Coefficients problems for families of holomorphic functions related to hyperbola
S. Kanas V. S. Masih A. Ebadian
math.CV 30C45, 30C80


Coefficients problems for families of holomorphic functions...  …]
Coefficients problems for families of holomorphic functions related to hyperbola

S. Kanas, V. S. Masih and A. Ebadian]S. Kanas^1, V. S. Masih^2 and A. Ebadian^3
^1 University of Rzeszow, Al. Rejtana 16c, PL-35-959 Rzeszów, Poland 
skanas@ur.edu.pl

^2 Department of Mathematics, Payame Noor University, Tehran, Iran
masihvali@gmail.com; v_soltani@pnu.ac.ir

^3 Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran
ebadian.ali@gmail.com

[
    [
    December 30, 2023
=====================




We consider a family of analytic and normalized functions that are related to the domains ℍ(s), with a right branch of a hyperbolas H(s) as a boundary. The hyperbola H(s) is given by the relation 1/ρ=( 2cosφ/s)^s   (0<s≤ 1, |φ|<(π s)/2). We mainly study a coefficient problem of the families of functions for which zf'/f or 1+zf”/f' map the unit disk onto a subset of ℍ(s).  We find coefficients bounds, solve Fekete-Szegö problem and estimate the Hankel determinant.


 [2010 Mathematics Subject Classification: 30C45, 30C80]
[Key words and phrases: univalent functions, subordination, starlike and convex functions, domain related to hyperbola, conic sections]
[1 Corresponding author] 




§ INTRODUCTION AND DEFINITION

Geometric interpretation of some properties of functions represents one of the major goal in geometric function theory of one complex variable. In this theory the description of geometries in succinct mathematical terms, establishing close links between certain prescribed and analytically expressed properties are the most desirable.
A specific relations occur with the families of domains contained in a right halplane, where the rigour of analytic reasoning one closely blends with the geometric intuition. The halfplanes, circular and angular domains, and domains bounded by conic sections have been popular and investigated so far, see for example <cit.>, and also <cit.>. The relations between analytic functions and those domains relay on the inclusions of the image of some analytic expressions of the unit disk 𝔻 in those domains.


Very recently a new subfamily of domains contained in a right half plane and  related to a hyperbola

    H(s)={ρ^ıφ:ρ=1/2cosφ/s^s,   -π s/2<φ<π s/2}

was defined <cit.>.  It has been assumed that 0<s≤ 1. Hyperbola  H(s) intersects the real axis at (u,0)=(2^-s,0) and has a slope angle to the real axis equal (π s)/2, see Fig.1. In a limiting case s=1 the hyperbola becomes a line that intersects a real axis at (u,0)=(2^-1,0). The detailed description of geometric behavior of H(s) and connections with the other curves of the right halfplane was given in <cit.>.

Let ℍ(s) be a domain with H(s) as a boundary. ℍ(s) is symmetric with the real axis, and starlike with respect to 1∈ℍ(s).  The family  {ℍ(s), 0< s ≤ 1}  constitutes a new subfamily of domains contained in a right halfplane which  is not reduced to any family of domains considered until now. Even the hyperbolas that occurred in the consideration of conic sections ∂Ω_k  (0≤ k < ∞) (see <cit.>) for no choice of parameter k and s  reduce to ℍ(s).




Fig. 1. The hyperbola H(s) for s=1/2.


By ℋ we will denote the class of functions  f holomorphic in the open unit disc  𝔻={z∈ℂ: |z|<1} of the complex plane ℂ, and with the power series

    f(z)=z+∑_n=2^∞a_nz^n     z ∈𝔻,

and by 𝒮  its the most important subclass consisting of univalent functions.

A comprehensive geometric characteristics of members of 𝒮 is still not known. The definitely much  known are functions from various subclasses of 𝒮, such as 𝒮𝒯, 𝒞𝒱, i.e. functions  which are starlike with respect to the origin and convex, respectively.

For further considerations we denote by 𝒫  the class of analytic functions that maps the origin to the point 1 and has a positive real part. The class 𝒫 has the significant relations with the majority of subclasses of 𝒮, and is known as the Carathéodory class.
We have

    𝒫={p(z)= 1+p_1z+p_2 z^2 +⋯,   p(z) >0, z∈𝔻}.

Now, let us denote

    (z):=1/1-z^s= ^-sln(1-z)    0<s≤1,  z∈𝔻,

where the branch of logarithm is determined by (0)=1. The function   maps the unit disk onto a domain ℍ(s) and play a role of extremal function in a class 𝒫()={p∈𝒫: p≺}.
The power series of  has the form

    (z)   =1+∑_n=1^∞q_nz^n=1+∑_n=1^∞ss+1⋯s+n-1/n! z^n=1+∑_n=1^∞(s)_n/n! z^n,

where (a)_n is known as the Pochhamer symbol; (a)_n=Γ(a+n)/Γ(a).
The necessary and sufficient condition which describes the members of 𝒫() was given in <cit.>.
 Let p∈𝒫. A function p∈𝒫() if and only if

    p(z)^1/s-1 <p(z)^1/s  z∈𝔻.




By 𝒮𝒯_hpl(s) we denote the subfamily of 𝒮 consisting of the functions f, satisfying the condition

    zf'(z)/f(z)≺(z)   z∈𝔻,

and by 𝒞𝒱_hpl(s) the subclass of univalent functions f such that

    1+zf”(z)/f'(z)≺(z)   z∈𝔻,

where ≺ denotes the symbol of subordination.

Let Φ_s,n∈𝒮𝒯_hpl(s) be given by

    zΦ'_s,n(z)/Φ_s,n(z) = 1/1-z^n^s  z∈𝔻,   n=1, 2,  ….

Then, the functions Φ_s,n(z) is of the form

    Φ_s,n(z)=zexp∫_0^z𝔮_s(t^n)-1/t dt=z+s/nz^n+1+(n+2)s^2+ns/4n^2z^2n+1
    
       +4n^2s+(9n+6n^2)s^2+(2n^2+9n+6)s^3/36n^3z^3n+1⋯.

Especially for n=1,

    Φ_s(z):=Φ_s,1(z)=z+s z^2+3s^2+s/4z^3+4s+15 s^2+17s^3/36z^4+⋯.

Also, let K_s,n∈𝒞𝒱_hpl(s) be given by

    1+zK”_s,n(z)/K'_s,n(z) = 1/1-z^n^s  z∈𝔻,   n=1, 2,  ….

From the above, we find that

    K_s,n(z)   =∫_0^zexp∫_0^w𝔮_s(t^n)-1/t dtdw= z+s/n(n+1)z^n+1
        +(n+2)s^2+ns/4n^2(2n+1)z^2n+1
      +4n^2s+(9n+6n^2)s^2+(2n^2+9n+6)s^3/36n^3(3n+1)z^3n+1⋯.

Especially for n=1,

    K_s(z):=K_s,1(z)= z+s/2z^2 +3s^2+s/12z^3+ 4s+15s^2+17s^3/144z^4+⋯.





§ PRELIMINARY RESULTS


In order to achieve our aim  we recall some definitions  and lemmas that will be useful in the next part of our paper.


By ℬ we denote the class of analytic selfmappings of the unit disk, that maps the origin onto the origin, i.e.

    ℬ={ω(z)∈ℋ, ω(0)=0, |ω(z)|<1, z∈𝔻}.

The class ℬ is known as the class of Schwarz functions.


 For the function ω∈ℬ and ω(z)=w_1z+w_2z^2+⋯ it holds
 
    w_2   =ξ (1-w_1^2),
    
     w_3   =(1-w_1^2)(1-|ξ|^2)ζ-w_1(1-w_1^2)ξ^2,

 for some complex number ξ, ζ with ξ≤ 1 and ζ≤ 1.



If ω(z)=∑_n=1^∞w_nz^n∈ℬ, then for real numbers t it holds
 
    w_2-t w_1^2≤{[               -t     for   t ≤-1,;                1 for   -1≤ t ≤ 1,;                t     for   t ≥ 1. ].

When t<1 or t>1, the equality holds if and only if ω(z)=z or one of its rotations. If -1<t<1, then equality
holds if and only if ω(z)=z^2 or one of its rotations. Equality holds for t=-1 if and only if ω(z)=zx+z/1+xz 0≤ x≤ 1 or one of its rotations while for t = 1, equality holds if and only ω(z)=-zx+z/1+xz 0≤ x≤ 1 or one of its rotations.



We  say that f is subordinate to F  in 𝔻, written
f(z)≺ F(z)  (or f≺ F), if there exists a function ω∈ℬ such that f(z)=F(ω(z))  (z∈𝔻) (see, for example  <cit.>). If  F is univalent in 𝔻, then f≺ F holds  if and only, if f(0)=F(0) and f(𝔻)⊂ F(𝔻).


Let  g(z)=∑_n=1^∞B_nz^n be analytic and convex univalent in 𝔻. If h(z)=∑_n=1^∞A_nz^n is analytic
in 𝔻 and satisfies the  subordination h≺ g in 𝔻, then

    A_n≤B_1    n=1, 2, ….





Let q,n ≥ 1. By H_q(n) we denote the q^th Hankel determinant of the form

    H_q(n) =
    
    a_n     a_n+1   …    a_n+q-1
    
    a_n+1    a_n+2   …    a_n+q
    ⋮   ⋮   ⋱   ⋮
    
    a_n+q-1    a_n+q   …    a_n+2q-2    a_1=1,

where a_k (k=1,2,...) are the coefficients  of the Taylor series expansion of function f of the form (<ref>).

H_q(n) was defined by  Pommerenke <cit.>, and for fixed q,n the bounds of |H_q(n)| have been studied for several subfamilies of univalent functions. The Hankel determinants H_2(1)=a_3-a_2^2 and H_2(2)=a_2 a_4 - a_3^2,  are well-known as Fekete-Szegö and second Hankel determinant functionals, respectively. Further, Fekete and Szegö  introduced the generalized functional
a_3-λ a^2_2, where λ is some real number.
We will give the sharp upper bound for the some Hankel determinants in the subclasses of univalent functions related to hyperbola H(s).





§ COEFFICIENT BOUNDS


This section is devoted to the general problem of coefficients in a class 𝒮𝒯_hpl(s) and 𝒞𝒱_hpl(s). We present the estimates of coefficients for elements of 𝒮𝒯_hpl(s) and  bounds of logarithmic coefficients, but we also solve  the Fekete-Szegö problem and a very popular the Hankel determinant |H_2(2)|  problem.

First we investigate the sharp order of growth for the coefficients of functions in 𝒮𝒯_hpl(s) and 𝒞𝒱_hpl(s) with 0<s<1/2.

Assume that the function f of the form (<ref>) belongs to the class 𝒮𝒯_hpl(s) 𝒞𝒱_hpl(s), respectively with 0<s<1/2. Then a_n=1/n (a_n=1/n^2, resp.) for n=1, 2, 3, ….


From <cit.>, the functions 1/(1-z)^s for 0<s<1/2 is in the class H^2, the Hardy class of analytic functions in 𝔻. Now, from the results in <cit.> we conclude the theorem.


Using properies of the functions p∈𝒫() we can obtain more satisfactory bounds for |a_n|.


Assume that the function f of the form (<ref>) belongs to the class 𝒮𝒯_hpl(s). Then

    a_n≤(s)_n-1/(n-1)!    n=2, 3, 4, ….




Let f∈𝒮𝒯_hpl(s). Then,  from Definition <ref>, there exists p∈𝒫() such that

    p(z)=zf'(z)/f(z)=1+p_1z+p_2z^2+…≺(z).

From the above relation we find the following relation between the coefficients of f and p as follows

    a_2=p_1   and   na_n=p_n-1+a_2p_n-2+⋯+a_n-1p_1+a_n   n=3, 4,  ….

Applying Theorem <ref> we deduce that |p_n|≤ s  n=1,2, 3, …,  then |a_2|=|p_1|≤ s, moreover by (<ref>)

    a_n≤s/n-11+ ∑_k=2^n-1a_k    n=3, 4,  ….

We next proceed by induction. Then, we have to prove that

    |a_n+1|≤(s)_n/n!  n=2, 3…,

assumming that the inequality |a_n|≤ (s)_n-1/(n-1)! holds for n≥ 2. Applying (<ref>) we observe that

    [                             |a_3|                                 ≤   s2(1+|a_2|) ≤ s2(1+s)=s(s+1)2!,;                             |a_4|                                 ≤ s3(1+s+s(s+1)2!) = s(s+1)(s+2)3!, ]

and finally

    [                                                       |a_n+1|                                                             ≤ sn(1+s+s(s+1)2!+s(s+1)(s+2)3!+⋯ +s(s+1)· ... · (s+n-2)(n-1)!);                                                                                                                           =                             s(s+1)(s+2)· ... · (s+n-2)(n-1)!. ]

Therefore

    |a_n+1|≤(s)_nn!,

and the assertion holds. Applying the Principle of Mathematical Induction, the formula (<ref>) is proved.


If f  of the form (<ref>)  belongs to 𝒞𝒱_hpl(s), then

    a_n≤(s)_n-1/n!    n=2, 3, 4, ….



 Let f∈𝒞𝒱_hpl(s). Using the fundamental Alexander relation between 𝒞𝒱_hpl(s) and 𝒮𝒯_hpl(s), we have  zf'∈𝒮𝒯_hpl(s), which gives that na_n satisfies  (<ref>), and then the assertion follows.


 
Let f∈𝒮𝒯_hpl(s) be given by (<ref>). Then

    |H_2(2)|=a_2a_4-a_3^2≤s^2/4.

The inequality is sharp; equality holds if f is a rotation of  Φ_s,2, given by (<ref>).


Let the function f of the form (<ref>) be in the class 𝒮𝒯_hpl(s). Then there exists a function ω∈ℬ,  ω(z) = w_1 z+w_2z^2+⋯,  such that

    zf'(z)/f(z)=1/1-ω(z)^s    z∈𝔻.

We have

    zf'(z)/f(z)=1+a_2z+2a_3-a_2^2z^2+a_2^3-3a_2a_3+3a_4z^3+⋯,

and

    [                           11-ω(z)^s=1+sw_1z                                           +                      (sw_2+s(s+1)2w_1^2)z^2;                                                                                       + (sw_3+s(s+1)w_1w_2+s(s+1)(s+2)6w_1^3)z^3+⋯. ]

Comparing the coefficients of z, z^2 and z^3  of both sides of the series expansion of (<ref>), we obtain

    a_2 =s w_1,
    a_3 = s2w_2+3s+12w_1^2,
    a_4 = s3w_3+5s+22w_1w_2+17s^2+15s+412w_1^3.

Hence, we have

    |H_2(2)|=a_2a_4-a_3^2=s^2/124w_1w_3-3w_2^2+(s+1)w_1^2w_2+7+6s-13s^2/12w_1^4.

Using Lemma <ref>, we write the expression w_2 and w_3 in terms of w_1. Since the class and H_2(2) are invariant under the rotation, then without loss of generality  we can assume that x=w_1, with 0≤ x≤ 1. Thus

    |H_2(2)|=s^2/12|4x1-x^21-|y|^2ζ-4x^21-x^2y^2-3y^21-x^2^2
    +(s+1)x^2y1-x^2
    +7+6s-13s^2/12x^4|

where y , ζ are complex number with |y|≤ 1 and |ζ|≤ 1. Using the triangle inequality, we obtain

    |H_2(2)|≤s^2/12{4x1-x^21-|y|^2+4x^21-x^2|y|^2+3|y|^21-x^2^2
    +(s+1)x^2|y|1-x^2
    +7+6s-13s^2/12x^4}.

Now, we begin by analyzing a behavior of h(|y|), that represent an expression in a curly brackets of the right hand side of the above inequality. Since 
    h'(|y|)=2x-1^2x+13-x|y|+(s+1)x^21-x^2≥ 0

  the function h(|y|) is increasing  on the interval  [0,1] so that h(|y|) attains its  greatest value at |y|=1, i.e. h(|y|)≤ h(1). Consequently

    |H_2(2)|≤s^2/12-13s^2+6s+17/12x^4+(s-1)x^2+3≤s^2/4.

The function Φ_s,2 with its rotation, where Φ_s,2 is given by (<ref>), shows that the bound s^2/4 is sharp.


 Let h∈𝒞𝒱_hpl(s), and h(z) = z+d_2z^2+d_3z^3+⋯. Then

    |H_2(2)|=d_2d_4-d_3^2≤s^236(1+9s^2/8(s^2+3s+6)).



Let the function h(z) = z+d_2z^2+d_3z^3+⋯ be in the class 𝒞𝒱_hpl(s). Then there exists a function ω∈ℬ,  ω(z) = w_1 z+w_2z^2+⋯,  such that

    1+zh”(z)h'(z)=11-ω(z)^s    z∈𝔻.

We have

    1+zh”(z)h'(z)=1+2d_2z+6d_3-4d_2^2z^2+12d_4-18d_2d_3+8d_2^3z^3+⋯

  Comparing the coefficients of the both sides of the series expansion of (<ref>), we obtain

    [                                      d_2                                 =s w_12,;                                      d_3                      = s6w_2+3s+12w_1^2,;                                      d_4 = s12w_3+5s+22w_1w_2+17s^2+15s+412w_1^3. ]

Thus, we have

    d_2d_4-d_3^2=s^23632w_1w_3-w_2^2+3s+24w_1^2w_2+2+3s-s^28w_1^4.

Using Lemma <ref>, we write the expression w_2 and w_3 in terms of w_1. Since the class and H_2(2) are invariant under the rotation, then without loss of generality  we can assume that x=w_1, with 0≤ x≤ 1. Thus

    d_2d_4-d_3^2=s^2/36|32x1-x^21-|y|^2ζ-32x^21-x^2y^2-y^21-x^2^2
    +3s+24x^2y1-x^2
    +2+3s-s^2/8x^4|

where y , ζ are complex number with |y|≤ 1 and |ζ|≤ 1. Using the triangle inequality, we obtain

    d_2d_4-d_3^2≤s^2/36|32x1-x^21-|y|^2-32x^21-x^2|y|^2-|y|^21-x^2^2
    +3s+24x^2|y|1-x^2
    +2+3s-s^2/8x^4|.

Now, we begin by analyzing a behavior of h(|y|), that represent an expression in a curly brackets of the right hand side of the above inequality. Since 
    h'(|y|)=1-x^2x+12-x|y|+3s+24x^21-x^2≥ 0

  the function h(|y|) is increasing  on the interval  [0,1] so that h(|y|) attains its  greatest value at |y|=1, i.e., h(|y|)≤ h(1). Consequently

    d_2d_4-d_3^2≤s^2/361+3s4x^2-s^2+3s+6/8x^4=:g(x).

Since g(x)  attains its maximum at x=(3s)/(s^2+3s+6), thus (<ref>) follows. 



Let f∈𝒮𝒯_hpl(s) be given by (<ref>). Then the following  sharp inequalities for Fekete-Szegö functional hold

    a_3-λ a_2^2≤{[      s^2λ-3s+14s              for        λ≥3s+34s,;               s2              for 3s-14s≤λ≤3s+34s,;     -s^2λ-3s+14s              for        λ≤3s-14s. ].



Form (<ref>), we have

    a_3-λ a_2^2=s2w_2-w_1^22sλ-3s+12.

The assertion is proved by the application of  Lemma <ref> with t=2sλ-(3s+1)/2. Equality is attained in the case 3s-1/4s< λ < 3s+3/4s for f(z)=μΦ_s,2(μ z), where Φ_s,n is  given by (<ref>), and by the function f(z)=μΦ_s,1(μ z)  for λ > 3s+3/4s or λ<3s-1/4s (here μ is an unimodular constant). For x∈[0,1] we define the normalized analytic functions f_x and g_x by

    zg'_x(z)g_x(z)=𝔮_sz(z+x)1+xz  and  zf'_x(z)f_x(z)=𝔮_s-z(z+x)1+xz

When λ =3s+3/4s  equality holds   for f(z)=μ f_x(μ z). If  λ=3s-1/4s  equality holds   for f(z)=μ g_x(μ z), where |μ|=1.





Let h∈𝒞𝒱_hpl(s), and h(z) = z+d_2z^2+d_3z^3+⋯. Then we have sharp inequalities

    d_3-λ d_2^2≤{[  -s^24λ-3s+13s            for      λ≤3s-13s,;             s6            for 3s-13s≤λ≤s+1s,;   s^24λ-3s+13s            for        λ≥s+1s. ].


Reasoning along the same line as in the proof of Theorem <ref>, we get

    |d_3-λ d_2^2|=s6|w_2-3sλ-3s-12w_1^2|.

Setting μ=3sλ-3s-12 in Lemma <ref> we obtain the assertion.
The inequalities are sharp for the functions

    f(z)= {[              μ K_s,2(μ z)    for  3s-13s< λ < s+1s,;                μ K_s(μ z) for  λ∈-∞, 3s-13s∪s+1s,∞,;                 μF_x(μ z)              for  λ=s+1s,;                 μG_x(μ z)          for  λ = 3s-13s, ].

where K_s,2 and  K_s are given by (<ref>) and (<ref>),  μ is an unimodular constant and the functions F_x and G_x 0≤ x≤ 1 are defined by

    1+zG”_x(z)G'_x(z)=𝔮_sz(z+x)1+xz  and   1+zF”_x(z)F'_x(z)=𝔮_s-z(z+x)1+xz.



Now, we find the bounds of the Fekete-Szegö functionals of z/f(z) and f^-1(z) when f is an element of the class
𝒮𝒯_hpl(s) or 𝒞𝒱_hpl(s).

Let the function F be defined by

    F(z)= zf(z)=1+∑_n=1^∞b_nz^n   z∈𝔻,

for f∈𝒮 given by (<ref>).

Let f∈𝒮𝒯_hpl(s),  and let F(z)=z/f(z) be of the form (<ref>). Then the following sharp inequalities hold

    b_2-λ b_1^2≤{[     s^2s-14s-λ            for       λ≤s-34s,;             s2            for s-34s≤λ≤s+14s,;     s^2λ-s-14s            for       λ≥s+14s. ].



For given f∈𝒮𝒯_hpl(s) with the power series (<ref>) let F(z)=zf(z). An easy computation shows that

    F(z)=zf(z)=1-a_2z+a_2^2-a_3z^2+⋯    z∈𝔻.

Equating (<ref>) and (<ref>), we obtain

    b_1 =-a_2,  
    b_2 = a_2^2-a_3,

and then 

    b_2-λ b_1^2=a_3-(1-λ)a_2^2.

Assertion now follows by application of Theorem <ref> with 1-λ.
The inequalities are sharp, the equality holds for the functions

    f(z)= {[                μΦ_s,2(μ z)     for  s-34s< λ < s+14s,;                  μΦ_s(μ z) for  λ∈-∞, -s-34s∪s+14s,∞,;                  μf_x(μ z)              for  λ=s-34s,;                  μg_x(μ z)            for  λ = s+14s, ].

where Φ_s,2 and  Φ_s are given by (<ref>) and (<ref>),  μ is an unimodular constant and the functions f_x and g_x 0≤ x≤ 1 are given by (<ref>).



Let h∈𝒞𝒱_hpl(s)  and h(z)=z+d_2z^2+d_3z^3+⋯. Also, let z/h(z)=1+∑_n=1^∞h_nz^n. Then the following sharp inequalities hold.

    h_2-λ h_1^2≤{[  s^24λ+13s        for     λ≥13s,;         s6        for -1s≤λ≤13s,; -s^24λ+13s        for    λ≤ -1s. ].



Let h∈𝒞𝒱_hpl(s). A simple computation shows that
z/h(z)=1+∑_n=1^∞h_nz^n=1-d_2z^2+(d_2^2-d_3)z^3+⋯. Then

    h_2-λ h_1^2=d_3-1-λd_2^2.

Assertion now follows by application of Theorem <ref> with 1-λ. The inequalities are sharp for the functions

    f(z)= {[          μ K_s,2(μ z)    for  -1s< λ < 13s,;            μ K_s(μ z) for  λ∈-∞, -1s∪13s,∞,;             μF_x(μ z)           for  λ=-1s,;             μG_x(μ z)         for  λ = 13s. ].

where μ is an unimodular constant, the functions F_x and G_x 0≤ x≤ 1 are given by (<ref>),  K_s,2 and  K_s are given by (<ref>) and (<ref>).



Now, let the function f^-1 be defined by

    f^-1(w)=w+∑_n=2^∞A_nw^n     |w|<r_0(f),

where r_0(f)  is the Koebe constant of the class 𝒮𝒯_hpl(s) (or 𝒞𝒱_hpl(s), respectively). By the Koebe constant of a class ℱ we mean the largest disk {|w|<d} that is contained in f(𝔻) for every f∈ℱ. The problem of Koebe constant was
solved in <cit.>.  Then

    f^-1f(z)=z     z ∈𝔻    and     ff^-1(w)=w     |w|<r_0(f).

The inverse function f^-1 is given by

    f^-1(w)=w-a_2w^2+2a_2^2-a_3w^3-5a_2^3-5a_2a_3+a_4w^4+⋯.





Let f∈𝒮𝒯_hpl(s) be of the form (<ref>),  and let f^-1(z) be given by  (<ref>). Then the sharp inequalities hold

    A_3-λ A_2^2≤{[      s^25s-14s-λ              for        λ≤5s-34s,;               s2              for 5s-34s≤λ≤5s+14s,;      s^2λ-5s-14s              for        λ≥5s+14s. ].



Equations (<ref>) and (<ref>) give

    A_2 =-a_2,  
    A_3 =2a_2^2-a_3.

Applying  (<ref>) to (<ref>), we obtain

    A_3-λ A_2^2=a_3-(2-λ)a_2^2.

Assertion now follows by application of Theorem <ref> with 2-λ.
The inequalities are sharp for the functions

    f(z)= {[                 μΦ_s,2(μ z)    for  5s-34s< λ < 5s+14s,;                   μΦ_s(μ z) for  λ∈-∞, 5s-34s∪5s+14s,∞,;                   μf_x(μ z)              for  λ=5s-34s,;                   μg_x(μ z)            for  λ = 5s+14s. ].

where Φ_s,2 and  Φ_s are given by (<ref>) and (<ref>),  μ is an unimodular constant, and f_x and g_x 0≤ x≤ 1 are given by (<ref>).



Let h∈𝒞𝒱_hpl(s) with h(z)=z+d_2z^2+d_3z^3+⋯. If h^-1(w)=w+∑_n=2^∞B_nw^n, then the sharp inequalities hold

    B_3-λ B_2^2≤{[   s^24λ-3s-13s            for      λ≥3s+13s,;             s6            for s-1s≤λ≤3s+13s,;   s^243s-13s-λ            for        λ≤s-1s. ].


 We use the  same reasoning as in the proof of Theorem <ref>.
Applying relations (<ref>), we obtain

    B_3-λ B_2^2=d_3-(2-λ)d_2^2.

Assertion now follows by application of Theorem <ref> with 2-λ. The inequalities are sharp for the functions

    f(z)= {[              μ K_s,2(μ z)    for  s-1s< λ < 3s+13s,;                μ K_s(μ z) for  λ∈-∞, s-1s∪3s+13s,∞,;                 μF_x(μ z)              for  λ=s-1s,;                 μG_x(μ z)          for  λ = 3s+13s. ].

where K_s,2 and  K_s are given by (<ref>) and (<ref>),  μ is an unimodular constant, and  F_x and G_x 0≤ x≤ 1 are given by (<ref>).


Acknowledgements

The authors thank the editor and the anonymous referees for constructive and pertinent suggestions.


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