On Booth lemniscate of starlike functions
R. Kargar A. Ebadian J. Sokół



Assume that Δ is the open unit disk in the complex plane and
𝒜 is the class of normalized analytic functions in
Δ. In this paper we introduce and study the class

    ℬ𝒮(α):={f∈𝒜: (zf'(z)/f(z)-1)≺z/1-α z^2,   z∈Δ},

where 0≤α≤1 and ≺ is the subordination relation.
Some properties of this class like differential subordination,
coefficients estimates and Fekete-Szegö inequality associated
with the k-th root transform are considered.


R. Kargar, A. Ebadian and J. Sokół]
       Rahim Kargar, Ali Ebadian and Janusz Sokół 

Department of Mathematics, Payame Noor University, Tehran, Iran
       rkargar@pnu.ac.ir (Rahim Kargar)

Department of Mathematics, Payame Noor University, Tehran, Iran
       ebadian.ali@gmail.com (Ali Ebadian)

 University of Rzeszów, Faculty of Mathematics and Natural
         Sciences, ul. Prof. Pigonia 1, 35-310 Rzeszów, Poland
       jsokol@ur.edu.pl (Janusz Sokół)

On Lattice Calculation of Electric Dipole Moments and Form Factors of the Nucleon
    [
    18 november 2016
=================================================================================




§ INTRODUCTION



Let 𝒜 denote the class of functions f(z) of the
form:

    f(z)=z+ ∑_n=2^∞a_nz^n,

which are analytic and normalized in the open unit disk Δ={z∈ℂ :
|z|<1}. The subclass of
𝒜 consisting of all univalent functions f(z) in
Δ is denoted by 𝒮. A function f∈𝒮
is called starlike (with respect to 0), denoted by
f∈𝒮^*, if tw∈ f(Δ) whenever w∈ f(Δ)
and t∈[0, 1]. Robertson introduced in <cit.>, the class 𝒮^*(γ) of starlike functions of order γ≤1,
which is defined by

    𝒮^*(γ):={ f∈𝒜:  ℜ𝔢{zf'(z)/f(z)}> γ,  z∈Δ}.

If γ∈[0,1), then a function in 𝒮^*(γ) is univalent. In particular we put
𝒮^*(0)≡𝒮^*. We denote by 𝔅 the class of analytic functions w(z) in
Δ with w(0) = 0 and |w(z)| < 1, (z ∈Δ).
If f and g are two of the functions in 𝒜, we say
that f is subordinate to g, written f (z)≺ g(z), if there
exists a w∈𝔅 such that f (z) = g(w(z)), for all
z∈Δ.

Furthermore, if the function g is univalent in Δ, then
we have the following equivalence:

    f (z)≺ g(z) ⇔ (f (0) = g(0)   and   f (Δ)⊂ g(Δ)).

We now recall from <cit.>, a one-parameter family of functions as follows:

    F_α(z):=z/1-α z^2=z+∑_n=1^∞α^nz^2n+1     (z∈Δ, 0≤α≤1).

The function F_α(z) is starlike univalent for 0≤α<1.
We have also F_α(Δ)=D(α), where

    D(α)={x+iy∈ℂ:  (x^2+y^2)^2-x^2/(1-α)^2-y^2/(1+α)^2<0},

when 0≤α<1 and

    D(1)={x+iy∈ℂ:  (∀ t∈ (-∞,-i/2]∪ [i/2,∞))[x+iy≠ it]}.




The Persian curve (cf. <cit.>) is a plane algebraic curve of order four that is the line
of intersection between the surface of a torus and a plane parallel to its axis. The equation in rectangular coordinates is

    (x^2+y^2+p^2+d^2-r^2)^2=4d^2(x^2+p^2),

where r is the radius of the circle describing the torus, d is
the distance from the origin to its center and p is the distance
from the axis of the torus to the plane. We remark that a curve
described by

    (x^2 + y^2)^2 - (n^4 + 2m^2)x^2 - (n^4 - 2m^2)y^2 = 0     (x, y)≠(0, 0),

is a special case of Persian curve that studied by Booth and is called the Booth lemniscate <cit.>.
The Booth lemniscate is called elliptic if n^4 > 2m^2 while, for n^4 < 2m^2, it is termed hyperbolic. Thus
it is clear that the curve

    (x^2+y^2)^2-x^2/(1-α)^2-y^2/(1+α)^2=0     (x, y)≠(0, 0),

is the Booth lemniscate of elliptic type (see figure 1). Two other special case of Persian curve are Cassini oval and Bernoulli lemniscate.

A plane algebraic curve of order four whose equation in Cartesian coordinates has the form:

    (x^2 + y^2)^2 - 2c^2 (x^2 -y^2) = a^4 -c^4.


The Cassini oval is the set of points such that the product of the
distances from each point to two given points p_2=(-c,0) and
p_1=(c,0) (the foci) is constant. When a ≥ c√(2) the
Cassini oval is a convex curve; when c<a<c√(2) it is a curve
with "waists" (concave parts); when a=c it is a Bernoulli
lemniscate; and when a<c it consists of two components. Cassini
ovals are related to lemniscates. Cassini ovals were studied by G.
Cassini (17th century) in his attempts to determine the Earth's
orbit.

The Bernoulli lemniscate plane algebraic curve of order four, the equation of which in orthogonal Cartesian coordinates is

    (x^2 + y^2)^2 - 2a^2 (x^2 -y^2) =0,

and in polar coordinates

    ρ^2=2a^2 cos 2 ϕ.


The Bernoulli lemniscate is symmetric about the coordinate origin, which is a node with tangents
y=± x and the point of inflection. The product of the distances of any point M to the two given points
p_1=(-a,0) and p_2=(a,0) is equal to the square of the distance between the points p_1 and p_2.
The Bernoulli lemniscate is a special case of the Cassini ovals, the lemniscates, and the sinusoidal spirals.
The Bernoulli spiral was named after Jakob Bernoulli, who gave its equation in 1694.





In <cit.>, the authors introduced and studied the class ℳ(δ) as follows:


Definition A. Let π/2≤δ<π. Then the function
f∈𝒜 belongs to the class ℳ(δ) if f
satisfies:

    1+δ-π/2 sinδ<
        ℜ𝔢{zf'(z)/f(z)} <
        1+δ/2sinδ     (z∈Δ).


By definition of subordination  and by (<ref>), we have that f∈ℳ(δ) if and only if

    (z f'(z)/f(z)-1)≺ℬ_α(z):=
    1/2isinδlog(1-z/1-ze^-iδ)
         (z∈Δ),

where π/2≤δ<π. The above function ℬ_α(z) is convex
univalent in Δ and maps
Δ onto Γ_δ={w: (δ-π)/(2sinδ)<ℜ𝔢{w}<δ/(2sinδ)},
or onto the convex hull of three points (one of which may be that point
at infinity) on the boundary of Γ_δ. In other words, the image of Δ may be a vertical
strip when π/2≤δ<π, while in other cases, a half strip, a trapezium, or a triangle.

It was proved in <cit.>, that for α<1<β, the following
function P_α,β:Δ→ℂ, defined by

    P_α,β(z)=1+β-α/πi log(1-e^2π
       i1-α/β-αz/1-z)     (z∈Δ),

maps Δ onto a convex domain

    P_α,β(Δ)={ w∈ℂ: α<
        ℜ𝔢{w}<β},

conformally. Therefore, the function P_α,β(z) defined
by (<ref>) is convex univalent in Δ and has the form:

    P_α,β(z)=1+∑_n=1^∞ B_n z^n,

where

    B_n=β-α/nπi (1-e^2nπ
    i1-α/β-α)     (n=1,2,…).


The present authors (see <cit.>) introduced the class 𝒱(α,β) as follows:

Definition B. Let α<1 and β>1. Then the
function f∈𝒜 belongs to the class
𝒱(α,β) if f satisfies:

    α<
        ℜ𝔢{(z/f(z))^2 f'(z)} <β     (z∈Δ).

Therefore, by definition of subordination, we have
that f∈𝒱(α,β) if and only if

    (z/f(z))^2 f'(z)
      ≺ P_α,β(z)     (z∈Δ).




Motivated by Definition A, Definition B and using F_α, we
introduce a new class. Our principal definition is the following.


  Let f∈𝒜 and 0≤α< 1.
Then f∈ℬ𝒮(α) if and only if
  
    (zf'(z)/f(z)-1)≺ F_α(z)     (z∈Δ),

  where F_α defined by (<ref>).

In our investigation, we require the following result.



We have that f∈ℬ𝒮(α) if and only if

    f(z)=zexp∫_0^z F_α(w(t))-1/t dt     (z∈Δ),

for some function w(z), analytic in Δ, with |w(z)|≤
|z| in Δ.



From (<ref>) it follows that there exists a function
w(z), analytic in Δ, with |w(z)|≤ |z| in Δ,
such that

    z(f'(z)/f(z)-1/z)=F_α(w(z))     (z∈Δ),

or

    z(logf(z)/z)'=F_α(w(z))    
          (z∈Δ).

This gives (<ref>). On the other hand, it is a easy calculation
that a function having the form (<ref>) satisfies condition
(<ref>).



Applying formula (<ref>) for w(z)=z gives that

    f_0(z)=z(1+√(α)z/1-√(α)z)^1/√(α)     (z∈Δ),

is in the class ℬ𝒮(α).




Let F_α(z) be given by (<ref>). Then

    1/α-1< ℜ𝔢{F_α(z)}< 1/1-α     (0≤α<1).



If α=0, then we have
-1<ℜ𝔢{F_α}=ℜ𝔢(z)<1. For 0<α<1,
the function {F_α} does not have any poles in
Δ and is analytic in Δ, thus looking for
the min{ℜ𝔢{F_α(z)}:  |z|<1} it is sufficient
to consider it on the boundary ∂
F_α(Δ)={F_α(e^iφ):φ∈ [0,2π]}. A
simple calculation give us

    ℜ𝔢{F_α(e^iφ)}=(1-α)cosφ/1+α^2-2αcos2φ    (φ∈[0,2φ]).

So we can see that ℜ𝔢{F_α(z)} is
well defined also for φ=0 and φ=2π. Define

    g(x)=(1-α)x/1+α^2-2α (2x^2-1)     (-1≤ x≤ 1),

then for 0< α<1, we have g'(x)>0. Thus for -1≤ x≤
1, we have

    1/α-1=g(-1)≤ g(x)≤ g(1)=1/1-α.

This completes the proof.

We note that from Lemma <ref> and by definition of subordination, the function f∈𝒜 belongs to
the class ℬ𝒮(α), 0≤α<1, if it satisfies the condition

    1/α-1< ℜ𝔢(zf'(z)/f(z)-1)< 1/1-α     (z∈Δ),

or equivalently
    
    α/α-1< ℜ𝔢(zf'(z)/f(z))<2-α/1-α     (z∈Δ).

It is clear that ℬ𝒮(0)≡𝒮(0,2)⊂𝒮^*, where the class 𝒮(α, β),
α< 1 and β > 1, was recently considered by K. Kuroki
and S. Owa in <cit.>.



If f∈ℬ𝒮(α), then

    zf'(z)/f(z)≺ P_α(z)      (z∈Δ),

where

    P_α(z)=1+2/π(1-α)i log(1-e^π
        i(1-α)^2z/1-z)     (z∈Δ),

is convex univalent in Δ.


If f∈ℬ𝒮(α), then it satisfies (<ref>) or
zf'(z)/f(z) lies in a strip of the form (<ref>). Then
applying the definition of subordination and  the function
(<ref>), we obtain (<ref>) and (<ref>).


For the proof of our main results, we need the following Lemma.








   
   






  






(See <cit.>)
Let q(z)=∑_n=1^∞C_nz^n be analytic and
univalent in Δ, and suppose that q(z) maps Δ
onto a convex domain. If p(z) = ∑_n=1^∞A_nz^n is
analytic in Δ and satisfies the following subordination

    p(z)≺ q(z)     (z∈Δ),

then

    |A_n|≤ |C_1|     n≥ 1.









§  MAIN RESULTS


The first main result is the following theorem.


Let f∈𝒜 and 0≤α<1. If f∈ℬ𝒮(α) then

    logf(z)/z≺∫_0^z P_α(t)-1/t dt     (z∈Δ),

where

    P_α(z)-1=2/π(1-α)i log(1-e^π
        i(1-α)^2z/1-z)     (z∈Δ)

and

    P_α(z)=∫_0^z P_α(t)-1/t dt     (z∈Δ),

are convex univalent in Δ.





If f∈ℬ𝒮(α), then by (<ref>) it satisfies

    z{logf(z)/z}'≺ P_α(z)-1.

It is known that if ℱ(z) is convex univalent in Δ, then

    [f(z)≺ℱ(z)]⇒[∫_0^z f(t)/t dt≺∫_0^z ℱ(t)/t dt]

and

    F(z)=∫_0^z ℱ(t)/t dt,

is convex univalent in Δ. By Corollary <ref>, we know that
P_α(z)-1 is  convex univalent in Δ. Therefore,
applying (<ref>) in (<ref>) gives (<ref>) with convex univalent
P_α(z).




If f∈ℬ𝒮(α) and |z|=r<1, then

    min_|z|=r|expP_α(z)|≤|f(z)/z|
          ≤max_|z|=r|expP_α(z)|.



Subordination  implies

    f(z)/z≺expP_α(z)

and expP_α(z) is convex univalent. Then
(<ref>) implies  (<ref>).





We now obtain coefficients estimates for functions belonging to the class ℬ𝒮(α).


 Assume that the function f of the form (<ref>) belongs to the class ℬ𝒮(α) where 0≤α≤ 3-2√(2). then |a_2|≤ 1 and
 
    |a_n|≤1/n-1∏_k=2^n-1(k/k-1)     (n=3,4,…).



  Assume that f∈ℬ𝒮(α). Then from Definition <ref> we have
  
    p(z)≺ 1+ F_α(z)=1+z+α z^3+⋯     (z∈Δ),

  where
  
    zf'(z)=p(z)f(z).

  We note that F_α is convex function for 0≤α≤ 3-2√(2) (see <cit.>).
  If we define p(z)=1+∑_n=1^∞p_nz^n, then from Lemma <ref>, we have
  
    |p_n|≤ 1.

    Equating the coefficients of z^n on both sides of (<ref>), we find the following relation between the coefficients:
  
    na_n=p_n-1+a_2p_n-2+⋯+a_n-1p_1+a_n.

  Making use of (<ref>) and (<ref>), we get
  
    |a_n|≤1/n-1∑_k=1^n-1|a_k|     |a_1|=1.

  Obvious that, from (<ref>) we have |a_2|≤ 1. We now need show that
  
    1/n-1∑_k=1^n-1|a_k|≤1/n-1∏_k=2^n-1(1+1/k-1)     (n=3,4,…).

  We use induction to prove (<ref>). If we take n=3 in the inequality (<ref>), we have |a_2|≤ 1, therefore the case
  n=3 is clear. A simple calculation gives us
  
    |a_m+1|    ≤1/m∑_k=1^m|a_k|=1/m(∑_k=1^m-1|a_k|+|a_m|)
       ≤1/m∏_k=2^m-1(1+1/k-1)+1/m×1/m-1∏_k=2^m-1(1+1/k-1)
       =1/m∏_k=2^m(1+1/k-1),

  which implies that the inequality (<ref>) holds for n=m+1.
  From now (<ref>) and (<ref>), the desired estimate for
|a_n|  (n = 3,4,…) follows, as asserted in (<ref>).
This completes the proof.


The problem of finding sharp upper bounds for the coefficient
functional |a_3-μ a_2^2| for different subclasses of the
normalized analytic function class 𝒜 is known as the
Fekete-Szegö problem.


Recently, Ali et al. <cit.> considered the Fekete-Szegö
functional associated with the kth root transform for several
subclasses of univalent functions. We recall here that, for a
univalent function f(z) of the form (<ref>), the kth root
transform is defined by

    𝔉(z)=[f(z^k)]^1/k=z+∑_n=1^∞b_kn+1z^kn+1    
     (z∈Δ).

Following, we consider
the problem of finding sharp upper bounds for the Fekete-Szegö
coefficient functional associated with the kth root transform for
functions in the class ℬ𝒮(α).

In order to prove next result, we need the following lemma due to Keogh and Merkes <cit.>.
Further we denote by 𝒫 the well-known class of analytic functions p(z) with
p(0) = 1 and ℜ𝔢(p(z))>0, z∈Δ.

Let the function g(z) given by

    g(z)=1+c_1z+c_2z^2+⋯,

be in the class 𝒫. Then, for any complex number μ

    |c_2-μ c_1^2|≤ 2max{1,|2μ-1|}.

The result is sharp.



Let 0≤α<1, f∈ℬ𝒮(α) and 𝔉 is the
kth root transform of f defined by (<ref>). Then, for any
complex number μ,

    |b_2k+1-μ
     b_k+1^2|≤1/2kmax{1,|2(μ-1)/k+1|}.

The result is sharp.


Using (<ref>), we first put

    1+F_α(z)=1+∑_n=1^∞ℬ_n z^n,

where
ℬ_1=1, ℬ_2=0, ℬ_3=α, and etc.
Since f∈ℬ𝒮(α), from Definition <ref> and definition of
subordination, there exists w∈𝔅 such that

    zf'(z)/f(z)=1+F_α(w(z)).

We now define

    p(z)=1+w(z)/1-w(z)=1+p_1z+p_2z^2+⋯.

Since w∈𝔅, it follows that p∈𝒫. From (<ref>) and
(<ref>) we have:

    1+F_α(w(z))=1+1/2ℬ_1p_1z+(1/4ℬ_2p_1^2+
     1/2ℬ_1(p_2-1/2p_1^2))z^2+⋯,

where ℬ_1=1 and ℬ_2=0. Equating the coefficients of z and z^2 on both sides of
(<ref>) and substituting ℬ_1=1 and ℬ_2=0, we get

    a_2=1/2p_1,

and

    a_3=1/8p_1^2+1/4(p_2-1/2p_1^2).

A computation shows that, for f given by (<ref>),

    𝔉(z)=[f(z^1/k)]^1/k=z+1/ka_2z^k+1+(1/ka_3-1/2k-1/k^2a_2^2)z^2k+1+⋯.

From equations (<ref>) and (<ref>), we have

    b_k+1=1/ka_2   and   b_2k+1=1/ka_3-1/2k-1/k^2a_2^2.

Substituting from (<ref>) and (<ref>) into (<ref>), we
obtain

    b_k+1=1/2kp_1,

and

    b_2k+1=1/4k(p_2-k-1/kp_1^2),

so that

    b_2k+1-μ
     b_k+1^2=1/4k[p_2-1/2(2(μ-1)/k+
     2)p_1^2].

Letting

    μ'=1/2(2(μ-1)/k+2),

the inequality (<ref>) now follows as an application of Lemma
<ref>. It is easy to check that the result is sharp for the
kth root transforms of the function

    f(z)=zexp(∫_0^zF_α(w(t))/tdt).



Putting k=1 in Theorem <ref>, we have:


(Fekete-Szegö inequality) Suppose that f∈ℬ𝒮(α) and 0≤α<1. Then, for any
complex number μ,

    |a_3-μ
     a_2^2|≤1/2max{1,|2μ-1|}.

The result is sharp.


It is well known that every function f∈𝒮 has an inverse f^-1, defined by f^-1(f(z))= z, z∈Δ and

    f(f^-1(w))=w     (|w|<r_0;  r_0<1/4),

where

    f^-1(w)=w-a_2w^2+(2a_2^2-a_3)w^3-(5a_2^3-5a_2a_3+a_4)w^4+⋯.



Let the function f, given by (<ref>), be in the class ℬ𝒮(α) where 0≤α<1. Also
let the function f^-1(w)=w+∑_n=2^∞b_nw^n be inverse of f. Then

    |b_2|≤ 1,

and

    |b_3|≤3/2.



  Relation (<ref>) give us
  
    b_2=-a_2   and   b_3=2a_2^2-a_3.

  Thus, we can get the estimate for |b_2| by
  
    |b_2|=|a_2|≤ 1.

  For estimate of |b_3|, it suffices in Corollary <ref>, we put μ=2. Hence
the proof of Corollary <ref> is completed.




















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