n=2∞βQ(n,β,l)anzβ+n-1,(l≥0,z∈U),
where 𝒬(n,β,ℓ)=((β+ℓ)/(β+ℓ+n-1))^m.
A function f∈ℳ is said to be in the class USL(α,k) (k-uniformly starlike Functions of order α) if it satisfies the condition:Re{zf′(z)βf(z)β-α}>k|zf′(z)βf(z)β-1|,(0≤α<1,k≥0),z∈U
and is said to be in the class UCV(α,k) (k-uniformly convex Functions of order α) if it satisfies the condition:Re{1+zf′′(z)βf′(z)β-α}>k|zf′′(z)βf′(z)β|,(0≤α<1,k≥0),z∈U.
Indeed it follows from (1.4) and (1.5) thatf∈UCV(α,k)⟺zf′∈USL(α,k).
The interesting geometric properties of these function classes were extensively studied by Kanas et al., in [4, 5], motivated by Altintas et al. [6], Murugusundaramoorthy and Srivastava [7], and Murugusundaramoorthy and Magesh [8, 9], Atshan and Kulkarni [10] and Atshan and Buti [11].
Now, we define a new subclass of uniformly convex functions of complex order.
For 0≤α<1, k≥0, υ∈ℂ∖{0}, we let 𝒩mℓ(α,β,k,υ) be the class of functions f satisfying (1.2) with the analytic criterion:Re{1+1υ(1+z(Imlf(z)β)′′(Imlf(z)β)′-α)}>k|1+1υ(z(Imlf(z)β)′′(Imlf(z)β)′)|,z∈U,
where Imℓf(z)β is given by (1.3).
2. Main Results
First, we obtain the necessary and sufficient condition for functions f in the class 𝒩mℓ(α,β,k,υ).
Theorem 2.1.
The necessary and sufficient condition for f of the form of (1.2) to be in the class 𝒩mℓ(α,β,k,υ) is
∑n=2∞(β+n-1)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l)an≤(k-α)+(1-k)(β+|υ|),
where 0≤α<1, k≥0, υ∈ℂ∖{0}.
Proof.
Suppose that (2.1) is true for z∈U. Then
Re{1+1υ(1+z(Imlf(z)β)′′(Imlf(z)β)′-α)}-k|1+1υ(z(Imlf(z)β)′′(Imlf(z)β)′)|>0,
if
1+1|υ|((β-α)-∑n=2∞(β+n-1)(β+n-α-1)Q(n,β,l)an|z|n-11-∑n=2∞(β+n-1)Q(n,β,l)an|z|n-1)-k[1+1|υ|((β-1)-∑n=2∞(β+n-1)(β+n-2)Q(n,β,l)an|z|n-11-∑n=2∞(β+n-1)Q(n,β,l)an|z|n-1)]>0,
that is, if
∑n=2∞(β+n-1)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l)an≤(k-α)+(1-k)(β+|υ|).
Conversely, assume that f∈𝒩mℓ(α,β,k,υ), then
Re{1+1υ(1+z(Imlf(z)β)′′(Imlf(z)β)′-α)}>k|1+1υ(z(Imlf(z)β)′′(Imlf(z)β)′)|,Re{1+1υ((β-α)-∑n=2∞(β+n-1)(β+n-α-1)Q(n,β,l)anzn-11-∑n=2∞(β+n-1)Q(n,β,l)anzn-1)}>k|1+1υ((β-1)-∑n=2∞(β+n-1)(β+n-2)Q(n,β,l)anzn-11-∑n=2∞(β+n-1)Q(n,β,l)anzn-1)|.
Letting z→1- along the real axis, we have
1+1|υ|((β-α)-∑n=2∞(β+n-1)(β+n-α-1)Q(n,β,l)an1-∑n=2∞(β+n-1)Q(n,β,l)an)>k[1+1|υ|((β-1)-∑n=2∞(β+n-1)(β+n-2)Q(n,β,l)an1-∑n=2∞(β+n-1)Q(n,β,l)an)].
Hence, by maximum modulus theorem, the simple computation leads to the desired inequality
∑n=2∞(β+n-1)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l)an≤(k-α)+(1-k)(β+|υ|),
which completes the proof.
Corollary 2.2.
Let the function f defined by (1.2) belong to 𝒩mℓ(α,β,k,υ). Then,
an≤(k-α)+(1-k)(β+|υ|)(β+n-1)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l),
where 0≤α<1, k≥0,υ∈ℂ∖{0}, with equality for
f(z)β=zβ-β(k-α)+(1-k)(β+|υ|)(β+n-1)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l)zβ+n-1.
3. Radii of Convexity and Close-to-Convexity
We obtain the radii of convexity and close-to-convexity results for f functions in the class 𝒩mℓ(α,β,k,υ) in the following theorems.
Theorem 3.1.
Let f∈𝒩mℓ(α,β,k,υ). Then f is convex of order δ(0≤δ<1) in the disk |z|<r=r1(α,β,k,υ,n,δ), where
r1=infn≥2[(2-δ-β)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l)(3-δ-β-n)[(k-α)+(1-k)(β+|υ|)]]1/n-1.
Proof.
Let f∈𝒩mℓ(α,β,k,υ). Then by Theorem 2.1, we have
∑n=2∞(β+n-1)[(β+n-1+|υ|)(1-k)+(k-α)](k-α)+(1-k)(β+|υ|)Q(n,β,l)an≤1.
For 0≤δ<1, we need to show that
|zf′′(z)βf′(z)β|≤1-δ,
and we have to show that
|zf′′(z)βf′(z)β|≤(β-1)-∑n=2∞(β+n-1)(β+n-2)an|z|n-11-∑n=2∞(β+n-1)an|z|n-1≤1-δ.
Hence,
∑n=2∞(β+n-1)(3-δ-β-n)(2-δ-β)an|z|n-1≤1.
This is enough to consider
|z|n-1≤(2-δ-β)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l)(3-δ-β-n)[(k-α)+(1-k)(β+|υ|)].
Therefore,
|z|≤{(2-δ-β)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l)(3-δ-β-n)[(k-α)+(1-k)(β+|υ|)]}1/n-1.
Setting z=r1(α,β,k,υ,n,δ) in (3.7), we get the radius of convexity, which completes the proof of Theorem 3.1.
Theorem 3.2.
Let f∈𝒩mℓ(α,β,k,υ). Then f is close-to-convex of order δ(0≤δ<1) in the disk |z|<r=r2(α,β,k,υ,n,δ), where
r2=infn≥2[(β+n-1)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l)(k-α)+(1-k)(β+|υ|)]1/n-1.
Proof.
Let f∈𝒩mℓ(α,β,k,υ). Then by Theorem 2.1, we have
∑n=2∞(β+n-1)[(β+n-1+|υ|)(1-k)+(k-α)](k-α)+(1-k)(β+|υ|)Q(n,β,l)an≤1.
For 0≤δ<1, we need to show that
|f′(z)βzβ-1-1|≤1-δ,
and we have to show that
|f′(z)βzβ-1-1|≤(β-1)+∑n=2∞β(β+n-1)an|z|n-1≤1-δ.
Hence,
∑n=2∞β(β+n-1)(2-δ-β)an|z|n-1≤1.
This is enough to consider
|z|n-1≤(2-δ-β)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l)β[(k-α)+(1-k)(β+|υ|)].
Therefore,
|z|≤{(2-δ-β)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l)β[(k-α)+(1-k)(β+|υ|)]}1/n-1.
Setting z=r2(α,β,k,υ,n,δ) in (3.14), we get the radius of close-to-convexity, which completes the proof of Theorem 3.2.
4. Extreme Points
The extreme points of the class 𝒩mℓ(α,β,k,υ) are given by the following theorem.
Theorem 4.1.
Let
f1(z)β=zβfn(z)β=zβ-β(k-α)+(1-k)(β+|υ|)(β+n-1)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l)zβ+n-1,
for n=2,3,4,….
Then, f∈𝒩mℓ(α,β,k,υ) if and only if it can be expressed in the form:
f(z)β=∑n=1∞Υnfn(z)β,
where Υn≥0 and
∑n=1∞Υn=1.
Proof.
Suppose that f can be expressed as in (4.2). Our goal is to show that f∈𝒩mℓ(α,β,k,υ). By (4.2), we have that
f(z)β=∑n=1∞Υnfn(z)β=Υ1f1(z)β+∑n=2∞Υnfn(z)β=Υ1f1(z)β+∑n=2∞Υn(zβ-β(k-α)+(1-k)(β+|υ|)(β+n-1)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l)zβ+n-1)=∑n=1∞Υnzβ-∑n=2∞βΥn(k-α)+(1-k)(β+|υ|)(β+n-1)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l)zβ+n-1=zβ-∑n=2∞βΥn[(k-α)+(1-k)(β+|υ|)](β+n-1)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l)zβ+n-1.
Now,
∑n=2∞(β+n-1)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l)(k-α)+(1-k)(β+|υ|)×Υn[(k-α)+(1-k)(β+|υ|)](β+n-1)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l)=∑n=2∞Υn=1-Υ1≤1.
Thus, f∈𝒩mℓ(α,β,k,υ).
Conversely, assume that f∈𝒩mℓ(α,β,k,υ). Since
an≤(k-α)+(1-k)(β+|υ|)(β+n-1)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l)(n≥2),
we can set
Υn=(β+n-1)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l)(k-α)+(1-k)(β+|υ|)an(n≥2),Υ1=1-∑n=2∞Υn.
Then,
f(z)β=zβ-∑n=2∞βanzβ+n-1=zβ-∑n=2∞βΥn[(k-α)+(1-k)(β+|υ|)](β+n-1)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l)zβ+n-1=zβ-∑n=2∞Υn(zβ-fn(z)β)=zβ(1-∑n=2∞Υn)+∑n=2∞Υnfn(z)β=Υ1f1(z)β+∑n=2∞Υnfn(z)β=∑n=1∞Υnfn(z)β.
This completes the proof of Theorem 4.1.
5. Integral Means
In order to find the integral means inequality and to verify the Silverman Conjuncture [12] for f∈𝒩mℓ(α,β,k,υ), we need the following definition of subordination and subordination result according to Littlewood [13].
Definition 5.1 (see [13]).
Let f and g be analytic in U. Then, we say that the function f is subordinate to g if there exists a Schwarz function w, analytic in U with w(0)=0, |w(z)|<1 such that f(z)=g(w(z))(z∈U). We denote this subordination f≺g or f(z)≺g(z)(z∈U). In particular, if the function g is univalent in U, the above subordination is equivalent to f(0)=g(0), f(U)⊂g(U).
Lemma 5.2 (see [13]).
If the functions f and g are analytic in Uwith g≺f, then
∫02π|g(reiθ)|ηdθ≤∫02π|f(reiθ)|ηdθ,η>0,z=reiθ,0<r<1.Applying Theorem 2.1 with the extremal function and Lemma 5.2, we prove the following theorem.
Theorem 5.3.
Let η>0. If f∈𝒩mℓ(α,β,k,υ) and {Φ(α,β,k,υ,n)}n=2∞ are nondecreasing sequences, then, for z=reiθ and 0<r<1, one has
∫02π|f(reiθ)β|ηdθ≤∫02π|f2(reiθ)β|ηdθ,
where
f2(z)β=zβ-β(k-α)+(1-k)(β+|υ|)Φ(α,β,k,υ,2)zβ+1,Φ(α,β,k,υ,n)=(β+n-1)[(β+n-1+|υ|)(1-k)+(k-α)]Q(n,β,l).
Proof.
Let fof the form of (1.2) and
f2(z)β=zβ-β(k-α)+(1-k)(β+|υ|)Φ(α,β,k,υ,2)zβ+1,
then we must show that
∫02π|1-∑n=2∞βanzn-1|ηdθ≤∫02π|1-β(k-α)+(1-k)(β+|υ|)Φ(α,β,k,υ,2)z|ηdθ.
By Lemma 5.2, it suffices to show that
1-∑n=2∞βanzn-1≺1-β(k-α)+(1-k)(β+|υ|)Φ(α,β,k,υ,2)z.
Setting
1-∑n=2∞βanzn-1=1-β(k-α)+(1-k)(β+|υ|)Φ(α,β,k,υ,2)w(z),
from (5.7) and (2.1) we obtain
|w(z)|=|∑n=2∞Φ(α,β,k,υ,2)(k-α)+(1-k)(β+|υ|)anzn-1|≤|z|∑n=2∞Φ(α,β,k,υ,n)(k-α)+(1-k)(β+|υ|)an≤|z|<1.
This completes the proof of Theorem 5.3.
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