
This paper is devoted to considering the attainability of minimizers of the L2-constraint variational problem
mγ,a=inf{Jγ(u):u∈H2(RN),∫RN|u|2dx=a2},
where
Jγ(u)=γ2∫RN|Δu|2dx+12∫RN|∇u|2dx+12∫RNV(x)|u|2dx−12σ+2∫RN|u|2σ+2dx,
γ>0, a>0, σ∈(0,2N) with N≥2. Moreover, the function V:RN→[0,+∞) is continuous and bounded. By using the variational methods, we can prove that, when V satisfies four different assumptions, mγ,a are all achieved.
Citation: Cheng Ma. Normalized solutions for the mixed dispersion nonlinear Schrödinger equations with four types of potentials and mass subcritical growth[J]. Electronic Research Archive, 2023, 31(7): 3759-3775. doi: 10.3934/era.2023191
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This paper is devoted to considering the attainability of minimizers of the L2-constraint variational problem
mγ,a=inf{Jγ(u):u∈H2(RN),∫RN|u|2dx=a2},
where
Jγ(u)=γ2∫RN|Δu|2dx+12∫RN|∇u|2dx+12∫RNV(x)|u|2dx−12σ+2∫RN|u|2σ+2dx,
γ>0, a>0, σ∈(0,2N) with N≥2. Moreover, the function V:RN→[0,+∞) is continuous and bounded. By using the variational methods, we can prove that, when V satisfies four different assumptions, mγ,a are all achieved.
One of the most fascinating areas of Complex analysis is the study of geometric characterization of univalent functions in the open unit disc U. Because of the challenging problem in studying the class S(1−1) of univalent functions in U as a whole, several subclasses of it emerged. The most studied of these are the classes CCV, SST, CCV(β) and SST(β)(0≤β<1) of convex functions, starlike functions and, convex and starlike functions of order β, respectively. Since the image domains of U plays a significant role in their geometric characterization, various subclasses of S(1−1) have been receiving attention in different directions and perspectives (see [4,6,20,23,28,32,33,36,37,38]). For this reason, Ma and Minda [17] gave a unified treatment of both SST and CCV. They considered the class Ψ of analytic univalent functions ψ(ς) with Reψ(ς)>0 and for which ψ(U) is symmetric with respect to the real axis and starlike with respect to ψ(0) such that ψ′(0)>0. They initiated the following classes of functions that generalized and unified many renowned subclasses of S(1−1):
S∗ST(ψ)={f∈A:ςf′(ς)f(ς)≺ψ(ς)} |
and
CCV(ψ)={f∈A:(ςf′(ς))′f′(ς)≺ψ(ς)}, |
where A is the class of analytic functions f(ς) of the form
f(ς)=ς+∞∑n=2δnςn. | (1.1) |
If
ψ(ς)=1+2π2(log1+√ς1−√ς)2, |
then CCV(ψ)=UCV is the Goodman class of uniformly convex functions [8,30], which was later modified and examined by Kanas and Wisniowska [13,14]. Similarly, SThpl(s)=S∗(1(1−ς)s), CVhpl(s)=C(1(1−ς)s)(0<s≤1), are made-known by Kanas and Ebadian [15,16], respectively. These consist of functions f∈A such that ςf′(ς)/f(ς) and (ςf′(ς))′/f′(ς) lie in the domain bounded by the right branch of a hyperbola
H(s)={σeiθ∈C:σ=1(2cosθs)s,|θ|<πs2}. |
More special families of Ma and Minda classes can be found in [3,9,10,24,25,26,31,34,39].
Recently, Kanas et al [18] introduced novel subclasses STL(s) and CVL(s) of SST and CCV, respectively. Geometrically, they consist of functions f(ς)∈A such that ςf′(ς)/f(ς) and (ςf′(ς))′/f′(ς) lie in the region bounded by the limaçon defined as
∂Ls(U)={u+iv:[(u−1)2+v2−s4]2=4s2[(u−1+s2)2+v2]},s∈[−1,1]−{0} | (1.2) |
as shown in Figure 1 for different values of s. s=0.35,0.5,0.6,0.71,0.75 and 1 corresponds to blue, red, green, gray, yellow and black. Some novel properties of these classes were derived in [18].
Motivated by this present work and other aforementioned articles, the goal in this paper is to continue with the investigation of some interesting properties of the classes STL(s) and CVL(s). To this end, the sharp bounds of the Hankel determinant, subordination conditions as well as some radius results for these novel classes are investigated.
To put our investigations in a clear perspective, some preliminaries and definitions are presented as follows:
Denoted by W is the class of analytic functions
w(ς)=∞∑n=1wnςn,ς∈U | (2.1) |
such that w(0)=0 and |w(ς)|<1. These functions are known as Schwarz functions. If f(ς) and g(ς) are analytic functions in U, then f(ς) is subordinate to g(ς) (written as f(ς)≺g(ς)) if there exists a Schwarz function w(ς)∈W such that f(ς)=g(w(ς)),ς∈U.
Janowski [12] introduced the class P(A⅁,B⅁),−1≤B⅁<A⅁≤1 of functions p(ς) satisfying the subordination condition
p(ς)≺1+A⅁ς1+B⅁ς, |
or equivalently, satisfying the inequality
|p(ς)−1−A⅁B⅁r21−B⅁2r2|≤(A⅁−B⅁)r1−B⅁2r2,|ς|≤r(0<r<1). | (2.2) |
As a special cases, P(1,−1)≡P and P(1−2β,−1)≡P(β)(0≤β<1) are the classes of functions of positive real part and that whose real part is greater than β, respectively (see [7]).
Definition 2.1. Noonan and Thomas [22] defined for q≥1,n≥1, the qth Hankel determinant of f(ς)∈S1−1 of the form (1.1) as follows:
Hq(n)=|δnδn+1…δn+q−1δn+1δn+2…δn+q−2⋮⋮⋮⋮δn+q−1δn+q−2…δn+2q−2| | (2.3) |
This determinant has been studied by many researchers. In particular Babalola [2] obtained the sharp bounds of H3(1) for the classes SST and CCV. By this definition, H3(1) is given as:
H3(1)=|δ1δ2δ3δ2δ3δ4δ3δ4δ5|=δ3(δ2δ4−δ23)−δ4(δ4−δ2δ3)+δ5(δ3−δ22),δ1=1, |
and the by triangle inequality,
|H3(1)|≤|δ3||δ2δ4−δ23|+|δ4||δ4−δ2δ3|+|δ5||δ3−δ22|. | (2.4) |
Clearly, one can see that H2(1)=|δ3−δ22| is a particular instance of the well-known Fekete Szeg¨o functional |δ3−μδ22|, where μ is a real number.
Definition 2.2. [18] Let p(ς)=1+∞∑n=1cnςn. Then p∈P(Ls) if and only if
p(ς)≺(1+sς)2,0<s≤1√2,ς∈U, |
or equivalently, if p(ς) satisfies the inequality
|p(ς)−1|<1−(1−s)2. |
Demonstrated in [18], was the inclusion relation
{w∈C:|w−1|<1−(1−s)2}⊂Ls(U)⊂{w∈C:|w−1|<(1+s)2−1}. | (2.5) |
It is worthy of note that the function Ls(ς)=(1+sς)2 is the analytic characterization of Ls(U) given by (1.2). Also, Ls(ς) is starlike and convex univalent in U for 0<s≤12. Furthermore, Ls(ς)∈P(β), where β=(1−s)2,0<s≤12.
Definition 2.3. Let f∈A. Then f∈STL(s) if and only if
ςf′(ς)f(ς)∈P(Ls),0<s≤1√2. |
Also, f∈CVL(s) if and only if
zf′(ς)∈STL(s),0<s≤1√2. |
Moreover, the integral representation of functions f∈STL(s) is given as
f(ς)=ςexp(∫ς0p(t)−1tdt),p∈P(Ls), |
while that of g∈CVL(s) is given as
g(ς)=∫ς0f(t)tdt,f∈STL(s). |
Furthermore, the extremal functions for each of the classes are given by
Ψs,n(ς)=ςexp(∫z0Ls(tn)−1tdt),Ψs,n(ς)∈STL(s)=ςexp(2snςn+s22nς2n),n=1,2,3,…=ς+2snςn+1+(n+4)s22n2ς2n+1+…. | (2.6) |
and for Ks,n(ς)∈CVL(s),
Ks,n(ς)=∫ς0Ψs,n(t)tdt,Ψs,n(ς)∈STL(s),n=1,2,3,…=ς+2sn(n+1)ςn+1+…. | (2.7) |
Lemma 3.1. [1] If w∈W is of the form (2.1), then for a real number σ,
|w2−σw21|≤{−σ,forσ≤−1,1,for−1≤σ≤1,σforσ≥1. |
When σ<−1 or σ>1, equality holds if and only if w(ς)=ς or one of its rotations. If −1<σ<1, then equality holds if and only if w(ς)=ς2 or one of its rotations. Equality holds for σ=−1 if and only if w(ς)=ς(x+ς)1+xς(0≤x≤1) or one of its rotations while for σ=1, equality holds if and only if w(ς)=−ς(x+ς)1+xς(0≤x≤1) or one of its rotations.
Also, the sharp upper bound above can be improved as follows when −1≤σ≤1 :
|w2−σw21|+(1+σ)|w1|2≤1(−1<σ≤0) |
and
|w2−σw21|+(1−σ)|w1|2≤1(0<σ<1). |
Lemma 3.2. [16] If w∈W is of the form (2.1), then for some complex numbers ξ and η such that |ξ|≤1 and |η|≤1,
w2=ξ(1−w21) |
and
w3=(1−w21)(1−|ξ|2)η−w1(1−w21)ξ2. |
Lemma 3.3. [19,Theorem 3.4h,p. 132] Let q(ς) be univalent in U and let θ and φ be analytic in a domain D containing q(U) with φ(ω)≠0, when ω∈q(U). Set Q(ς)=ςq′(ς)⋅φ(q(ς)),h(ς)=θ(q(ς))+Q(ς), and suppose that either
(i) h(ς) is convex, or Q(ς) is starlike,
(ii)
Reςh′(ς)Q(ς)=Re(θ′(q(ς))φ(q(ς))+ςQ′(ς)Q(ς))>0. |
If p(ς) is analytic in U with p(0)=q(0),p(U)⊂D and
θ(p(ς))+ςp′(ς)φ(p(ς))≺θ(q(ς))+ςq′(ς)φ(q(ς))=h(ς), | (3.1) |
then p(ς)≺q(ς), and q(ς) is the best dominant in the sense that p≺t⇒q≺t for all t.
Lemma 3.4. [29] Let h(ς)=1+∞∑n=1cnςn,G(ς)=1+∞∑n=1dnςn and h(ς)≺G(ς). If G(ς) is univalent in U and G(U) is convex, then |cn|≤|d1|, for all n≥1.
Lemma 3.5. [11] Let w∈W. If |w(ς)| attains its maximum value on the circle |ς|=r at a point ς0∈U, then we have ς0w′(ς0)=kw(ς0), where k≥1.
Throughout this work f(ς) is taken to be of the form (1.1) while w(ς) is of the form (2.1). In the next sections, the main results are presented.
In this section, we assume 0<s≤12. First, we establish a few auxiliary results whose applications will be needed hereafter.
Lemma 4.1. Let f∈STL(s). Then
|δn|≤(2s)n−1(n−1)!,n≥2. | (4.1) |
Proof. From the definition of f∈STL(s), we have
ςf′(ς)f(ς)=p(ς),p∈P(Ls), | (4.2) |
where
p(ς):=1+c1ς+c2ς2+c3ς3+…. | (4.3) |
Comparing the coefficients of ςn in (4.2), it follows that
(n−1)δn=cn−1+δ2cn−2+δ3cn−3+⋯+δn−1c1. | (4.4) |
It is obvious from Lemma 3.4 and the fact that Ls(ς) is convex for 0<s≤12 that
|cn|≤2s,n≥1. | (4.5) |
Using this result in (4.4), we obtain
|δn|≤2sn−1n−1∑m=1|δm|,δ1=1,n≥2. | (4.6) |
We need to show (4.1) by Mathematical induction. For this reason, assume (4.1) is true and proceed to prove
|δn+1|≤(2s)n(n)!,n≥2. |
From (4.6),
|δ2|=|c1|≤(2s)1,|δ3|≤2s2(1+|δ2|)≤(2s)22!,|δ4|≤2s2(1+|δ2|+|δ3|)≤(2s)33!, |
and finally,
|δn+1|≤2sn(1+(2s)1+(2s)22!+(2s)33!+⋯+(2s)n−1(n−1)!)=(2s)nn!. |
Therefore,
|δn+1|≤(2s)n(n)!,n≥2. |
Hence, by Mathematical induction, we have the desired result.
In view of Theorem 4.1 and the definition of functions in CVL(s), we are led to the following result.
Lemma 4.2. Let f∈CVL(s). Then
|δn|≤(2s)n−1(n)!,n≥2. |
Lemma 4.3. Let f∈STL(s). Then
|δ2δ4−δ23|≤19s412. |
The bound 19s412 is sharp for the function
Ψs,1(ς)=ς+2sς2+52s2ς3+73s3ς4+…. | (4.7) |
Proof. For f∈STL(s),
ςf′(ς)f(ς)=(1+sw(ς))2, | (4.8) |
where w∈W. Comparing coefficients of ς,ς2 and ς3 in (4.8), we arrive at
δ2=2sw1,δ3=s(w2+52sw21)andδ4=23sw3+83s2w1w2+73s3w31. | (4.9) |
By Lemma 3.1, we obtain
|δ2δ4−δ23|=4s23|w1(1−w21)(1−|ξ|2)η−(1−w21)(3−w21)ξ2+14sw21ξ(1−w21)−1916s2w41|. |
Let x=w1,ξ=y with 0≤x≤1 and |y|≤1. Then the triangle inequality gives
|δ2δ4−δ23|≤F(x,|y|), |
where
F(x,|y|)=4s23(x(1−x2)(1−|y|2)+(1−x2)(3−x2)|y|2+14sx2|y|(1−x2)+1916s2x4), |
and
∂F∂|y|=4s23(2(1−x2)(3−x2−x)|y|+14sx2(1−x2))>0. |
This means that F(x,|y|) is increasing on the interval [0,1]. So,
F(x,|y|)≤4s23((1−x2)(12+x2(s−4))+1916s2x4):=F(x), |
where
F′(x)=2x(s+8)+19s2x3>0, |
which implies that F(x) is an increasing function of x on [0,1]. Consequently,
|δ2δ4−δ23|≤4s23F(1)=19s412. |
Lemma 4.4. Let f∈CVL(s). Then
|δ2δ4−δ23|≤s29. |
The bound s29 is sharp for the function
Ks,2(ς)=ς+s3ς3+3s220ς5+…. | (4.10) |
Proof. From the definition of f∈CVL(s) and (4.9), it is easy to see that
|δ2δ4−δ23|=|16s2w1w3+19s3w21w2−19s2w22−19s4w41|. |
The rest of the proof follows as in Theorem 4.3.
Lemma 4.5. Let f∈STL(s). Then
|δ2δ3−δ4|≤2s3. |
The bound 2s3 is best possible for the function
Ψs,3(ς)=ς+23sς4+718s2ς7+…. | (4.11) |
Proof. From (4.9), a computation gives
δ2δ3−δ4=2s2w1(w2−76sw21)−83s2w1(w2−158sw21)−23sw3. |
Employing Lemma 3.2, we write the expression for w3, and applying the triangle inequality together with Lemma 3.1, we obtain
|δ2δ3−δ4|≤14s23+2s3[(1−x2)(1−|y|2)+x(1−x2)|y|2], | (4.12) |
where we have taken w1=x,ξ=y with 0≤x≤1 and |y|≤1. Let H(x,|y|) represents the right side of (4.12). Then
∂H(x,|y|)∂|y|=−4s3(1−x2)(1−x)≤0. |
Thus,
H(x,|y|)≤H(x,0):=H(x), |
where
H(x)=14s2x3+2(1−x2)s3andH′(x)=2s3(7s−2x). |
It is clear that H(x) attains its maximum value at x=7s2. Thus, H(x)≤H(7s2)=2s3. Consequently,
|δ2δ3−δ4|≤2s3. |
Lemma 4.6. Let f∈CVL(s). Then
|δ2δ3−δ4|≤s6. |
This bound cannot be improved since the function
Ks,3(ς)=ς+16sς4+118s2ς7+… | (4.13) |
attains the equality.
Proof. Using the definition of f∈CVL(s) and (4.9), we find
δ2δ3−δ4=14s3w31−13s2w1w2−16sw3. |
Let w1=x(0≤x<1) and ξ=y with |y|≤1. Then applying Lemma 3.2 and following the procedure of proof as in Theorem 4.5, we arrive at the desired result.
Lemma 5.1. Let f∈STL(s). Then for a real number μ,
|δ3−μδ22|≤{s2(5−8μ)2,forμ≤5s−28s,s,for5s−28s≤μ≤5s+28s,s2(8μ−5)2,forμ≥5s+28s. |
It is asserted also that
|δ3−μδ22|+(μ−5s−28s)|δ2|2≤s,5s+28s<μ≤58 |
and
|δ3−μδ22|−(μ−5s+28s)|δ2|2≤s,58<μ<5s+28s. |
These inequalities are sharp for the functions
{¯λΨs,1(λς),forμ∈(−∞,5s−28s)∪(5s+28s,∞),¯λΨs,2(λς),for5s−28s≤μ≤5s+28s,¯λPx(λς),forμ=5s−28s,¯λQx(λς),forμ=5s+28s, |
where |λ|=1 and
ςP′x(ς)Px(ς)=Ls(ς(x+ς)1+xς),ςQ′x(ς)Qx(ς)=Ls(−ς(x+ς)1+xς),0≤x≤1. |
Proof. From (4.9), we have
|δ3−δ22|=s|w2−s(8μ−5)2w21|. |
Then using Lemma 3.1 with σ=s(8μ−5)2, we obtain the required result. For μ=1 in Theorem 5.1, we deduce the following sharp result.
Corollary 5.1. Let f∈STL(s). Then
|δ3−δ22|≤s. |
Lemma 5.2. Let f∈CVL(s). Then for a real number μ,
|δ3−μδ22|≤{s2(5−6μ)6,forμ≤5s−26s,s3,for5s−26s≤μ≤5s+26s,s2(6μ−5)6,forμ≥5s+26s. |
It is asserted also that
|δ3−μδ22|+(μ−5s−26s)|δ2|2≤s3,5s−26s<μ≤56 |
and
|δ3−μδ22|−(μ−5s+26s)|δ2|2≤s3,56<μ<5s+26s. |
These inequalities are sharp for the functions
{¯λKs,1(λς),forμ∈(−∞,5s−26s)∪(5s+26s,∞),¯λKs,2(λς),for5s−26s≤μ≤5s+26s,¯λPx(λς),forμ=5s−26s,¯λQx(λς),forμ=5s+26s, |
where |λ|=1 and
(ςP′x(ς))′P′x(ς)=Ls(ς(x+ς)1+xς),(ςQ′x(ς))′Q′x(ς)=Ls(−ς(x+ς)1+xς),0≤x≤1. |
Proof. Using the definition of CVL(s) and (4.9), we get
|δ3−δ22|=s3|w2−s(6μ−5)2w21|. |
Then using Lemma 3.1 with σ=s(6μ−5)2, we obtain the desired result. For μ=1 in Theorem 5.2, we deduce the following sharp result.
Corollary 5.2. Let f∈CVL(s). Then
|δ3−δ22|≤s3. |
Theorem 5.3. Let f∈STL(s). Then
|H3(1)|≤s236(2s+1)(57s3+12s2+46s+34) |
Proof. The proof is immediate from (2.4), Lemma 4.1, Lemma 4.3, Lemma 4.5 and Corollary 5.1.
Theorem 5.4. Let f∈CVL(s). Then
|H3(1)|≤s2540(2s+1)(12s2+6s+33) |
Proof. The proof is straightforward from (2.4), Lemma 4.2, Lemma 4.4, Lemma 4.6 and Corollary 5.2.
Theorem 6.1. The CVL(s)-radius for the class SST(β) (where β=(1−s)2) is given by
R1={12+2s−s2+√s4−4s3+6s2−4s+7,fors≠√2−1√2,15,fors=√2−1√2. | (6.1) |
This radius is sharp for the functions given by
f0(ς)={z(1−z)2s(2−s),fors≠√2−1√2,z1−z,fors=√2−1√2. | (6.2) |
Proof. Let f∈SST(β). Then
ςf′(ς)f(ς)∈P(β),β=(1−s)2. |
This means that there exists w∈W such that
ςf′(ς)f(ς)=1+2(1−β)w(ς)1−w(ς), |
Let
ςf′(ς)f(ς)=p(ς). | (6.3) |
Then by Schwarz lemma,
|p(ς)−1|≤2(1−β)r1−r. | (6.4) |
It follows from logarithmic differentiation of (6.3) that
|(ςf′(ς))′f′(ς)−1|≤|p(ς)−1|+|ςp′(ς)p(ς)|. |
It is known from [27] that for p∈P(β),
|ςp′(ς)p(ς)|≤2(1−β)r(1−r)(1+(1−2β)r). |
Using this result together with (6.4), we write
|(ςf′(ς))′f′(ς)−1|≤2r(1−β)(2+(1−2β)r)(1−r)(1+(1−2β)r). |
We need to show that
|(ςf′(ς))′f′(ς)−1|≤1−β. |
However, it holds if
3(1−2β)r2+2(2+β)r−1≤0. |
Let T(r)=3(1−2β)r2+2(2+β)r−1. Then T(0)=−1<0 and T(1)=6−4β>0 such that T(0)T(1)<0. Thus, there exists R1∈[0,1] such that
3(1−2β)R21+2(2+β)R1−1=0. | (6.5) |
Therefore, 3(1−2β)r2+2(2+β)r−1≤0 for all r<R1 and R1 is the smallest roots of (6.5) given by (6.1).
For sharpness, we consider the functions f0(ς) defined by (6.2). At the point z=R1, we have
|(ςf′0(ς))′f′0(ς)−1|=1−(1−s)2. |
Theorem 6.2. Let f∈STL(s). Then f∈CCV for all z in the disc |ς|<R2, where R2 is the positive roots of the equation
s3x5−3s2x4+(3−s2)sx3−(1−3s2)x2−(3s+2)x+1=0. | (6.6) |
Proof. For f∈STL(s),
ςf′(ς)f(ς)=(1+sw(ς))2, |
where w∈W. Therefore
Re(ςf′(ς))′f′(ς)≥Re(1+sw(ς))2−2r|w′(ς)1+sw(ς)|≥(1−sr)2−2r(1−|w(ς)|2)(1−r2)(1−sr), |
where we have used the extension of Schwarz lemma (see [21]). Thus,
Re(ςf′(ς))′f′(ς)≥(1−sr)3(1−r2)−2r(1−sr)(1−r2)>0 |
if (1−sr)3(1−r2)−2r>0. Let T(r)=(1−sr)3(1−r2)−2r. Then T(0)=1>0 and T(1)=−2<0 with T(0)T(1)<0. Therefore, there exists R2∈[0,1] such that (1−sR2)3(1−R22)−2R2=0. Hence, (1−sr)3(1−r2)−2r>0 holds for all r<R2, where R2 is the smallest positive roots of (6.6).
Theorem 6.3. Let p∈P. Then p∈P(Ls) for all z in the disc
|ς|<2s−s22+2s−s2,0<s≤√22. | (6.7) |
Proof. Let p∈P. Then
|p(ς)−1+r21−r2|<2r1−r2,r∈(0,1). |
We want to prove that
|p(ς)−1|≤1−(1−s)2. |
Now,
|p(ς)−1|≤|p(ς)−1+r21−r2|+2r21−r2≤2r1−r2+2r21−r2=2r1−r<1−(1−s)2 |
if (6.7) is satisfied. To show that the radius cannot be improved, we consider the function
p0(ς)=1+ς1−ς. |
Then for ς=2s−s22+2s−s2,
|p0(ς)−1|=|2ς1−ς|=2s−s2, |
which shows that equality is attained for (6.7).
Corollary 6.1. The STL(s)-radius and CVL(s)-radius for the classes of starlike and convex functions are given by (6.7).
Theorem 7.1. If p(ς) is analytic in U with p(0)=1 and satisfies the condition
Re(ςp′(ς)p(ς))<2ss−1, | (7.1) |
or
Re(ςp′(ς)p(ς))>2ss+1, | (7.2) |
then p(ς)≺(1+sς)2 for −1<s<0.
Proof. Let p(ς) be defined by
p(ς)=(1+sw(ς))2 | (7.3) |
Clearly, w(ς) is analytic in U with w(0)=0. To prove our result, it is required to show that |w(ς)|<1 for all ς∈U. From (7.3), a simple calculation gives
ςp′(ς)p(ς)=2sςw′(ς)1+sw(ς). |
Suppose there exists a point ς0∈U such that
max|ς|≤|ς0||w(ς)|=|w(ς0)|=1. |
Then by Lemma 3.5, w(ς0)=eiθ and ς0w′(ς0)=kw(ς0). Thus,
Re(ς0p′(ς0)p(ς0))=Re(2sς0w′(ς0)1+sw(ς0))=2k(1−Re(11+seiθ))>−2k(s1−s)≥2ss−1. |
This contradicts (7.1). Therefore, there exists no ς0∈U such that |w(ς0)|=1. Thus |w(ς)|<1 in U, so that p(ς)≺(1+sς)2 for −1<s<0.
Similarly,
Re(ς0p′(ς0)p(ς0))=2k(1−Re(11+seiθ))<2k(s1+s)≤2ss+1, |
which contradicts the assumption (7.2). Hence, the proof is completed.
Following the discussion demonstrated by Sharma et al in [35] for Theorem 3, we present the following results.
Theorem 7.2. Let −1<B⅁<A⅁≤1, 0<s≤1√2 and p(ς)=1+A⅁ς1+B⅁ς. Then p∈P(Ls) if and only if
1−2s+s2≤1−A⅁1−B⅁≤1+A⅁1+B⅁≤1+2s−s2 | (7.4) |
or, equivalently, if and only if
A⅁≤{2s−s2+(1−s)2B⅁,forB⅁(B⅁−A⅁)≤0,2s−s2+(1+2s−s2)B⅁,forB⅁(B⅁−A⅁)≥0. | (7.5) |
Proof. The proof follows the techniques presented in [35,Theorem 3] For B⅁=0,A⅁=0 and B⅁=−A⅁, we give the following consequences of Theorem 7.2.
Corollary 7.1.
(i) p(ς)=1+A⅁ς∈P(Ls)⟺0<A⅁≤2s−s2.
(ii) p(ς)=1/(1+B⅁ς)∈P(Ls)⟺(s2−2s)/(1+2s−s2)≤B⅁<0.
(iii) p(ς)=(1+A⅁ς)/(1−A⅁ς)∈P(Ls)⟺0<A⅁≤(2s−s2)/(2+2s−s2).
Corollary 7.2. Let −1<B⅁<A⅁≤1 and consider
ςf′(ς)f(ς)=1+A⅁ς1+B⅁ςand(ςf′(ς))′f′(ς)=1+A⅁ς1+B⅁ς. |
Then f∈STL(s) and f∈CVL(s), respectively if and only if conditions (7.4) or (7.5) is satisfied
Applying Corollary 7.1 along with the integral representation for the classes STL(s) and CVL(s), respectively, we present the following examples.
Example 7.3.
(i) For 0<A⅁≤2s−s2,
f1(ς)=ςexp(A⅁ς)∈STL(s)andf2(ς)=exp(A⅁ς)−1A⅁∈CVL(s). |
(ii) For s2−2s1+2s−s2≤B⅁<0,
f3(ς)=ς1+B⅁ς∈STL(s)andf4(ς)=1B⅁log(1+B⅁ς)∈CVL(s). |
(iii) For 0<A⅁≤2s−s22+2s−s2,
f5(ς)=ς(1−A⅁ς)2∈STL(s)andf6(ς)=ς1−A⅁ς∈CVL(s). |
Theorem 8.1. Let −1≤B⅁<A⅁≤1,0<s≤12 and p(ς) be analytic in U with p(0)=1 such that
1+ρςp′(ς)≺1+A⅁ς1+B⅁ς(ρ∈R∖{0},ς∈U). | (8.1) |
If
|ρ|≥A⅁−B⅁2s(1−s)(1−|B⅁|), | (8.2) |
then p∈P(Ls).
Proof. Let q(ς)=(1+sς)2,0<s≤12. Then q(ς) is convex univalent in U. Consider the functions ϕ(ω)=ρ and θ(ω)=1. These functions are both analytic in a domain containing q(U) with ϕ(ω)≠0. A computation shows that
Q(ς)=ρςq′(ς)=2ρsς(1+sς)andh(ς)=1+ρςq′(ς)=1+2ρsς(1+sς). |
Further,
ReςQ′(ς)Q(ς)≥1−2s1−s>0 |
and
Reςh′(ς)Q(ς)=ReςQ′(ς)Q(ς)>0. |
Using Lemma 3.3, the subordination condition
1+ρςp′(ς)≺1+ρςq′(ς) |
implies p(ς)≺q(ς). To complete the proof, it suffices to prove that the circular disc 2.2 is contained in the region bounded by the curve h(eiθ)(θ∈[0,2π)). To this end, we must show that
|h(eiθ)−1−A⅁B⅁1−B2⅁|≥A⅁−B⅁1−B2⅁. |
Now,
|h(eiθ)−1−A⅁B⅁1−B2⅁|=|2ρseiθ(1+seiθ)+B⅁(A⅁−B⅁)1−B2⅁|≥2s|ρ|(1−s)−|B⅁|(A⅁−B⅁)1−|B⅁|2. |
Thus,
2s|ρ|(1−s)−|B⅁|(A⅁−B⅁)1−|B⅁|2≥A⅁−B⅁1−|B⅁|2 |
if (8.2) is satisfied.
Theorem 8.2. Let 0<s≤12 and p(ς) be analytic in U with p(0)=1 such that
1+ρςp′(ς)≺(1+sς)2(ρ∈R∖{0},ς∈U). | (8.3) |
If
|ρ|≥2+s2(1−s), | (8.4) |
then p∈P(Ls).
Proof. Following the same arguments as in the proof of Theorem 8.1, we arrive at where to show that
(1+sς)2≺1+2ρsς(1+sς):=h(ς). |
To achieve this, it is enough to show that the domain bounded by the limaçon is inside the region bounded by the curve h(eiθ)(θ∈[0,2π)). As a result, we need to find ρ for which
|h(eiθ)−1|≥(1+s)2−1. |
Now,
|h(eiθ)−1|=2|ρ|s|1+seiθ|≥2|ρ|s(1−s)≥2s+s2 |
if
|ρ|≥2+s2(1−s). | (8.5) |
Theorem 8.3. Let −1≤B⅁<A⅁≤1,0<s≤1√2 and p(ς) be analytic in U with p(0)=1 such that
1+ρςp′(ς)≺(1+sς)2(ρ∈R∖{0},ς∈U). |
If
|ρ|≥(2s+s2)(1+|B⅁|)2A⅁−B⅁, | (8.6) |
then
p(ς)≺1+A⅁ς1+B⅁ς. |
Proof. Let q(ς)=1+A⅁ς1+B⅁ς. We have that q(ς) is convex univalent in U. Therefore, following the method of proof in Theorem 8.1, we arrive at where to show that
(1+sς)2≺1+ρ(A⅁−B⅁)ς(1+B⅁ς)2:=h(ς). |
For this, we need to establish that the region bounded by the limaçon lies inside the domain bounded by the curve h(eiθ)(θ∈[0,2π). A simple observation of (2.5) suggests it suffices to show
|h(eiθ)−1|≥(1+s)2−1. |
Now,
|h(eiθ)−1|=|ρ|A⅁−B⅁|1+B⅁ς|2≥|ρ|A⅁−B⅁(1+|B⅁|)2. |
But
A⅁−B⅁(1+|B⅁|)2≥(1+s)2−1 |
provided (8.6) holds. This completes the proof.
Theorem 8.4. Let 0<s≤1√2 and p(ς) be analytic in U with p(0)=1 such that
1+ρςp′(ς)p(ς)≺(1+sς)2(ρ∈R∖{0},ς∈U). | (8.7) |
If
|ρ|≥(2−s)(1+s)2, | (8.8) |
then
p(ς)≺(1+sς)2. |
Proof. Let q(ς)=(1+sς)2. Then q(ς) is convex univalent in U. The function ϕ(ω)=ρ/ω and θ(ω)=1 are analytic in the domain containing q(U). Set
Q(ς)=ςq′(ς)ϕ(q(ς))=ρςq′(ς)q(ς)=2ρsς1+sς |
and
h(ς)=θ(q(ς))+Q(ς)=1+ρςq′(ς)q(ς)=1+2ρsς1+sς. |
Then
Reςh′(ς)Q(ς)=ReςQ′(ς)Q(ς)>11+s>0. |
From Lemma 3.3, the differential subordination
1+ρςp′(ς)p(ς)≺1+ρςq′(ς)q(ς),ς∈U |
implies p(ς)≺q(ς). To finalize the proof, we need to prove
(1+sς)2≺1+2ρsς1+sς, | (8.9) |
which is equivalent to showing
Ls(U)⊂h(U). |
It is easy to see that the transformation h(ς)=1+ρςq′(ς)q(ς) maps U onto the disc D(a,r), where
a=1−(1+2|ρ|)s21−s2<1andr=2|ρ|s1−s2. |
Therefore (8.9) holds if and only if
1−1−(1+2|ρ|)s21−s2<2|ρ|s1−s2, |
which implies
|ρ|>(1+s)(2−s)2. |
We choose to omit the proof of the next theorem since it follows the same argument as in Theorem 8.3.
Theorem 8.5. Let −1≤B⅁<A⅁≤1, 0<s≤1√2 and p(ς) be analytic in U with p(0)=1 such that
1+ρςp′(ς)p(ς)≺(1+sς)2(ρ∈R∖{0},ς∈U). |
If
|ρ|≥(1+|A⅁|)(1+|B⅁|)(2s+s2)A⅁−B⅁, | (8.10) |
then
p(ς)≺1+A⅁ς1+B⅁ςς∈U. |
Remark 8.1.
(ⅰ) If we put p(ς)=ςf′(ς)/f(ς) and p(ς)=(ςf′(ς))′/f′(ς) in Theorem 8.1-8.5, we obtain the conditions on ρ for which the respective subordination conditions (8.1), (8.3) and (8.7) imply f∈STL(s) and f∈CVL(s).
(ⅱ) We note that our condition 0<s≤12 cannot be relaxed in Theorem 8.1 and Theorem 8.2. Otherwise, starlikeness of Q will not be achieved. As such, the proof of the theorems will be extremely difficult to obtain via Lemma 3.3.
The Ma and Minda classes of functions are the comprehensive generalization of the classes SST and CCV. These classes are vital in (GFT) because of their importance in science and technology. To this end, continuous studies of their subfamily, which are related to Limaçon domain were investigated. Coefficients bounds, Fekete Szeg¨o inequality as well as the upper bounds of the third Hankel determinants for these subclasses were derived. Finally, the techniques of differential subordination were also used to obtain some restrictions for which analytic functions belonged to these families. In addition, to have more new theorems under present examinations, new generalization and applications can be explored with some positive and novel outcomes in various fields of science, especially, in applied mathematics. These new surveys will be presented in future research work being processed by authors of the present paper.
The authors would like to thank the Rector of COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research and academic environment. The fourth author is supported by UKM grant: GUP-2019-032.
The authors declare that they have no conflicts of interest.
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